ry an eet erry ~ 4H miateiteit a inlsepoiecasesety weet aes re lonhl reruns sahamiabereiiebe: e Speer irae Per eens Cees Ceara ee errr eres CORNELL | UNIVERSITY LIBRARY FROM Cornell Univ ydraulic Tit LOW of water through or AY DRAULICS. HYDRAULICS. THE FLOW OF WATER THROUGH ORIFICES, OVER WW ft oT i Ss, AND THROUGH OPEN CONDUITS AND PIPES. BY HAMILTON SMITH, Jr., Member Am. Soc. of C.E., and Am. Inst. of MAE. RIGHTS OF TRANSLATION RESERVED. NEW YORK: JOHN WILEY AND SONS, 15, ASTOR PLACE. LONDON: TRUBNER AND CO., 57 & 59, LUDGATE HILL, E.C. 1886. HYDRAULICS. THE FLOW OF WATER THROUGH ORIFICES, OVER WEIRS, AND THROUGH OPEN CONDUITS AND PIPES. BY HAMILTON SMITH, Jr. 1886. CONTENTS. PAGES INTRODUCTION _... ne las en bg og ati ae i 1-4 Authorities, p. 1. Remarks, p. 3. NOMENCLATURE . : dt a an ain .. 95-10 General rbinitinas p. 5. Definition wf enbols, p. 6. CHAPTER I. PROPERTIES OF WATER ... . is fas ste sa Lhe 16 Pressure. Amount of compressibility, Grassi, et cet., p. 11. Hydrostatic pressure does not affect flow in pipes—Dubuat, Robinson, Darcy and Smith, p. 11. Impurities. Specific gravity of water from springs, rivers and lakes, p. 12. Probable effect upon the flow by impurities in the water, p. 12. Hear. Changes in specific gravity by changes in temperature, p.13. Table L., relative densities and weights of a cubic foot of water with 7’ from 14° to 212°, computed from Rossetti, p. 14. Effect of changes in temperature on flow through orifices and in pipes, p. 16. CHAPTER Il. THEORY OF HYDRAULICS ae . 7-2 PROGRESS OF THE SCIENCE oF Hyprautics. From Torricelli to neg, pp. 7. 19; Dubustis ian ing propositions, p. 18. GENERAL Propositions. Orifices, weirs and conduits, p. 19. VaLuE or (2¢)%. Formula of U.S. Coast and Geodetic Survey, p.19; Table II., value of (2 )% in different latitudes and at various elevations, p. 20. ForMULE apopTep. Orifices—p. 21: Table III, ratio < for vertical rectangular orifices, p. 21 ; Table IV., ratio 2 for vertical circular orifices, p. 22. Wetrs—General formula for vertical rectangular weirs, p. 22. Open Conduits and Pipes—Chezy formula, p. 23. PROPERTIES OF CIRCULAR CONDUIT, PARTLY FILLED. Table V., p. 24. CHAPTER III. FLOW THROUGH ORIFICES 25-67 Previminary Remarks. Early experiments, p. 25. Methods seek p. 26, ExperimentaL Data. JLesbros: Description of apparatus, p. 27; Table VI., forms of approach for feeding canals, p. 29; Table VII., vertical rectangular orifices, full contraction and free discharge, p. 31; Table VIII. (Poncelet and Lesbros) flow with full contraction, p. 34; Table IX., flow with contraction more or less suppressed, p. 36. Hamilton Smith, Jr.: Tables X. and XI., flow through orifices with free discharge, pp. 38-39 ; Table XII, flow with submerged discharge, p. 40. £ilis: methods, p. 40; Table XIII., flow through rectangular il CONTENTS. PAGES orifices, free discharge, p. 41; Table XIV., flow through circular orifices, free discharge, p. 42; Table XV., flow through submerged orifices, p. 43. Weisbach: flow through circular orifices, p. 43 ; comparison flow of water, mercury and oil, p. 44; Unwin: effect of changes in temperature, p..44. Francis: submerged discharge, p. 44. Steckel: flow through annular openings, et cet., p. 45. Bazin: flow through orifices placed side by side, p. 45. Castel: flow through convergent mouth-pieces, p. 46. Bornemann : flow under sluice gates, p. 47. Bossut and others: Table XVI., various experiments, p. 48. Errect oF TEMPERATURE. Results found by author and Professor Unwin, p. 49. ConDiTION oF EpcrEs. Results found by author, p. 50. IRREGULAR SuppLy. Experiments of author at Holyoke, p. 51. Variations 1n Water. Unknown changes in water may possibly notably affect discharge through small orifices, or for low heads, p. 52. ORIFICES IN A THIN Wati.—Talues of ¢. Discussion experimental data, p. 53; general propositions, p. 56; Table XVIL., co-efficient ¢ for square vertical orifices, p. 58; Table XVIIL, co-efficient ¢ for circular vertical orifices, p. 59; co-efficients for rectangular orifices, p. 60. MeasuREMENT OF H. Whether from plane of orifice, or from contracted vein, p. 60. Suape or Escaping Vein. Recurring forms of jet, p. 60. SuBMERGED Discuarce. Discussion of data, p. 61; general proposition, p. 62. QUICKSILVER AND OIL. Résumé of author’s experiments, and conclusions from same, p. 62. OrniFices In A TH1n Watu. Dynamic effect, and general conclusions, p. 63. Conrraction Mopir1ep. Various degrees of suppression, Table XIX., Lesbros’ experiments, and conclusions from same, pp. 64-66 ; Bidone and the author, p. 67. CHAPTER IV. VELOCITY OF APPROACH is any ass _ .. 68-88 GENERAL ConsIpERaTIons. Formula he= 35, p. 68. EXPERIMENTAL Data. Lesbros: his experiments untrustworthy, p. 69. Jteley and Stearns : apparatus, p. 69 ; methods of discussion, p. 70; results of experiments with 5-ft. suppressed weir, Tables XX. and XXI., pp. 73-75 discussion of these results, p. 75; results with weirs having end contraction, Table XXII, p. 77. J. B. Francis: his formula, p. 78; results with weirs having nearly full end contraction, Table XXIII, p. 79; discussion of Francis’ results, p. 79. Castel: methods employed for weir experiments, p. 80; results with suppressed weir, 2.4 long, Table XNIV., p. 81; analysis of these results, p. 82. Other Authorities : allusion to other experiments, p. 83. Conciusions. (General discussion, p. 83; Table XXV. giving approximate corrections. for additional head due to velocity of approach, p. 85. Formuta. Bernoulli, Boileau, Francis, et cet., compared with expression used by author, p. 86. CHAPTER V. FLOW OVER WEIRS ... 89-164 PRELIMINARY STATEMENTS. Formula to be used, p. 89; early experiments, p. 89 ; dangers of error in experiments for orifices and weirs, compared, p. 90. WEIRS WITH LENGTHS GREATER THAN .65. Experimenta, Dara—Supprussep Werrs. Lesbros: results with 1=.656, Table XXVL, p. 92. Francis: apparatus, p. 92; details of experiments, p. 93; results with J=10., Table XXVIT, p. 94. Hteley and Stearns : apparatus, p. 94 ; results with 7=19., Table XXVIII, p. 95; results with J=5., Table XXIX., p. 95. Weirs witH Contraction. Poncelet and Lesbros: results with 1=.656, Table XXX., p. 96. CONTENTS. Lesbros: results with /=.656 (end contraction slightly modified), Table XXXI., p. 97. Francis details of experiments, p. 97 ; results with 7=10. and /=4., Table XXXII, p. 99. Hamilton Smith, Jr.: results with 1=2.6, Table XXXIIL, p. 100. Meley and Stearns : apparatus, p. 100; results with 7=3.0 and /= 2.3, Table XXXIV., p. 102 (this table also includes a number of experiments with one end contraction suppressed). Lesbros : results with /= 1.97 (crest of weir .164 wide), Table XXXV.,, p. 104. Welrs wit Contraction SUPPRESSED ON ONE oR MORE SIDES, WIT VARIOUS Forms oF Approacu. Lesbros: Table XXXVI, bottom contraction suppressed, p. 105; contrac- tion nearly suppressed on one side, Table XXXVIL, p. 105; contraction suppressed on bottom and nearly suppressed on one side, Table XXX VIIL., p. 106; contraction nearly suppressed on both sides, Table XX XIX., p. 106; contraction suppressed on bottom and nearly suppressed on both sides, Table XL., p. 106; inclined sides of canal, Table XLI., p. 107. Surrace Curve at Weir. Lesbros: central surface curve, suppressed weir, Table XLIL, p. 108. Experiment No. 129 deduced from these results, p. 108. Poncelet aid Lesbros : curves with weir having full contraction, Table XLIIL, p. 109. Zesbros: curves with various forms of approach, p. 110; deduction Experiment No.°130, p. 110. Preley cred Stearus: curves at suppressed weir, 7=5.0, Table XLIV., p. 111. Cusfl: description of experiments, p. 111. Herschel: curve at Holyoke dam, Connecticut River, p. 112. Boileau: p. 112. Hf —H, (H=Heap arove Surrace Curve, anp #,=Heap in Prange or Weir.) Lesbros, &c. : weirs with contraction, p. 112; suppressed weirs, p. 113 ; various forms of approach, p. 113. General remarks as to surface curve and /—ZJZ/,, p. 114. Errect oF Suppression. General conclusions, p. 115. Complete Suppression: discussion and comparisons, pp. 116-118. Partial Suppressiun : comparisons and discussions, pp. 118-121. Conclusions : propositions and illustrations, pp. 121-123. DETERMINATION OF Qo-EFFICIENT OF DiscHARGE FOR WEIRS WITH FULL CoNTRACTION, AND FOR SUPPRESSED WeEIrRs. Discussion experimental data, general propositions, and de- termination of c¢, pp. 123-126; comparison by experiments of Francis, Table XLV., p. 127, and by experiments of Ellis, Table XLVI.,"p. 128. Weirs with one End Contraction suppressed: experiments Fteley and Stearns, Table XLVII., p. 129. Conclusions : remarks, p. 129; directions for proper use of co-efficients, p. 130; Table XLVIIL, giving values of co-efficients for weirs, p. 132. formule in Use: Lesbros, p. 133, Boileau, p. 133, and analysis of some of Boileau’s experiments, p. 134; Simpson and Weisbach, p. 135; Francis, p. 136; Fteley and Stearns, p. 136; Graéff, p. 137; modifica. tion of Francis’ formule, p. 137. > Weirs with LENGTHS LESS THAN .65. Experiments. JLesbros: 1=.066, Table XLIX., p. 138; Castel: apparatus, pp. 138 and 140; results with / from .033 upwards (embracing some experiments with / greater than .65), Table L., p. 139, and Table LI., p. 141; discussion of these results, p. 142; criticism of Castel’s experiments, p. 144. SUBMERGED WEIRS. Formuta. Dubuat: p. 145. Experiments, Jteley and Stearns: apparatus, p. 146; results, 2=5.0, Table LIL, p. 148. Francis: apparatus and computation, p. 149; results, 7=about 11.1, Table LIII., p. 151. Conciusions. Formula proposed by Francis and author, p. 152. General discussion, pp.153-156. iii PAGES iv CONTENTS. Broap anp Rounpep Orzsts. EAGHES Broap Crests. Fteley and Stearns: apparatus, p. 156; results, Table LIV., p. 157. Discus- sion of these results, and comparison with Lesbros’ experiments, p. 158. Rounpep Crests. Fiteley and Stearns: apparatus, p. 159; results, Table LV., p. 160. MEASUREMENT oF H. Experiments. Boileau: p. 160. Francis: p. 161. Mteley and Stearns : methods, p. 161; results, Tables LVI. and LVIL., pp. 162-163. General Observations: p. 163. CHAPTER VI. THE FLOW OVER WEIRS AND THROUGH ORIFICES COMPARED 165-169 Contraction sy Surrace Curve at Weir. Comparison contraction on upper part of sheet from a weir, with normal contraction from a square orifice, by Lesbros’ determinations, pp. 165-168. GENERAL COMPARISONS AND Suaaestions, p. 168. CHAPTER VII. FLOW THROUGH OPEN CONDUITS ae ee si ... 170-198 Cuezy Formuna. Dubuat’s proposition and Chezy formula, p. 170. ExperiMEnTAL Data. Darcy-Bazin: canals of Marseilles and Craponne, Table LVIIL., p. 171; apparatus, p. 172: results with experimental conduits; Table LIXa., rectangular, p. 173 ; Table LIXs., trapezoidal and triangular, p. 175; Table LIXc., semi-circular, p. 176; small rectangular conduit, Table LX., pp. 176-177; sluice-ways, Table LXI., pp. 177- 178; canals, Table LXII., pp. 178-180; rectangular pipes, Table LATII., pp. 180-181. Fteley and Stearns: Sudbury Conduit; description and methods, p. 182; results normal flow in conduit, Table LXNIV., p. 183; surface inclinations less and greater than bottom slopes (motion not uniform), Table LXV., pp. 184-185 ; various conditions inner surface, Table LX VI., pp. 185-186 ; various inclinations, Table LX. VIL. pp. 186-157. Canats. Cunningham: results with Ganges Canal, p. 188. Hamilton Smith, Jr. : results with ditches in California, p. 188. Rivers. L/winphreys and Abbot: Mississippi River results, Table LN. VIJII., pp. 189-191. Com- ments on such experiments, p. 191. Conciusions. fect of changesin A (condition of surface): comparison of experiments and deductions, pp. 191-193. Zffect of changes in velocity (v) and size (v): contrast of experi- ments, p. 193. Propositions in regard to Chezy formula. General laws and apparent exceptions, pp. 194-195. fect of form: comparison flow through square, semi-circular, et cet., conduits, p. 195. : FormuLm in Usr. Chezy, Dubuat, et cet., et cet., p. 196. Discussion of these formule, p. 197. CHAPTER VIII. FLOW THROUGH PIPES a ae ‘ah ai ... 199-276 Formut#. Geueral discussion as to Chezy formula, p. 199. Effect of temperature, p. 201. ExpERIMENTATION. Methods to determine total head, length, diameter, quantity and effective head, p. 201. SHort Pipes. Bossut, et ce’. pp. 203-205: Table LXIX., flow through vertical short pipes (Bossut), p. 204; table from D’Aubuisson, p. 205. Co-efficients adopted, p. 206. Di- vergent Adjutages: Francis, pp. 206-207, and Table LXX., p. 207, giving flow through submerged tubes ; discussion, p. 208. Very Narrow Pipes. Poisewille: apparatus, p. 209; experiments, conclusions and formule, pp- 210-213 ; effect of temperature, p. 212; flow of other liquids, p. 213. Mayen and Jacobson, p. 213. ExperimentaL Dara. Couplet: description of 5-inch pipe, p. 213; details of experiments with CONTENTS. Vv PAGES this pipe, Table LXNXL, p. 214; his other experiments described and discussed, pp. 214- 216. Bossut: description pipes, p. 216; details, Series I. and II., Table LXXII,, p. 217; Series ITT. and IV., Tables LXXTII. and LXXIV.,, p. 218. Dubuat: results with tin pipes, Table LXXV., p. 219. English Authorities : Provis, p. 219; Leslie, Duncan, Simpson, Bidder, et cet., pp. 220-221. Seotch Authorities: Jardine, Leslie and Gale, pp. 221-222; Crawley, Colinton, and Loch Katrine pipes, Table LX XVI. »p. 222. Darcy: apparatus, p. 223; Series I.-NXII. with experimental conduit pipes, Table LX XVIL., pp. 225-228. Tees methods, p. 228 ; Hamburg mains (new and old pipes), Table LX XVIIL, pp. 229-230; Bonn pipe, Table LXXIX., p. 231. Lampe, Danzig Main: methods and apparatus, p. 231; piezometric determinations, Table LX XX., p. 233 ; results, Table LX XNJ., p. 234. Kirkwood: Croton and Jersey City mains, Table LXNXIL, p. 235. Tubbs, Rochester Pipe: description, p. 235; results, Table LX XXIII. p. 236. Stearns, Rosemary Pipe: description, p. 236; results, Table LXXXIV., p. 237. Hamilton Sinith, Jr., Cherokee Pipe: description, p. 237 ; results, Table LXXXV., p. 238. Werw Almaden Pipes : Table LAX XVI, pp. 239-240. Bloomfield and Texas Creck Pipes: Table LXXXVIL, p. 241. Clarke: Dorchester Bay tunnel pipe; description, p. 241; results, Table LXXXVIIL, p. 242; Moon Island conduit or pipe, results, p. 243. Other Experiments : allusion to same, p. 243, Doss H =e CORRECTLY REPRESENT EFFECTIVE Heap h? p. 244. Direct Proof: Bossut, Dubuat and Smith (New Almaden), pp. 244246. Proof by Pivzo- meters: Darcy, pp. 246-253; some of Darcy’s results, Table LXXNINX., pp. 249-252, Conclusion : p. 253 ; methods of proposed experiments, p. 254. Prezometers. Dubuat’s views, p. 254; Darcy, p. 255. Jills: apparatus, p. 255; results and pare 259, Conctusioss. Variation in n, in v=n (rs), eunsed by changes in \, D and r. New Almaden experiments ; description of pipes, p. 260; effect of v, p. 261; effect of D, p. 261; effect of A, p.261. Discussion of Data: Couplet, Bossut and Dubuat, p. 262; Scotch authorities, p- 263 ; Darcy, pp. 263-265 ; Iben, et cet., p. 265; author, p. 265; Darcy-Bazin and Fteley- Stearns (open conduits), p. 266; list of selected experiments, p. 267 ; ae with Plate XIV., pp. 267-268. Determination values of Co-efficient n: 267-272 ; values adopted for smooth pipes, Table XC., p. 271; general remarks, pp. ss i 272, conclusions, pp. 255-256. Discussion : piezometers and their errors, pp. FormuL In Use. Dubuat, Prony, et cet., et cet., pp. 272-273 ; comparison of values of v deduced by several formule, Table XCI., p. 274; suggested formule, p. 275 ; Professor Reynolds’ experiments, effect of temperature, and low velocities, pp. 275-276; Professor Unwin, experiments with discs, p. 276. CHAPTER IX. EXPERIMENTS WITH ORIFICES AND WEIRS (CALIFORNIA) —... 277-289 Mivers’ Incu. Description of this method of measuring water, pp. 277-278. CoLumpia Hitt Experiments. Apparatus: description and methods, pp. 278-281. Orifices— Small Heads : modules for miners’ inch, Tables XCII. and XCIII., pp. 281-282; circular orifices, Table XCIV., p. 283. Weir: results with J/=2.6, Table Xov,, . 284. OriFices—GReEAT Heaps. Worth Bloomfield and French Corral: Bene and apparatus, pp. 285-286 ; results, Table XCVL., p. 287; discussion errors and results, p. 288. CHAPTER X. EXPERIMENTS OF FLOW THROUGH PIPES a se ie .. 290-315 New Aumapen. Apparatus and methods, p. 290. Description of pipes, pp. 293-298. Formule adopted, p. 298. Results, Table XCVIL., pp. 299-301. Mr. J. B. Randol, p. 302 vi CONTENTS. PAGES Norra Bxoomrrenp. Apparatus and methods, pp. 302-304. Measurement of Q; by weir gauging, p. 304; results of same, Table XCOVIII, p. 306. Velocity of stones sent through pipes, p. 307. Temperature, et cet., p. 307. Results of flow, Table XCIX., p. 308. Remarks, p. 309. Humsve Pirz. Description, p. 309. Results, Table C., p. 310. VELocity STONES AND WooprENn Biocxs. Table and remarks, p. 311. Texas Oruex Pips. Description, p. 311. Determination of @, p. 313. Results, Table CL, p. 314. CHAPTER XI. EXPERIMENTS WITH ORIFICES, 1884-5... i we ... 316-361 Hotyoxe. Apparatus: description and methods, pp. 316-319. Free Discharge, 1885: methods, pp. 319-321 ; results, Table CII., pp. 322-339 ; chances of error, pp. 340-342. Submerged and Free Discharge, 1884: leakage, p. 342; results, Table CIII., p. 344; remarks as to submerged discharge, p. 846; acknowledgment to Holyoke Water Power Company and Mr. Clemens Herschel, p. 347. GREENPOINT. Apparatus: description and methods, pp. 347-351; verification of Table I. (Rossetti), p. 350. Normal Discharge ; constant heads: methods, p. 351; results, Table CIV., p. 352. Dropping Heads: methods, p. 352; results, Table CV on p. 353, and table on p. 354. Effect of Temperature: description, p. 354; results, Table CVI, p. 355: expansion of orifice and tank, pp. 355-356. Effect of Oiling Orifice: Table CVIL, p. 356. Flow of Quicksilver: Table CVIIL, p. 357. Flow of Oil: Table CIX., p. 358. Capil- larity: water, quicksilver, and oil in glass tube, pp. 358-359. Chances of Error: discus- sion of same, p. 359. Determination of c: remarks, p. 360. Mr. T. F. Rowland, p. 361. CORRECTIONS AND NOTES—WEIRS. (CHAPTER V.) oe ae ies ... 361-362 Corrections. Apparatus for Fteley and Stearns 5-foot weir, p. 361; Mr. Stearns’ opinion in regard to these experiments, p. 361. SUPPRESSED WeIRS. Notes as to free discharge, and discharge confined by prolongation sides of canal, p, 362. Measurement or H. Danger of error from effect velocity of approach, p. 362. Weirs with contraction preferred to suppressed weirs, p. 362. Suorr Weirs. Experiments. Donkin and Salter, 7=.125, p. 362. Il. 2: Vive ay Weeds 4 TL, Sie SOU ~ NUE XIV. gp SVL oXvVil VII. XI. ATLAS. . Forms of approach used by Lesbros for Orifices and Weirs. Figs. 1 to 12. . Poncelet and Lesbros, and Lesbros. Diagrams showing results with rectangular vertical Orifices, full contraction, and free discharge. Hanulton Smith, Jr. Results with vertical Orifices, full contraction, and free discharge. Experi- ments at Holyoke, Greenpoint, and Columbia Hill. Hamilton Smith, Jr. Flow through circular vertical Oritice of water, quicksilver, and oil. . Hamilton Simth, Jr. Flow through Submerged Orifices, Holyoke, 1884. Diagram showing results of standard experiments of flow over Weirs. Poncelet and Lesbros, et cet., et cet. Diagram showing values of co-eflicient ¢ for Weirs, having either full contraction, or contraction suppressed at both ends. Lesbros. Results with weirs having various forms of approach as shown on Plate I. Castel. Results with short Weirs; (/ from .033 to 2.232). Darcy-Bazin. Results with rectangular Open Conduits, having various conditions of inner surface. Fig. 1. Darcy-Lazin. Results with rectangular Open Conduits, illustrating effects of changes in velocity and size. Fig. 2. Fteley-Stearns. Results with Sudbury Conduit. Bossut, Dubuat, and Hamilton Smith, Jr. Experiments with small smooth pipes. Hamilton Smith, Jr. New Almaden experiments with small Pipes of various kinds. Diagram showing results of standard experiments with Pipes, by Couplet, et cet., et cet Also showing values of co-efficient in Chezy formula, for smooth pipes. . Figs. 1-6. Forms of nozzles used at North Bloomfield and French Corral, for Orifice experiments with great heads. Figs. 7-8. Apparatus at Columbia Hill for experiments with Weir and Orifices. Figs. 1-3. Apparatus for New Almaden experiments with Pipes. Figs. 4-12. North Bloomfield Pipes. Plan, section, and arrangements for measuring volume of flow Apparatus at Holyoke for experiments with Orifices. INTRODUCTION. N this volume will be discussed the flow of water through orifices, over weirs, and through open conduits and circular pipes. The authorities chiefly relied upon in establishing the values of the co-efficients of discharge or velocity are as follows : For Oririces.—Poncelet and Lesbros, “ Hupériences hydrauliyues sur les lois de écoulement de Peau.” Tome IIL., Savants étrangers, V Académie des Sciences, 1832. Lesbros, ‘ Hxpériences Hydrauliques” Tome XII, Savants étrangers, ? Académie des Sciences, 1852. T. G. Ellis, “ Hydraulic Experiments with Large Apertures at Holyoke, Mass., 1874;” Transactions Am. Soc. of C\E., February, 1876. A large number of original experiments by the author, many of which were made with the co-operation of Mr. Clemens Herschel. For Weirs.—Poncelet and Lesbros, and Lesbros, as above. J. B. Francis, “ Lowell Hydraulic Experiments,” New York, 1868. Fteley and Stearns, “ Haxperiments on the Flow of Water, made during the Construction of Water Works for Boston;” Transactions Am. Soc. of CE., January, February, and March, 1883. A number of experiments by the author. For Pirzs.—Couplet, “ Recherches sur le mouvement des caux ;’ Mémoires de ? Académie des Sciences, 1732. Bossut, “ Traité théorique et expérimental @Hydro- dynamique, par M.?Abbé Bossut, Paris, 1786.” Dubuat, “ Principes d Hydraulique, Paris, 1786 ;” and later edition of same work printed in 1816. Lampe, “ Untersuch- ungen tiber die Bewegung des Wassers in Rohren ;” Der Civilingeniew, Vol. XLX., 1873. Stearns, “ Experiments on the Flow of Water in a 48-inch Pipe;” Transactions Am. B 2 INTRODUCTION, Soc. of C.E., January, 1885. Hamilton Smith, Jun., “ Flow of Water through Pipes ;” Transactions Am. Soc. of C.E., April, 1883.* For Orex Conpvurrs.—Darcy and Bazin, “ Recherches Hydrauliques, entreprises par M. H. Darcy, continuées par M. H. Bazin. Paris, 1865” (also published in Tome XIX., Savants étrangers, Académie des Sciences, 1865). Fteley and Stearns ; very valuable unpublished experiments with a large conduit at Boston, Mass., the results of which have been very kindly communicated to the author by those gentlemen. The effort has been made to critically examine all the recorded experiments which have been made with weirs and pipes by German, French, English, and American authorities. It is only by careful analysis that it is possible to separate the most reliable experimental data from the great mass thus far contributed by savants and engineers, and in which, it is almost needless to observe, there is a great amount of chaff. In discussing the laws governing the movement of water under various conditions, many of the views expressed by us have been stated before, either in the same form or in cognate forms. It would needlessly encumber this volume to attempt to refer in detail to the theories suggested in the many works upon Hydraulics, which are in part or in whole repeated here. Our final conclusions, however, possess a fair claim to originality, as many of them considerably differ from the views which have heretofore obtained. Especial care has been taken to accurately give, with sufficient detail, the results of the experiments from which these final conclusions have been drawn. We have endeavoured to state these results in a perfectly candid spirit, and have in no instance rejected data which seemed to us to possess reasonable claims to accuracy, because they appeared to be antagonistic to our final conclusions. Original authorities have almost exclusively been consulted, and not unfrequently errors, either in typography or in the reductions of the author, have been corrected. The danger of quoting secondhand is well shown by the fifty-one experiments with pipes by Couplet, Bossut, and Dubuat, given by Prony in his “ Recueil de cing Tables, Paris, 1825.” Prony makes two blunders in Nos. 10 and 11, giving in both cases the erroneous length of 138.5 inches, instead of 737 inches, the correct values ; his final reductions, however, are based upon the correct lengths. These errors, with many additional ones, are generally repeated by English authorities ; in no work thus far published in the English language are these experiments given correctly. In spite of the care exercised in the preparation of the material for this volume, it is hardly possible that the results as here * Some of these experiments have been re-calculated by the author, so that the results, as given in this volume, differ slightly from those given in the paper published by the Am. Soc. of C.E.. + These errors were made by Prony in his original table, in his ‘‘ Recherches Physico-Mathématiques sur la théorie des eaux-courantes,” Paris, 1804. INTRODUCTION. 3 given can be altogether free from error. Should the reader discover any such inaccuracies, he will confer a favor by communicating them to us. The reader, however, who compares our data with the original authorities, must keep in mind that we have often corrected original errors, and also that seeming small errors in the determination of the final co-efficients arise from the omission of unnecessary decimals in the tabulated results. Perhaps some statement should be made of the reasons which induced us to use the English foot as our unit of measure, instead of adopting the metric system, which would have involved no more labor, as such a large proportion of our material has been derived from French authorities. In the first place, for the quantities we chiefly have to deal with, the metre is much too long for a convenient unit; volumes in cubic metres being especially objectionable on account of the necessary long decimals required to express most values of Q. In the next place, this work is designed to be of value particularly to English- speaking scientists and hydraulicians. In order to thoroughly comprehend a given experiment, the values should at once make a direct impression upon the mind of the reader. Few persons have had more experience in examining metrical data, and in transforming them into English expressions, than the author; but we must confess, that in order to obtain a quick and clear idea of the given sizes, especially as to linear mea- surements, it is always necessary to first, either mentally or upon paper, translate the metre into the foot. We fancy that this is true of almost every one who has not been educated from childhood to think in French measures. The formula, Q=c 3 (2gh)'" Ch, for rectangular vertical weirs, has in all cases been used. As (2g)'is nearly constant, the form adopted by Mr. Francis and others of Q=c' lh’, and c’=$c (2 9)’, is somewhat more convenient, but the longer expression is the correct one, and the values of ¢ deduced by it are of much advantage in facilitating the comparison of the respective co-efficients of discharge for orifices and weirs. We have not attempted to discuss the laws governing the distribution of velocities in open conduits and pipes, as we have not experimentally examined this question with sufficient care to enable us to speak with authority upon it. In order to be a competent critic as to the value of experimental data, one should possess a thorough knowledge of all the many details necessary in order to secure trustworthy results. We have hence almost exclusively confined our discussion to those branches of the science of Hydraulics, where we know by practical experience the proper methods of investigation which should be followed by the experimenter. No attempt will be made to discuss the retarding effect of angles or curves in con- duits and pipes. The experimental data at hand are entirely insufficient to permit a satisfactory analysis of this quite complicated subject ; in fact, about the only experi- 4 INTRODUCTION. ments of value are those made by Bossut and Dubuat with small pipes. To the con- structing engineer the effect of bends should have but little practical importance, as in building a costly conduit or pipe he would be grossly at fault should he construct curves sharp enough to notably retard the flow. The selected experiments for orifices, weirs, open conduits, and pipes are numbered consecutively for each order. In the concluding chapters of this volume will be found a detailed description of -the experiments made by the author with orifices, weirs, and pipes. We are under many obligations to Mr. Ross E. Browne, Mr. A. Fteley, and Mr. F. P. Stearns for valuable suggestions and criticisms ; we also have to thank Professor W. C. Unwin, M. H. Bazin, Herr Iben, and Professor Dr. Lampe for material and suggestions. NOMENCLATURE. Where not otherwise expressly stated : All linear distances will be given in English feet ; All measures of area in English square feet ; All measures of capacity in English cubic feet ; All temperatures by the Fahrenheit scale. French measures have been reduced to English as follows : 1 metre = 3.2809 English feet.* 1 French inch (systéme ancien) = .027 0699 metre t = .088 814 English foot. The unit of time is 1 second. The word orifice always signifies an opening, the upper side of which is covered by the hquid in the feeding reservoir ; when not qualified, it means an opening pierced ina ‘thin wall,” the escaping jet only touching the regular line formed by the inner edge or corners, with perfect interior contraction. The term suppression of contraction always means for an orifice or a weir, placing the side or sides of the feeding canal or reservoir in line at right angles to the side or sides of the opening; thus, contraction suppressed on two sides of « rectangular weir indicates that the axial line of the feeding canal is normal to the plane of the weir, and that the width of the feeding canal is the same as the length of the weir. Partial suppression signifies that one side or more of the feeding canal is so near the corresponding side or sides of the opening as not to allow full or normal contraction as the water enters the plane of the opening. These terms must not be confounded with total suppression of contraction which can be only accomplished for an orifice by having the form of approach similar or nearly similar to that of the contracted vein. It is manifest that for a weir there never can be total suppression of contraction. * Prof. W. A. Rogers, of Cambridge, U.S.A., has determined this ratio to be 3.28086, with the possibility that it may be more nearly 3.28085. + Ratio adopted by Prony. 6 NOMENCLATURE. The expression open conduit signifies a conduit in which the upper surface of the water is exposed to the air; a conduit enclosed with walls on all sides becomes a pipe when the conduit is full. The expression hydraulic-grade line signifies: for open conduits or streams, the surface axial line; for pipes, in general the straight line uniting the inlet and outlet ends of the pipe, the ordinate of the end point being the difference in elevation of the two ends, or H, and the abscissa the total length, or J. Strictly speaking, the ordinate for this point should be the “ frictional” head, or 4h. SYMBOLS. Unless otherwise expressly stated, the following characters will always have these significations : a=area. For rectangular orifices, 9 @=/ w. is 5 weirs, west th i S conduits, a=dw. » circular orifices, a=D Z , a % pipes (being mean area of the pipe for its entire length), a=D’ 3 s d,—area of water section in feeding canal for a weir at the measuring point for H, the section being vertical and normal to the axial line of the canal. a,=d F. b =co-efficient for h,, in correction for velocity of approach for orifices and weirs, in the expression h=H+b - =IT+ 0 hp C’=approximate co-efficient of discharge for orifices and weirs. For rectangular or circular orifices; with the head measured from centre of the orifice to the surface of the still water in the feeding canal or reservoir, Q=C (29 H) "a. », weirs ; with no allowance for increased head due to velocity of approach, Q=C% (2g H)" 1H. e=correct co-efficient of discharge for orifices and weirs. For rectangular vertical orifices ; Q=cl 3 (2 9) : (Hf, — ym » ¢ireular, triangular, and irregularly-shaped vertical or inclined orifices ; with formule based upon the proposition that each successive horizontal layer of water passing through the orifice has a velocity due to its respective head. * Where for an orifice the velocity of approach is notable, Hy and Ht must be corrected for the additional head due to this velocity. NOMENCLATURE.—Sympso.s. 7 For rectangular vertical weirs; Q=c 3 (2 gh) ae c, =co-efficient for rectangular weirs, with full contraction on the three sides (bottom and two ends). c, = co-efficient for weirs, with contraction suppressed on both ends. c’,= co-efficient for weirs, with contraction suppressed on one end. c; =co-efficient for a weir of infinite length, and being nearly the same as the co-efficient c’,. c, = co-efficient for weirs, when there is partial suppression of contraction on any one side. d=vertical depth of water. For open conduits; mean vertical depth in axis of conduit. D= diameter. F =width of a rectangular feeding canal, of uniform section, for an orifice or a weir. g=acceleration of gravity ; being the velocity acquired by a body falling freely in vacuo at the expiration of the first second of its fall; g is slightly variable in different latitudes, and at different elevations above sea level. G =inner depth of a rectangular vertical weir, measured from the crest of the weir to the bottom of the feeding canal ; also the inner depth below the lower edge of a vertical orifice. H=wmeasured head. For rectangular or circular, vertical, orifices ; the vertical elevation above the centre of the orifice, of the surface of the comparatively still water, determined ata point several feet up-stream from the orifice. » rectangular vertical weirs; the vertical elevation above the crest of the weir, of the surface of the comparatively still water, determined at a point several feet up-stream from the weir. », pipes ; the vertical difference in elevation between the surface of the water at the inlet and at the outlet. Where pipes discharge into the air ; the vertical difference in elevation between the surface of the water at the inlet, and the centre of the lower or discharge end of the pipe. H,=head measured from bottom of rectangular or circular, vertical orifices to surface of still water. H,=head measured from top of rectangular or circular, vertical orifices to surface of still water. H,=head measured from crest of a submerged weir, or from the centre of a submerged orifice, to surface of still water up-stream. H, =head measured from crest of a submerged weir, or from the centre of a submerged orifice, to surface of water down-stream. H,,=vertical height from crest of a rectangular vertical weir, to mean surface of the water in the plane of the weir. 8 NOMENCLATURE—Syrmpo.s. h=eftective head. a 2 : », submerged orifices ; difference in elevation between surface up-stream and. surface down-stream. h=H,—H,. ,, submerged weirs ; difference in elevation between surface up-stream and sur- ioe fom dy 2 For orifices and weirs; H corrected for velocity of approach. h=/7+ b: uw face down-stream, corrected for velocity of approach. h= ,, open conduits; difference in elevation of surface for a certain length, after the regimen of flow has been established. » pipes; the “frictional” head, being either H corrected for losses due to primary contraction and to imparting velocity (had — <3 , or, the oe difference in elevation of surface in piezometric tubes after the regimen of flow has been established. h’=for pipes the loss* of head due to contraction at the entrance, and to imparting 2 2 go h,—head due to velocity of approach for orifices and weirs. /,= velocity. f’= 2 a 24 vu h’ = effective additional era for orifices and weirs, for proper corrections for velocity Lend B= +h’, of approach, h’,= =p! dy L=horizontal distance from the side of a vertical rectangular orifice or weir, to the corresponding side of the rectangular feeding canal. (=distance on one side ; L’=distance on other side; hence 0+/+L/=F) /=length. For rectangular orifices and weirs ; horizontal distance between vertical sides or ends. open conduits ; a certain length along their course after the regimen of flow has been established. pipes ; either the total length measured along the line of the pipe, whether it be horizontal, inclined, straight, or curved, from the inlet to the outlet;7 or where piezometers are used, the length between the terminal piezometers. (The upper piezometer being placed at a point below where the regimen of flow has been established.) 2 a? * Looking at the flow through pipes from a dynamic point of view, the effective head is oe which produces the velocity of the escaping jet from the lower end of the pipe; this is expressed approximately by 2 ase It is, however, more convenient, in discussing the flow through pipes, to consider the “‘ frictional” head as the effective head. + Where an inlet funnel-shaped mouth-piece of the form of the contracted vein is attached to the pipe, this mouth-piece will not be included in; where the form of a mouth-piece is considerably dissimilar from the contracted vein, then its length will be included in J. NOMENCLATURE.—SyYmMpoLs. 9 l’=total length of a pipe when h is determined by piezometers. | . . . . . 1 > m= co-efficient for mean velocity in circular pipes.» = 1 (+) ,and 2m=n. n=co-efficient for mean velocity in open conduits and pipes in Chezy formula of r=n (psy. o=co-efficient of contraction for orifices, or for very short pipes. p=wetted perimeter. For rectangular orifices ; p=217+2 w. » circular ss p=D a, » rYectangular weirs; p=/4+2h. “si - open conduits ; p=w+2d. » eircular pipes; p=D a. q=absolute quantity of water held by any measuring vessel. @=quantity of water discharged in one second of time, through an orifice, over a weir, or through a conduit or pipe. Q=4. A=radius, h=5- : : ; u ‘ : r=: hydraulic mean radius of open conduits or pipes. c= for circular full pipes, dD — ae _D. | : Wy ee 1s ‘ : h s=sin of hydraulic inclination for open conduits or pipes. s= 7: T=temperature in degrees of Fahrenheit scale. de 3 _ _ Centigrade scale. t=time in seconds, generally indicating length of time in measuring qg for the particular experiment. wu =maximum velocity per second in an open conduit or pipe. v» =mean velocity per second. For orifices at smallest section ; v= a Q rectangular weirs; v= ik D 3? open rectangular conduits ; v=. w Q D™ 4 ”? » fullcircular pipes; v= 10 NOMENCLATURE.—SymMpo.s. v,= mean velocity of approach in the feeding canal for an orifice or a weir, as the water passes the measuring point for HH. For rectangular canals of uniform section; v,= wee iv = width, For rectangular vertical orifices; the vertical distance between the horizontal sides. ,, rivers or streams of irregular section ; in general the surface width. A=condition of wetted surface of an open conduit or pipe, so far as relative rough- ness or smoothness 1s concerned. A°= very smooth surface, such as glass. A® = maximum degree of roughness for conduits. A} ct cet. = degrees of roughness, varying from A° to A, An increased value of A hence signifies an increased degree of roughness. B=angle of convergence or divergence of the side of an interior or exterior adjutage, with the axial line of an orifice, or short pipe. a=ratio of circumference of a circle to its diameter; 7=3.141592, and ™ =.785398. 4 1 CHAPTER I. PROPERTIES OF WATER. PRESSURE. Warer, when subjected to great pressures, is appreciably compressed; when the pressure is removed it resumes its original form, thus being perfectly elastic. The most reliable experiments indicate that its compression is in direct proportion to the pressure. Canton, Sturm, Regnault, Oersted, and Grassi have determined that the compressibility of water is from .000 040 to .000 051 for one atmosphere. M. Grassi obtained the following results* with distilled water ; T Maximum Pressure in Mean Compressibility for , Atmospheres. 1 Atmosphere. 32° 7.4 000 0502 35° 10.0 -000 0515 51° 5.1 -000 0480 56° 8.4 .000 0476 79° T.2 -000 0455 128° 6.3 .000 0440 These experiments showed that compression was in direct proportion to the pres- sure, and, strange to say, that compressibility diminished with increasing temperatures. Hence, for each foot of pressure, distilled water will be diminished in volume from .000 0015 to .000 0013. This is so minute a change that it can be neglected in the con- sideration of our future experimental data. Dubuat,+ by observing the oscillations of water in different siphons, believed he had demonstrated that the amount of hydrostatic pressure upon the interior walls of a pipe had no noteworthy effect upon the flow through the pipe. Dr. Robinson,{ in a more satisfactory manner, demonstrated the same truth. He used a bent tube, having its axis throughout in the same plane, swinging on hollow trunnions to an inlet and an outlet tank. The outlet end was plugged, and the inlet tank filled to a certain height ; the siphon was placed in a horizontal position, the plug * Ann. de Chim. et de Phys., III., 31, 1851. + Principes @ Hydraulique, 1816, Vol. IL, p. 42. { Ency. Brit. Article Rivers. 8th Editiun. 12 PROPERTIES OF WATER.—PRESSURE. withdrawn, and the time noted when the height of the surface of the water in the two tanks became identical. The experiment was then repeated, with the siphon ma vertical position. There was no appreciable difference in the two times; if any difference, the time was shorter for the siphon vertical, This was possibly due to a shght increase in a, caused by the greater pressure. Darcy, who considered Dubuat’s experiments as not entirely conclusive, found that the flow through a pipe with similar hydraulic heads, was practically the same with small as with considerable hydrostatic heads.* These particular determinations are fully confirmed by all of M. Darcy’s experiments with pipes; the pressure was always greater upon the longitudinal half of the pipe adjoining the inlet, than upon the other half, and the indicated piezometric heads for the two halves show no evidence of any disturbing effect, caused by this difference in pressure. Our pipe experiment, No. 356, was with an “ inverted siphon,” the deepest point in its longitudinal section being 760 feet below the hydraulic-grade line ; the discharge through this pipe appears to be normal, that is to say, fairly agreeing with other experi- ments, where the pipes sustained but little pressure. Hence, we can assume, that the loss of head due to fiietion, cross curreits, et cet., as the water passes through a pipe, is not appreciably affected by the amount of pressure to which the interior walls of the pipe are subjected. The question of pressure will, therefore, have no bearing upon our future discussions. Imevurirties. The water of springs, rivers, and lakes is always slightly heavier than pure (distilled) water, owing to inorganic matter carried either in solution or in suspension. The following statement gives the specific gravity of the water of a number of springs, rivers, and lakes ; Rivers. Garonne Sp. Gr. 1.000 149 Boisgaraud (D’Aubuisson). Thames at Twickenham (1847) 1.000 3 Watt’s Dictionary of Chemistry. Mississippi (filtered) 1.000 25 Riddell. Springs. Holywell, Malvern 1.001 2 Watt's Dictionary of Chemistry. Bradford Moor coal-pits 1.000 78 i i 5 Artesian well, Trafalgar-square 1.000 95 r 5 " Carlsbad 1.004 97 Z “4 Cheltenham 1.006 4 ; 5% ; Lakes. Dead Sea 1.172 Thick oil, passing through an orifice, has a much larger co-efficient of discharge than water; hence it is probable that water carrying in suspension a very large * Recherches expérimentales relatives au mouvement de l’eau dans les tuyaux. 1857, pp. 84—86, and p. 12. PROPERTIES OF WATER —Inmpurities. 13 quantity of clayey sediment will have a slightly larger co-efficient of discharge than pure water, either through an orifice or over a weir. For conduits and pipes, it is most probable that for very small values of, or very low velocities, muddy water will flow with a slower velocity than clear water ; the increased viscosity of the water due to the sediment in suspension will, in all probability, with such small values of » or v appreciably retard the flow. Whether or not with considerable values of 1 and v, the impurity of water has any notable effect per se upon the flow is uncertain. A very impure stream, like that generally flowing through a sewer, will with ordinary velocities soon make a slimy deposit upon the walls of the conduit or pipe, thus increasing the value of A, and the flow may consequently be indirectly greatly retarded by the impurities. We are rather inclined to the opinion that very minute and unknown changes in the water sometimes affect notably the flow through small orifices with low heads, and perhaps by analogy notably affect the flow over weirs with such low heads as .2 and less. Possibly this may be due to varying quantities of impurities, such as greasy particles, in the water. Heat. Water is greatly affected by changes in temperature. The following table, deduced from a table compiled by Rossetti,* gives the specific gravity and the absolute weight in English avoirdupois pounds of a cubic foot of distilled water for each degree of the Fahrenheit scale from 14° to 212°. The given weights are, of course, in air. The table of Rossetti is based upon experiments made by himself, Kopp, and others, and embodies the most accurate determinations thus far made upon the density of water with various temperatures. According to Kupffer,} the weight of a cubic centimetre of distilled water at 39.2° is not exactly one gramme. If this be so, our given weights of a cubic foot are too low by about ygdputh part. Rossetti considers that the most probable temperature of maximum density is about 4.07° Cent., or 39.33° Fahr. . The density of water below freezing point (32°) was determined, by taking advantage of the remarkable property of water of remaining unfrozen when kept perfectly quiet, while the temperature is being reduced from above the freezing point to —10° Cent.. * Annales de Chimie et de Physique. IV. Series, Vol. 17, 1869. + Vide Units and Physical Constants. J. D. Everett. 14 PROPERTIES OF WATER.—Hzar. TABLE I. Relative Densities, and Weights of a Cubic Foot, of Distilled Water. Fahrenheit Scale. Computel frone Rossetti’s Deductions from his own and other EBapervments. | | Temper) Relative Weight of | temper. Relative WORKOF ‘Temper Relative os . y- | Foot. ature. ensl vy. Foot. | ature. y Foot. 14 99814 47 99087 G24 16 80° 669) 62.217 15° | 99831 | 48" 99983 62.413 | sic | 99654 » 62,208 16° | 99846 i 49° 99979 62.411 | 82° 99639 | 62.199 17 * 99860 | 50° 99975 62.408 g3° 99624 «62.189 18° 99873 pI 99970 62.405 | 8H | 99608 | 62.179 19° | 99886 | | 59° 99964 62.402 | 86° | .99592 62.169 20° | .99898 ae 99958 | 62.398 86" 99576. «62.159 aye 99910 b opae 99952 «= «62.394 | 87° =.99560 62.149 23° 99920 BS 99946 » 62.390 | gx? 99544 62,139 23° 99930 56° 99939 —«2.386.° 89° «| U527 Ss 6.129 24° 99938 57° 99931 | 62.381 90° 99510 62.118 25° 99947 58" 99924) 62.377 gio | .yedy2 62.107 26° | .99954 59° 99916 ; 62372; 92 , .99474 + 62.096 are 99961 60° 99907 , 62.366 93° 99456 «62.084 ag 99968 | 61° 99898 | 62.360 g4° | 99437 | 62.073 29° 99973 7 99889 62.345 gx° | (99418 62.061 30° 99979 | 63° 99880 62.349 96 » .99399 | 62.049 B1° 99983 | 64° «| «199869 «62.342 97° | 99379 , 62.036 32° 99987 | 62.416 | 65° 99859 «62.336 93° | .99359 | 62.024 33° 99990 | 62.418 | 66° 99848 62.329 99° 99339 «62.011 34° 99993 | 62.420 | 67° o9x37 | 62.322 | 100° | 99318 | 61.998 35° 99996 | 62,421 68" 99826 62.315 101° «99298 «61.986 36° 99997 | 62492 | 69° : 9814 62.308 * 109° | .99977 | 61.973 37° 99999 | 62.423 | 70° 99802 | 62.300 } 103° .99256 | 61.960 38° | .99999 | 62.423 | 71° 99790 | 62.293 | 104 | 99235 | 61.947 39.3° | 1. 62.424 | 72° 778, 62.285 105" | 90214 | 61.933 40° 99999 | 62423 | 73" 99765 62.277 «| «106" ~, .99193 | 61.920 41° 99999 | 62.493 | 74° 99752 | 62.269 107". .99171 | 61.907 42° 99998 | 62493 | 75° 99739 | 62261 10s 99149 | 61.893 43° .99997 | 62.422 || 76° 99726 | 62.253 | log? | .99127 | 61.879 44° 99994 | 62.420 | 77° 99712 62.244 110° | .99105 | 61.865 45° 99992 | 62.419 | 78° 99698 | 62.235 11° | .99082 | 61.851 46° 99990 | 62.418 | 79° 99684 62,227 112° | .99060 | 61.837 PROPERTIES OF WATER.—HeEat. ' Weight of TABLE I.—continued. Temper- | Relative veut | Temper- | Relative ature. | Density. | Rina: | ature. | Density. 113° 99037 | 61.823 | 147° | 98134 114° 99014 | 61.809 148° | 98104 115° 98991 | 61.794 149° | 98074 116° 98968 61.780 150° | 98043 7 8944 | 61.765 151° .98013 118° 98920. 61.750 152° | 97982 119° 98895 61.734 153° | 97952 120° 98870 | 61.719 154° 97921 121° 98845 | 61.703 155° .97889 122° | .98830 ' 61.687 || 156° | .97857 123° 98794 61.671 157° 97826 124° .98768 61.655 158° 97794 125° 98741 61.638 159° .97762 126° 98714 61.621 160° 97729 127° .98688 61.605 | 161° | .97697 128° 98661 61.588 162° 97664 129° 98634 61.571 163° .97631 130° -98608 61.555 164° .97598 131° 98582 61.539 165° 97565 132° .98556 61.523 166° 97531 133° 98530 61.506 167° .97498 134° 98503 61.490 168° 97465 135° .98476 61.473 169° 97431 136° 98449 61.456 170° 97397 137° 98421 61.439 171° .97363 138° .98394 61.422 172° .97330 139° 98366 61.404 173° .97296 140° .98338 61.386 174° 97262 141° 98309 61.368 175° 97228 142° 98280 61.350 176° 97194 143° 98251 61.332 177° .97160 144° .98222 61.314 178° £97125 145° 98193 61.296 179° 97091 146° 98164 61.278 180° .97056 Weight of a Cubic Foot. 61.249 61.241 61.222 61.203 61.184 61.165 61.146 61.126 61.106 61.086 61.067 61.047 61.027 61.006 60.986 60.966 60.945 60.925 60.904 60.883 60.862 60.842 60.821 60.799 60.778 60.757 60.736 60.715 60.694 60.672 60.651 60.629 60.608 60.586 ho _ Temper- ature. 181° 182° 183° 184° 185° 186° 187° 188° 189° 190° 191° 192° 193° 194° 195° 196° 197° 198° 199° 200° 201° 202° 203° 204° 205° 206° 207° 208° 209° 210° 211° 212° Relative Density. 97021 96986 96950 96915 96879 | 96843 | 96808 96772 | 96737 .96701 .96665 .96629 96593 -96556 .96519 .96482 96445 -96408 .96371 .96333 .96295 .96257 96219 .96180 .96141 .96102 .96063 .96024 95984 £95945 95905 -95865 Weight of a Cubic Foot. 60.565 60.543 60.520 60.498 60.476 60.453 60.431 60.409 60.387 60.365 60.342 60.320 60.297 60.274 60.251 60.228 60.205 60.182 60.159 60.135 60.111 60.088 60.064 60.040 69.015 59.991 59.966 59.942 59.917 59.893 59.868 59.843 16 PROPERTIES OF WATER.—HEat. It will hereafter be shown that the flow of water through a small orifice was quite appreciably diminished by an increase in the temperature from 48° to 132°. Hence it is probable that for both orifices and weirs, an increase in 7’ wil] somewhat diminish the flow. In the ordinary ranges of 7 met with in practice, however, this effect will be so very slight, that it is not worth while to make 7'a factor in the formula, which expresses the discharge. Changes by variation in Z will probably only be appreciable with small orifices, or with very low heads for orifices or weirs. With glass tubes of very small diameter, Poiseuille and Hagen have shown that 7 is a most important factor, the discharge being increased threefold by an increase in 7 from 0° Cent. to 45° Cent. . Experiments with pipes of large diameter are not precise enough to determine whether or not 7 has any notable effect on the flow through such pipes. With r=.25 and over, it is likely, with ordinary temperatures and velocities, that 7 need not be considered. We conjecture though, that with very low velocities in either conduits or pipes of considerable size, where the resistance to the How in a notable degree is caused by the adhesion of the water to the surrounding walls, that changes in T may then appreciably affect the flow. This supposition is to some extent confirmed by the experiments lately made by Professor Reynolds. Experiments made by Professor Unwin with discs rotating rapidly in water of various degrees of temperature, show that the friction rapidly diminishes with an increase in temperature; this perhaps may indicate that with considerable velocities in pipes or conduits changes in temper- ature affect the flow. CHAPTER II. THEORY OF HYDRAULICS. Tue science of Hydraulics dates its origin from the great discovery of Torricelli, enunciated in his “ De Jot Gravium Naturaliter Accelerato, 1643,” that the velocity of a fluid passing through an orifice in the side of a reservoir ix the same as that which would be acquired by a body fulling in vucno® from the vertical height, measured from the surface of the fluid in the reservoir to the centre of the orifice. This proposition, known as the theorem of Torricelli, is expressed by v= (29 A). Upon it rests the whole theory of water actuated by the force of gravity. Mariotte made many experiments illustrating the truth of this theorem, the results of which were published after his death, in 1686. From this date for nearly a century Hydraulics engaged the attention of the greatest mathematicians of the age, being discussed by Newton, Daniel and John Bernoulli, Euler, Maclaurin, and d’Alembert. But these great men investigated the science only as geometers, and their labors resulted in but little advantage to its development. In 1738 Daniel Bernoulli published in his “ Hydronamica” his famous equation, which gave rise to many bitter controversies, all of which were practically barren of good, as the reasoning of the disputants was chiefly based upon mathematical abstrac- tions, instead of resting upon a firm foundation of experimental facts. The elder Michelotti made a large number of careful experiments,{ especially with orifices, in which he showed that the velocity of the escaping jet measured at the smallest section of the vena-contiacta was substantially the same as that due to (2 9 H)*, and that the velocity in the plane of the orifice was about ,%?,ths of the * «A heavy body, falling freely.” + Sperimenti idraulici, et cet. Turin, 1767 and 1771. 18 THEORY OF HYDRAULICS. maximum velocity of the escaping vein; the diameter of the venc-contracta hence being about 7gyths of that of the orifice. Calling o the co-efficient of contraction, we hence have approximately, v=o (2.9 H)*, o having a nearly constant value of .62. The Abbe Bossut published, in 1771, his first work upon Hydrodynamics, and his final edition in 1786. His reflections were founded upon his own careful experiments, and he may be said to have been the first to place the science of Hydraulics upon a proper footing. The Chevalier Dubuat, using the experimental data of Bossut, reinforced by many experiments of his own, in 1786 published his complete work, “ Principes @' [1ydraulique,” in which he discusses with wonderful clearness and ability the laws governing the flow of water through orifices, over weirs, and in natural and artificial conduits. He had in 1779 announced, in a preliminary edition, his great discovery of the law of uniform motion in conduits, which he thus states: “ Quand Peau coule wiiformément Jans un lit quelcouque, la force accélératrice que Voblige & couler est éyale a& la somme des résis- tances qwelle essuie, soit par sa propre viscosité, soit parle frottement du lit.” From the experimental data which he had at hand he framed, by a most beautiful course of reasoning, a formula for the flow in uniform channels, as follows, in English feet ; ie 88.51 () we) _ 0894 (14.0298). ( =) — Hyper. Log. €. + 1.6 s BY Dubuat unfortunately came to the very erroneous final conclusion that the character of the wetted surface, A, has no appreciable effect upon the discharge. He thus expresses this opinion: ‘‘ Les molécules d’eau s’introduisent dans les pores de la paroi, et remplissent toutes les petites cavités de sa superficie. | B76 L5 9953 | 5. | .9996 57 9610 80 9823 1.6 9959 9) 6. 9997 D8 | 9627 | 83 | 9845 1.8 9968 | 8. 9998 59 9643 | 90 | 9863 2.0 9974 10. 9999 | For vertical circular orifices, approximate formule only can be used to express our * This formula is deduced as follows: Assume two rectangular weirs, having each the length, J, of the orifice, and the respective heads of Hyand H;, The difference between the discharge of these two weirs will represent the discharge for the orifice having the width Hp — Ht =w. QO =cFl (2g Heth. v= Cc 3 l (2 gy'le Ht Ale Q =c8l (2g): (Hy — Arh) This supposition conforms with our second general proposition. 22 THEORY OF HYDRAULICS.—FormuLs. Orijices. second general proposition. In Table IV, will be found the ratio =, which will answer all our purposes in obtaining the value of c. TABLE IV. Vertical Circular Orifices. Ratio of i C H c H a D © D c D c 5 .9604* 1.95 994.8% 2.9 9983 6 9753 13 9953 2.3 9984 625 9774# 1 9960 2.4 9986 5 9823 15 9965 x 2.5 9987* | 75 9849% 1.6 9969 3. 9991 8 9867 it 9973 3.5 9994+ 875 9892 * Ls 9976 a 9995 9 9897 1.9 9978 45 9996 | 1. 9918# 9. 9980" 5. 9997* | a 99330 | ld 9982 10. 1+ | 1.2 9944 | The values of O which have no asterisk attached, have been obtained by interpolation, and some of c them may be fully .0001 in error. In using Tables ITI. and IV., it must be kept in mind that, when reducing the value of C’ as determined by experiment to the correct co-efficient +, C’ must be multiplied by a ; on the other hand, when (/ is to be computed from a table of values ¢ of ce, c must be multiplied by . : Werrs. The only weirs which we propose to consider are vertical rectangular ones, the sills or crests being horizontal. The formula which will be used, when no corrections for velocity of approach are necessary, is, Y@=C 3 (2g H) 1 H. This expression is in accordance with our second general proposition, for: dividing ) the line H into an indefinitely great number, n, of equal parts, distant ‘ H,., et cet. D from the summit of 1, the last number of the series being H " | H, the discharge will NM ; THEORY OF HYDRAULICS.—FormuLez. Weirs. 23 CS ire. « oa be represented by Q=Cl H (2 g)* x H*% <” = . . The sum of the members of this series divided by their number is 3. French hydraulicians, commencing with Dubuat, have, as a rule, used the height, //,,, of the water in the plane of the weir, in formulating expressions for the discharge over weirs. We believe, however, that this is a vicious method ; we prefer to consider the surface curve of the escaping sheet, from the measuring point in still water to the plane of the weir, as the upper portion of the contracted vein; this assumption will be fully discussed hereafter, in tracing the analogies between the discharge over weirs and through orifices. French authors, when only considering the head H, generally have used the co-etticient C’= 3% C. American authorities have generally used C” = 3 (2 g)4% C’; assuming (2 7)%=8.020 as a constant, C” = 5.347 C. We prefer the longer, and theoretically correct expression, especially for facilitating comparisons of the values of C or c, for weirs and orifices. Oren Conpurts AND Pipss.~ Using the Chezy formula for circular full pipes ; nae DBD RB" =o (re)® =n (= *] ee bo “Dr “x i Dh wher Age ge tiv? ou Us Paty ae) , Q = Dyn (9 3)" = 3997 1. (p°}) ‘ If we consider that (2g)”* being a variable, should be a function in the equation, making f = Gg , we have, v = f (2 gi's)* In the following table are given the properties of a circular conduit partly full. The table is based upon a radius of unity; the given arc is that of the wetted surface p; dis the axial depth; w is the surface width. 24 THEORY OF HYDRAULICS.—CircuLaR CONDUITS. TABLE V. Properties of a Circle eee a Radius are Unity; partly filled with Water. Are, Dp | a | w a | rar’ i Ave. p as d | w i a r | are 360° | 6.283 | 2. 0 (31416 5 © 2.2214 | 180° 3.142 I lo Iasvog}.5 [1.1107 350° | 6.109 | 1.9962) .1743 3.1411 5142 2.2625 170° 2.967 9128 1.9924 1.3967 4707 | .9583 340° | 5.934 1.9848| 3473 | 3.1381 .5288 ' 2.2820 | 160° 2.793 '.8263 1.9696 1.2253 |.4388 .8116 330° | 5.760. 1.9659 5176 3.1298) 5434 2.3072 150° “2.618 7412 Bases 1.0590 : 4045 6735 320° | 5.585 1.9397 ‘6840, 3.1139/.n575 |2.3251 | 140° 2443 6580 | 1.8794 .9008 | .3685 5465 310° | 5.411 | 1.9063) 8452 | 3.0883 |.5708 , 2.3332 | 130° 2.269 .5774 1.8126, .7514 | 3312 | 4324 308° 5.376/1.8988 .8767|3.0818 .5733 19.3334 120° 2094 .5 |1.7321! .6142 .2933 3326 300° | 5.286 1.8660 1. eee 2.8290 | 110° | 1.920 |.4264 |1.6383 4901 |.2553 | .2476 290° 5,061 1.8191 1.1472! 3,0006).5928 2.3103 y 100° 1.745 .3572 1.5321 3803 2179 1775 280° 4.887 1.7660 1.2856 2.9359 }.6008 2.2755 || 90° | 1.571 62929 142 2854 |.1817 1217 270° 4.712 LTOTL 4142 2.8562 6061 2.2236 80° 1.396 '.2340 1.2856 2057 1473 .0790 260° | 4.538 | 1.6428 1.5321 2.7613|.6085 2.1540 | 70° 1.222 .1809 |1.1472 1410 |.1154 ' 0479 257° | 4.485 1.62251.5652 2.7299 .6086 2.1207 | 60° L047 .1340)1. | .0906 .0865 .0266 250° 4.363 ,1.5736 1.6383 2.6515|.6077 |2.0670 || 50°: .873 (0937 | 8452 .0533 |.0611 i .013 240° 4.189, 1.5 1.7321, 2.5274 6034 1.9632 40° | 698.0603 , C840 .0277 .0396 0035 230° | 4.014 (14226), 8126 2.3901|.5954 1.8143 30° | 524 |.0341 ' 5176 0118 .0225 .0018 220° 3.840! 1.3420). 8794 2.2413 5837 1.7123 20° B49 01523473 00352 0101 .00035 210° | 3.665 | 1.2588/1.9319 2.0826 5682 1.5699) 10° | 175.0088 | 11743 00044 0025 .00002 200° | 3.491 | 1.1736|1.9696 | 1.91 e j1.4199 |, oF; 0 0° 0} O 0 0 190° | 2.316 1.08721.9924 | 1.7449 .5262 | 1.2657 | | | From the preceding table it will be obscrved that with a circular section 7 is greatest with a wetted are of 257°, when it ix .6086 with D=2; it has the value of P=. when the are is either 360° or 180°. Assuming that the discharye is approxi- mately in proportion to a7”, it will be greatest when the wetted are is about 308° ; roughly speaking, a pipe will discharge 5 per cent: more water when filled to 3 9ths of its diameter than it will when completely filled; also, the discharge from a circle will be double that from a semi-circle of the same diameter. CHAPTER III. FLOW THROUGH ORIFICES. Mariorte, Bossut, Michelotti, and other physicists have investigated the flow of water through orifices of different shapes; the results of their experiments are given in brief in the many text books treating upon Hydraulics. These investigators, with heads varying from a few inches to 25 feet, proved conclusively the truth of the fundamental principle, that the velocity of a jet varies substantially as (2 g h)*. With square-edged orifices, with the jet escaping freely into the air, through a thin side—so that the only contact was with the inner sharp corners of the orifice—it was found that the discharge was represented by Q= Cu (2 g H)%, the co-efficient C having a nearly constant value of .62. This discharge of only ;%2,ths of that due to the effect of gravity, was shown to be almost entirely caused by the contraction or convergence of the fillets or veins of water as they formed into place; the jet after its escape from the orifice assuming a much smaller section, termed the vena-contracta, whose velocity was nearly that due to (2 g H)*%. Michelotti found that the diameter of circular jets, measured at the smallest section of the venc-contracta, was about 7,ths of the diameter of the orifice; hence the area of the smallest section was about 362,ths of that of the orifice. He determined that the co-efficient of discharge, C, of these orifices was from .60 to .62; thus showing that the mean velocity of the vena-contracta at its smallest section was almost exactly (2 g H)*%. Hence some authors have assumed that the co-efficients of contraction and discharge (efflux) are identical. It was also shown that an increased discharge could be obtained by rounding the inner corners of an orifice, and that by adding a trumpet-shaped mouth-piece, converg- ing towards the orifice, the value of C’ could be increased to .95 or slightly more; @., Q=very nearly a (2 9h)”, the area « being taken at the outer or smallest section of the orifice. These early experiments were sufficiently precise to demonstrate these general principles, but were not made with enough care to warrant the deduction of the minor laws of variation in C, due to changes of form in the orifices, differences of H, variation E 26 FLOW THROUGH ORIFICES. in T, et cet.. For instance, comparing the discharge through square and circular orifices with D=side of the square, with full contraction and equal heads, some experimenters have found C larger for the square section than for the circular, while others have found just the reverse.* We hope, by the aid of the experiments made by MM. Poncelet and Lesbros, those made by M. Lesbros, and the very careful experiments lately made by ourselves, to be able to draw final trustworthy conclusions as to the general effect upon Cor ¢c, of variation in section, head, and temperature. The ratio of the area of the contracted section to the area of the orifice is the co-efficient of contraction, The ratio of the mean velocity at the contracted section to the theoretic velocity is the co-efficient of velocity. The co-efficient of discharge (often called co-efficient of efflux) is the product of these ratios. Weisbach has found that with a head of 3.4 metres the co-efficient of velocity was .978 for a circular orifice with full contraction, and we can assume that this co-efficient is always less than unity. The co-efficients of contraction and velocity can be obtained by direct measurements of the contracted vein, and the parabolic curve of the jet; from these quantities the co-efficient of discharge, C or c, can be readily deduced. This is an objectionable method, and should never be employed where accuracy is desired. Even with a circular jet it is difficult to make a fairly accurate measurement of the contracted section, owing to the slight vibrations in the jet ; the cruciform section of the jet from a rectangular orifice can never be measured with any reasonable degree of accuracy. As we do not intend to discuss the dynamic effect of jets from orifices, we will entirely neglect the co-efficients of contraction and velocity, and only obtain the co-efficient of discharge, which, with properly conducted experiments, may be determined with great accuracy. In all of the experiments about to be given the velocity of approach was so inconsiderable that no correction for v, need be applied. Hence # is always practically identical with h. The value of the correct co-efficient of discharge, ¢, will generally be given. ExperiMentaL Data. Leshros.—Vertical Rectangular Orifices, In the year 1827 MM. Poncelet and Lesbros commenced at Metz an elaborate series of experiments upon the discharge through rectangular vertical orifices of various sizes, the largest being .656 square. These experiments appear to have been executed with a care before unknown in experimental Hydraulics. The results were published in the Mémoires of the Academy of Sciences, Savants étrangers, Paris, 1832. M. Lesbros, during the years 1828-1835, continued these investigations, repeating some of the experiments, and making very many additional ones. His results are pub- lished in the same Mémoires, Paris, 1852. The experiences gained in conducting the first series of experiments doubtless * In the elaborate treatise upon Hydraulics, lately published by M. Graéff, the author assumes that both theory and experiment demonstrate the flow will be greater through a circular orifice than through a rectangular one of the same area. M. Graétf roughly places the value of C at .60 or .61 for rectangular openings, and at .64 for circular openings. It will be seen hereafter that our experimental results lead to very different conclusions. Vide ‘‘ Traité d’Hydraulique, par M. A. Graéff, Paris, 1883.” Tome deuxitme, p. 24; and ‘ Essai sur la théorie des eaux courantes,” by M. Boussinesq. ORIFICES.—ExperIMENTAL Data. Lesbros. 27 enabled M. Lesbros to continue his labors with still greater accuracy, and hence these later experiments are probably somewhat more reliable than the first. A careful analysis of the results obtained by him impresses one with the belief that his investigations were conducted with care, and that his stated facts are honestly given. Lesbros used various forms of approach for the canal feeding the orifices experi- mented upon; the forms of approach which were used in the experiments hereafter selected are shown by Figs. 1 to 12 on Plate I. Reference will be made in the follow- ing tables and diagrams to the particular figure which represents the form of the feeding canal for the several experiments quoted. In all of the experiments selected from Lesbros there was a perfectly free discharge into the air; he also made a large number of experiments with the discharge into an uncovered canal or flume, but which will not be discussed by us, owing to the uncer- tainty attending such submerged discharges, due to the irregular or curved form of the axial surface line in the canal. The quantity discharged was directly measured in vessels of ample size. The height of the surface of the still water in the feeding reservoir was measured by a vertical rod, sharply pointed at its lower end, placed at a distance of 11.48 feet above the orifice, where the water was practically stagnant, or unaffected by the discharge. Hence in these experiments the additional head or force due to the velocity of the water, as it passed the measuring point for H, need not be considered. In some of the forms of approach, particularly those shown by Figs. 6 and 7, Plate I., a portion of the head, H or h, was absorbed by “ friction” and adhesion with the sides of the feeding canal, and also by primary contraction as the water entered the mouth of the canal. The height of the surface of the water at a distance of only .0656 foot above the orifice, was also determined, by which these losses of head were indicated with more or less accuracy. The largest orifice, .6562 square (2 decimetres), was cut in a copper plate, having a thickness of .013 ft. (4 mm.) with beveled outer sides, firmly fixed to a wooden frame. In order to obtain orifices with the same length and smaller width,* a copper gate of the same thickness (.013) was employed, sliding on the inner side of the fixed plate, and moved vertically by means of an iron rod. The lower edge of this gate, which formed the upper side of the orifices of less width than .6562, was sharply beveled on its inner face. The widths of the opening were determined by a scale attached to the iron rod, and also by templates. The smaller fixed orifice was cut in a copper plate, .013 thick, the opening being 1.9685 x .0656 (.6 m. x.02 m.), and the plate being firmly fastened to a wooden frame. It was first placed with its long side horizontal, 7= 1.9685 and w=.0656; it was then * It must be be kept in mind that for vertical rectangular orifices, the length J is the horizontal side, and the width w the vertical side or height. 28 ORIFICES.—ExperRIMENTAL Data. Lesbros. reversed, with J=.0656 and w=1.9685. In this last position, in order to obtain smaller widths, a copper gate was employed, sliding on the inner face of the fixed plate. In regard to the foregoing methods, we may remark: that the two fixed copper plates were altogether too thin; any swelling of the wooden frames on which they were mounted would very likely have resulted in distorting the form of the orifice. The experiments made by the aid of the gates, especially those where ” was small, have the following dangers or chances of experimental error: warping of the copper gate under considerable pressures ; this was in part guarded against by stiffening the outer side of the gate by a wooden block, supported by a screw ; the inner bevel of the lower edge of the gate was objectionable, as it doubtless somewhat added to the flow, compared with the normal flow through an opening with a plane inner face, such as was the case with the two fixed plates; when the gates were employed, the methods of measuring we cannot be considered as exact. It is much to be regretted that MM. Poncelet and Lesbros did not employ fixed brass plates, at least .02 thick, for each dimension of orifice experimented upon. The several dimensions of these orifices could have been accurately determined by means of a delicate measuring or comparing apparatus. We hence regard the experiments with the two fixed plates—orifices .6562 x .6562 and 1.9685 x .0656—as much the most reliable of the series. Enrors of measurement in the larger orifice would be less apt to notably affect the values of the co-efficient c, than in the smaller orifice. When the gates were employed, errors in w are probably nearly constant for a given width. It is, however, almost imposssible to immovably hold in place, even for a few hours, such a gate; changes in the temperature of the comparatively long iron moving rod, as M. Lesbros has pointed out, would appreciably change the width of the opening ; there is always more or less “Jost motion” in such apparatus, and the lower cdge of the gate doubtless wax sometimes appreciably out of parallel with the lower edge of the fixed plate. In the following table are given the distances from the edges of the two fixed orifices to the bottom and sides of the respective feeding canals employed, as shown by Plate I. These distances were the same not only for the two fixed orifices, but also for the orifices formed by the two sliding gates, ORIFICES.—ExprrimentaL Data. Lesbros. 29 « TABLE VI. Distances from Sides of Orifices to respective Sides of Feeding Canals in some of Lesbros’ Experiments. | Distances. Length | - soe ~| Number of , (Horizontal) | Lower Edge | Vertical Sides of Orifice Width of Figure, of Orifice. | of Orifice to to respective Sides Approach on vide Bottom of of Canal. Inner Face | Plate I. Canal. fae eee aa of Orifice. 1 G is | baled =P! a a ee L772 5.709 | 5.709 (12.074 Fig. 1 1.772 LetG2 5.709 8.137 oh ee 1.772 1.772 L772 4250 | i 8 | 0 | 5.709 5.709 | 12.074 ze 0 066 5.709 re 78s 6562 0 .066 .066 TSS “nO 0 : 0 0 656 og «T | Lae .066 5.709 6200) « 2% ! Li .066 066 788 | lg 1.772 0 0 656 » 10 , 0 | 066 066 788 tt | L772 066 066 788 , 12 | 1 0656 | L772 6.004 6.004 12.074 ~ 4 | | 1.9685 1.772 5.053 5.053 12.074 ll | 1 For Figs. 11 and 12 the side approaches were inclined at an angle of 45°, vide Plate 1. Owing to the swelling of the wooden sides of the canals, the above distances slightly varied from time to time. These sides were of planed plank, tongued and grooved. In the following table are given the results of Lesbros’ experiments, with the forms of approach shown by Figs. 1, 2, and 3, Plate I. Liesbros assumes (2 g)” as 8.0227, which will be taken as its value in all the following reductions, both from Lesbros, and Poncelet and Lesbros. In the table, H is the height from the centre of the orifice to the surface of the still water in the reservoir, taken at a point 11.48 feet above the plane of the orifice ; /Z,, is the height from the centre of the orifice to the surface of the water, measured at a point .0656 foot above the plane of the orifice, or up-stream. The values of the co-efficient Care deduced from H/, and are generally the 30 ORIFICES.—EXPERIMENTAL Data. Lesbros. means of several determinations with constant values of H. The co-efficient c has been deduced from C, by aid of Table ITI. The experiments made by the author with orifices are numbered from 1 to 155 inclusive, and will be given after our résumé of Lesbros, and Poncelet and Lesbros. Hence we begin with No. 156 in the following table. Lesbros measured his heads to .0001 metre. In the transfer to feet we have given them to .0001 foot, for the purpose of accurately comparing the differences between and H,,, which are often minute. ORIFICES.—ExperimentaL Data, Lesbros. 31 TABLE VII. Lesbros.—Flow through Vertical Orifices, with Free Discharge into Air, and Full, or very nearly Full ‘ ee : 3 ( Contraction. LH, or effective Head, measured 1148 Up-stream from Orifice, Os on ee Iyé | Means. mo Meme nlm | ty | le a= | . | | i ¢ sce af 2 | 6562 6562 -5.5113 | 5.5096 | 4.9008 | .6043)] 5 54 | joa me 3 * BS ‘ 5AS83 5.4866 ; 4.8711 ated ! 1357 3 ! * : 4 2.9905 | 2.9886 | 3.6135 | 6019 2.99 | 6052 158 a ae 5 1.3370 | 1.3399 ; 24024 | 6015 ; 134 | .6030 159 6-8 . » | 317" out | Larse | 5962 | .83 | .6002 160 | 9-11 : , | .3970| .3842; 129507 | 5746 | 40 | .5946 | | | | | | : ES 3) 65626562 «5.8272 | 5.8239 | 5.0147 | .6014] 13 ae 2, ‘i 5.7682 | 5.7649 | 5.0132 6043 1614] 14 e Wok iw 5.4706 5.4673 4.8832 | 604441 5.60 | .6032 | oy. OF gy be «= Stew! Sater stsezo | cot] | ! El) Gee Vl ake ey » 5.4663 5.4630 | 4.8738 | .6035 J , | 162 | 718 | | yy, B.0BNL 3.0348; 3.6518 | 6065 3.04 + .6068 163 | 19-20, > , » «+ 13980 1.4009 | 24617 6027 | 140 ' .6042 164 21-22 i OR eg ge Ok R78. 18685 .5960 | 62 6001 165 | 2326, «| 6 | | 4039! «39041 1.9611 | 8745 | 40.5935 | | ! | ! | | 166 | 311-313] 2 | 6562 | 16404) 5.6471 | 5.6487 1.2661 | .6170 | 5.65 6170 167 3143815, | om | j 29866, 1L5TOL, 6814 | 6304 | 1.57 — .6305 168 316317 =, yg i Ss« 7H «1.0860 ' 5692 | 6320 1.09 | .6321 169 318319! 4, 6} 4} 78667379! 4696 | 6336 | 74 | 16339 io |soami| =, fF, | 1138 | O807 | 1788 G62 | us| 6308 | | 7 | ' | lf] 282381 3 6562 | .16L04) 5.7531) 5.7527 Laid | 6167 = 5.75 | 6167 7 a%s%| ,, : : 3.2500 3.2207 9724 66265 8.23 | 6265 178 327-3391 e 1.6290 1.6385 6965 | 6319 | 1.63 ° .6320 ivi ‘sa0a5x! . 6447 6559! 4400 | 6346 | 64 | 6350 | | 178 | 661-662 | 2 6502 03281] S8TLL S.8THL: 2571 | 6142 | SAT , .6148 176 603-664, ,, 7 ; 22801) S200k Osa ee 8S 177 "665-666 | ee ‘ 1.6355 1.6401 1427 | 6461 | 164 | .6461 32 191 192 193 194 195 196 197 198 199 200 201 202 203 957-959 Lesbros’ Nos. | 667-669 | 670-671 | ‘652673 674-676 677-679 680-681 682-685 | 948 949-950 951 952-953 954-956 960-961 962-963 964-966 967-968 969-970 971-973 974-976 977-978 | 979-980 981-982 983-985 986-988 989-991 992-995 996-999 Figure. Plate I. lo ” TABLE VII.—continued. 6562 , 03281: ” | ” 656 2 032 81, ” ” ” | ” ” | ”? + | 1.9685 | .065 62° i ” | ” i | | | 06562, .6562 | | ” | et \ \ 1 | ” ” | | | .06562| .16404'. | | ” oP) | 7% 0 ' *} . ” | i 065 62. .065 62 us i ” | ” a H 7989 .0804 6.1553 38.2530 1.5404 219 io lo 4 5.6169 5.6054 3.3514 3.3416 .8022 4.8410 2.4000 LO778 5.4971 2.9118 1.6388 .6480 5.7432 | 3.1579 1.8849 8957 5.7924 3.2071 1.9341 9449 4331 | ORIFICES.—ExPERIMENTAL Data. Lesbvros. Means. H,, Q C h c 8019 | .1015 | .6572 80 | .6572 0801 | .0342 ) .6992 080 | .7004 6.1573. 2632 6142 6.16 | .6142 3.2560 se 6305 | 3.25 | .6305 1.5335 | .1396 | .6511 L54 . .6511 8265 | 1032 .6588 82.6588 3793 | .0705 | .6685 372.6685 LOSS? Ue) ge. | | ean 1.5274 | .6225 J 1.1915 | .6281 irl casat| B38 | 6368 List7 ) 6254 S87 .6343 80 | .6343 1.4963 | 6956 | 484 .6267 1.0096 | 6289 | 240 | .6335 6565} 6103 | 1.08 | .6380 5016 | .6193 | 5.50 | .6194 3735 | 6336 | 2.91 | 6340 2830 | .6399 | 1.64 i .6410 1794 | .6451 65 | 6524 4977 4.6172 § B74 | 6172 09712, 6329 3.16 | .6329 07565) 6380 . LS8S — .63x0 05273) .6451 | 90 | 6453 | 05194 .6164 ' 5.79 | .6164 03913 .6326 | 3.21 | .6396 030 58 .6366 | 1:98 | 6966 O26 GF ° 8h ) Aisy 014 83) .6525 6527 | 433 ORIFICES.—ExprerimMentaL Data. Lesbros. 33 Comparing the heights H and Jf, in the preceding table, it will be observed that their differences follow no gencral order. We hence feel inclined to attribute these differences chiefly to experimental error, except for the lowest heads, where the surface in the reservoir was not very far above the top of the orifice. In this latter case the surface immediately above the orifice must have been appreciably depressed. In the second series (orifice .6562 x .6562, Fig. 3) the velocity of approach in the feeding canal was much greater than for any other of the serics ; the effective head due to this velocity could not have amounted to more than .0006, or .0002 metre, a quantity barely appreciable with the methods of measurement employed. In the following table are given the results of the experiments made by Poncelet and Lesbros in 1828, the form of approach being the same as Fig. 1, Plate I. There was hence for these experiments almost perfect contraction. For the orifices, where « was less than .6562, the same objections apply, so far as great accuracy was concerned, which we have stated in regard to the Lesbros experiments where the sliding gate was used. The second column in this table, headed “ P. and L.,” gives the number of deter- uninations by Poncelet and Lesbros, the head being constant and ( the variable; (’ has been deduced from the average value of (, as was also the case with the Lesbros experiments. The head due to velucity of approach for all these experiments was not appreciable ; i.¢., at the measuring point for H,. As before, we revard the differences between and H,, given in the first series, as chiefly due to experimental error, except for Nos. 212 and 213; for these experiments the given depression at the orifice is much greater than that shown by Lesbros, with the same orifice, and Figs. 2 and 3. Poncelet and Lesbros.—Flow through Vertical Orifices, with full Contraction. ORIFICES.—EXPERIMENTAL Data. Leshvos. TABLE VIIL measured 11.48 Up-stream from Orifice. (im all ones a, 2 lw (2g H)% A | | Means. 4 | No. a ae H dh; i Cc ; 0: 5 w I @Q C pl alee er Pee ete | | 204 1 6562 4.8295 L.N2S8 L574 6026 4.83) .6028 224 3 16404 .1529 "2083 | 6168 295 12! ,, 14,6992 4.6986 4.5140|.6028 4.70 '.6030 225 2 4, 1191, 1814 6088 206 [1] ,, |4.3144'4.3187 4.3319] 6037 |4.31 |.6039 | , | beta aor la! . \a0tet w Ole! 41701 sotd 4.02) ,c094 1292 FOS 8 E5808 ORD 6225 208 |3| ,, '3.1251)3.1241 3.6906|.6044 3.13).6047 |777 ; . oe ae 209 |6| ,, |1.8668'1.8655 2.8437 6025 /1.87).6083 |agg/ 7°» = 1.5688 ane on mol?! 4. “Kela0 ieienaaree| moor ieilgor | oe “S2BS AOsT “828 211\3} ,, | .7940| 7897 1.8324 |.5953 | 79) .5997 Pe a «are a 212 /1) ,, | 5249) .5141 1.4706 |.5876 | 525 5989 |280 po ie 213 4 » | 4003) 8727 1.2527 5732 ' 400}.5927 231 3 yp ERS Wed 1 E285 |i | | | | |» 1/,065624.5939 4603 .6217 214 |2 .3281 [5.1031 some Gti BIG) GU ||” ance: ny ane 215.2 4, (8.8045 1.9315, .6152 3.30|.6153 ly | AABTE dod 624d 216 |3 (1.5807 1.8403 |.6172 11.58].6175 io5 13] | 3.9398 3934 6335 27 2 f 3740 .6422 | 6080 | 374} 6132 9342 , ‘1.9861 2517 6425 ais jal, | .2067 4653/9929 207 6111 a55 | | 3655 ser anes | | | | 1936 2 4, | .0666|.05x9 6604 [1 164 04/5.5450 1.2571 | 6182 | | ; Jl BSesT 1.2549 an psdl errs 237 1.032 814.5998 2298 6204 ara |. | (esse Lona e171{| | loa 2! , |e gpn4!ae31 +6191 1], (5.5339 1.2550 ae sg 3.2576 1964 6301 220 |2) ,, (3.5460 1.0164 6250 3.55|.6250 240 |3) ,, 1.6386 1419 |.6426 291 16) ,, {1.5650 | 6786|.6281 |1.56].6282 241 2! ,, 6398 /.090 27 .6534 992 16| , | .6972 4544 |,6302 | .70].6306 242 4) ,, | .1936),051 36 .6758 2 (8, | er | 3199, 6288 |.347|.6303 1243 3), | 0571 .028 71) .6958 Effective Head Means. 68 | 186 066 *Oa21 6411 6433 ORIFICES.—EXpERIMENTAL Data. Leshros, 35 Comparing the three curves formed by plotting, with 4 and ¢ as co-ordinates, the three series of experiments with the respective forms of approach shown by Figs. 1, 2, and 3, it will be seen that they closely agree. Any divergence between these curves is not in excess of probable experimental error. Hence we can draw the conclusion that the narrowing of the feeding canal, from a width of 12.1 as in Fig. 1, to the width of 4.2 as in Fig. 3, had, at the utmost, a very slight effect upon the co-efficient of dis- charge. Therefore with an orifice .66 square, when the sides of the feeding canal are 1.77 from the respective sides of the orifice, the discharge will be practically the same as though these distances were infinite. M. Lesbros notes that with the highest heads employed for Fig. 2, Experiments Nos. 156-160, 166-170, and 175 to 179, the vein after its escape from the orifice con- verged a little towards the prolonged direction of the face of the canal nearest the orifice. This phenomenon indicates that the side of the canal nearest the orifice had some effect upon the discharge. But it may be remarked, this is a test of marvellous delicacy ; we are of the opinion, judging from our own experiences, that such a vein could be perceptibly inclined after its escape, without affecting the co-efficient of discharge more than .000 05, a quantity much below the limits of appreciation. M. Lesbros draws some general conclusions from the variation of the curves—form of approach Figs. 1, 2, and 3, with size of orifice constant—which we do not consider tenable. He did not, in our judgment, sufficiently recognise the danger, or one may say certainty, of experimental errors. The escaping jet or vein, for Experiments 188-190, with the high and narrow orifice took a remarkable shape, with flanges at the summit and base. Vide Plate VI. of Lesbros. The contracted section of the jet from the larger fixed orifice (.6562 square) was measured with much care, in order to obtain the co-efficients of contraction and velocity. There are so many mechanical difficulties in the way of an accurate measurement of the jet from a square or rectangular orifice, and so many intricate theoretical considerations to be taken into account, that we do not feel inclined to give much weight to these determinations. The results of a number of the Lesbros experiments, with various forms of approach as indicated by Figs. 4 to 12 inclusive, Plate I., are given in the following table. The length of the orifice was in all cases .656. 36 ORIFICES.—EXPpERIMENTAL Data. Leshros. TABLE IX. Lesbros.—-Flow through Vertical Orifices, with the Contraction more or less suppressed by the Sides of the Feeding Canal. Orifice .656 x .656 5.9 6215.3 637 5.9 6615.8 670 5.5 611 5.8 6a7'5.3 1637 5.6 6215.0 “61 5.6 .623 2.9 .637|5.5 .661,5.5 1.671 3.0 .612 3.0 .629 3.0 |.639 3.9 .642 3.0 |.612 4.7 afe5 It. 636 4.2 663 4.6 |.675 1.5 .610 1.3 633 1.5 ‘641 2.6 OLE 86 609 | 4.2 625 68 635 2.6 .666 2.9 om 83.607 da 633. 20 01 1.5 646 40 610 | i aT 624 dt .673.12 1692. 41 G07 42 649 at! 658) 49 649. | 2.5 ical eR, : 1 t : i 1.3 6241 73 695 | | 93.624 | | : | | joa A & © ! : , i | = NY : Orifice .656 x 328 Orifice .656 x .164 Fig. +. | Fig. 5. | Fig. 6. [ s 7. | Fig. 9. || Fig. 4 | Fig. 5. | Fig. 6. | Fix. 9. = et Peery eee Wee (orc hh pene ! ae eee soot | h | ec | h | clh' eth « h ec | h ) oe hre|lh)e h|\e rea, Sat ea oe hs a ma ye 6.1 6425.9 653 6.0 670 6.0 6796.0 .629 . 1 664 3.8 S670 6.0 676 6.1 634 5.0 O445.4 |654/4.1 673) 1 31 600.2 lees Me 671/42 1.679 3.2 1.635 4.2 645/26 |.656 3.1 .G75. 1.5 °.633|i4.3 '.666 3.0 '.673/2.9 1680/1.6 |.637 I | \ I \ 3.3 647! .67).658 2.1 .678' 71'636'3.3 ‘60 1.6 676 [1.8 682! .82}.642 3 : : | : i | r y | 2.8 618] 33.652 87 681 | 29 667 93.676] .75 1.6871! 12 L663 1.6 .648 | 37.689 1.6 |.668, .20|.672] .45 |.689 67 649 | .69 L671! .26 | 693 39.644) | | | Po | a8 |.670 a Me “s = = 4 eee settine » MSD | Orifice .656 x .098 | - Orifice 656 x. 033 | Fis 4. Fig. 5. | Fig. 6. | Fig 9. i Fig 4. | Fig. 6. | Fig 6 oe Fig. & | Fig. 9. pak a ar aS ee c | h li h c h c | h ¢ h c | i | c h cy 6.0 “675 wale 677 6.0 |.680 6.2 ae 694 5.8 -695 6.0 re 6226.1 i: | | 4.3 675 ee | 681 /4.3 .682|2.9 1.641 4 4.9 6983.7 ‘70 (4.3 r 701 3.3 634 3.3 662, 238 e8 l) 683 2.9 .686 11.6 648 3.1 .701 1.4 AOE Teel STOLL. = |.649 |1.6 “669 1.7 680 [1.6 a 1.6 .692} .85).655 12.0 .703] .78'.708'1.7 1.701] .71|.660| .71 685 | ! ! 55 ree 81 ee 73 7694 08 |.679 1.3 .707| .26 | 719) .76 107 31.677) 28 .710 | * Or 96 | 22 687 | 22 O85 222 7698 A9 716 .06 er) 26.716) .04).710) 04 .766 | c | h | e¢ | 1 ORIFICES.— EXPERIMENTAL Data. Lesbos. 37 In some of these series of experiments there was a notable loss of head from primary contraction, as the water entered the feeding canal from the reservoir proper. This was especially the case with the forms of approach represented by Figs. 6 and 7. The “ frictional” loss of head in the canal could hardly have been appreciable, even for Fig. 7, where contraction was suppressed on three sides of the orifice. The given co-efficients for these two forms of approach (Figs. 6 and 7) are doubtless appreciably too low. Error of this kind could have been avoided by having the upper entrance of the feeding canal trumpet-shaped ; had this been done the effect of velocity of approach could have been readily determined, by a comparison of the height //, and the surface height a foot or so above the orifice. The difference between H and /Z,, as given by Lesbros, approximately represents the head due to velocity of approach, of which only a portion is lost, plus the head absorbed in overcoming the primary contraction, all of which is lost.* TTamilton Sinith, Jun. The experiments with orifices made by the author, are described in detail in Chapters IX. and XI. It will only be necessary here to give a résumé of them, stating the approximate sizes and heads, and the co-efficients c. * With comparatively high velocities of approach, as was the case with Fig. 7, the water ‘‘ piled up” immediately above the inner side of the orifice, thus taking an abnormal level at the point where Hw was measured. It is possible that in some of these experiments this abnormal elevation of the surface at Hw, may have fully represented the total head H minus the loss by primary contraction. 38 ORIFICES.—ExperimentaL Data. Hamilton Sivith, Jun. TABLE X. Smith.—Flow through Vertical Orifices with Free Discharge into Air. California Experiments, 1874-1876. s Bs zB : ee h c Size. ‘2 Remarks. os & 3| 6.8 = AlAs az 1; 3 583 6161} Rectangular, w= .167; ¢=4.17 Wood.| Full contraction. a} 9 1.00 | .5988 a i100; TH106 y. | « : 3) I 1.33 | .6087| Circular, D= .253 Iron. | ,, a 4, 1 1.20 | .6041 3 D= .419 4 A a BY) oh 1.09 | .5929 4 D= .66 af ‘i s 6) 1 1.05 | 5913 3 D=1.01 i Bi a Vortex above orifice. 7 322.3 1.040 | Circular nozzle. Least D=.053) _,, Converging mouth, cide Fig. A, Pl XY. 8 314.5 {1.004 \ nue 3 9 3193 986 >? 3 3 M ” a} ” ” , ” ” 10 316.1 {1.006 3 . » C, ‘ ‘Li 3327 1.007 | ; » D=.087 \ C’ 12 336.0 [1.005 J “= a 13 317.9 {1.011 1 14 315.6 1.014 J > ” ” D=.102 ” ” ” ” ” D. ” ” | 15 316.3 615 | Circular ring. D=.060, Steel. | Contraction slightly suppressed, Fig. E. 29 14 o12\6 ae ; 3 a D=.085) _,, 9 somewhat ,, ie, 17 313.2 662 | 1s 127.3 647 " 7 D=.182 4 ” , x @ G The co-efficients ¢ for Experiments Nos, 2, 6, and 5 are in all probability within a small fraction of the truth. Nos. 1, 4, and 3 are less reliable, owing to danger of error in the measurement of w and D. Nos. 7 to 18 inclusive can be only considered as approximations, because for them ( was indirectly determined by the flow over a weir. For Nos. 7 to 14 it is altogether improbable that c exceeded .995 or .998 at the utmost. The chances of error in these experiments are discussed in Chapter IX. They possess, however, considerable value in proving for very great heads, that with converging mouth-pieces ¢ has a value of about 1, and that with full contraction ¢ will be about .60 for small circular orifices. They demon- strate conclusively the truth of our fundamental proposition, the theorem of Torricelli, In the following table is given a summary of the experiments made in 1885, under normal conditions. Smith—Flow through Vertical Orifices with Full Contraction and Free Discharge into Air, 1885, Lo wr a mow ww We Mm -I a} of Number Hee | Determinatio ' 1 { ' TLS. ! He wou eR em We Hee www Wwonrw Ye oO RP We ew wr w wo wb www w ORIFICES.—ExperimMentaL Data. Hamilton Sinith, Jun. 39 739. 6495! 243°. 6208 13.19 | 6264 185.6525 Circular in brass. 59 190 | .0611 | 200 | 6475 203 | .6481 240 | 6438 | | 283.6336: 282 | 6453 382 | 6473, 283 6457 | .335 | .6330 | 336 | .6376| | 401 | 6377, 437 | .6301, | 536 | .6265 | 720 6199. | .910 6160) 929 | 6194) i174 | 6113, 2.73 6070. ls.s7_ | .coso! 4.63 | .6051 | | 129 6337 | 264 | 6288, 457 | 6155 661.6120) 900 | .6096 | 1.73 | 6042 1.87 | .6028 2.05 | .6038 3.18 | 6025 Description. Circular in steel. D=.020 T= 10° ~ 46° D=.050 T=19 —54. Circular in brass. D=.100 No. +1 known to be defective. T for Nos. 40-47 = 49°— 51°, fT for Nos. 56-58 = 62° — 62.5°. TABLE XI. | B38 | No | 2-8 h ee Se | 5S nae 47 3 4.60 | 60 | 3 1.80 | ! 3 1 181 , | él | 3 | 2.81 | 62 3 | 4.68 | oY) Bey 313 | | 64 3 | .457 | 65 2 | 651 | 66 3 | S77 67(2) 1° 170 | 6S o 159, 69 { Ser 70 3 | 3.70 71 3 | 4.63 72 3 | 181 73 2 480 | es 2 | .677| 75 3 | 939 | 76 3) 171 | a ar eee | we! 3 : 3.74 79 3 | 4.59 80 2° Bel 81 2 442 82 3 | .665 es | 8 | oF 84 3 | 188 85 1 | ara 86 2 | 2:83 87 3 | 3.75 88 3 | 4.70 c Description. 6061’ GQircular in iron. 6041 | D=.100 | 6033; 7=60.5-63°. .6026 | 6410 | Square in brass. 6354 | .050 x .050 62x60 P=51°- 52°. 6238, .6149 No. 67 somewhat 6157. defective. 6127 6113: 6097 .6292 Square in brass. 6184 .100 x .100 16157 | 7'=49°—50". 6139 | 6084, 6076. .6060 | 6065 | .6476 Rectangular in brass. 6361, 1=,300; w=.050 6312 | T=49° - 52°. 6280. 6203 | 6180 6184 6176 | 6168 | 40 ORIFICES.—-EXPERIMENTAL Data. Huauilton Smith, Jun. The foregoing experiments were made with much care, and under conditions favorable to great accuracy. Those with the smallest circular orifice are the least trustworthy, owing to greater danger of comparative error in the measurement of D. TABLE XII. Suith._—Flove through Vertical Submeryed Orifices, pierced in Brass Plates, with Full Contraction. Holyoke, W884. Depth of submergence from .57 to .73. Circular, D=.05 Circular. D=.10) Syuare. .05x.05 | Syuare. 10x .10 || Rect.. 30 x .05 No| & | © | No. | # ye Bing bw | Bo] od | e No} hc 97 | 4.08 ; 6016! 100 | 3.97 oan 4.06 | 6068, 112 | 3.95 Oe 119 | 2.77 | .61x8 98 , 2.16 | 6041 101 | 3.57 .5987 1110] 2.21 ae 113 | 3.11 ee 120 | 1.63 ; .6207 99 | 487, 6183 102 2.99 5989; 111 ee 114 i 232 | 6040] 121 | 614 6219 ! 103° 258.5997 | 115 | 152 6055 : | | 104 | 2.00 eo | 116 ' .771 .6053 | , 105 | 1.51 | .6006 | | 117 | 410: .6091 | | 106 | 985) .6025 | Lie 07, 21 107 645.6027 | . 108. 250. 6048 The above experiments are considerably less reliable than those in Table XI., for the reasons given in Chapter XI. Nox. 119-121 do not fairly represent submerged discharge, owing to the thickness of the sides of the orifice, which formed a slight divergent adjutage, and thereby increased abnormally the co-efficient c. A number of other experiments, illustrating effect of temperature, et cet., et cet., will be described in Chapter XI. They need not be summarized here. Ellis. Mr. T. G. Ellis, in the year 1874, made at Holyoke, Massachusetts, an extensive series of experiments with vertical and horizontal orifices of large sizes, with heads up to 18 feet. His results were published in the Transactions of the Am. Soc. of C.E., February, 1876. The discharge or Y, which in some cases reached the large amount of 48 cubic feet per second, was measured over a sharp crested weir, the flow being computed by the weir formula of Mr. J. B. Francis. This indirect method of obtaining the value of ( adds considerably to the chances of experimental error, and these experiments hence cannot be considered as exact as most of those of Lesbros and the author. The Francis ORIFICES.—Experimentat Data. Ellis. 41 formula, as will be hereafter pointed out, gives results pretty near the truth for the measuring weirs used by Ellis, and therefore the resulting co-efficients of discharge from his orifices should give quite smooth curves, when his results are shown graphically. There are, however, frequent irregularities in these curves, which indicate rather inaccurate methods of observation. The final results given by Ellis can therefore only be considered as approximative, although probably without errors of very serious consequence. They are the best authority to be had for large orifices with high heads. The experimenter states that in all cases the heads above the orifices were suffi- ciently large to prevent the forming of vortices. In the following tables we will only give the approximate heads, and the correct co-efficient «, deduced from the average values of C' for nearly similar heads given by Ellis. These represent the means of several hundred determinations. TABLE XIII. Ellis. —Flow through Vertical Rectangular Orifices, nearly full Contraction and free Discharye into Air, Rigid Iron Plates. 1=2.00 . w=2.00 =2.00 . w=1.00 1=2.00 . w=.50 1=1.00 w=1.00 No h c No h c No. h c No. h c 244 2.07 616 247 1.81 599 256 | 1.42 612 | 266 1.49 ST 245 3.05 .600 248 3.03 .600 | IST 2.91 612 267 3.69 599 246 3.54 .608 249 4.66 598 258 4.75 608 | 268 4.80 599 250 5.67 OT 259 6.36 608 269 5.49 596 251 6.87 598 | 260 1 8.54 607 270 6.72 599 252 7.69 598 | 261 | 9.63 606 27) 9,80 596 253 8.48 599 | 262 . 11.56 604 |! 272 9.90 .602 254 9.65 .600 263 | 13.51 604 | 273 12.00 600 255 11.31 605 264 | 15.06 602 | 274 13.63 601 265 16.97 600 | 275 15,13 601 | | 276 17.56 597 In the above experiments there appears to have been a partial suppression of bottom contraction for the 2. x2. orifice, the horizontal floor of the discharging basin being .50 below the bottom edge of the orifice ; for the other orifices the suppression of bottom contraction was less. G 42 ORIFICES.—ExXPERIMENTAL Data. Allis. TABLE XIV. Elliis.— Flow through Vertical Circnlur Orvifices, nearly full Contraction, and five Discharge into Air, Rigid Iron Plates. D=2.00 D=1.00 | D=.50 No. h | ale | h ¢ No. | h c | | a77 | 177 595 | 284 | 115 | 578 | 294 | 3.15 | .600 278 | 2.60 | .596 || 285 | 2.36 | 589 || 295 | 4.16 | 602 279 | 447 le 286 | 4.81 | 590 | 296 | 6.35 | .605 aso | 5.83 | .610 | 287° 7.97 | .586 | 297 7.30 | .601 981 | 6.93 |.613 || 288 ; 7.92 | 589 298 | 8.01 | .601 282 | 8.34 / 613 289 | 10.88 594 || 299 | 9.06 | .602 283 | 9.64 , 615 290 | 12.48 ee 300 | 10.51 | .601 291 | 14.13 | 595 | 301 | 11.97 | .600 292 15.66 oe 302 12.98 | .601 293 | 17.72 | .600 || 303 . 1447 | .601 | | 304 15.46 | .601 ! | | 305 | 15.85 | .605 | | 306 | 17.26 | 596 Contraction appears to have been somewhat suppressed on the bottom for the largest of these three orifices, and very slightly suppressed for the smallest orifice. The following experiments were made with a submerged discharge, the two vrifices, which for free discharge had been placed in a vertical position, now being placed in a horizontal position. The converging mouth-piece for the 1.x 1. orifice was formed by an inner facing of wood .5 thick bolted to the iron plate and curved on each of the four sides in the shape of a quarter of an ellipse, whose semi-diameters were .50 and .33, with the larger diameter at right angles to the plane of the plate. Contraction was probably nearly perfect for the plates without the convereent adjutage. ORIFICES.—ExpertmentaL Data. Ellis. 43 TABLE XY. Ellis.—Flow through Horizontal Submerged aes Circular D = 1.00 Square 1.00 x 1.00 Square 1,00 x 1.00 Nearly full Contraction.|Nearly full Contraction. || Curved Mouth-piece. Ey ae OP oe (ie ST, I No. h ¢ No. | h | ¢ No. | c | i ra Me a 307 2.60 | 607 | 316 2.32 | 600 |} 323 3.04 | .952 308 4.71 | 590 | 317 3.92 | 602 || 324 5.77 | 946 309 6.41 | 606 | 318 | 7.99 | 606 | 325 | 10.54 | .943 310 8.10 | .599 | 319 . 11.58 | 605 | 326 13.57 | .943 311 8.80 | 600 | 320 14.31 | 611 | 327 1822 O44 | 312 | 12.09 | 600 | s2t | 16.22 | 606 | 312 | 1495 | .601 | 332 , Lado | .806 | 314 | 16.29 | .602 | 315 | 18.66 | .599 | | | | General Ellis measured the head upon his measuring weir, by means of a tube placed near the inner face of the weir. This is a vicious method, as will be hereafter shown, but in these experiments the velocity of approach was never large, and serious errors should not have resulted. Note our remarks on some weir determinations by the same authority, Chapter V. Weisbach. Herr Julius Weisbach, in his “ Lehrbuch der Ingenieur,” n “ Der Civilingenieur, ” Ath revised edition, and Vol. X., gives the following values of ¢ for circular orifices “in a thin wall” ; | H Dp | 066 | 33 | 82 2.0 3.0 45. | 340. 033 | .711 | .665 | .637 | .628 | .641 | .632 | .600 066 | 629 | .621 10 | 622 | 614 | 13 | 614: .607 ; i For an orifice, with D=.033, and a well rounded mouth-piece : 1 =.066, h=1.64, c=.967; h=11.5, c=.975; h=56, c=.994; h=338, c=.994. He states that c is much greater for long narrow orifices than for those which are circular or regular, and that ¢ is about 14 a free discharge. 959% per cent. smaller for a submerged than for t4 ORIFICES.—EXPERIMENTAL Data. Weisbach. He found the following values for c, with the three given liquids, with a head drop- ping from 1.08 to .30 (?), the opening in each case having a diameter of about .02 ; | = | Thin Plate. Conoidal Convergent Cylados ae with Mouth-piece. t=3 | Water... 709 942 885 | Mercury ... 670 989 .900 : _ j-430 with 7=54" | /.363 with T=54° Rape seed oil’ 674 { \.665 ,, 7=102" l.604 « 7=102° | Urwin. Professor W. C. Unwin, in the “ Philosophical Magazine” for October, 1878, gives the following results illustrating effect of changes in temperature. With a head dropping from 1.47 to .98 in a vertical cylindrical reservoir, and with an opening in each case having a diameter of .033 ; Thin Plate. Convergent Conoidal Mouth-piece. | T t ¢ a a) | 149.6 593 (89. | 986 am 149.4 | 594 We" Agee 991 140° 1488 | .596 130° 90 975 | 65° 149.2, 595 ee | 93 943 aie 148.0 | .600 | Oa 949 | ae | 600 | | Professor Unwin states that the given values of ¢ may possibly be in error 2 or 3 per cent., from errors in measurement of H and D. Francis, Mr. J. B. Francis has made a number of experiments with orifices, an account of which will be found in his “ Lowell Hydraulic Experiments.” The values of Q were unfortunately obtained by weir measurement, and often with minute heads. Such determinations are especially unreliable. With a converging cycloidal mouth-piece, and a minimum or outer diameter of .1018, with a submerged discharge he found a value of .815 for c, with /=.034; this value steadily increased with increasing heads, until it was .944 for // =1,52. With a circular submerged orifice, D=.1017 with full contraction, he found values ORIFICES.—EXxpermMENTAL Dara. Francis. 45 of ¢ of from .56 to .59, with a maximum head of 1.5. In all probability for these experiments @ was under estimated fully 5 per cent. . Steckel, Mr. R. Steckel* has lately made a large number of experiments with small orifices under various conditions. With a horizontal circular orifice in a thin plate, D=.032, with H measured from the plane of the orifice, he found, with free discharge into air and full contraction, the following results; H= 4.2, c=.621; H=2.4,c=.628; H=1.0, ¢=.628. With a horizontal circular orifice, with D=,033, having an iron cylinder of D=.015 placed normal to the centre of the orifice, thus forming an annular opening, and with nearly constant heads of .25 above the plane of the orifice, he obtained the following results : With the cylinder removed ... as ve 508 ab we €=.673 » 9, base of the cylinder .025 above plane of oritice we 6S. 673 Gis ttn, bie sss, ces as .0004 ,, oe tS als we CS ATB9 oe a 3 in Se age. 3 vce C= 26 Sg gm Ga is .0083 below ., 4, 4; we €=.685 ” y yo» ” .017 3, ve er .. =¢=.680 For the first two experiments, the area taken for establishing c is the area of the orifice ; for the last four experiments, « is the area of the ring. The foregoing results are doubtless chiefly due to suppression of contraction by the introduction of the cylinder. Bazin. For the experiments with open conduits, Darcy-Bazin series,{ (J was dctermined by the flow through one or more of 12 similar adjoining orifices. Each orifice was .656 x .656, with a constant head of 2.624 above the centre of the orifice. The co- efficients of discharge for these orifices, C, were determined by using a section of the experimental conduit as a measuring vessel. The following values of C were obtained ; One gate open ... sma pan a ban we C =.633 Two gates ,,_... ee we aa Me ee C=.642 Three ,, 4, «.- ‘ we ue Gs “5 C =.646 Four <3 4) «6s ne a3 bs ie see C=.649 Five to twelve gates open... ss sti he C =.650 The value of C appears to have increased when two or three adjoining gates were opened, aside from the effect of velocity of approach, which for only three gates open must have been barely appreciable. * Essay on the Contracted Liquid Vein. A paper read before the Royal Society of Canada. Ottawa, 1884. t Recherches Hydrauliques. Darcy-Bazin. First part, page 61. 46 ORIFICES.—EXPERIMENTAL Data. Bazin. We have ourselves found similar results with several orifices 1. x 1.06 placed a few inches apart. Castel, M. Castel made a number of experiments with conical convergent tubes, which are described by M. D’Aubuissonin the “ Annales des Mines, 1838.” The tubes were of smoothly polished metal; @ was determined by the measurement of y in vessels of known size. In the following statement we give the general results obtained; the heads employed varied, for each given experiment, from .2 m. to 3.0 m., but as there was only a very slight difference* in the value of C with varying values of H for the same adju- tage, it will only be necessary to give the mean values of C. Castel-D’ Aubuisson.—Converginy Mouth-pieces. i=length of adjutage. C=co-eflicient of discharge at smallest section of adjutage. Given angle=angle of prolonged sides. All measures metrical. 1=.040 D-=about .0155/1=.035 D=about .0155|2=.050 D-=about .020 2B C 2B | Cc | 2p | Cc 0 8290 gd’ | 929 250° | Old 1° 36° 866 10° 28” 945 5° 26" 930 3° 10° 895 | 12° 42 951 6° 54! 938 4°10’ 912 16? 02’ 940 ie sor |) ou 5° 26 924 19° 06’ 926° 1210 | .950 7°52! 2 340’ | 936 8” 58" 934 l=.030 D=about.0153 15° 02! ; 949 10° 20’ i SS a aie ee 12° 04’ 942 ze S hs 23° 4! 930 13° 24’ 946 eee we 33°52 | 920 14° 28" O41 | = SS ee 16° 36’ 938 ?=.100 D=about .020 19° 28" 924 | : SS. a ha ur" 52 965 ae on 14° 12 958 ae hee 16° 34 951 40° 20’ 870 48° 50’ 847 | * If there was any difference due to variation in H, it was aslight increase in C—say .001 or .002—for the greatest head compared with the smallest head. ORIFICES.—EXpERIMENTAL Data. Castel 47 Castel also determined the co-efficient of velocity for the jets from these mouth- pieces. Note our remarks on the experiments of Castel, given in the Section on Short Weirs, Chapter V. Borneimeainar. Herr K. R. Bornemann has investigated the flow under sluice gates placed across a rectangular flume or canal. He has published an account of his elaborate series of experiments in “ Der Civilingenieur,” Vol. XXVI., 1880. These experiments were necessarily often complicated by a high velocity of approach, as the length of the orifice was very nearly the width of the canal. The following sketch shows the points where H, and H, were measured. ag ,2 Adding for velocity of approach 3s . or -effective head h=H,—H,+ a , the 2y co-efficients ¢ range in 63 experiments from .668 to 1.015. In general, the high values > 2, a of ¢ are when a is a large part of h. These abnormal values of ¢ are due in part to an uy under estimate of the effect of velocity of approach, and still more to the incorrect point taken for the measurement of the lower head H,. It seems to us clear that this head should be measured in the lower portion of the wave as shown at a in the sketch. In any event it should not be measured on the crest of the lower water, which is piled up by the energy of the water escaping from the orifice. Such an escape presents many interesting mathematical problems. Unfortunately, so far as experimental proof is concerned, it is not practicable to measure the head at a with any reasonable degree of accuracy, owing to the boiling of the water. This same kind of discharge has been investigated by Boileau and by Lesbros. We regard the investigation of the flow through such orifices as of no value for the practical gauging of water. The quantity of flow can be determined by almost any other method with greater accuracy. Bossut and Others. In the following table are given the results of experiments with orifices “in a thin wall” made by Bossut, Castel, and others. Several of these determinations have been taken from the “ Traité d’Hydraulique” of D’Aubuisson. 48 ORIFICES.—EXPERIMENTAL Data, Bossut and Others. TABLE XVI. Flow through Orvfices. ‘ CIRCULAR. RECTANGULAR. Authority. D : H | C Remarks. || Authority. | Size. H | Cc Remarks. Bossut ... | .044 | 12.5 614 Horizontal Bossut ... | .089 x .022 12.5 | .612 | Horizontal “s 089 | 12.5 | .617 Z - .089 x .089 13.5 | 617 ‘ a 1S, 12.5 | .618 , é Ms 78x78 | 12.5 | 618 ! i a 044 | 9.6) .613; Vertical Castel ... | .033x.033 | .16 .655 j 089 | 9.6 | 617 . Michelotti | .089 x .089 | 12.5 | .607 4 O44 13 .616 x - 089 x .089 | 22.4 ' .606 089! 4.3 3 619 ” i, 178x.178 = 7.3 | 603 ae f _ Surface 3 we fe ote 3 .089 | .052 649 l inolines ie o A178 x 178 | 1.2.6 | 603 : wards orlice ; Castel .. | .033' 21° .673 | a! 178% 178 | 939 602 033° 1.0 .654 7 27 x2 74 | 616 4 049 | 145 ' 1632 : ee 12.6 | .619 : 049 | a8 617 : | 27x27 | 224] 616 a O98 55.629 | #,4| Bidone ... coe L1' 620 Michelotti | .18 | 72 607 | oy Z 0380x ).121 0 Ll | 6200 = ae | Tes | 613! si {43s Ld} 626 a | 53 6.9 | .619 | | | | 53 | 12.0 =| | | by t | For the horizontal orifices of Bossut HZ was measured from the plane of the orifice. Bossut, however, was of the opinion that for a vertically descending jet, H should be measured from the section of the contracted vein, after its escape from the orifice. Rennie and other experimenters have made a great number of determinations of the value of ¢ for small orifices, but their results are still more discordant than the ones given in Table XVI. Their errors probably chiefly arose from inaccurate measurements of the size of the orifice, and sometimes doubtless from a shght rounding of the inner corners. The orifice, with D=.089 and H=.052, of Bossut was the old standard French pouce d’eau, being 1 French inch in diameter, with a constant head of 1 line above the top of the opening. Mariotte and Couplet had before determined the flow through this standard or module ; their resulting values of Care considerably higher than that found by Bossut. Errect or TEMPERATURE. Before attempting to determine what values should be assigned to c for orifices, it ORIFICES.—Errect or TEMPERATURE. 49 will be well to first discuss what effects, if any, are produced upon the discharge by changes in the temperature of the water, by slightly altered conditions of the edges of the orifices, by an irregular feeding supply, and by possible variations in the character of the water. In regard to the effect of temperature we have a number of experiments described in Chapter XI., and those made by Prof. Unwin. The Greenpoint experiments with a circular orifice, D=.020, with full contraction and with a head dropping from 3.2 to .56 in a cylindrical reservoir, gave the following results.—Nos. 125-142, Table CVI. ; With a temperature of about 48° the most probable time of discharge was 253.8"; as the temperature was increased from 48° the times became longer, until at 130° ¢ was about 257.5". As c is in the inverse ratio of ¢, this shows that for this quite small orifice an increase of T from 48° to 130° diminished ¢ about 14 per cent.. We do not conceive it possible that experimental error could have resulted in such a marked difference. Prof. Unwin with a similar but larger orifice, D=.033, with a head dropping from 1.5 to 1.0, found that ¢ was diminished about 1 per cent. by an increase in 7 from 61° to 205°. For a much larger circular orifice, D=.10, « had a value of .6013 with T'= 49°, and of .6014 for 7’=62.5°—vide Experiments Nos. 47 and 58, Table CII. This indicates that for such an orifice, with h=4.6, a change in J of 13.5° has no appreciable effect. It was thought, however, that a change in 7 of 25° or 30° for this same orifice, would change « for the same head, perhaps .0005, or about 5th of 1 per cent.. Also, that for a smaller orifice, D=.05, an increase in 7 of 30°, for heads less than one foot, notably diminished the flow. The proof in regard to the foregoing is, however, not absolute.* We are of the opinion that with orifices “in a thin wall,” the effect of ZT increases as « and h diminish. That is to say, the smaller the head and the smaller the orifice, the greater the effect which will be produced by changes in 7; ¢ diminishing as T is increased. With a large head such as 10 feet, it is not probable that a change in T of 50°— which can be assumed to be the maximum range in general practice—will cause any appreciable variation in ¢ for any orifice larger than D=.02, or .02x.02. For a large orifice, such as D=1., or 1.x 1., it is doubtful if such a change in 7 will have any noticeable effect upon c, with any head, no matter how small. With a short convergent mouth-piece, with D=.033, with a head dropping from 1.5 to 1., Prof. Unwin found a definite increase of flow by raising 7 from 60° to 190°. Weisbach, with a lower head, and a convergent mouth-piece, found with rape-seed oil * Note our remarks on experiments made in 1884, at Holyoke, Chapter XI. 50 ORIFICES.—EFrrect or TEMPERATURE. that an increase of 7 from 54° to 102° increased the flow about one half. Our experiments and those of Weisbach prove that the co-efficient ¢, with such mouth- pieces, will be practically unity with great heads such as 300 feet. In our experiments, Nos. 7 to 14, the temperature of the water was about 60°. Now there is no likelihood that an increase in Z7'to 200° would with such a head have appreciably increased the flow, which, with the lower temperature, was already at a maximum. With sucha great head, even with a very viscous liquid such as oil, we fancy that variation in 7’ would have no notable effect upon c. For the flow of water through converging mouth-pieces, we can conclude that an increase in 7’ will increase ¢ in the inverse ratio of « and h. With a large orifice, as, for instance, D=1., it is doubtful if a change in TJ of 50°, will, for any head, have an appreciable effect upon c. ConpiTIon or EDGES. In Table CVII., Nos. 143—148, Chapter XI, is given an account of experiments made with the inner face and edges of a circular orifice (D=.020) well wet with a mineral oil of fair body. The flow was diminished at first nearly 2 per cent. from the normal flow with the orifice perfectly clean ; this diminution became less and less as the escaping jet washed the oil away. With a much larger orifice, D=.10, with a normal condition of the plate ¢ had a value of .6013, with h = 4.60 (No. 47, Chapter XI.). The inner face and edges of the orifice were then well wet with sperm oil, and with the same head ¢ was .5996—No. 54. The experiment was repeated, and c had a value of .6004—No.55; the plate was then examined, and the inner edges found to be nearly free from oil. These experiments show that even for a very small orifice, when reasonable care is taken to keep the inner edges clean, no abnormal flow will occur from sheht particles of greasy matter adhering to the edges; also with a large orifice, such as D=1., even if the inner edges should be wet with oil, it would have no appreciable effect upon c. We regard the diminished flow in these experiments as being perhaps entirely due to the very thin film of oil, which reduced the section of discharge. Experiments Nos. 59 to 62, Chapter XI., were made with an orifice in an iron plate having as nearly as was possible the same diameter as the orifice in a brass plate with D=.10. The edges of this orifice appeared in August, 1885, when it was compared by a microscope of pretty high power with the orifice in brass, to be somewhat more per- fect, the edges being better defined with less rounding. The co-efficient ¢ should hence have presumably been very slightly higher for the orifice in brass than for the one in iron. By comparing the values of ¢ for Nos. 59-62 with the curve on Plate III. for D=.10 (brass plate), it will be noticed that they are about ,),,th higher, instead of being lower, as was expected. This is not a large difference, and may possibly be due ORIFICES.—Conpition oF EDGEs. 51 to experimental error. These experiments show that for orifices in either iron or brass, there is no notable change in the value of , due to differences in the two metals. So far as very slight imperfections or roundings of the inner edges of an orifice are concerned, it is apparent that the resulting effect upon ¢ will be in the inverse ratio of the area. IRREGULAR SuppLy. Careful experiments, with the circular orifice D=.10, were made by us at Holyoke, to determine what effect, if any, was produced by an “irregular” feeding supply. For these experiments, the normal arrangement is shown by Fig. 1, Plate XVII., where a screen or rack enclosed the feeding water entering the tank A from the lower end of the iron supply pipe. When this screen was in place, for the largest head of 4.6 the escaping jet appeared to be perfectly true always for a distance of .2 from the orifice ; after this distance the jet sometimes began to twist, and at other times appeared to be perfect for a distance of 1.5 from the orifice. (Experiments Nos. 46, 47, and 58, Chapter XI.) With the rack removed, and a vertical partition substituted, 3.92 high, placed across the tank in the same place as the vertical portion of the rack and open on top, the jet twisted more than it did when the enclosing rack was employed, and at times seemed to be perceptibly imperfect near the orifice. (Nos. 53 and 52.) With a vertical partition 1.86 high, the jet was ragged and twisting. (No. 51.) With a vertical partition .93 high, the jet was ragged and twisting, but not apparently as much so as with No. 51. (No. 50.) With no protection whatever for the escaping water from the iron supply pipe, the jet was exceedingly ragged and twisting, being very far from smooth immediately at the orifice, and apparently being more ragged than in any of the other experiments. (Nos. 48 and 49.) The following statement shows the values of « for the foregoing experiments ; Hea | Partitions, OPEN on Top. = 7 N tion. h Bee 3.92 high. LSGhigh. | 98 high, eee 6025 6020 6026 6039 6025 3.2 { 07} goed \ 6038 \ 6024 6024 J 6020 >.6021 6037 J 6024 J 6023 6013) re | 6015 | cor oe ae | 6010 ae 6012 6017 § 6013 oe 6008 6014 +6014 | goo7 6015 52 ORIFICES.—IRREGULAR SUPPLY. Tt will be observed that there was no variation in the values of ¢ of consequence, except with the partition .93 high, when it was about ,j9th higher than for the other experiments with the same head of 3.2. These results were exceedingly surprising to us, as it seemed almost certain that with such a very ragged and distorted jet which escaped when there was no protection whatever over the supplying stream, there must be a notable variation in ¢ from its value when the jet was nearly perfect. From the foregoing experiments it will be seen what a wonderfully delicate test of the evenness of supply is furnished by the form of a jet escaping from an orifice, carefully pierced “in a thin wall.” It is perhaps a misnomer to call the supply an “irregular” one, in the preceding experiments. The supply was very nearly regular, but the particles of water evidently found their way towards the orifice in an irregular manner. VARIATIONS IN WATER. The experiments made at Holyoke in 1885 with a circular orifice, having a diameter of .05, showed, with heads of less than one foot, astonishingly large variations in the deduced values of ¢ for nearly equal heads, considering the great care employed. The variation in temperature for these experiments only amounted to 3°; this may have been a slightly disturbing cause, but there is no likelihood that it had sufficient influence to account for the very irregular curve formed by plotting the results, with 4 and ¢ as co-ordinates. These irregularities are shown by Tables XI. and CII., and graphically by Plate III. The large number of determinations for this orifice with low heads were made for the purpose of endeavoring to fully trace these discrepancies. The chances of experimental error in these experiments will be fully discussed in Chapter XI. Taking all these chances of error into account, it seems to us almost impossible that they could have amounted to more than 755th of « for the low heads of .2 to .8, and to more than y4,th of « for a head of .8. It will be observed that there are variations in the meccius of c of over jJgth, with h=.28 (Nos. 24-27), and with the comparatively large head of .9, a variation of {,;th. For Experiments Nos. 84 and 35, consisting of 9 determinations with nearly equal heads, « ranged from .6152 to .6200, showing a variation of ;45th. These discrepancies seem to us to be beyond the limits of possible experimental error, For the circular orifice in a brass plate of .10 diameter, being 4 times larger, the values of c are exceedingly close for the largest head employed. In 13 determinations with this head (4.6), ¢ ranged from .6007 to .6017, as will be noticed by reference to the statement in the preceding section. The chances of experimental error for these last experiments were very little less than for the .05 circular orifice with h=.9, and yet for the latter orifice the variation in « is five times greater. An examination of Table CII. will show that variation in the deduced values of ¢ for nearly equal heads, as a rule, increases as both « and / diminish. ORIFICES.—VaRiIAtTiIoNs IN WATER. 53 From these results we can conclude that with small orifices, such as D=.05 and less, and with heads of less than 1., it is practically impossible to obtain single results with certain accuracy. Also, this uncertainty appears to disappear either with larger orifices or with greater heads. We conjecture that these excessive variations in « are due to some unknown change in the character of the water. Possibly it may be partly due to the varying quantities of greasy matter carried by the water. We make the foregoing suggestions in a tentative spirit, for we fully realize how easy and natural it is to underestimate the chances of experimental error. These errors sometimes aggregate in one way in the most surprising manner. We trust that some future experimenter may be able to definitely decide whether or not our conjecture is well founded. OriFices 1x A THIN WALL. Values of ¢ for Orifices tv a thin Wall. Our first task is to endeavor to determine what values shall be given to c, for vertical orifices with full contraction ; with a free discharge into the air; with the inner face of the plate, in which the orifice is pierced, plane ; and with sharp inner corners, so that the escaping vein only touches these inner edges, which constitute, as nearly as may be possible, a mathematical line. To this form, following the French expression, we give the name of “an orifice in a thin wall.” In considering the weight to be given tu the experimental data at hand, we think the experiments made by ourselves, and given in Table XI., should rank first. They were made under very favorable circumstances for a high degree of accuracy, as the orifices were pierced in rigid plates of metal, with very nearly perfect inner edges, and whose dimensions were determined with an unusual degree of care by delicate micro- meter measurement. The measuring vessels were also practically free from leakage, and of unchanging form. We trust that our readers who carefully analyze these experiments, as given in detail in Chapter XI., will not regard this preference for our own work as egotistiec. Next in authenticity will come the Lesbros experiments with his largest orifice, .66 x .66, in a fixed copper plate, and after these our experiments Nos. 2, 6, and 5 in Table X. The experiments with the five orifices in brass plates, and with the small orifice in steel, made under normal conditions in March and May, 1885, have been plotted on Plate III., with ¢ as ordinates, and h as abscissee. For the five larger orifices (brass plates) Thad a range from 48° to 54°; for the smallest orifice 7 ranged from 40° to 46°. Hence, for all these experiments the variation in T was so slight, that it could not have notably affected the general results. 54 ORIFICES.—In a THIN WALL. Values of e. From the plotted experimental points, six curves have been drawn on Plate ee representing the most probable curve for c, for each of the orifices. These curves rarely vary more than yj/5,th part from the mean experimental value of ¢, except for the .05 cireular orifice with / less than 1.. Examining these curves, we see that for each of them the value of c decreases as the head increases,* and that they become asymptotic in nature with the higher heads, indicating that with sufficiently high heads ¢ will practically become constant. The curves all begin to rise very rapidly for heads less than .3. As before stated, for these minute heads the danger of experimental error is comparatively very large ; hence, we cannot be nearly as certain in regard to the forms of the beginning of the curves, as for their forms with h=1, and upwards. Comparing the curves for the three circular orifices, having the respective dia- meters of .02, .05, and .10, it will be observed that the smaller the orifice the higher is the curve, and also as the heads increase the curves constantly approach each other. This is especially noticeable in the curve for the smallest orifice, Experiments Nos. 123 and 132, From these facts it seems evident that for these three circular orifices, ¢ will have practically the same values when / is sufficiently great. Probably with 4=50 or thereabouts, ¢ will not differ more than .002 for the three, having a value of near .595. Comparing the two curves for the square vrificcs, we see similar results, the curves constantly approaching each other ash increases. It will also be noticed for heads above .6, where the curves begin to be fairly reliable, that for the same head, the difference in « between the .1 x.1 and .05 x.05 curves, is very nearly the same as between the curves for the circular orifices, D=.10 and D=.05. For these two square orifices we can assume that ¢ will have a value of about .600, with h = 50. Comparing the curves for the .1x.1 and D=.1, and those for .05x.05 and D=.05, it will be seen that for all heads above .6, ¢ ix about .005 hicher for the square than for the respective circular orifice. If there be any change in this difference as fh increases, it is that the difference increases. Hence with great heads we may fairly assume that ¢ will always be larger for the square than for the circular orifice. Comparing the rectangular orifice, 7=.3 and w=.05 with the square orifice, .05 x .05, it will be observed that the curves continually diverge as /; increases. Hence with evreat heads, the rectangular orifice will always have larger values of ¢ than the square orifices ; with /=50, ¢ for the rectangular orifice will probably have a value from .610 to .612. On the same plate are plotted the 6 Columbia Hill experiments, Nos. 1-6. Comparing No. 2 for an orifice nearly square with 7= 1.06 and w=1.00, with No. 6 for * Experiment No. 7), with the orifice .1 x .1, appears to be slightly divergent from this law, but not beyond the limit of possible experimental error. ORIFICES.—In a THIN WatL. Values of c. 55 a circular orifice with D=1.01, we have the respective values of « of .5988 and .5913 with a head in both cases of about 1.0. Both these values seem to accord very fairly with the Holyoke results. Nos. 5, 4 and 3, with circular orifices having the respective diameters of .66, .42 and .25 are not so harmonious ; assuming the Holyoke results as correct, ¢ for No. 3 should have been about .603, instead of its given value of .6087, thus showing an apparent error of about 1 per cent.; ¢ for No. 4 should have been about .601 instead of .604, thus showing an apparent error of about } per cent. No. 1, with a long rectangular opening (/= 4.17 and w=.17), is in accord with the Holyoke .3 x .05 orifice. The considerably lower value of ¢ for No. 1 compared with the curve for the -3 x .05 orifice at the same head, being fairly attributable to the much greater width Git 2.05). We were unpleasantly surprised after the computation of the Holyoke work, to find these disagreements with the Columbia Hill experiments. Although these defective results were obtained by single determinations we regarded them as quite reliable, and thought the given values of ¢ should be within ,i,th of the truth. As the Holyoke experiments were conducted with a greater degree of care than those made at Columbia Hill we must regard the latter as defective, when the comparative results are contradictory. The heavy dotted curve on the same plate, No. III., represents the most probable curve for ¢ for the Lesbros orifice, .66 x.66. (Experiments Nos. 156-160, 161-165, and 204-213.) The lowering of the values of ¢ with the small heads appears to be due to the comparatively small depth of water above the summit of the orifice, or more strictly to the surface inclination towards the orifice. Computing c’ from /7, for experiments 210-213, Table VIII., we have the following results ; No. 210 H=1,3140 H, =1.3120 e=.6017 e’ =.6025 3 7940 7897 5997 6014 «a 2 5249 5141 5982 6050 = 218 4003 3727 5927 6186 From the foregoing we see that with « deduced from the head immediately above the orifice, its value rises with diminished heads. The curve for the Lesbros orifice is apparently about .003 too high, in comparison with the Holyoke .10 x .10 orifice, and the Columbia Hill 1.06 x 1.00 orifice. The Ellis experiments give about the following results ; Square, 2.x 2, H=2.1 to 3.5 c= .607 5 Tse H=3.7 to 18. c=.600 Rectangular, 2. x 1. fH =3.0 to 11. c= .600 - 2.x 5 iT =1.4 to 17. c=.606 Circular, D=2 H=1.8 to 10. c= .606 9 D=1. H=2.4 to 18. es 2 D=.5 H=2.1 to 17. e=.601 2» They approximately indicate that for such large orifices, with heads from 3. to 17., ¢ remains constant; also that probably c is smaller for circular than for square orifices, 56 ORIFICES.—In a THIN WALL. Values of ¢. with D=side of square. These results, considering their chances of experimental error, are fairly in accord with our experiments and those just referred to from Lesbros. The experiments quoted from Weisbach, for circular orifices, show results for the smallest diameter agreeing pretty-well with our Holyoke curves; his values of ¢ for the three other orifices appear to be 1 per cent. or more too high. The experiments of Prof. Unwin* show considerably lower values of ¢ than we found. The experiments of Bossut, Michelotti, Rennie, et cet. et cet., generally show larger values of « than those given by us. On Plate IT. are plotted most of the experiments of Lesbros and Poncelet-Lesbros, given in Tables VII. and VIIL, with ¢ and / as co-ordinates. Comparing these results with the curves drawn on Plate III. for the Holyoke experiments, they present wide divergencies ; the values of « for all the Lesbros orifices less than .66 x .66, being higher than is indicated by Plate III. for similar orifices and the same heads. For instance, with a square orifice .066 x .066, with h=1.93, Lesbros finds a value of ¢ of 6366. (Experiment No. 201.) According to our Holyoke curves, for such an orifice and head, c should be about .612. This is a very great difference, amounting to nearly 4 per cent.. We can suggest no reason for this disagreement, unless it be the errors resulting directly and indirectly from the use of the sliding gate employed by Lesbros for decreasing the height of his orifices. This conjecture is in part confirmed by the fact that with his small orifice in a fixed plate, /=1.97 and w=.066, the values of ¢ appear to differ not more than 1 per cent. from our Holyoke results? Our Chapter on Weirs was written before the Holyoke experiments had been made, and in our determination of the co-efficients of weir discharge, we have made considerable use of Lesbros’ experiments with weirs, which seemed to us reliable within about 1 per cent. in the values of «. It was hence especially disagreeable to find our orifice experiments differing as much as 4 per cent. from those made by an authority we had so largely used. The Holyoke experiments were made with such care, the resulting curves are so sym- metrical in themselves, and so harmonious one with another, that we cannot conceive it possible that they are wrong, and those of Lesbros are right. Considering the experimental values shown on Plate ITT. as authentic, we can now enunciate the following general propositions for orifices in a thin wall, with « not over 1. and any side not less than .01. For similar forms, iw matter how much a may differ, with a great head such as 100 feet, the co-efficient of discharge, ¢, will be practically the same. * In these experiments ¢ was the important quantity, for the determination of effect of changes in temperature. The value of ¢ was only determined incidentally, as its exact value had no bearing upon the experiments. + The values of ¢ for this Lesbros orifice should probably be a little higher than those for the Holyoke .3x .05 orifice, the heads being the same. ORIFICES.—In 4 THIN Wai. Velues of e. 57 Each form with great heads will have a distinctive and coustent calue of e. With great heads, e will have a value of about 592 for cfreular forms ; about 598 for square forms ; aud about .610 for rectangular forms with b= 6 uw, It is quite probable that the foregoing propositions will apply to orifices of any size, the feeding reservoir always beine large enough to insure perfect contraction. The Lesbros experiments, as given on Plate IL., indicate that with great heads ¢ will be about the same for all his orifices. M. Graéff states that with an iron gate or valve at the Furens dam, ¢ is constant under great pressures, with the valve in different positions. "ide “ Traité d’Hydraulique,” 1883. In the following tables, constructed from the curves on Plate III. as a basis, are given the values of ¢ for square and circular orifices, for areas up to 1., and with heads from .3 to 100. The heads must be measured some distance from the orifice, whenever the vertical height of the orifice is greater than ,- The given values of ¢ are very nearly identical if, : ‘ ‘ with C, when - or 7, is over 4; the co-efficient C’can be readily obtained by the use of Tables III. and 1V. To insure a high degree of accuracy the orifice must be carefully pierced through a rigid metallic plate. 58 ORIFICES.—In a THIN Watt. Values of ¢. TABLE XVII. Values of Co-rfficient ¢ for Square Vertical Orifices, with sharp Edges, full Contraction, and free Discharge in Air. Nots.—T7o obtain Co-eficient C,in G@=Ca(2 gh)4, use Table ILI, Head ! from LENGTH OF THE SIDE OF THE SQUARE. Centre of 7 ane ee ee ee, ; 10) 12 | 15 | 20 40 | 60 | MO) 10 3 (2) 642. 632 | 624! 617 | .612 | 4 643 | 637° 628.621 | 616 611 ; 5 648 , 639 ' 633 | 625; 619 614, .610 | 605. 801 | 597 6 660) .645 | 636.630 | 623 617 618! 610! 605 601) 598 546 7 | 656} 642: 683 | 628 621 | .616 | 612 609 | 608 | 602 | 599.598 596 8] 652.639} .631 | 625 | 620) 615 611 ; 608 | 605} 602 | .600 | 598.597 9 |} 650} 637 629 .623 619 | eu | 610 | 608 | 605, .603 | 601) 599 598 1.0 || .648 | .636 | 628 | 622, 618, .613 | 610 | 608 | .605 .603 | .601 ; .600 | .599 1.2 | 644 633 | .625 | .620 | 616 | 611 | 609 .607 } 605, .604 | 602 601 | .600 14 || 642 | .630| .623/ 618) 614 610 es 606 | .605 | 604 602.601 601 16 640.628 621 617.613 | .609 | .607 | .606 | a0 605 | 603} .602 601 Ls 638 | .627 620 | 616 612 | 609 | .607 .606 | .605, 605, 603 .602 .602 2.0 637 | .626 | 619 | .615 i 608 06 606 | .605 | 605.604 602.602 25 | 684 624° 617 613} .610| .607 |) .606. .606 | 605 | .605 ie 603 , .602 3.0 632 622 616! 612 .609| .607 | 606 606 | 605 | 605 604 | .603 . .603 3.5 630.621 | 615 | 611 | 609! 607 606.605 | 605 605 604) .603 602 4 628 619 | 614) 610, 608} .606 | .606 | 605, .605 .605 603 603 602 5. | 626! 617° 613 610! .607 | .606 oe 605 | .604 | 604 | .603 | .602 ° 602 6 623 616 | .612 | .609 | sor | .605 | 605.605 604) .604 603 | 602 / 602 7 621, 615 .611 | .608 | 607 605 fe 604 ! 604 | 604 | 603 | .602 602 8 619.613 | .610' 608 | .606 | 605 604 604 ee | 603 | 603, .602 .602 y 618 | 612 609 © 007 606 | 604 604 604 603.603 | 602 602 601 10. | 616 611 | .608 | .606 | .605 604} 604 603 1.603.603, 602 | 602.601 20. | .606 | .605 | 604 .603 | 602 602 | .602 | .602 | 602 | 601} .601 | .601 | .600 0. (®)|, .602 | 601 | 601 | .601 | .601 600 | .600 | 600 | .600» .600 | 599 1599 1.599 100. (%) | ; «288 , .b98 O98, 598 98 O98 OS 98 ogs | ORIFICES. In A THIN WaLL. Vulues of e. 59) TABLE XVIII. Values of Co-efficient ¢ for Circular Vertical Orifices, with sharp Edges, full Contraction, and free Discharge in Atr. Nore.—7 obtain Co-efficient C,in Q=Ca (2 gh)¥, use Table IV. Head from . ates Centre of | | Oritice. | .02 | 08 O-+4 05 | .O7 | 10 | AD) 15 | 20 | .40 | .60 .80 | 1.0 DIAMETERS. | ; 621 » 613 .GO8 | Le 641 628 | 620 | .615 | .610 | 606 .604 | .602 - 600 98 | 596 . 594 | 592 14 | 638 | 625 | .618 .613 609 | .605 .603 see es 599 596 | 594 | .593 L6 ° 676 624 |.617 812 | 608 | 80S | 002 | BU | 600 ~ 209 " OT | 0D | aot 1.8 | 634 .622 | 615 °.611 | 607 | 604 . 602 | .601 | 599 | 599 O97 ; 595 | .595 2.0 632 | .621 | 614 | .610 | .607 | 604 . 601 | .600 : 599 | 599 597 | 596 | .595 2.5 .629 | .619 | .612 608 | .605 | .603 ' 601 600 599 599 | 59S | 597 | .596 27 | 617 | 611 | .606 | .604 | 603 | .601 | .600 .599 | 599 | .598 | .597 | .597 625 | .616 | 610 | .606 | 604 | 602 | 601 | .600 | 599 | .599 23 | 614 | 609 | 605 | 203 | .602 “600 299 899 | B98 | SOF | BOT | 596 621 1.613 | .608 | .605 , .603 | .601 — 599 | 599 ) .598 | 598 597 | 596 | .596 618 | 611 | 607 | .604 | .602 | 600 | .599 + 599 | 598 | .598 | 597 596 | 596 | 616 | .609 | .606 | .603 | .601 | 600 599 | 599 | .598 | 598 A9T | 596 | .596 8, 614 | 608 | .605 | .603 | .601 | .600 | 599 | 598 | 598 | .597 30 . baz 628 * | | A | 637 | 631 | 624 | 618 | 612 | .606 | | | | a (G43 G38 627 621_GI_ 610 | .605 - 600 596 92 6, .655 eo 630 624 618 | 613 | 609 | 605 | 601 596 , 593 | .590 7 | 651 | 637 | .628 .622 ' .616 | .611 ‘ .607 | 604 | .601 597 | 594 | 591 | .590 8 . 648 634 | .626 | 620 | 615 | 610 606 | 603 601.597 | 594 | 592 | 591 9 | 646 632 | 624 | 618 ; 613 | 609 | 605 | 603 | 601 | 598 | 595 | .593 | .591 Lo. | BW «68 | 203 ny 612 | 608 605 | .603 600 | 598.595 | 593 | 591 Bl or Su ae oS a | 596 | 596 | .596 9. 613 | .607 | .604 | .602 | .600 | .599 | 599 598 | 597 | 597 | 596 | 596 | .595 10. ' .611 | .606 | 603 | .601 | 599 | 598 | 598 .597 | 597 | 597 | .596 | 596 | .595 596 | 595 | .594 20. .601 | .600 | 599 | 598 | .597 | .596 oe 596 |} 596 | 596 50. (2) .596 | 596 | 595 | .595 | 594 | 594 | 594 | 594 | 594 | 594 100. (2) .593 | .593 | 592 | 592 | 592 | 592 | 592 | 592 | 592 | 592 | 592 | 592 | .592 cae | We have not sufficient data to warrant the construction of a table, giving the co-efficients for rectangular orifices. We conjecture that an orifice haying w=1., and various lengths greater than a, will give values of ¢ not very much greater than for an orifice 1.x 1.. The experiments of Ellis, Nos. 247-255, seem to verify this conjecture. If this be true, an orifice with #=.01 and /=6 1, should show a greater increase of c, compared with an opening .01 x .01, than does our orifice of .3 x .05 compared with the 60 ORIFICES.—Iy a THIN WaLL. Values of ¢. one .05x.05. If the orifice .8x.05 had been increased in length considerably, say to 1., the resulting co-efficients would probably only have been a trifle larger. The following general principles must be kept in mind, when it is required to assume a co-efficient of discharge for rectangular orifices, whose least dimension is less than 0.8. Zhe co-efficient ¢ increases as the least dimension dvercases ; this rate of fnerease is yreatest with very low heads, aud but slight with such a considerable head us 10. Fore rectangular orifier ¢ will have a larger value than for a square orifice, awhoase side is equal to the least dimension of the veetangle ; as the ratio between the two sides of the rectangle ‘nereases, the least dimension being constaut, the cuerease in € becoines less aed less rapid, By keeping these principles in view, a pretty accurate guess, as to the proper value of ¢ for rectangular orifices, can be made by a study of Plate ITI. Measurement or 1. With a horizontal oritice, should H he measured from the plane of the orifice, or from the proper section of the contracted vein? We had hoped to have experimentally demonstrated this problem at Holyoke, but the investigations made there with vertical orifices consumed sv much time, that we could not make the desired experiments with horizontal openings. The proper way to determine this interesting question is, to first obtain the curve for c with a viven orifice placed vertically, and then to ascertain the discharge when it is placed in a horizontal position. It is most probable that A should be measured from the contracted vein. Now if this be so, should not H for a vertical orifice be measured also from the section of the contracted vein in order to obtain accurate theoretical results? With small heads the jet begins to drop appreciably immediately after its escape from the plane of the opening. Hence were H measured from the contracted vein for such small heads, its value would be greater than when measured from the centre of the orifice, and the deduced value of ¢ would consequently be smaller. SHare or Escartna VEIN. A very singular phenomenon was noticed in Experiment No. 40, with the circular orifice (D=.10) with the lowest head. The jet was perceptibly flattened soon after its escape from the orifice, and then enlarged and diminished three times before the jet finally struck the floor, some 7 feet vertical below the orifice. The section of the descend- ing jet, however, became smaller and smaller, as it was accelerated by gravity. That is to say, looking at the jet sideways, the eye being normal to the plane of the jet, the swelling in the jet was at a point where the stricture was seen by the eye in the plane of the jet. The following sketch illustrates this phenomenon ; the heavy line indicat- ing the outline of the jet from the side pomt of view, and the dotted lines its outline from the front point of view. ORIFICES.—Suapr or Escapina VEIN. 61 Vertecal Orifice This peculiar form was doubtless caused by the velocity of the particles of water escaping from the bottom of the orifice, bemg much greater than that of the particles at the top of the orifice. Why the descending jet, falling freely in air, should maintain this recurring elliptical form, is an interesting problem for mathematical enquiry. Some- what similar phenomena have been discussed by Bidone, Buff, Magnus, Savart, and lately by Lord Rayleigh in the Proceedings of the Royal Society, Vol. XXIX., p. 71, 1879. Lord Rayleigh is of the opinion that the explanation for such recurring forms given by Buff is the correct one, which is that it is caused by capillary action. In this connection we are reminded of a singular phenomenon in long, uncovered, rectangular, inclined “shoots” for water, where the wave action is rhythmic. In one of these “shoots” at North Bloomfield, California, the length was about 1500 feet witha steep incline; the width of the “shoot” or wooden flume was about 3} feet; the supply of water was regular, and sometimes amounted to 70 cubic feet a second. The water soon after its entrance at the head of the ‘‘shoot” assumed a wave motion, each wave being of great length and separated from the succeeding wave by quite a little interval of time. It would be an interesting philosophical study to investigate such wave movements and determine in what manner they are affected by changes in quantity, inclination, and depth. Such “shoots” are very common in the Sierra Nevada of California, and the phenomenon could there be readily investigated, under widely varying conditions. SUBMERGED DISCHARGE. Experiments Nos. 97 to 121, Table XII., were made with the 5 brass plates at Holyoke, placed in a vertical position. The details of these experiments are fully given in Chapter XI.; the resulting values of c will be found on Plate V. The solid curved lines represent the most probable values of c, for the orifices with a free discharge ; the dotted curved lines represent the most probable value of ¢ for the submerged discharge. The values of ¢ (for submerged discharge) for the .3 x .05 orifice are abnormally high, owing to the comparatively thick divergent sides of this orifice. For the other orifices c is also slightly too high from a similar cause. 62 ORIFICES.—SUBMERGED DISCHARGE. Examining Plate V. it will be observed that the dotted curves with increasing heads drop pretty rapidly for the two smaller orifices, and much more slightly for the two larger orifices (.1x.1 and D=.10). The values of c for submerged are in all cases less than those for free discharge, for these four orifices ; this difference becomes less marked as the orifices increase in size, and as the heads increase. The maximum difference between c for submerged and for free discharge is .0189 for the orifice 05 x .05, with h=.35; this is about 3 per cent.. The minimum difference is for the orifice .1 x .1, with / =3.95, when it ix.0014, or about 1th of 1 per cent... The retarda- tion of discharge caused hy submerged discharge is probably measured by the wetted perimeter p, while the quantity is measured by « (/)*. We can hence generalize as follows, with reasonable safety ; The co-epicwnt ¢ for submerged orifices will be always less than for the same orifice with free discharge, h betng in both cases the same, but with great heads this difterence will become tnappreciable, This difference diminishes as the size of the opening increases, so that for orifices as large as 1.x 1., or D=1., probably vt becomes (nappreciable, except perhaps for quite small heads. In speaking of orifices with free discharge, we mean those having a sufficient head above the summit of the opening to insure full contraction. An orifice 1.x 1., witha head of say .2 above the summit of the opening, will have an abnormally low value of c, especially if AZ be measured several feet back from the orifice. The same orifice sub- merged one or two feet, and with the same head of .7, or .7 =$+.2, will in all probability have a higher value of ¢, than that deduced from the free discharge as above. The Ellis experiments with submerged orifices, 1. x 1. and D=1. indicate thatc is about 1 per cent’ higher than for free discharge. Weisbach states that c for submerged will be about 1} per cent. smaller than for free discharge. Francis thinks that c for submerged is about 5 per cent. less than for free discharge. Steckel, with horizontal orifices, found that c for submerged was +1 to 5 per cent. less than for free discharge > in his experiments, however, ¢ for the free discharge was deduced from // measured from the plane of the orifice- (JUICKSILVER AND OIL. On Plate ITV. will be found curves for ¢ for the discharge, through the sane circular vertical orifice having a diameter of .02, of thick oil, water and quicksilver. These experiments—Nos. 152-155, 122-130, and 149-151—are fully described in Chapter XI. The experimental data for the water and quicksilver are sufficient to establish the curves for c, with h from .5 to 3. For the oil only one value of ¢ was determined, being that fora head dropping from 3.2 to .58 in a cylindrical reservoir. Nos. 122 to 124, were made with constant heads for water, and are much more reliable than Nos. 125 to 130, which were made with “dropping” heads. It will be observed that Nos. 126 and 129 for the “dropping” heads, give slightly too low values for ¢, compared with the curve deduced from the constant heads. ORIFICES.—QUICKSILVER AND OIL. 63 . The form of the curve for the quicksilver does not differ greatly from that for the water. This establishes the interesting fact that quicksilver in its flow from such an orifice follows the same law as water; ie., ¢ diminishes as the head increases, and the curve becomes asymptotic with large heads. With great heads, such as 100 and over, It is probable that quicksilver for circular orifices may have a practically constant value for c of about .580 or .590. It is possible, however, that with such great heads quicksilver may have the same co-efficient as water, which for circular orifices we have assumed would be about .592. Comparing the three liquids, it will be seen that the flow of the most viscous liquid was much the greatest, and that of the least viscous the least. Hence we can say for ordinary heads, such as 4 feet and less, that the more viscous a liquid the greater will be the value of c. Sufficient liquidity of course must be maintained, in order that the flow shall be uniform of any very viscous liquid. It would be very interesting to determine the co-efficient of discharge of thick oil from an orifice with a head of 100 feet or over. Tt is not at all improbable, that with such a head it would be about the same as for water. If this should prove to be the case, the proposition might be warranted that with great heads, and the same form of orifice, all liquids have the same value of c, disregarding the retarding effect of the air, which would be notable with very light liquids. The very large value of ¢ found by us for the oil, tends to show that, especially with low heads, if water has much oily matter in it, thus increasing its viscosity, the quantity of flow may be somewhat changed from what would be discharged with purer water. Weisbach found that quicksilver had a lower value of ¢ than water, which agrees with our results. For rape-seed oil he found ¢ slightly higher than for the quicksilver, and considerably lower than for the water. In our experiments—Nos. 152 to 155—the only uncertainty as to reasonable accuracy, was caused by the jet of oil in places adhering more or less to the divergent sides of the orifice. The jet, however, at no time filled the divergent “tube” formed by the sides of the orifice, and it is not likely that the dripping along the lower side added largely to the flow, and very possibly it did not increase the flow at all. In concluding this discussion of the flow of liquids through a thin wall it may be remarked, that in our judgment, Weisbach and others have placed the loss of energy in jets escaping from a thin wall at too high a figure. With large heads, such as 100 feet and over, we conceive that the loss of energy is inappreciable. Some experiments made by us, with a head of about 300 feet, showed that a jet escaping from a ring, with nearly full contraction, was slightly more effective than a jet from a converging mouth-piece, where ¢ was practically unity. Authors, when discussing the flow through orifices in a thin wall, very frequently speak of losses by “ friction”—this seems to us an incorrect phrase. The inner edges of O+ FLOW THROUGH ORIFICES—In a THIN WALL. a perfect orifice form a geometrical line, on which there can be no friction proper. With low heads there may be losses by cohesion of the liquid particles, and by eddying move- ments, but these loxxes can hardly be large. The outgoing water feeds the escaping vein in a uniform manner, so that there can be no considerable losses by eddies. This is in part indicated by our experiments Nos. 48 and 49, where an “irregular” supply, which distorted and greatly twisted the escaping jet, had no appreciable effect upon the co-efficient of discharge. It seems clear to us that the loss of cnervy for a square or circular orifice must diminish as h or « increases. For the loss will be approximately measured by p, while the quantity is measured by « (h)”. The law of discharge is just the reverse, for ¢ is greatest with the smallest orifices and the least heads. This is perhaps due to the form of contraction being moditied by the narrow limits of the sides, and which modification disappears with the great velocity due to high heads. The diminished dischareve of circular orifices, compared with square ones, leads to the conclusion that the inereased discharge results from the peculiar form of contraction at the four corners of the square opening. But if this be so, one would suppose that in a rectangular opening, having its least dimension equal to the side of a square, the value of ¢ would be less than that for the square, as the influence at the corners must be proportionately less ; we have seen that just the reverse of this is true. Contraction Mopirtep. The effect of partial or complete suppression of contraction on the sides of an orifice, by bringing the side or sides of the feeding canal so close to the respective side or sides of the orifice as to interfere with contraction, can be determined pretty fairly by the Leshros experiments, with the forms of approach shown by Figs. 4-12, Plate I. The results of these experiments are given in Table IX. In the following table are given the valucs of c, with h=1., 2., 3., and 5., for the forms of approach shown by Figs. 1, and 4 to 12 inclusive; these valucs have been obtained by interpolation from the viven values of « in Table IX., and from Plate IT. The orifice was square, in a fixed plate, the side being .656. ORIFICES.—Conrraction MopIFIEp. 65 TABLE XIX. Lesbros.— Values of « for carious Heads, with a Vertical Orifice 656 x 656. Forms of Approach variously modified, as shown by Figures on Plate I Free Discharge. | | | Distance to Sides of | Canal, from Respec- | Heads, in Feet. tive Sides of Orifice. } i ia | 7 7 Remarks. 2) @) Z v fio 2a 30 50 | iy Mae fe og 601 | 604 | .605 .603 | Full contraction. 4 QO 5.7 57 624 , .624 624 624 | Suppressed on bottom. 5 0 066 5.7 : 636 | 636 .637 .637 5 eS 5, and nearly on one side. e |] of 066 | .066 | 681 | 670 | .665 662 | 7 ae 2 , _ both sides. t 0 0 _ 0 695° .685 | .677 .673 . Suppressed on both sides and bottom. 8 . L& | .066 5.7 .608 ! .610 : 612.611 | Nearly suppressed on one side. 9 | 18 .066 .066 . 635° 631: 629 627°, j , both sides, 10 | 18 0 0 » 645.640 | .639 | 637 | Suppressed on both sides. | 11; 0! 066 .066 | .647 645 O14 641 12) 18 | .066 066 609 | 611) 612 611 Fr Figs. 11 and 12 the side approaches were inclined at an angle ‘of 45°; vide Plate As before remarked, in some of the forms of approach—notably Fig. 7—there was a considerable loss of head, due chiefly to primary contraction as the water entered the mouth of the feeding canal, the head H having been measured in the still water above the canal; hence the foregoing results do not sufficiently indicate the increased flow caused by suppression of contraction. The head producing the velocity of approach in the feeding canal is, of course, included in the measured head HZ. By reference to Table XIX., it will be seen that as suppression becomes more and more complete, the value of ¢ increases. We can show this clearly by the following statement of the percentage of increase above ¢ for full contraction, with different degrees of suppression for the 4 heads. Fig. Amount of Suppression. M=)., H=2.. ees i Bs H=5 Contraction nearly suppressed on one side.. 1.2 1.0 1.2 1.3 3 suppressed on one side.. ee ae ise ie a 38 | 3.30) 3.1) 3.5 fe nearly suppressed on two sides ee ie Pee goa|| hie 4.5 4.0 4.0 5 : $9 suppressed on one side and nearly suppressed on another side | 5.8 5.3. 5.3 5.6 10 | 4 ss two opposite sides ane 2 ie | 7.3! 6.0 56 5.6 6 | 3 " one side and nearly suppressed on two other sides} 13.3 | 10.9 9.9 9.8 c 4 ‘ feegidey ews ee, TS 66 ORIFICES.—Conrraction MoDIVIED. These results are fairly uniform, showing a constant increase in c as suppression of the wetted perimeter increases, and also that as the heads increase the effect of sup- pression diminishes. This latter result, especially for Fig. 7, is in part produced by the loss of head spoken of. The side approaches, placed at an angle of 45°, show a discharge considerably less than when the sides are normal to the orifice, as will be seen by comparing the results in Table XIX.; those of Fig. 11 with Fig. 6, and those of Fig. 12 with Fig. 9. For the Lesbros orifices with /=.656, and w=.328 and less, we can compactly show the effect of four forms of approach in the following statement, which gives the percentage of increased discharge caused by suppression for each orifice.* It is based upon the results given in Tables VITI. and IX. o ! i g i 6 Orifice. Fig. 4 Fig. 9 Fig. 5 Fig See | Heads. | Heads. Heads, Heads. I w | the | 6. | 1 3. | 5 a UN ge | Boy | B | 6 Fhe! I so eer 24 eee A ees nd) | .656 | 3.8 3.1 31 tr Be 40 4.0 5.8 | 5.3 | 5.6 13.3 9.9 9.8 328 7 5.4 5 5.4 |) 3.2 | 2.4 | 3.1 7.0 Ae te HO) 10.7 9.8 | 10.0 656. 164 | 6.3 6.5 74 | 16 L4 2.4 T3 Ted 8.4 | 8.9 | 8.6 9.5 098 | 7.8 7.6 8.5 3.4 21 2.4 8.1 8.8 9.2 | 9.6 | 9.2 9.5 .033 8.9 | 11.1 | 124 " 4.5 5.1 5.6 85 | 11 ) 12.2 | 85 | Le | i eae |: t | The above percentages show fairly reasonable results for the form of approach, Fig. 4; they constantly increase as w diminishes, because the ratio of suppression to the wetted perimeter, », constantly increases ; for w=.656, calling the suppressed line S, we have“ =.25, while for w=.033, : =.48. The results for Fig. 9 do not seem to be so satisfactory ; in this form of approach there was full bottom contraction, and Z and L/ were each .066; therefore as w diminishes it is almost certain that the percentages should decrease ; especially for the smallest value of w, .033, it seems to us that there must have been nearly perfect contraction, and hence for this width the percentage should have been barely appreciable. As we have before observed, these Lesbros experiments with w less than .656, where the sliding gate was employed, are probably much less exact than those made with the orifice .656 x .656. * Let ¢ be the co-efficient of discharge, with full contraction, for any one of the three heads, and for any one of the five orifices; taken from Table VIII. Let c’ be the co-ef. for the same head and orifice, and for any one of the four ce forms of approach ; taken from Table IX. Then will be the percentage of increase as given in the statement. ORIFICES.—Contractrion MOopiriep. 67 Bidone’s experiments showed* that the increased discharge by partial suppression of contraction of rectangular orifices, was nearly in proportion to 15.2 per cent. for com- plete suppression, or S=p. This for square orifices would give ; Contraction suppressed on one side, 3.8 per cent. increase. si 5 » two. sides, 7.6 33 11.4 These determinations for one and two sides agree closely with Lesbros’ results, with h=1.; for three sides Bidone’s percentage is less than that of Lesbros. With circular orifices, Bidone found that partial suppression up to {ths of the perimeter, increased the discharge in the ratio of 12.8 per cent. for complete suppression. If contraction be suppressed completely, the approach becomes a tube; this form will be hereafter discussed in the Chapter on Pipes. ” 2” ” ” ” three Xe ” With convergent mouth-pieces, the co-efficient ¢ will be about .95 for small heads, when the mouth-piece is formed with care. For very high heads ¢ practically becomes unity, even when the mouth-piece has straight converging sides, with the angle f rather small. It also seems established by our experiments (Nos. 15-18), that with great heads the channel of approach need not be as large in proportion to the orifice, in order to produce complete contraction, as with small heads. The results of Lesbros in general confirm this proposition. Experiments Nos. 89 to 96, Chapter XI., were made by bisecting an orifice with £=.30 and w=.05, by brass vertical sheets of various thicknesses. A very thin sheet (Nos. 89-92) had very little effect either upon the co-efficient c, or upon the form of the escaping jet. A sheet .04 thick, which reduced the area from .015 to .013, increased the value of ¢ nearly 1 per cent.. These experiments (Nos.94-96) illustrate the effect of an incomplete suppression of contraction, as the escaping jets united at a short distance from the plane of the orifice. * Vide D’Aubuisson’s ‘‘ Traité d’Hydraulique.” 68 CHAPTER IV. VELOCITY OF APPROACH. Wuen, for either an orifice or a weir, the feeding stream passes with an appre- ciable current the point where the surface height of the water is measured, then an additional force due to this velocity of approach presents itself for consideration. This additional force will evidently increase the discharge. In all the experiments we have selected of the discharge through orifices, the velocity of approach wax so inconsiderable that it could be neglected without sensible error, and in general with orifices the feeding canals are made so large in proportion to the quantity discharged, that v,—the mean velocity in the feeding canal as it passes the measuring point for the head—is an inconsiderable factor. With weirs, how- ever, and especially with those having end contractions suppressed, +, is often a considerable quantity, and in extreme cases may largely increase the discharge. Hence, before discussing the flow over weirs, it becomes necessary to determine what ettect Yq produces. The head, h,, required to produce 7, is approximately represented in +,=(2g/,)", 2 Vv, orh,=—. 2y This expression is not strictly accurate, as the threads of water in the cross section of the canal at the point for H, have widely varying velocities. If we could divide this section into a number, 2, of minute equal sections, and determine the velocities 7,,7,, ct cet., of the several sections, the sum of the squares of the several velocities, divided by », would much more nearly give the frue value of v,°. This result would unquestionably somewhat differ from the square of the mean velocity. It is sometimes 2 . . * Vg assumed that the true velocity head is necessarily greater than rr The question to be solved is, what portion of the kinetic energy, due to the head i, for the section «,, produces a useful effect at the opening of discharge ? As to this point there has been a great diversity of opinion among hydraulicians and mathematicians. Some have thought that the entire kinetic energy of the water, as it passes the measuring point for H, should be considered in computing the effective head, /. Others substantially make / =/7+h,, thus assuming that the additional forec VELOCITY OF APPROACH. 69 is that due to the velocity of a section of the feeding stream, having the same area as the opening, the velocity of this section being taken as 7,. This subject has engaged the attention of many great minds during the past two centuries, and the discussions and controversies in regard to it, if collected, would fill several good-sized volumes. If the same amount of thought and time, devoted to these almost fruitless discussions, had been employed in ascertaining what the /vcfs really were, we would know a good deal more about the effect of velocity of approach than we do at present. EXPERIMENTAL Data. Lesbros, (1.25 w,) . Lesbros made h= H+ 1.56 h,=H+ , in a number of his experiments, where the crest of the weir was placed at various heights above the bottom of the feeding canal. The quantity 1.25 v, was assumed by him to be the central surface current in the canal. These experiments of Lesbros have, however, but little value, as the problem was greatly complicated by a primary velocity as the feeding water escaped from an orifice with considerable pressure, into a feeding canal of comparatively short length. Fteley and Stearis. Messrs. Fteley and Stearns have lately made a large number of experiments for the purpose of determining the effect of velocity of approach. A full description of these investigations has been published in the Transactions of the Am. Soe. of C.E., pp. 1-118, 1883. From their results we will, in a great measure, draw our final conclusions. The experiments made by these gentlemen bear every evidence of great care, skill, and honesty. The data they have given illustrating the effect of velocity of approach, and their determinations of the quantity of flow over weirs of various kinds, constitute one of the most valuable contributions to experimental Hydraulics, which has been given to the world in late years. They placed a horizontal sharp-crested weir, 5 ft. long, with vertical sides, at the end of a horizontal rectangular open flume or canal, having the same width as the length of the weir; the end contractions were hence entirely suppressed. The plane of the weir was normal to the axial line of the canal. This canal had a false bottom, moving vertically, but always retaining its horizontal position, extending from the weir a distance of 17 feet up-stream; this false bottom could be placed at will at the respective distances of .5, 1., 1.7, 2.6, and 3.56 feet below the crest of the weir. A constant volume of water was admitted to the upper end of the canal, and the surface height above the crest was measured, as the section of the feeding stream was 70 VELOCITY OF APPROACH.—ExreriMEnts. Fteley and Stearns. changed by lowering or elevating the false bottom of the canal, thus forming one series of experiments. The supply of water was then increased, and another series of surface heights measured. There were 21 series of experiments thus made, Q being constant for each series, with H ranging from .1930 to .9443, when there was the maximum inner depth, G', of 3.56 below the crest. With the greatest value of Q ; Hf, with G =3.56, was .9443 re ae As the false bcttom was raised for each series, the section of the feeding stream was diminished, and consequently », increased. Hence by comparing the varying values of H—observed lead above crest—with the varying values of h,, deduced from (g, 1t was thought that the effective value of h,, or h,', could be determined. The measuring hook-gauge, reading to .0001, determined the surface elevation of the water above the crest, at a point on the side of the canal 6 feet above the weir. The discharge was free into the air, except that the sides of the canal were con- tinued beyond the crest; this projection, however, not extending lower than the crest. There were 94 experiments made with end contraction suppressed. In such experiments it seems apparent to us that there are three causes to be taken into consideration, which produce the changes in H, with varying values of (@, @ remaining constant. They are; First.—That force due to »,, which increases 1,* and hence lowers H. Seconp.—The partial suppression of contraction, caused by bringing the bottom of the canal nearer the crest, which changes the shape of the escaping contracted vein, and increases 7, and hence lowers JZ. Tuirp.— As the section of the feeding stream diminishes, there results an increased amount of loss of head between the measuring point for Hand the crest, due to increased resistance between these points; this decreases +, and hence elevates H. This is repre- sented to some extent by the diminished value of + in the canal, 7 = “*, and is ) further affected by the increase in 1, ; an approximate determination of the amount of this increased resistance presents a most complicated problem. We have seen with orifices, that the effect of suppression by straight approaching sides at right angles to the plane of the orifice, adds as much as 15.6 per cent. to the discharge, when suppression extends to three sides of a square orifice ; and with con- traction partially suppressed on two opposite sides—orifice .656 square, and channel of * It must be kept in mind that v is the velocity in the plane of the weir, having a section with the length, Ul, of ( a a A We will in no case regard v as the actual velocity in the plane of the weir, which would be expressed by PS ‘ the weir, and the height H or h, depending upon our conception of the effect of 7’; hence v= VELOCITY OF APPROACH.—ExpErimEents. Feeley and Stearus. 71 approach .787 wide—the discharge was increased 53 per cent.; vide Table XLX. ef seq., the normal discharge being with full contraction. It is apparent that with weirs, sup- pression or partial suppression of contraction will produce similar results; this will be directly shown hereafter, in our analysis of weir experiments. The third cause is, with the 94 experiments now under consideration, of minor importance compared with the first and second causes, and its separate consideration can be omitted without producing serious errors. In certain cases, however, this increased loss of head, probably chiefly produced by adhesion of the water to the three sides of the feeding canal, may produce a greater effect in elevating H, than the lowering effect upon H of complete suppression on both sides and the bottom of the weir; this will be pointed out in our future discussion of some of Lesbros’ weir determinations. We will now proceed to analyze the experimental results in these 21 series of Fteley and Stearns, with the view to determine as nearly as may be possible, the effect of the first cause, and the combined effects of the last two causes, in increasing the velocity «. We will first assume that as 7 increases, the inner depth below the crest remaining constant, the percentage of increased velocity caused by partial suppression will increase; that is to say, with a constant inner depth of .5 below the crest, partial suppression will result in increasing the value of c—co-efficient of discharge—more and more as H is increased. This assumption has already been shown to be true for orifices, and will be hereafter dwelt upon in discussing the effect of partial suppression upon the discharge over weirs. We will also assume, that with the maximum inner depth of 3.56 below the crest, the effect of partial suppression is barely sensible, even with the maximum value of .944 for H. We have seen with an orifice .656 square, that the discharge was not affected sensibly when the sides of the feeding canal were placed 1.77 from the sides of the orifice; hence it seems safe to suppose, that with a weir having an inner depth 4 times greater than the depth upon the weir, the effect of partial suppression will be very sheht. This question will be more elaborately discussed in the following Chapter on Weirs. Messrs. Fteley and Stearns, in their reduction of these experiments, propose a 2 variable factor b, inh = H+b a9) b having a range from 1.33 to 1.87, and representing the entire effect of variation in G upon ZH. After several trials we found that by giving 6 a constant value of 14, we obtained results fairly agreeing with our conception of the problem. That is to say, we assume 2 that 4 Pa_ or (1.15.47 v4)" 29 29 y,)2\ % charge, due to the velocity of approach; hence Q=c 2 (2 q)” (¥ + Se) ' approximately represents the additional head and area of dis- 72 VELOCITY OF APPROACH.—Exreriments. Fieley and Stearns. ‘ (1 (1.1547 1)? 29 increase in v, diminished both the observed head and the observed area. The following table has been constructed upon this basis, showing the results from the 94 experiments. The various columns represent ; =e 2(2g)*h'"1. For, it is obvious in these experiments, that an 1st Coliiin.—The number of series and experiment. 2nd Columi.—-G@, or the inner depth from the crest of the weir to the bottom of the canal. 3rd Columi.—tT, or measured head above crest of weir, at a point six feet up-stream from weir. 4th Colin. hy = ie. This has been taken from Fteley and Stearns, and was deter- mined by them with approximate accuracy, by computing () for each series by their formula for weir discharge ; for each experiment, 1, = (G a i 5th Colimn.—h, =h,b, b having a constant value of 11. 6th Coluinn.—The assumed value of /, or T+h,’, being the effective head, taking only into consideration the effect of velocity of approach. The figures in small type at the beginning of each series, give the assumed value of /. for that series, with the effect of partial suppression entirely eliminated. 7th Columu.—The percentage of increased discharge, or more strictly the percentage of increased value of the co-efficient of discharge, c, assumed to be due to our second and third causes combined. This computation is based upon the principle that the co-efficient of discharge will vary inversely as /)":; the percentage is hence obtained by comparing the 3 power of / at the head of each series, with the same power of the succeeding values of h—column 6—for that particular series. Note our remarks in last section of this chapter, in regard to the inaccurate conception embodied in formula here used—(=c 2 (2 g)% hile 1. VELOCITY OF APPROACH.—Exvrriments. Fteley and Stearns. 73 TABLE XN. Iteley and Stearns.— Weir Experiments, Showing Effect of ducreasing v%, also attended with Partial Suppresston of Bottom Contraction, Weir with End Contractions suppressed. a ee 4 h 6 i 1 | 2 | 3 | 4 5 6 7 Be ef |) My Ne oe ag @ | mw | ko he a I 1931 aoe 4277 | 1 | 3.56 | 1930 |.0001 |.0001 |.1931 | 0 25 | 3.56 |.4263 |.0009 |.0012 | .4275 | 0.1 2 | 170 |.1924 |.0004 |.0005 |.1999 | 9.2 26 | 9.60 |.4257 |.0015 ;.0020 |.4277 0 > | 1.00 |.1913 |.0009 }.0012 |.1925 | 0.5 |, 27 | 1.70 |.4230 |.0030 |.0040 4270 0.2 4 50 |.1884 |.0027 |.0036 |.1920 | 0.9 28 | 1,00 |.4154 |.0068 |.0091 |.4245 | 1.1 | 29* | 50 |.3997 |.0169 |.0225 | 4399 1.9 I. 2690 | 5 | 3.56 '.2685 |.0002 |.0003 |.2688 01 | VIT. | | Poe 6 | 2.60 |.2685 |.0004 |.0005 |.2690 | 0 30 | 3.56 |.4933 |.0013 |.0017 |.4950 | 0.1 7 | 1.70 |.2676 |.0009 |.0012 |.2688 | 0.1 31 | 2.60 |.4923 | 0022 | .0029 |.4952 | 0 & | 1.00 |.2649 |.0022 |.0029 '.2678 0.7 | 32 | 1.70 |.4886 |.0044 |.0059 |.4945 | 0.2 9 5O | 2595 (0060 | 0089 2675] 08 | 33 | 1.00 |.4784 |.0097 |.0129 |.4913 | 1.2 34 50 |.4597 |.0230 .0307 |.4904 |) 1.5 If. 3368 | 10 | 3.56 |.3361 |.0004 |.0005 |.3366 | 0.1 | VIIE. | 5169 11 | 2.60 |.3358 }.0008 |.0011 |.3369! 0 | 35 3.36 '.5148 ].0015 |,0020 |.516% ) 0 12 | 1,70 |.3341 '.0016 |.0021 |.3362 | 0.3 | 36 | 2.60 5132 |.0025 |.0033 |.5165 | 0.1 13 | 1,00 |.3308 |.0038 |.0051 |.3359 | 0.4 . 37 | 1.70 |.5088 |.0050 |.0067 |.5155 | 0.4 14* | .50 |.3188 |.0100 |.0133 |.3321 | 21 38 | 1.00 1.4965 |.0108 | .0144 |.5109 | 1.7 | 39% | .50 |.4706 |.0256 |.0341 |.5047 | 3.5 Ty. . 3578 j 15 | 8.56 |.3570 |.0005 |.0007 | 3577 | 0 i. 5647 16 | 2.60 |.3561 |.0009 | .0012 |.3573 | 0.2 40 | 3.56 |.5620 |.0018 |.0024 |.5644 | 0.1 17 | 1.70 |.3549 }.0019 .0025 |.3574 | 0.9 41 | 2.60 |.5596 |.0031 |.0041 |.5637 | 0.3 18 | 1.00 |.3495 | .0044 |.0059 |.3554 | 1 42 | 1.70 °.5550 |.0062 |.0083 |.5633 | 0.4 19 50 |.3404 |.0114 |.0152 |.3556 | 0.9 43 | 1.00 |.5409 |.0132 |.0176 |.5585 | 1.6 44 50 |.5187 |.0302 |.0403 |.5590 , 1.5 vy. 4232 20 | 3.56 |.4219 |.0008 |.0011 |.4230 | 0.1 xe ee 21 | 2.60 |.4205 |.0015 |.0020 | 4225 | 0.2 45 | 3.56 |.5994 |.0022 | .0029 |.6023 | 0.1 22 | 1.70 |.4183 |.0030 | .0040 |.4223 | 0.3 46 50 |.5509 1.0345 |.0460 |.5969 | 14 23 | 1.00 |.4106 |.0067 |.0089 |.4195 | 1.3 24 50 |.3976 |.0165 |.0220-).4196 | 1.3 ? 74 VELOCITY OF APPROACH.—Experiments. Fteley and Stearns. TABLE XX.—continued. | | 1 Ae ll\| at ! | Bap ey A | i 3 4 | 8 6 | 7 Serres) | Pageeerrees eae eee, ‘ | | ’ Per |j iy bet | Per No | ¢ | w]e |W | -m i a Wo | Hl] ht | OM | cant NI, 6177 | L. | | .7922 | | 47 | 3.56 |.6143 |.0024 |.0032 |.6175 0 | 72 | 356.7859 |.0046 |.0061 '.7920 0 48 | 2.60 |.6120 |.0040 ).0053 /.6173 | O.1 | 73 | 1.70 .7695 ].0142 |.0189 | .7884 0.7 49 | 1.70 |.6059 |.0077 |.0103 |.6162° 0.4 | ! ul | | ! | 50 | 1.00 |.5904 |.0162 |.o216 |.e120 14 | *VEL Haake ai pie || 0: | ees | wre] .0180 gids. 6a “AE || BOR Gee ORS OO ee 15 2.60 8278 .0088 .0117 oo 0 XIL eee | 76 | 1.70 | S147 |.0163 | .0217 A30k 0.6 52 | 3.56 |.6741 |.0030 |.0040 |.6781 | 0 | 77 | 1.00 ,.7875 1.0322 |.0429 |.s304 | 1.6 53 | 2.60 |.6707 | .0051 | .0068 ee o2 | 48 50.7409 .0667 |.0889 '.8298 | 1.7 54 | 1.70 | 6629 |.0097 |.0129 |.6758 , 0.6 | | : | 65 | 1.00 |.6443 |.0201 0268 |.6711 | 1.6 | VUE oo | | 79 , 3.56 ,.8651 .0059 |.0079 |.8730 | 0 56* | 50 |.6179 |.0435 | .0580 |.6759 | 0.5 i | 80 | 2.60 8595 /.0097 |.0129 |-8724 , O.1 XIII. 7018 i 81 1.70 '.8467 |.0178 |.0237 ).8704 ' 6.5 st | 3.56 |.6971 |.0033 |.oo44 |.7015 | 01 | : | 58 | 2.60 |.6917 |.0056 |.0075 |.6992 | 06 | “1% | 9909 59 | 1.70 |.6836 |.0106 |.0141 6977 0.9 | Ree ee ee eee no | tow lieese anise poss eter a3 SR RO ey 010) [Olas aeRO Oa six | .50 |.636+ |.0466 |.0621 .9983 . 07 St ae 8593 |.O187 1.0249 82) 09 : 85 ! 1.00 | .8293 |.0366 | .0488 AT8l 19 XIV, 7203 | x6 50 ..7765 0752 |.1003 ;.8768 | 2.1 62 3.56 .7153 |.0036 |.0048 |.7201 | os ! 63 | 2.60 7117 |.0059 |.0079 |.7196 | 0.1 | SX - 64 | 1.70 |.7018 |.0113 |.0151 7169 Gyo ee ee xe | aso: | gage |-coan | exorcise. ae POR | 2.60 9181 |-O114 |.0152 |.9333 0 pet ae seer pian mene leneee cae $9 . 1.70 |.9011 |.0209 |.0279 .9290 | 07 | | 90 | 100 8668 |.0406 |.0541 [9209 2.0 Ve 7815 | | 67 3.56 1.7753 |.0044 |.0059 |.7x12 01 | SXF | “9847 63 ‘9.60 |-7718 |.0073 |.0097 |7215 9 | 82, 3.98 ,-9443 |.0075 |.0100 |.9543 © 0.1 es Udell ace eee aa lpeeey cud | 92 | 1,70 1.9215 |.0220 |.0293 |.9508 | 0.6 70 | 1.00 ,.7376 |.0275 |.0367 |.7743, 14, 98 | 100 -NNBE | .0426 |.0568 | 9423 | 2.0 71.50.6922 .0589 0777 weai ag 7 7 20 ‘i OEP [tlds 3 a | i The maximum velocity of approach was in Experiment No. 94, when it was 2.35 ; in this experiment h,’ is not quite one seventh of H. a The final results given in the preceding table, can be compactly stated as follows ; VELOCITY OF APPROACH.—Exprrments. Fleley and Stearns. 75 TABLE XXI. Pteley and Stearns.—Weir Experinents. Effect of Partial Suppression of Bottom Contraction, less the Loss by Additional Friction and Adhesion between Measuring Point for H and the Crest. Showing Percentage of Increase of c, due to above Causes combined. G@ = Inner Depth below Crest. Series. Maximum = Welncipt He cae || ang | L170, 1.00 | 50 I, | x | 02 | Os . 0.9 Il. 268 0.1 0 , 01 O07 O28 TIT, BG) 0d © | 83 0.4 ce Yj) BF ay) 0.2 0.2 lo) 0.9 ye, 22 : 0.1 02 0.3 | 13 1.3 VI. 426 O01 @ if C2 " 1.9% VIL 493 0.1 0 i o2 ie is VIII. 515 0 G1: @4 7 iF 3.5% i os . ai 03 | O04 1.6 1.5 xX .600—sO | ed Kl 614 | 0 G1} o4 ie 0.3% XU, 64 © | @€2 0886 | 16. o5* KI. B97 * 04 06 | 09 | Ls 0.7* RIV Sb ® | Gl * a 19 2.6# sy 15 0.1 0 0.5 14 2.9 XVI .786 0 0.7 XVIL.@ 82 0 0 ; 06 : 16 17 XVII. 865 | O 0.1 | 6 XIX. 881 | 0 Ol. | U8. 29 om SX. oa | 9G 0 | 07 2.0 | SEL 04. | oa EG) Boo aa An asterisk is attached to those experiments, in the preceding tables, which the authors state were imperfectly made, and hence more or less unreliable. An examination of Table XXI. will show that the percentage of increase in the co-efficient ¢ is fairly uniform for the several series—disregarding the imperfect experi- ments. With the lower values of Q, and hence lower values of /7, the assumed effect of our second and third causes combined, is less than 1 per cent., with the minimum inner depth of .5 below the crest; with the larger values of H, this effect increases to a little more than 2 per cent.. These percentages of increase in « agree reasonably well with the experiments of Lesbros with orifices, and with the conclusions stated hereafter, in regard to the increased discharge over weirs with partial bottom suppression, compared with full bottom contraction, h being constant in both casex. 76 VELOCITY OF APPROACH.—Experiments. Fteley and Stearns. Especially with the lower values of 77, it must be kept in mind, that we are dealing with very slight differences in the computation of the percentage of increase in c, and therefore cannot expect much greater uniformity—taking into consideration the element of experimental error—than the given results indicate. Somewhat more satisfactory results could have been obtained by making the factor } a variable, having a value of 1.86 for Series I., and gradually decreasing to 1.22 or 1.20 for Series XXI. In fact, from theoretical considerations, it seems to us that ) must be a variable, diminishing with ©. The given percentages of increased discharge due to partial suppression, for Nos. 78, 86, and 94 appear to be about 1 per cent. too low; for these experiments “c was less than for any of the others. ft A careful study of these tables cannot result otherwise, than in impressing us with the definite belief, that in any event H+h, gives too small a value for h, and that H+(14xh,) can be assumed as the value of / in our equation, without very serious error, for weirs with end contractions suppressed. If we assume that h=H1+h,, in Q=c 2 (29)* h'® 7, we would have the following percentages of increase in the value of c, to attribute to our second and third causes combined, the inner depth, or G, being constant at .5. Series I, a oe 1.5 per cent. with H=.2 3 II. ang ais 1.9 sj i. LSS i Vv. a oe 3.1 55 » H=.4 5 X. _ aie 41 ai 5 HLS 6 3 XV. oe oh 5.6 i 3 2S « DER ee sis 6. ‘i a SD 3 Se ni ais 2 a sy LS It is to say the least most improbable that such an increase in ¢ as is here indicated could be caused by partial suppression ; this is especially true for Series I. and II., where H was only about one-half as much as (7. These last computations indicate the general accuracy of our first assumption, that the percentage of increase due to partial suppression, @' remaining constant, increases with H. In fact with any other hypothesis, a satisfactory analysis of these 94 expe- riments would not be possible. Experiments were also made with the same canal and false bottom, with weirs having the respective lengths of 3, 3.3 and 4 feet, the end contractions being more or less suppressed. These experiments indicate that b should increase as the length of the weir diminishes, the width of the canal remaining constant. We will assume 1.4 as the constant value of b for the shortest weir, with smaller constant values as the length increases. VELOCITY OF APPROACH.—ExperimMents. Pleley und Slearus. 77 TABLE XNII. Feeley and Stearns.— Weir Experiments. Showing effvct of increasing v,, also attended with Partial Suppression of Bottom Contraction. Weirs of various Lengths ; End Contractions more or less suppressed. Width of Canal, ° Percentage No ae @ | oe” BL od | tos in| b(') htt ek 3 : ! _ = See | ! ( | | oe ! | 95 (| 3.56 8702 — .0020 ) () 0028 | s730. 1 | Bae 96 | | 17 | .8612 .0058 || || 0081 » .x693 0.7 2.26 a a eae eee igo, oa ft +) ois; scat 15 2.27 | | 98 L| BO oe [J 0302 | 612: 21 | 2.00 XXIII | 5778 | ! OG 45 3.56 | .5765 | .0008 |, 001L | .5776 0.1 | 202 100 | 1. 5647 | .0057 0079 . .5726 Ld; 2.35 101 | 5) GST4 0126 ol) are | Ge | 1.65 ct .8093 102 | | 3.56 | 8062 | .0020 .0028 | .8090 bl | 200 103. | 2.6 | .8036 .0033 0046 .8082 0.2 | 1.99 104 + 0 3.3 | L74| 17 | .7969 .0061 |} 1.394} .0085 | .8054 0.7 2.16 105 | iE 7850 | .0120 .0167 | .8017 14 2.09 106 | | 5 | .7677 = .0239 | .0382 | .8009 16, 178 i | : | : XV. ; . | .9350 107 3.56 | .9307 , .0029 | .00£0 | 9347 ' 0.1 ie 108 eg 9024 | .0162 0225 | .9249 16 j 2.10 109 | oh) PTR. 30809 (p° ‘} 0430 | .9205 2.4 | 191 XXXVI. ; | .7080 | 110 ‘ ‘ ec 28) | Oe ae) Nucaa 0029 | .7077 Ob get eu 1 6639 0283 |f 7 \ 0388 | .7027 ii | Ln The last column in the table is the value of b, in the equation Q=c 3 (24) (w +) »2\ 4% = l ( HT + 0) , the entire variation in HZ being attributed to velocity of 2g | 29 approach. The figures in small type in this column are not the result of experiment, but were assumed by Messrs.Fteley and Stearns, in order to obtain the value of H, with ‘ffect of velocity of approach entirely eliminated. Examining the preceding table, we see results which generally agree with our con- ception of the causes of the changes in H. Experiments Nos. 100 and 101, however, 78 VELOCITY OF APPROACH.—Experments. Fteley wid Stearns. appear to be flatly contradictory. More harmonious results could have been obtained by making ) a variable increasing with H, as was also the case for the weir with end contractions suppressed. From these experiments we can fairly assume, that for weirs having full contrac- tion, b can be given a constant value of 1.4, without danger of serious error in the resulting value of ¢. Hence for such weirs, J / 2 Q=c 3 (29) (H+ ak i (H+ 0.188 1) . Messrs. Fteley and Stearns propose a table of values of 6, which will doubtless give good results when applied to weirs having the same forms of approach as those employed in their experiments. But if the form of approach be changed, then appreciable errors may result from using their co-efficients of correction for OF 7 . This will inevitably be the case in all formule, such as those of Weisbach, et cet., where a distinct 4g conception of the separate effects of velocity of approach and suppression of contraction is not kept in view. J. B. Francis. Mr. Francis has made a few experiments, from which the effect of v, in increasing the discharge can be deduced. They are to be found in his ‘“ Lowell Hydraulic Ex- periments,” and will be given in detail in our next chapter. The experiments embraced in our Weir Nos. 49, 52, and 54 (Francis’ Nos. 11-33, 56-61, and 72-78) were with a weir having nearly complete contraction, the inner depth below crest being 4.6, the width of the canal being 13.96, and the length of the weir being nearly 10.00. The experiments embraced in our Weir Nos. 51, 53, and 55 (Francis’ Nos. 36-43, 62-66, and 79-841) were with the same weir, the inner depth, G, being 2.014, and the end contractions remaining unchanged. ‘Mr. Francis adopted, as a correction for velocity of approach, the following formula, h= [(# + ie ) i (5) i This formula rests upon the conception that the head /, which imparts the velocity in the feeding canal at the measuring point for H, accele- rates the flow at the weir, considering the vertical section of discharge, having the height H and the length /, as an orifice, with the head h, above the upper side of the < C “ 2 VD orifice. This formula gives a smaller value for hf, than the equation h=H+ ce = g H+hy. Mr. Francis’ final equation for discharge is hence, @=c 3 (2 g h)'2 Uh, h being obtained by his expression 2\ F/o Fa Pls 3 3 . , A z he | (#4. e a za Although this equation gives an algebraic result agreeing with the stated Ms 29 og S conception, still its form is objectionable, inasmuch as the head due to velocity of approach should only be considered as effective in accelerating the velocity of the escaping sheet, and not in increasing its area. The \ 3/o AVM 3 ee ey ts expression, @=c 1 3 (29)# [(# + a = (3) | is not open to this criticism. The following table shows the varying values of c in the foregoing experiments, VELOCITY OF APPROACH.—Experments. J. B. Francis. 79 with the values of h computed first by Mr. Francis, and then by our corrections deduced from the Fteley and Stearns experiments; 1e., = /1+b/,, with b ranging from 1.4 to 1.8. TABLE NNIITI. J.B. Francis —Welr Beperiments, with nearly full End Contractions, 1=9.997 (29)4=8090 e-...@ EE ag aa No. Weir Francis. Smith. No. | Experi- G H | @ t. | = ee Te ee Ce ee one ata ee h | c b h c 112 49 | 46 9977 | 32.580 | 1.0007 .6089 1.4 | 1.0019 .6078 113 51 2.014 | 1.0504 | 36.002 1.0640 .6137 1.3 | 1.0697 .6088 Lb 52 4.6 .7990 23.430 .8007 6118 lt 8014 | .6110 115 53 2.014 68269 | 25.041 i B47 6143 Lo .8380 .6107 . | 116 54 4.6 .6238 | 16.215 6246 ° 6146 a Fla 6251 + .6138 117 55 2.014 6493 | 17.340 .6536 | .6140 ! re 6553 ' 6116 ! | | 1 The value of ¢ should be in each of the three series of experiments just given, where H was nearly the same, slightly greater for the inner depth (G) of 2.014, than for the depth of 4.6, on account of the increased discharge to be expected from the partial sup- pression of bottom contraction with the shallower depth. It will be observed that the values of h given by Mr. Francis in these experiments, more fully satisfy this condition than the values of 4 given by the author. These experiments of Mr, Francis are entitled to the greatest respect, as we shall hereafter point out, but in them the corrections for velocity of approach are so minute, that not so much value can be placed upon them in determining the effect of »,, as upon our conclusions drawn from the experiments of Fteley and Stearns, where very much greater corrections for v, are required. Should, however, the large value of 2.05 or more for b be applied to the reduction of these Francis experiments, as has been suggested by Messrs. Fteley and Stearns, there would follow palpable inconsistencies in the deduced values of c, which would be altogether incompatible with our belief in the great accuracy of Mr. Francis’ work.* We feel warranted in assuming that these experiments, Nos. 112-117, indicate that b can be but little, if indeed any larger than 1.4 for weirs with full contraction, the section of approach being not much over eight times greater than the area of the weir. * The channel of approach for the Francis experiments with end contraction, was abruptly narrowed to a width of about 12.5 feet, just above the hook-gauges ; vide Plate XIV., ‘‘ Lowell Hydraulic Experiments.” It is apparent that it would have been better to have had a channel of uniform width for say 12 or 15 feet up-stream from the weir. This defective arrangement appears to be the only feature open to criticism in the Francis experiments. !' 80 VELOCITY OF APPROACH.—Experiments. J. B. Francis. Comparing Experiments Nos. 116 and 117, we see that with H about one-third of (7, the effect of partial bottom suppression of contraction was not appreciable, even with the small weight given to v, by Mr. Francis. This fully justifies our primary assumption that when G' = 38.56 and H=.944, the effect of partial suppression is barely sensible. The last 6 experimental values (Nos. 112-117) represent the means of 55 distinct determinations, in all of which Q was directly measured. This makes them much more reliable than single determinations ; if, however, the objectionable form of approach caused a slight error, this error would be constant, and would hence not be eliminated by repetition of the experiments. The respective heads for the three series somewhat differ ; by reference to our Plate VII. it will be seen that these differences in # produce a slight, but still appreciable, effect upon the values of the co-efficient c. However, the resulting variations in ¢ are not notable enough to change our general conclusions in regard to the effect of x, . Castel, Some experiments made by M. Castel with a weir with end contractions suppressed, should afford data by which the effect of velocity of approach can be determined. An account of these experiments is to be found in the “Mémoires de Académie de Toulouse,” Tome IV., 1837. The feeding canal employed was 19.5 long, 2.428 wide, and 1.772 high. The weir was formed by placing a wooden dam at the lower end of the canal, surmounted by a copper bar, having a thickness of .01; the height of the weir, or (', was varied from .10 to.74. The supply of water entering the upper end of the canal, was obtained from a vertical pipe-column, the quantity of flow being regulated by a valve. 68s 636 | 50 | 20 643 «| 44 644 636 646 645 | 654 ie 6! G50 650 649 641 650 | 649 | 657 13° "660 658 | 656 648 655 652 660 | 10 676 670 | 663 | O54 659 | 656 | 662 | In accordance with the views before expressed, with the same head as G! increases e should diminish. By reference to the foregoing summary it will be observed that the values of c, JZ being constant, follow no regular order with variation in (*. The results for G'=.738 appear to be especially irregular, compared with those for smaller values of (. Taken as a whole, the deduced values of ¢ do not sufficiently indicate the effect of partial suppression on the bottom. Assuming that the experiments are trust- worthy, this would show that our values of / are placed too high, or, in othcr words, that b should be somewhat lowcr than 13. More satisfactory results would have been obtained by giving }) a value of about 1.2 for the smaller values of (7. Although we regard the experiments of Castcl as entitled to some weight,* still we consider them as much less reliable than the data taken from Fteley and Stearns, and Francis. They possess, however, sufficient value to warrant their insertion here ; with very large values of /,’ in proportion to H, they show results fairly according with our preceding deductions. t * Note our remarks on Castel’s experiments given in Chapter V. in the section treating on short weirs. + In Experiment No. 118 ha’ is 21 per cent. of H, while with the highest velocity of approach in the Fteley- Stearns series—No, {%4—h,,' is only 14 per cent. of H. VELOCITY OF APPROACH.—ExperimMents. Other Authorities. 83 Other Authorities. This subject has been investigated experimentally by several savants, in addition to the three authorities we have made use of, but their results are discordant and often self-contradictory. This can be safely attributed to a large range of experimental error. ConcLusions. It is evident that the force due to velocity of approach, which causes an acceleration of velocity at the weir, must be less than the total kinetic energy of the water as it passes by the measuring point for H. For, with a weir with contraction suppressed on all three sides—or more exactly, a canal of uniform section with the water discharging from an open end—the effective head due to the velocity in the section at H will be diminished by the “ frictional” loss between H and the end of the canal, and also further by the changes in velocity of the horizontal layers of water as they feed the escaping sheet, these velocities being in proportion to the respective head for each layer. When there is total or partial contraction—the area at the weir being smaller than that of the canal—there will be in addition losses or absorptions of energy, due to diagonal and eddying movements of the water, as the various fillets make their way more or less directly towards the crest. When there is a weir proper it can be regarded as an obstruction, causing a notable additional variation of direction in the movement of the fillets of water. Any increase of such variation must inevitably result in increased loss of head. Hence, from theoretical considerations, we can assume that as contraction becomes more and more complete, or in other words as the distance increases of a side of the feeding canal from the respective side of the weir, the greater will be the proportional loss of energy between the measuring point for HZ and the weir. Any attempt to compute the allowance to be made for velocity of approach, from the total energy or vis-v7va of the water, as it passes the measuring point for H, presents so many complicated problems, that we have preferred to use our very simple form, which although not strictly accurate, gives sufficiently satisfactory results, when h,’ is not very large in proportion to h. | When it is required to measure the flow of water with great accuracy, the velocity of approach should be reduced to the minimum which is practicable. No serious error will result, where v, is less than one foot, by making for weirs with full contraction b=1.4, and for weirs with end contractions suppressed b=14. It is doubtless theoretically true that the larger a, is in proportion to «, the greater should be the value of b in our equation, but this is in a considerable measure compen- sated for by the fact, as before stated, that as a, increases, a being constant, the greater is the proportional loss of energy between the point where H is measured and the weir. In a feeding canal having a considerably larger section than the area of the weir, the various fillets of water in the section at the point for JZ have widely varying 84 VELOCITY OF APPROACH —Coneusions. velocities. It seems reasonable to suppose that the additional force produced by the velocity of approach, is chiefly that due to the velocity of the section of the feeding stream nearest or in line with the section of discharge, where the particles of water fairly begin to form into the escaping vein; this velocity is unquestionably greater than v,. For instance, with a weir with end contractions suppressed, with a very great inner depth below the crest compared with the head, with the measuring point placed the usual distance of 6 feet or so from the crest, the velocity at the bottom of the canal will be practically a7, and v, will be very low. At the same time, with a consider- able value of H, such as 1 foot, there will doubtless be an appreciable surface current at the measuring point, and very much more than is indicated by v,. With a weir with full contraction, having an unobstructed channel of approach of considerable length, the central surface current, which we conceive is the current which chiefly causes the increased flow at the weir, will be from 1.1 to 1.25 greater than ry. Hence we can pretty safely take for weirs with full contraction v7, as the central surface current just above the measuring point; in this case no correction of h, should be employed. In channels of approach having irregular sections it will probably be safer to take this surface current, rather than to compute h,’ from the mean velocity. Our given value of b of 1.4 for such weirs is expressed in h,’ ae al, so that assuming the surface current to be 1.2 times the mean velocity, the result is practically the same, whether the surface current be used for v, without correction, or if the mean velocity be used, and the resulting velocity head increased by the factor b.* The following table gives the values of hy, h,x 14, and h,x 1.4, for velocities of approach, v,, from .04 to 1.9. The values of h,, et cet., can be immediately obtained by this table for velocities up to 1., with an error not exceeding .0001. In practice v, will rarely exceed 1., and as before stated, where much accuracy in the measuremeut of water is desired, v, should be maintained at less than 1.. It must be kept in mind that these corrections are intended to compensate solely for additional head due to velocity of approach. Where G or Z is so small as to cause partial bottom or side suppression, then further allowance must be made for the increased flow caused by this suppression. The given corrections are intended especially for weirs; for orifices with full 2 contraction it is probable that, in @= Ca (2 9)” (#2 + vs) , 6 should have a some- what larger value than 1.4. * The ratio of the surface velocity to the mean velocity may be more or less modified by the use of screens, placed near the measuring point for H. Such screens may have the effect of producing a yreater velocity near the bottein, than at the surface. £705 .007 @ .0103 | .0108 | | .0167 jL9 VELOCITY OF APPROACH.—ConcLusions. 85 TABLE XXV. Corrections for Velocity of Approach, h=H+ my 29 = 64.36 uw? ‘ a ue og | és nt nd | ve | Ve v2 | ve ad v2 | ag 3 agp tal Se Bayi ay vy : x er lt aq Ve Fe bay Lee (a ! i ca (poms acs ——_ oh eae - ee ee = pees a kes aes, | ed 04 ooo 02 0 ! 0 38 002 24 .0030 | .0031 i 1 | 007 83.0104 | .0110 | 88 1.012 03 .0160 | .0168 05 |ooo wal .0001 | .0001 : 39 |.002 36 .0032 | .0033 ||.715.007 94) .0106 |.0111 | .885..012 ly .0162 | .0170 .06 |.000 06, .0001 | .0001 .40 a 49 .0033 |.0035 |.72 .008 Pad 0113. . 89 lov s1 31 .0164 |.0172 07 000 08 .0001 | .0001 .t1 Oe 61.0035 | .0037 sit pata a .0109 | .O114 1895) .012 £5, .0166 ,.0174 08 |.000 10 .0001 ;.0001 .42 .002 74] .0037 | .0038 |.73 .008 28 .0110 » OLLG .90 |.012 58) .0168 | .0176 .09 000 15 .0002 0002 43.002 87) .0038 | .0040 ; £735 008 39 0112 |.0118 | .905,.012 73) .0170 | .0178 10 -000 15 .0002 .0002 .44 -003 oY .0040 | .0042 \ 4 008 51 .0113 | .0119 ; 91 1.012 87 .0172 | .0180 11 .000 19 .0003 ‘0003 45 .003 14 .0042 | .0044 | .745.008 62 .0115 |.0121 | .915.013 01 .0173 |.0182 12, |,000 22.0003 | .0003 | 46 .003 29.0044 |.0046 .75 008 74).0117 |.0122 | .92 .013 15.0175 |.0184 13 -000 = .0004 (0004 | AT -003 43) .0046 | .0048 795, 008 86 .0118 |.0124 | .925/.013 29) .0177 | .0186 14 .000 30 .0004 0004 48 .00358 .0048 .0050 .76 .008 97|.0120 |.0126 |' .93 013 44.0179 |.0188 15 |.000 35'.0005 |.0005 |.49 |003 73\.0050 |.0052 765.009 09.0121 |,0127 | 935,013.58 .0181 |.0190 16 000 40 .0005 0006 .50 |.003 88.0052 | .0054 | .77 009 a1 0123 |.0129 . .94 |.013 3 .0183 | .0192 17.000 a, .0006 Dee | 51 ae 04) .0054 0057 .775.009 33 0124 |.0131 | .945/.013 87 .0185 | .0194 18 .000 50 .0007 (0007 | .02 004 20) .0056 | .0059 | 78 009 45 0126 | .0132 | 95 oe 02 ONT | .0196 19 000 56, .0007 | 0008 .53 |.004 36) .0058 | .0061 | .785.009 57; .0128 |.0134 ee 17|.0189 | .0198 .20 .000 62.0008 oe Ot poe 53) .0060 | .0063 ; 79 '.009 70 .0129 | .0136 |, -9 ; O14 a .0191 | .0200 21 | 000 68 .0009 | .0010 | .55 .004 70) .0063 | .0066 795.009 82 0131 | .0137 a O14 47 .0193 | .0203 22° 000 75) .0010 oot 56 004 87) .0065 | .0068 | .80 |.009 94 .0133 |.0139 | .97 ia 62) .0195 | .0205 23 000 er -OO11 | .0012 | .57 Ean 05.0067 | .0071 .805 .010 07) .0134 .0141 | sa) O14 a, 0197 peal 24 ee 90.0012 |.0013 | .58 .005 23 .0070 |.0073 .81 .01019 0136 |.0143 .98 1.014 99) .0199 | .0209 25 | 000 97 -0013 | 0014 |, .59 Loos al .0072 | .0076 4 815.010 32 0138 | 0144, 985.015 07) .0201 | .0211 26 | 001 05 .0014 ee | 60 .005 59.0075 | .0078 | .82 |.010 $5, .0139 | .0146 ] 99 ‘015 23) .0203 | .0213 27 001 13.0015 | .0016 | .61 |.005 78; 0077 | .0081 | .825..010 a -O141 | 014s | 995 015 .0205 | .0215 .28 .001 22 .0016 .0017 | .62 .005 97 .0080 | .0084 |.83 01070 .0143 | .0150 [1. "015 54.0207 1.0218 29 ‘001 31.0017 .0018 |) .63 006 17) .0082 | .0086 83 3 010 83 Ol44 |.0152 1.1 L018 80.0251 0263 .30 |.001 a) .0019 | .0020 | .64 eee oe .0085 | .0089 | | .84£ 1.010 2 .0146 |.0153 1.2 .022 37 .0298 | .0313 .31 |.001 49 .0020 | .0021 | .65 006 56 .0088 | .0092 | 845.011 09 .0148 |.0155 1.5 .026 26] .0350 | .0368 82 |.001 59 -0021 | .0022 | .66 006 17 .0090 | .0095 | 85 lou ” .0150 | .0157 jl4 .030 45) .0406 | .0426 O93 001 69 69. -9023 | .0024 | 67 foal oF .0093 | .0098 | poet 36) .0151 | .0159 nega Die 96) 0466 | .0489 3B4 ee 30 .0024 |.0025 | .68 ee 18 .0096 | .0101 | 86 O11 49) .0153 |.0161 ae .039 78 .0530 | .0557 85 001 90! .0025 | .0027 | .69 007 40! 0099 | .0104 | -865,.011 63) .0155 | .0163 ‘LT O44 90 .0599 | .0629 36 We 01.0027 -0028 || .70 |.007 61) .0102 |.0107 |) .87 .011 76! .0157 | .0165 hes eee . 0671 | .0705 37 002 13, -0028 | .0030 | | 879.011 90.0159 mG 09) .O748 | .O785 86 VELOCITY OF APPROACH.—ForMUL. FormMuLez. The famous and much disputed equation of Daniel Bernoulli is, for horizontal orifices ; ee ge e or H= | . hence, (apg) (Gd) ( % . . (A) Q=a (27 oaEaE ) = theoretic discharge. Te This formula applied to an orifice at the bottom of a prismatic reservoir makes @ infinite when a=, thus requiring an infinite supply into the entrance of the reservoir.* We believe that the theoretical accuracy of this expression is now generally admitted by mathematicians.t In it, however, neither friction or adhesion on the sides of the reservoir, eddying or diagonal movements caused by unequal velocities, nor the pheno- mena indicated by the form of the escaping vein, are taken into consideration. The fundamental proposition upon which the equation is based, is that the water moves in one mass with uniform velocity. But in the case of weirs, the escaping hori- zontal layers of water move at very different speeds, the velocity of the layers being in proportion to the square root of the head, and hence being 0 at the surface, disregarding the effect of velocity of approach. Therefore the equation is especially inapplicable to weirs. Boileau and others have followed equation (A). Boileau’s final equation for weirs with end contractions suppressed, given in his “ Traité de la mesure des caux cou- rantes,” 1s H\*% Sz (B) Q= Se a ee Lid Gy By. With this expression when (' =0, () becomes infinite. If we propose for vertical rectangular orifices, placed at the end of a feeding canal, Q 2 22. a, ? Aes 5 ; H+/.,= effective head at the orifice, there results, a g re * Tn such a case what will be the proper value of H? If we could measure it by a piezometric tube attached to the bottom of the reservoir, disregarding effect of frictional and eddy losses, would it not be 0? + Vide paper by Weisbach, in the ‘‘ Allgemeinen Maschinen-encyclopiidie,” ‘‘ Ausfluss.” VELOCITY OF APPROACH.—FormuL@. a7 (C) Q=c (29)¥ 12 ((Hy +h.) —[ H+ h.)"). Now, as in weirs £7, disappears, and H, becomes H, we have, (D) Qae (2 g)4 13 (LH +he}h—he) Mr. Francis has adapted equation (D), although in a different algebraic form, in making his corrections for v,. It is based rigorously upon the conception, that the force causing increased velocity at the weir, is that due to the energy of a section of the feeding stream, having the same area as the area at the weir (H/), and having the mean velocity, 7, of the feeding stream. re 2 Gee being the co-efficient of contraction at the orifice. This would approximately result for weirs, in 2) Q=c$(2g)* 1( LH4 2.6 hy] — [2.6 hy]"”) Mr. Neville proposes a modification of this expression by making h,’ = 05 ae 2 h, m equation (E) having its usual signification of 3°: = Y The formula which we have adopted is, (F) G=¢4 (29) (A+ a) tlae 2 (2 9)" th, b having a value of 1.4 for weirs with full contraction, and 14 for weirs with end con- tractions suppressed. This mode of expression is faulty, as it assumes that the head representing the effect of velocity of approach not only adds to the measured head, thus imparting additional velocity, but also adds to the section of escape, or area at the weir. We have used this equation on account of its simplicity ; it gives sufficiently satisfactory results when applied to our experimental data. The expression ©) Que (2 y)"13 ([H+bAT*—[ AT"), with } having larger values than in equation (F), would doubtless be more logical, but (G) would give but little, if indeed any closer results than (F), and would involve con- siderable additional labor in computations. We think it is apparent that theoretically our factor b should be a variable, increasing with fe. this view is fairly sustained by the experimental data given in this a “a eS ; i chapter. The expression () =b would roughly indicate the law of increment in b; a b would hence be unity with suppression on three sides, when a,=«, which is probably not far from the truth. Even if there were at hand data from which an expression could be framed-which would accurately state the rate of increment in b, there would 88 VELOCITY OF APPROACH.—ForMULz. result such a complicated formula for weir discharge, as to render it almost fatally objectionable for practical use. Formule, such as that of Boileau, those of Weisbach, et cet., are intended to apply not only for 7,, but also for suppression or partial suppression of contraction. When there is complete suppression on the three sides, these causes are very closely related to each other ; still we conceive that even then they should be considered not as a whole, but separately. The two phenomena are essentially distinct, and it seems to us the proper method of investigation to be followed, is to attempt, as we have done, to deter- mine what effect each cause has, in increasing the discharge. 89 CHAPTER V. FLOW OVER WEIRS. Ir has before been shown that for rectangular weirs, with vertical sides and horizontal crests, the discharge is correctly represented by, Q=c2 (2Q9h)* a, and a=1h; or, Q=c$(2g) tlh; ce being the co-efficient of discharge to be ascertained by experiment. All the following reductions are made with this formula to determine the varying values of c. When v, is of moment, the observed head, H, will be corrected in accordance with the results stated in our last chapter, on Velocity of Approach. It must be kept in mind, however, that H will only be corrected for the additional force due to v,; H will not be corrected for the increased discharge due to suppression, or partial suppression of contraction ; nor, for the diminished discharge caused by increased friction and adhesion on the sides of the feeding canal between the measuring point and the weir, when the canal was of so small a section as to make the effect from these causes sensible. As the water approaches the crest of a weir, the surface forms a notable curve, extending above the weir an increasing distance as the length of the weir and the depth upon the crest increase; several of the forms of this curve will be shown hereafter. Many experimenters have measured both —the vertical elevation above the crest to a point where the influence of this curve ceases to be appreciable—and also H,—the height from the crest to the mean surface line of the water in the plane of the weir— thinking that both heads should be known in order to accurately determine Q. The exact measurement of H,, adds largely to the labors of the experimenter,* and in properly constructed weirs its determination is of no value in obtaining Q. Hence only H will be considered, except in discussing some of Lesbros’ experiments, where there was a large loss of head between the measuring point for H and the weir. Dubuat, Eytelwein, and other early hydraulicians made a large number of experi- ments with weirs, but beyond proving that the formula Q=C 2 (2g H1)*1 H was * Owing to the irregular form of the surface at the weir, H,, can never be measured as exactly as H, where the water presents a very nearly horizontal section, nor can H,, be measured by the hook-gauge, the most delicate appliance yet invented for determining surface heights. . N 90 FLOW OVER WEIRS. approximately correct, or, that the velocity at the weir, 7, is very nearly in proportion to (2 g H)*%, these experiments are of no value. They fixed the general value of C'at .62 or thereabouts, but gave very contradictory minor results. For instance, with nearly full contraction, some of them showed that as H diminished below .5, C’ also diminished ; while others gave diametrically opposite results. The later and far more careful experiments of Lesbros, Francis, Fteley and Stearns, and ourselves afford sufficient data to deduce with very considerable accuracy the laws governing the discharge over weirs with the three simplest forms of approach, and with depths up to nearly 2 feet. It may be remarked at the outset, that the accurate determination of the cv- efficient of discharge, c, is far more difficult with weirs than with orifices. With a very small orifice the very exact measurement of its area becomes necessary if it be desired to establish ¢ within close limits, and hence very delicate measuring appliances must be used, but with an orifice with the least side above .5, the dimensions can be obtained with sufficient precision by ordinary scale measurement. The head used for gauge water by orifices is rarely less than .6 and generally 1. or more; with such values of H, a slight error in its measurement is unimportant. With weirs the great danger of error arises from imperfect determination of H, / being usually of a size where slight errors in its measurement will not sensibly affect c. Comparing the relative changes in ¢ caused by a small error in 1; Weir. With H= .1, an error of .001 will change ¢ 1.50 per cent. 55 wi (ele. ee ; 5; j 75 45 ii —@ i= 6 ‘i i 3 Ab ‘5 Orifice. ,, H= 6 i 3a 95 O83 ., ae 3 df= 7, sy i 33 050, ” pos 2 i e ‘3 025 5, 5 9 f= 10, i ‘i 5% 1008" 4; With a vertical gauge-rod pointed at the lower cnd, such as was used by Lesbros, probably .001 was the limit for accurate measurement of {, although his readings were to .0003 (.1 millimetre). With the much more perfect hook-gauge used by Francis and others, perfectly still water can be measured to .0002 with comparative certainty ; there are, however, such fluctuations of the surface of the water in practice, that the experi- menter may consider himself very fortunate if his limit of error in H does not exceed .0005, or even more. We should hence expect to see much greater experimental variations in the value of c for weirs, than was the case with orifices. The weir experiments of Mr. Francis, and Messrs. Fteley and Stearns were, however, executed with such care, and upon such a grand scale, that their resulting values of ¢ for heads above the crest of more than .5, are more reliable, than many of the values of « we have viven for orifices. It is apparent that the larger the opening, provided the measuring vessel for q be proportionately FLOW OVER WEIRS. 91 large, the more reliable should be the values of ¢; for in such case, errors of dimension become comparatively smaller. It is quite likely that with small heads, such as .2 and less, a change in the temperature of the water of 30° or 40° may have a notable effect upon the discharge ; it is also possible that with such small heads, there may be variations caused in the flow by unknown changes in the character of the water, such as we have conjectured may oceur with very small orifices, or with oritices with very small heads. EXPERIMENTS. The weirs we are first about to describe, were all rectangular, with vertical sides, with free or nearly free discharge into the air, and of lengths from .66* to 19. . They can be divided into three categories : First.— With both end contractions suppressed, by placing the weir at the end of a feeding canal of rectangular section, whose width was the same as the length of the weir; the plane of the weir being at right angles to the sides of the canal. In all cases the crest was sharp-edged, so that the escaping water only came in contact with the inner corner; the inner side, or end of the canal, below the crest was vertical, and of sufficient depth to give complete or nearly complete bottom contraction. Seconp.—With complete or nearly complete contraction on the three sides of the weir, the width of the feeding canal therefore being considerably greater than the length of the weir. The discharge into the air was perfectly free. The sides of the weirs were thin (except in one weir of Leshros), so that the escaping water only came in contact with the square inner corners of the three sides. Tuirp. — With contraction suppressed on the bottom or on one side ; with contrac- tion partially suppressed on one or both sides; and with various forms of approach as shown by Figs. 4, et cet., Plate I. First: Contraction SUPPRESSED AT BOTH EXyps. Leshvos. Lesbros, using his orifice in a fixed copper plate, of .6562 square as a weir, and with form of approach shown by Fig. 10, Plate I., obtained the following results. H was measured in still water at a point 11.48 feet above weir, and hence H=h,. * The discharge over similar short weirs, with / as small as .033, will be discussed hereafter. 03 lw SUPPRESSED WEIRS.—Experiments. Lesbros. TABLE XXVI. Lesbros.— Weir with End Contractions Suppressed. Perfectly free Discharge into the Air, Inner Depth below Crest 1.772. (2 g)t=8.0227 Fig. 10, Plate I. 2 , ta | E | |B Spee No. [Lesbros No. iT roar H—-Il, GQ | ’ c {Spey ; ee rE eas) cae thane 1 1764-6 8009 | .6549 | 1460 | 1.6158 | .6562 | .642 32 2 1767-9 | .5098 | .4958 | onto | 92991 ' ,, | .649 4.2 . 3 1770-1 | .3350 | .2891 | .0529 | .4424 r | 650 te 4 1772-4; .1857 | 1542 | 10815 | .1846 yo | ates 5b 5 1775-7 | 03917 ea eed 1/43 0627 OL72 | OLAS The escaping vein in Nos. 1 to 4 constantly enlarged horizontally; with No. 5 it diminished in length after escape from the weir. : There was a perceptible loss of head caused by primary contraction, as the water entered the narrow feeding canal from the main reservoir, where H was measured. Francis. Mr. J. B. Francis made in 1852, at Lowell, Massachusetts, a series of 88 expe- riments on weirs with various forms of approach, most of them with /=10., and a few with /=4.. His results are published in ‘‘ Lowell Hydraulic Experiments,” pp. 103-145, New York, 1868. These experiments were made upon a scale before unknown, and with great care and skill. The use of the Boyden hook-gauge enabled him to determine the surface elevation of the water with much greater accuracy than had before been possible with previous researches, where H was measured by a pointed descending gauge-rod. His measuring vessel or tank had a capacity of 12 138 cubic feet, with a depth of 9.5 feet, thus affording exceptional facilities for the absolute measurement of large volumes of water, with great exactness. The flow of water from the weir was very rapidly con- nected or disconnected with the flume leading to the measuring tank, by a swinging apron. ‘Times were determined to .1 of a second. The crest of the weir was of cast iron, ith of an inch wide on top, and vertical on the up-stream side. The canal was 13.96 feet wide, with its bottom 4.6 feet below crest immediately at the weir, and 5.048 feet below crest at the gauges. The canal was narrowed to a width of 9.992 feet a distance of 20 feet up-stream from the weir, for experiments with end contractions suppressed. The head upon the crest was measured by two hook-gauges, placed 6 feet. above the weir, on each side of the canal; these were enclosed by small boxes to diminish oscilla- tions in the surface of the water. These boxes were, perhaps, objectionable, as they SUPPRESSED WEIRS.—Exrerments. Fruneds. 93 may have slightly interfered with the normal flow of water to the weirs having full contraction. To determine effect of v,, a false bottom was placed in the canal, extending horizontally a distance of 23 feet up-stream from the weir, at an elevation of 2.014 feet below the crest. To obtain a shorter length, a false piece 2 feet long was placed in the centre of the 10-foot weir, thus forming two weirs each 4 feet long. He places (2 g)* at 8.0202, which value will be used in our reductions from all his experiments. These weir experiments of Mr. Francis still rank first in reliability, and will be accepted as unquestionable authority ; the only other weir determinations thus far given to the world, which bear comparison with them in regard to accuracy, being those of Messrs. Fteley and Stearns. The Francis experiments with end contraction suppressed, were with perfectly free discharge into the air—except Nos. (Francis) 51-55 as hereafter noted, with G=4.6. In the following transcript of these experiments, the values of hf are given as deduced by the Francis expression of h=[(H+h,)"*—h,'!}" ; the co-efficient c’ is obtained by the formula Q=c' 2 (2 g)% h'= 1. The length of the weir in all cases was 9.995. No oP) tel ge | c! ne nh, | ni! @ | o! 44 |.9867 |.0046 9912 , 32.909 6240 ' 51 | 1.0050 .0049 1.0097 33818 6237 45 |.9849 |.0046 9893 2.849) 6246 62 1.0060 0048 11.0106 sa.71, 6220 46 |.9745 |.0045 |.9788 | 32.362 .6254 |, 53 | 1.0052 0048 1 0098 33.727 | .6219 47 |.9762 |.0045 |.9805 32.430 6250 | 54 | .9926 .0047, .9971 33.088) .6219 y 5 | | ; [ 48 | .9760 |.0045 |.9803 32.436 6253 | 55 | 9924, 0047, 9968 33.069 6217 | 32.499) 6247 : | | 49 |.9777 |.0045 | .9821 Means | 1.0003; “11,0048 [33.494 | .6222 9769 |. 9812 | 32.460] .6249 ey: pons | eee eee) cr or fe 67 7362 002i 7382 (21.153 | .6241 Means |.9791 9834 (32.561 6248 | 68 | .8019 .0026 .8045 24.104, .6251 | | 69 8095, 0027 8121 hess 6251 70 ads 0028 .8176 24.688) .6249 2 | 71 ney .00 7] .8159 (24.558) .6236 23.790) .6246 | Means 7955! | ne | The foregoing means are calculated from 3 power of [7 and h. For Nos. 51-55 (Francis) the sides of the canal were extended past the weir, so that the escaping vein was confined to the same length as the weir ; access, however, was given for the entrance of air under the descending sheet of water. 94 SUPPRESSED WEIRS.—Experiments. Srancis. For our purposes the means of the three foregoing series can be taken; applying our corrections for h,, we have ; TABLE XXVIII. ranetsx.— Weir with End Contractions Suppressed. Depth on inner side of crest = 4.6. (29 ae =. ka a 7 ; 1 | eee | - “ Francis Temp. of | : ; 0 No. | No. Water. / H h be | b | hh, _| ¢é 6 | 4450 | we 9791~—«.0045 | 120060 ~—«.9851'| 9.995 | 32.561 | 6282 i S155 433° | 1.0003 | .0048 ie .0064 = 1.0067 | 5 | 33.494 .6205 x 67-71 7955 0026 ie | 0035 ©.7990 i, 23.790 6233 | Fteley and Stearns. Messrs. Fteley and Stearns experimented with two suppressed weirs, one with /= nearly 5., and the other with /=nearly 19.. The measuring basin for the largest volumes was a section of the new Sudbury conduit for the Boston water supply, and had a capacity of about 300 000 cubic feet, with a change in elevation of 3 feet; the area was therefore very large in proportion to the depth, and hence this basin was not as favorable for very precise measurements of @, as would have been one of smaller area but greater depth. The largest volume measured was (J=130 cubic feet, which is probably a larger quantity than has ever before been accurately determined. For Experiments Nos. 9 to 26 inclusive, () was obtained from this large basin; for Nos. 27 to 34, @ was obtained from a smaller section of the same conduit. The experiments with the large measuring basin are thought to be freer from errors in (J, than with the small basin. The crest of the 5-foot weir was a nickel-plated steel straight-edge, having a thickness of .0066 ; the crest of the 19-foot weir was a planed iron bar .02 thick ; the water in these experiments only touched the sharp inner edges of the crests. The length of the shorter weir varied somewhat during the experiments, which was caused by changes in the sides of the canal. The head upon the crest was measured by a hook-gauge placed below the weir, its lower end or hook determining the elevation of the water in a movable pail. This pail was connected by a rubber hose with an opening in a piece of plate glass, whose face was flush with the side of the feeding canal, and set at a point 6 feet above the weir, and about .4 foot (vertically) below the level of the crest; the orifice in the glass plate had a diameter of .04, with smooth sides, having its axis normal to the face of the plate. Those experiments given by these authors, which are stated by them to have been imperfectly made, will not be included in the following tables. All the reductions from the experiments of Fteley and Stearns will be made with our values of /,/’. SUPPRESSED WEIRS.—Exprniments. Fteley and Stearns. 95 TABLE NXVIII. Fteley and Stearns.— Weir with Bnd Contractions Suppressed. Escaping Vein confined by Prolongution of Sides of Canal, but ivhich did not exter below Level of Crest, Inner depth below crest, 6.55, b=1} (2 gy = 8.020 = 45 a = xo, | Means. ie Pee ok Ldn, me) I 0 - ... No. r : | | Water. . | | | | h | c 9 1. * ge | 1.6038 01097 0146 | 1.61N+ 18.996 | 130.117.6223 | ed ) ous 10 2 87 1.4546 | .008 14 ; 0113 | 1.4659! | 112.066.6217 1.47 | .6217 11 3 37° | 12981 .00621) .0083 | 13064! 94,192 6211 | 1.30 | .6211 1G 5 39° | 1.1456 ' 00440 .0059 | 1.1515 i Toe. 61K Lib: | 19s 6 43° | 9873 | 00291 | .0039 | .9912 62.061, .6192 13 { | ne. | i " | I! 99 | e194 7 1 38° | 9864 00291 ' .0039 ; .9903 . 62.023 .6197 2 14 a 46°) 181" 60173 0093 | ald : 46.760 6185 82 | 6185 15 | 9 | 44° | 6460 .00089 .0012 | .6472 is 32.685 6181 | 65 | .6181 16 | 10 | 41° | 4685 00035 ! .0005 | 4690 s 2O1TR .61NG , AT | 6186 TABLE XNIX. Fteley and Stearns.— Weir with End Contractions Suppressed. Escaping Vein eonfined by Prolongation of Sides of Canal, but which did not extend below level of Crest. Temperature of Water, 36°. Tunis depth below crest, 4 17. b=14 (24 gr =.) Gao hae pase 22 oo) —, aie us. No. Fa 8 Hl h, h, h q Q Gi esas = _ » No. | | h ¢ aa atetin ps ‘nemmanetl ot Neemmmiacnsienl det icjacumtigs actin heh natascteae Spee MS meeps 1.8198 .0064 .0085 .8283 5, 12750 6327 | iv { 82 |.6304 3 |.8118 0061 0081 S199 5, 12.466 hee 2 . 4 |.6761 .0037 |.0049 6810 5. 9.430 |.6277 |) Bare 18 { | i | h ga 1.6276 5 1.6713 .0037 | 0049 6762 4.997 9.322 6275 J \ 19 | 6 |.5203 |.0018 0024 |.5227 4.996 6.342 6283 52 he 20 7 |.4810 .0016 |.0021 .4831 4.999 5.766 6425 48 | 6425 & 4761 .0014 | .0019 | 4780 4.999 5.547 6280 it eae ; 272 9 |.4569 ee 0017 ,.4586 | 4.999 , 5.199 | .6264 id | | : 1 ‘OD 10 |.3890 , 0008 0011 !.3901 | 4.999 4.094 (6287 390 .6287 23 ! 12 |.3407 | .0006 0008 8415 [4.994 | 3.3540 6294 341 | 6294 24 | 13 7.3114 0004 .0005 | .3119 | 4.998 2.9355 |.6306 .312'.6306 25 14 ;.2598 0003 ,.0004 .2602 /4.999 2.2415 6319 260 .6319 26 | 15 |.2467 |.0002 |.0003 :.2470 | 4.999 | 2.0780 6338 247.6333 27 | 17 |.2190 |.0002 |.0003 |.2193 | 5. | 17474 .6305 — .220,.6365 18.2182 |.0002 |.0003 |.2185 | 5. 1.7211 |.6304 |) | 28 | | |\ 218 | 6432 19 |.2176 |.0002 |.0003 |.2179 | 5. L7837 pape W | | [Table continued on neat page. 96 SUPPRESSED WEIRS.—Exprriments. Fteley and Stearns. TABLE XXIX.—continued. " a & a, eel ee. RR op ' 6 ee 0. | ie be lb ) c | No. h é I | 20 '.1650 |.0001 |.0001 |.1651 | 4.998 | 1.1705 | .6529 29 4 ! - \ 164 6504 /(21 1.1627 |.0001 |.0001 |.1628 | 4.995 | 1.1367 |.6480 30 22 |.1444 1.0001 |.0001 |.1445 4.995 | .9469 |.G455 . 144) .6455 ; 23 /.1235 | 0 0 |.1235 |4.998 | .7495 |.6462 31 a 123 6515 D4 }|.1225 | 0 0 |.1225 |4.998 | .7526 |.6569 ep 2S i 39 | {2% |-1009] 0 0 | 1009 | 4.996 846 6829 | 101 Jabs 128 |.1008 | 0 0 |.1008 |4.996 | .5877 |.6875 33 29 |.0991 | 0 0 0991 |4.996 | .5498 1.6598 099} .6598 | af | 30 |.0746 0 0 0746 |4.996 | .3652|.6710 075 8710 | | | | | | It will be observed from the preceding tables that for heads above .3 (with one exception, No. 20) the values of ¢ are very uniform, while for heads below .22 there are variations from one to four per cent. with practically identical values of 1. These experimental discrepancies well illustrate our remarks at the beginning of this chapter in regard to the errors which may be expected to attend the weir measurement of water with small heads, even if the experiments are conducted by most careful and expe- rienced investigators with the aid of the most perfect measuring appliances. SECOND: COMPLETE, OR NEARLY Complete, ConTRacrion. Poncelet and Leshros. We will first take the results of Poncelet and Lesbros, with a weir .66 long; with inner depth below crest, 1.77, and with width of approach 12.1, thus having absolutely complete contraction on the sides. These experiments were made in 1828. TABLE XXX. Poncelet and Lesbros.— Weir with full Contraction. Discharge free into the Air. // measured 11.48 above weir ; hence Z=h. G=1.77 (2 g)4=8.0227 Figure 1, Plate I. ive | 9 | x, |\w#-m) 7 | @ . | L 2a ae ae CePA, (em eater ee | 35 | 6821 6273 | .0548 | .6562 | 1.1528 | 583 36 | .5351 4849 | .0502 5 -8098 589 | 37 | .B876 2082 | .0394 : | 4071 591 | 38 -1985 | .1686 .0299 » | 1864 | .600 | 3 1463 | 1207 | 0256 | ny 1194 608 | 40 | 0771 | 0577 .O194 | .04676| .622 Lesbros. In 1834 Lesbros obtained the following results, using forms of approach shown by WEIRS WITH CONTRACTION.—Expreriments. Lesbros. 97 Figs, 2 and 3, Plate I., and the same length of weir—using same orifice in a thin plate for a weir as in experiments given in Table XX VI. TABLE XXXI. Lesbros.— Weir with Full (2?) Contraction. Discharge free into the Air. H measured 11.48 above weir ; a> = ts (2 g)#=8., 02 27 Length always .6562. Fig. 2, Plate I. - ‘Fig. 3, Plate I. eau “E21ah SBT Gal fast ei Lesbros . Lesbros Nera BH) HH, Q c [No NG a HHH, Q ¢ 1666 | aed 1680 | | a0) 41 \/ 1667 | .5955| 0515 |.9476 -.9516| .590 | 45 |. 1681 | 3724. 0397 ee 4753 | .596 | 1668 ae l 1682 | 4719 1669 4541 1683 Lora 1670 4588 1684 | 0777 3625 0393 4565} .596 || 46 1070! 0987 | .0774| .630 | 1671 4564 1685 | | | .0767 i \ 1672 ! 4567 1686 | | 0777 1673 1612 | 1674 | 1597 43 1788| .0279 1618 .610 ;) 1675 1629 | 1676 1636 | 1677 oo) | 44 | 1678 | 0958} .0203 |.0654'-.0653/ G28 °° ( 1679 nae | : | | With Fig. 2, the vein after its escape from the weir with the higher heads converged a little towards the (prolonged) side of the canal, nearest the weir. With Fig. 3, the escaping vein had the same appearance as with Fig. 1. The head in the plane of the weir, or /Z,, in the Lesbros experiments was the mean height above the crest. It will be observed by looking at the foregoing values of @, that they vary in one case 24 per cent. for the same given head. This was probably more due to variations in H, than to errors in measuring g or J. Francis. Mr. Francis made 71 experiments with weirs having nearly full contraction, where he obtained Q by direct measurement of y.* We will first give the results of the 10 series, each series having nearly equal heads, with the reductions based upon the Francis [(H+h,)'? -hg? |"; cis obtained by Q=c' 3 (2g)*% Wl. The width of approach was in all cases 13.96; Zand L’ were hence each about 2 feet. As before, (2 g)# = 8.0202. The means are computed from the 3 formula of h = The discharge was in all cases free into the air. 3 power of HZ and h. * Lowell Hydraulic Experiments ; 2nd Edition, pp. 122-125. 98 WEIRS WITH CONTRACTION.—ExpERIMENTS. Francis. ls = 4 : “Na eee ted | @ ! é Francis @ and ea ar \ = _ : , i 1.5243 | 0092 |1.5330 61 282| .6040|] 34 = 0102) .0019 1.0121 25.988 | .5969 2 | 28 1.5504 0095 |1.5594 |62.569| 6011] 35 a 1.0262, 0020 1.0282 26.568 | .5959 3 bl a 1.5593 | .0097 1.5684 63.206 | .6020 itch Tae 1.0202 26.275 | .5964 4 1.5691 | .0097 |1.5783 \63.351 | .5977 : : 36 1.0280} .0140'1.0410 34.848) .6138 Ricans | 11.5508 1.5598 162.602| 6012 37 | SS 1.0372. .0143 1 1.0504 |, 293, 6133 5 1.2369! .0054 |1.2421 [45.089| .6094| 38 = 1.0445) .0146 1.0580 35.725, .6142 6 | 8 Lose 0055 1.2472 45.344 6091 / 39 | ~ 11.0449) .0146 1.0584 35, 766 6145 ei 1.2479} .0055 11.2532 15.678. 6091; 40 | _ 1.0460) .0146 1.0595 oa 6137 8 | 11.9508) .0055 1.2561 45.494) .6046/ 41 | & 1.0513] .0148 1.0649 36.072! .6141 9 | “ 1.2529 | 0056 1.2582 45.934 6089] 43 | i 1.0794) .017 1.0939 37.487 .6130 | & 1.2549 .0056 1.260 245.853 6064) 43 | © 1.0711} .0154 1.0853 37.051 6130 Means | ‘Lede: 1.2528 45.565 6079 Means 11.0504 | 1.0640 |36, 002.6137 11 -.9671| .0028, .9698 BLL 610156 | eS ©8186 0018. .8203 24.319 6123 12 11.0275 | .0033 /1.0807 33.942 6068) 57 | & | .8075' 0017: .8092 23,794: .6115 ig 2 1.0839 .0033 1.0372 34.287 6064 58 | wl | 7956 0017 .7973 23,276} .6117 ia 1.0331 | .0033 /1.0364 34.972 6077 | 59g 7769, 0016, 7784 22.480 6121 15 1.0406 | .0034 1.0439 34.655) .6079|| 60 | F : 8012) 0017, .8029,23,543 .6122 16 1.0373 | .0034 1.0406 |34.533' .6086| 61 | & | .7940: .0016' .7956'23.170' .6108 17 9632) 0028) .9659 30.957 .6101 | | + Stivers ge ! Means 7990 | .8007 23.430 .6118 18 9799 0029 .9787 91.538 | .6094) = Pn ae Bae 19 | 9795 .0029| .9823 131.658] .6084 paee je ee “145 20 | 2 | sas] .0030! .9917 32.144! 6089! 8 | ji 7872) 0073, 7941 23.258) .6149 21 | 4 | .9946| .0030/ .9975 32471. e097 | ° : = ee oor? SETS BeOS) oGl? 29 S1Br| 000 | S180 2eeniypong, AP) Sg Ne) Ae) seeRe Be eal ie 23 |__| 9280, .0025 .9304 29.193) .6085 mol ae oo GEO) SETS PEOOE| SGISS 24 = | 9462} .0026] .9488 30.090 .6091 Means 8269) 3347 25.041| .6143 250, & 11.0127) .0032 11.0158 33.300, .6085 72 ~ 5919! 0007! .5926!15.000! .6151 26 1.0116 .0032 |1.0147 33.316 6098) 73 | ts 5924 .0007 | .5931 115.027 | .6155 27 9949 / .0031| .9979 32.554 6110) 74 | & 6106) .0008| 6114 15.712] .6149 28 1.0336] .0034|1.0368 [34.414] 6098] 75 | + | ¢559/ .oo10l .e562 7.430) 6135 29 1.0556] .0035 1.0591 35.474) 6089] 76 © | 6430) .0009| .6439'16.984| .6149 30 1.0692) .0037 [1.0727 36.175] 6091] 77 | y | .6379| .0009' .6388'16.760| .6141 31 9837] .0029 | .9865 31.929} 6096] rg | 2 | gar 0009. .6346 [16.589 .6140 32 | 9782} .0029| .9810 31.671] .6098 — 33 | .9670| .0028 .9697 30.994 .eo72) e"rs eve , 6246 116.215 | .6146 Means 9977 1.0007 /82.580 | .6089 WEIRS WITH CONTRACTION.—Expveniments. Francis. 99 oe G a H a . Q ! y = G va Ut | fe . h | Q ! a : | 79 .6515 |.0045 |.6558 |17.425|.6139 | 85 t= 6694 | 0030 6723 14.220 | .6033 80 S | 6559 |.0046 | .6603 |17.613 | .6142 86 ~ 1.6790 |.0031 |.6819 |14.519 | .6030 81 | | .6598 |.0047 |.6643 117.772|.6141 ' sv | 4.6836 |.0031 |.6866 (14.664 | .6028 82 ~ | 6313 }.0042 |.6353 16.618/.6139 | 88 | = 6881 |.0032 |.6912 |14.7xK | .6019 83 = |.6425 |.0044 1.6466 '17.058|.6137 | Means| si 6801 | 6830 [14.518 6027 84 a1 | 6546 | .0046 | .6589 17.554!.6140 | : aml | Means | © | 6493 16536 17.340 .6140 | | Experiments (Francis) Nos. 34-35 and 85-88 with /=7.997, were made by dividing the weir with / = 9.997 into two weirs, each 3.9985 feet long. This was done by placing a dam or partition in the centre of the long weir, 2 feet in length. These experiments hence should be considered as showing the discharge over a weir with /=4., with con- traction rather imperfect at one end. Taking the means of the foregoing 71 determinations, we have the following table, with h,’ and h calculated according to our methods, and « deduced from our values of h. The results in this table will be adopted by us as authentic. The greatest difference between our values of ¢ and those of c’ deduced by the use of the Francis formula for velocity of approach, is .0049 for No. 51 (Francis Nos. 36-43), where c=.6088 and c’=.6137 ; this is a change of about 4th from the co-efficient given by Mr. Francis. TABLE XXXII. Francis.—Weirs with Contraction more or less complete. Width of Approach in all Cases 13.96. Discharge free into the Air. H measured 6. above Weir. h=H+b ae (2 7) = 8.0202 . ET es : — 2142 U8 eee SS . ein Bg Be et i Francis T | ‘ No. Nos. lof Water G H h, b h, h L Q ¢ 47 | 14 | 465° | 46 | 1.5508 | .0095 | 1.4 | .0133 | 1.5641 | 9.997 | 62.602 | 5988 48 | 5-10 | 485° | 46 | 1.2476 | .0055 | 14 | .0077 | 1.9553 | 9.997 | 45.565 | .6061 49 | 11-33 aie 4.6 9977 | 0030 | 14 | .0042 | 1.0019 | 9.997 | 32.580 | .6078 50 | 3435 | 48° | 46 | 1.0183 | .0020 | 1.4 | .0028 | 1.0211 | 7.997* | 26.975 | .5956 51 | 36-43 | 487° | 2.014 | 1.0504 | .o148 | 1.3 | .0193 | 1.0697 | 9.997 | 36.002 | .6088 52 | 56-61 46 7990 | .ool7 | 1.4 | .o024 | .go14 | 9.997 | 23.430 | .6110 53 | 62.66 | 487° 2.014 .8269 | .0083 | 13 | .o111 | .s3a0 | 9.997 | 25.041 | .6107 54 | 72-78 | 489° | 4.6 6238 | 0009 | 1.4 | 0013 | .6251 | 9.997 | 16.215 | .6138 55 | 79-84 | 48.7° | 2.014| .6493 | 0045 | 12 | .oo60 | .6553 | 9.997 | 17.340 | .6116 56 | 85-88 2.014} 6801 | .0031 | 1.4 | .0043 | .oxdt | 7.997* | 14.548 | .6009 * The co-efticient c, for Nos. 50 and 56, applies to a weir having a length of 4.0. 100 WEIRS WITH CONTRACTION.—ExpERiMEntTS. Francis. The channel of approach for the Francis experiments with end contraction, was narrowed to a width of about 12.5 feet, just above the hook-gauges. This change in the section of the feeding canal was objectionable ; in our judgment, however, errors, or rather discrepancies, arising from this defect would not be large. TTamilton Sinith, Jun. In Chapter IX. ix given a detailed description of 12 experiments made by the author. They are sufficiently exact when @ does not exceed 10.—Nos. 57-63. After that limit the measuring vessel for y, whose capacity was about 1300 cubic feet, was too small to afford very accurate results. The effect of velocity of approach, which was insensible for volumes less than @=10., became notable for larger values of @; for this weir it is difficult to decide what should be the proper correction for v,; probably C for Nos. 67 and 68 is about 4 per cent. higher than ¢; on this supposition the correct co- efficients, c, would be for No. 67, ¢=.581, and for No. 68, ¢=.578. The proper correc- tion for v, for Nos. 64 and 65 would be slight. was measured at a point 7.6 feet above the weir, vide Fig. 8, Plate XV. TABLE NXNIII. Smith.— Weir with Complete Coutraction, Lnner Depth below Crest 3.8. No Correction for vq. Free Discharge into dir, Temperature of Water from 50° to 60°. (2 g%=8.0177 Me 0 ae > & 0 C I | a7 sd|LGY | 2.586 B.582 6087 58 6168 |, 4.034 | 6032 59 =| .6470 | 4 4.338 6030 60 OWE og 4.567 6020 61 7072 : £950 | .6021 | 62 1.0681 e 8.988 | 5890* | 63 1.1063 . 9455, D878 | 64 1.2033 - 10.783 5910 65 1.3257 2.585 | 12241 | seo, | 66 1.5301 2 15.614 | 5918 67 L7195 18.19f 5840 | 68 1.7327 ze 18.318 | 5812 | * With No. 62 there was some little doubt as to whether termination of experiment was correctly determined or not. Feley aid Stearis. Fteley and Stearns indirectly determined the discharge over weirs with nearly com- plete contraction, and over weirs with contraction suppressed at one end. In one of WEIRS WITH CONTRACTION.—Experiments. Fteley and Stearns. 101 the weirs of the latter form, contraction was also partially suppressed at the other end, the distance from end of weir to side of feeding canal being only 1. . For these experiments @ was determined by the flow over the weir 5 feet long, with end contractions suppressed, and whose co-efficients of discharge have been shown by Experiments Nos. 17-34. To obtain the proper values of « for this 5-foot suppressed weir, we will slightly anticipate by referring to Plates VI. and VII., where the curve of ¢ for this weir has been drawn. The head, H, for the 5-foot or datum weir, was generally measured at the beginning and ending of each series of experiments, and did not greatly vary ; hence Q was nearly constant for each serics. HH was in all cases measured at a point 6 feet above weir. Tt will be noticed that a number of these experiments belong to our third category, but for convenience they are here included with those belonging to the second. The discharge was free into the air, except where contraction was suppressed on one or both ends; in the latter case the side, or sides of the feeding canal were extended, but only above level of crest. The crest was sharp-edged, being the one employed for Velocity of Approach Experiments, Nos. 1-111. The length was changed by putting in wooden false pieces ; to the irregular swelling of these wooden pieces is due the slightly varying values of /. The values of ¢ and Q in small type are based upon the 5-foot curve (suppressed weirs) on Plate VII. ; the slightly varying values of (y in ordinary type are interpolated, for each series, in accordance with Table XXVIII. of Fteley and Stearns. 102 Fteley and Stearns. —Weirs with more or less Contraction at one or both Ends. Inner Depth below Crest 3.56. WEIRS WITH CONTRACTION.—EXPERIMENTS. about 40°, TABLE XXXIV. (2. g)¥ =8.020 Fteley and Stearns. Temperature of Water Distances from Ends of Weir to ie Sides of Canal. | 7 | I | : | 24 a ae es 24 101 9 oH \ 3 4 2 8 70 4 | 1.0 1.0 5 | & if @ 6} 0 0 7 1.0 0 o a), 1.0 72 9 | 1.0 1.0 10 | 0 0 Lj; 0 0 (ie, 10 0 a2 18 0 1.0 nm, 48 | 1.0 1.0 7a | 18 0 Le 7¢ | 16 | 16 17 iT 0 18 | 0 0 7 | 2 | a 1.0 78 | 20 0 ig a | 0 0 22 | 0 0 7); 23 | 10 1.0 sO | 24 | 1.0 L7 e 25 | 0 L7 26 | 0 Ly | 7 jo 0 28 0 0 ze 29 1.0 0 ee 0 1.0 | 2303 .000 15 .2691 .000 15 2693 .000 15 .3301 |.000 14 .2304 |.000 15 | 1509 |.000 05 .1761 |.000 05 1763 |.000 08 | 9155 000 04 1509 .000 05 3942 .4244 |.000 84; ' 4978 5678 4245 .4308 6215 THIS 5764 5766 4302 OO 87 5115 5997 5996 .3368 .000 44 000 43 3944 .000 43 4843 000 41 ~ 4498 '.000 42 5824 000 39 .3369 .000 44 | .000 81) .000 78 .000 84; .000 87 .000 79 .000 75 .000 81 .000 81 .001 41 .001 35 .001 35 ly Ld 14 Ld 3 ly Lt 14 1.4 Ii 3B ia 1.4 14 1.4 1.4 1.4 13 li 14 1.4 i 3 jt ld lt lt 14 ig 13 1.4 1.4 oe ; 0001 | 0001) . 0001. 0002 0002. 0002 0002 0002 0006. .0006 .0006 .0006 , .0006 .0005 .0006 -0011 .0011 0011 .0011) . 0012 0011 0011 0011 0011 0012 | -0019 .0019 .0019 | Means. h d Q c h c | .0001 | 1510 | 5,0048 1.0127) .6450 » 1762 4.0062 1.013 6304) 176 638 1764 | 4.0064] 1.013} .6383 © 2156|3.0080 1.013] .6292' .216 .6292 1510 | 5.0048) 1.0127] .6450 .2305 5.0044 1.8820) .6356 2693 | 4.0063 1.482 .6287 | o69 gays 2695 4.0062) 1.883) 6283. | .3303 | 3.0081] 1.883 .6168 . .330 616s .2306 | 5.0044 1.8832 .6356 .3374 | 5.0045] 3.3021| .6297 i 3948 | 4.0063 3,302! 6214 a .395 |.6212 .3950 | 4.0061| 3.302] .6210 | | 4849 3.0080, 3.303, 6083 485 .6083 £504! 3.3110! 3.303] .6173 |) .450 .6173 5829 | 2.3132] 3.303] .6001 | .583 .G001 .3375 5.0045] 3-3035| .6297 : 4255 | 5.0043) 4.6614 .6277 .4989 4.0058 4.662) 6177 499 6177 5689 3.3107 4.663) 6139) 569 .6139 4256 5.0043) 4.6630) .6277 . .4320 | 5.0049] 4.7685! .6276 | 6226 |3.0070| 4.764] .6032| 623.6032 -T489 | 2.3125 4764, 94S 749 pee 5775 | 3.3104 4.763) 6132.) eosteane 5177 | 3.3104) 4.763] .61g9| f-°/8 8190 .4314] 5.0049] 4.7586! .6276 .5134 |5.0046 | 6.170 | .6268 | .6016 |4.0063! 6.172] .6175 l 6015 (4.0050) 6.172! .6179 | f°? fot" WEIRS WITH CONTRACTION.—Expreriments. Fteley and Stearns. 103 TABLE XXXIV.—continued. Distance from, Ends of Weir to | | Means. No. Is a Sides of Canal. : ae b h,, h 1 Q ¢ | , = | | | | Ie etd t 83 | 31 | 1.0 1.0 739800126, 14 | .0018) .7416/ 3.0070 6.172) 6011, 742.6011 8 | 32: 0 1.7 | 6860100130, 1.4 | .0018] .6878|3.3101} 6.172| 6114] .688|.6114 85 | 33 | 10 LT | $905,001 18 1.4 | .0017| .8922/ 2.3125] 6.172) .5924], .892 5924 | 34 0 0 511700141 1.0019 .5136) 5.0047) 6.173 | .ez68 | | 86 | 35 | 1.0 L7 | 9548 00141, 14 | 0020) .9568) 2.3126) 6.840| .5911| .957 5911 36 0 0 | .5477 00170 11 | 0023, .5500|5.0046) 6.840 | .6267 37 0 0 6010 .002 19 14, .0029| .6039 | 5.0040) 7.870 | .6268 sf 38 | 10 0 | .7055 00208 1.4 .0029| .7084) 4.0076) 7.871 .6161 | 709 6166 39 0 1.0 | .7064!.00208 1.4 | .0029/ .7093 4.0064) 7.871) .6151 83 | 40 | 1.0 1.0 | 871400193 14 | 0027] .8741 | 3.0104| 7.871] .5984]) .874| 5984 41 0 0 | 6011 00219 13 .0029 .6040/ 5.0040] 7.872 | .e26s | v0 42 0 1.7 | 806200198 1.4 0028 .8090 3.3112] 7.871| .6110 809 i 43 0 L7 | 8063001 98 1.4 .0028) 8091 3.3110) 7.860 .6101 44 0 0 .6002|.00219 12 .0029. .6031 5.0039] 7.854 | .6z6s . 90 | 45 | 1.0 1.0 | .8702 .001 93, 14 .0027 .8729/ 3.0098) 7.856] .5986/| 873 .5986 46 0 6925 00322 14 .0043 .6968/5.0042) o770 | eeze | Ww; 0 6924 .00322 12 0043. .6967 5.0042) 9.768 | .6278 18 | 0 1.7 | .9317.00287 1.4 .0040/ .9357| 3.3095, 9.759] .6093 91 { : . | a \ 935 6092 49 0 1.7 929700285 1.4 .0040! .9337]| 3.3095) 9.722) .6090 | 50 0 0 | 6897 00318 11 .0042/ .6939| 5.0042) 9.709 | .e278 51 1.0 0 | .9448/.00452 1.4 | 0063) .9511 | 4.0065/12.306 | .6193 | st ee | 3 1.0 | .9432 00452 1.4 .0063/ .9495 4.0043 12.306 | .6213 | Peis [53 0 0 8047 00480 12 0064, .8111 | 5.0038) 12.306 | 6207] [ Bt | 10 | 0 | 9460 .00452 1.4 .0063) 9523] 4.0065)12.306 | .6182 | ! Column /, in preceding table is taken from Fteley and Stearns, and is correct enough for our reductions, although obtained from slightly different values of Q, than those which we have adopted. As Q was indirectly determined in these experiments, the results cannot be considered to have equal weight compared with the other weir determinations of the same authors, where @ was obtained by absolute measurement. The results, however, shown by this table (XXXIV.) are doubtless near the truth, as the curve on 104 WEIRS WITH CONTRACTION.—Experments. Fteley and Stearns. Plate VII. for the 5-foot weir, which we have used, was deduced from experiments made under very nearly similar conditions. Lesbros. Lesbros with a square-edged weir, having a thickness of .164, and approach 12.1 wide (Fig. 1, Plate I.) obtained the following results : TABLE XXXV. Lesbros.—Thick-edged Weir with full Contraction. Discharge free into Air (except the Contact of the Vein on the Crest). H measured 11.48 above weir; hence 4=h. Inner depth below crest, 1.77. 2 g)4 = 8.0227 Fig. 1, Plate I. 3 | Secl og R (em) 3 Q ‘ es 0. | ! 93 1840 1.3806 | 1.2549 |.1257 | 1.9685 | 10.031 | 584 | 94 1841 1.3796 | 1.2540 |.1256 59 9.984 585 ; 95 1842-3 8711 7720 |.0991 ‘4 5.031 | .588 2877 | 96 1844-5 494 .0617 ‘i 1.3193 | 607 | The escaping vein did not touch vertical sides of weir, but attached itself more and more to the crest, as the heads diminished. TurrD: Conrraction SUPPRESSED ON ONE SIDE oR Borrom, AND VARIOUS Forms or APPROACH. Lesbros. A number of experiments in Table XXXIV. belong to this category, and particular reference will be had to them when we discuss the curves representing c. All the following experiments have been selected from Lesbros. It will be sufficient to give the means of Q, which were, with one or two exceptions, repeated several times. This repetition, however, as we have suggested before, does not seem to have checked the accuracy of the measurement of H, and in which generally lies much the greatest danger of error. The various forms of approach are shown by Plate I., and also by Table VI. For all these experiments H was measured in still water in the reservoir at a distance of 11.48 above the weir, and hence H is always equal to h. (2g)” is constant at 8.0227. The discharge was free into the air. Where there was contraction the edges were sharp, so that the escaping vein only came in contact with the inner corner lines. These WEIRS.-: Various DEGREES OF SUPPRESSION.—EXPERIMENTS. Lesbros. 105 experiments (97-128) were all made by using the orifice, .6562 x .6562 in a fixed copper plate, as a weir. Temperature of water is not stated; most of the experiments were made in the autumnal months, and a few in August. TABLE XXXVI. Lesbros.— Fig. 4, Plate I. Bottom Contraction Suppressed. Distances from Sides of Weir N : se to Sides of Canal. W | H, HH, 1 | 0 i ( 2 SS Bottom. Sides. 97 | 1687- 9 | | .6732 | .6152 .0580 | .6562 ae t it 607 98 1690-2 | |, 38934 ° .3383 .0551 | %5 | 530 4 : 613 99 | 1693-4 |} “0 | 571 5.71 | .2198 .1706 0499 4 2220 ) .614 100 | 1695-7 | 1014 | 0659 | 0355 | ,, .069 71 | 615 101 | 1698-9 |; | | .0528 ye .0305 .0223 | 5 025 34 | 59S The vein in all these err (97. 101) did not touch the chamfered ieee ive of the crest; for heads below .05 the vein attached itself to this chamfer (level). TABLE XXXVITI. 8, Plute I. Contraction nearly Suppressed on One Side. \Distances from Sides of Weir | No. — eee at ow oo Se. epee og 0G ‘ | | Bottom. | Sides, | | 102 1730-2 | | 7 6683 5997 | 0686 | 6562 | 1.1551; .602 103 | 1733-5 |) 4885 4331 | 0554 |, «| .7310] 610 lod 1736-8 |. 1.77 066 714 | 31362733 | .oso3 |, | 8791 | 615 105 | 1739-41 | I! 1667 144d | 10293), 1521 | .637 | 1742-4 | Ll 0659 | .0591 | 0068 | : | 03973 .669 The surface of the water in the reservoir was Sieaated a little higher on the side of the canal sieanest the weir than on the other side, and the escaping vein converged more or less towards the prolonged direction of this side (.066 from end of weir), according as 7 was more or less great. 106 WEIRS. Various DEGREES OF SUPPRESSION.—EXPERIMENTS. Lesbros. TABLE XXXVIIT. Lesbros.— Fig. 5, Plate I. Contraction Suppressed on Bottom, and nearly Suppressed on One Side. Distances from Sides of Weir | | No. — to Sides of Canal. | oH Ul, | pee 1 : Q | ; Bottom. Sides. | | | | 107 | 1700-1 6847 | 5984 | 0863 | 6562 | 1.2158) .611 108 | 1702-3 0 066 | 5.71’! 3566 | 2887 | .0679 | _,, 4547, .608 109 | 1704-6 \ | (! o702 0483 | 0269 | - 037 oe! D571 The surface of the water was higher on the side of the canal nearest the weir than on the other side, and the escaping vein converged’more or less towards the prolonged direction of this side (.066 from end of weir). For No. 109 the vein attached itself a little to the lower chamfer of the crest, on the end of the weir farthest from the side of the canal. TABLE XXXIX. Lesbros.—Fig. 9, Plate I. Contraction nearly Suppressed on Both Sides. Distances from Sides of Weir | see. sh \ to Sides of Canal. | H It, =m 1 Q - Bottom. | Sides. | . | 110 1745 | (. 7133 .6562 | 1.3505 | .639 111 1746-7 | | | | 6857 6014 | .0843 8 ! 1.2671 | .636 112 1748-50 | | | | 5085 4446 | .0639 x .8080 | .635 113 | 1751-3 |r 1.77 .066 0665 | .3835 | .3379 | .0156 » | s255" 630 114 1754-7 | | | | S20 .2900 | .0391 ‘3 | 4179 | .631 115 1758-60 | | .1693 1476 | 0217 5 | «1560 | .638 116 1761-3 , .0686 .0545 | 0141 oO .041 85) .664 The water in No, 110 covered the upper edge of the orifice, except for a length of .2 in the centre. This experiment, therefore, is just on the dividing line between that for an orifice and that for a weir. TABLE XL. Lesbros.—Fig. 6, Plate 1. Contraction Suppressed on Bottom, and nearly Suppressed on Both Sidvs. Distances from Sides of Weir | aT, a to Sides of Canal. W H, Ho, I | Q : Bottom. Sides. 117 1707-9 || | [| .8366 5036 3330 6562 | 1.5321) .570 118 1710-2 | .8340 5020 3320 sf 1.5258 | 571 119 1713-4 | .6460 3967 2493 5 1.046 2| 574 120 | 1718-6 || 9 igne 066 || 5089 | .3159 | .1880 ; 7223 | 575 131 1717-20 | 3392 22175 Oe 3966 | .572 122 1721-3 : .2014 .1207 0807 . A767 | 557 123 1724-6 | | 1326 0743 .0583 i .089 46) .528 124 1727-9 | | || .0702 .0364 .0338 + .031 40) .481 WEIRS. Various DEGREES OF SUPPRESSION.—EXPERIMENTS. Lesbvos. 107 For 117, 118 and 119 there was a strong boiling action at the escape from the weir, caused by the shock of the water against the narrow inner intervals on the ends of .066. See remarks in Surface Curve at Wezr as to the form of the surface for experiments 117-124. TABLE XLI. Lesbros,—Fig. 12, Plate 1. Dnuelined Sides of Canal. \Distances from Sides of Weir. | I | to Sides of Canal. | Ne oe | o Sides of Cana | We | H, | U-H, h 5 mii Ik og | | | Bottom. Sides. | 125 | 1778-9 |. i { 6545 | .5873 | .0672 | 6562 1.1246 .605 126 | 1780-2 | wes 066 | AT34 4200 | .0534 i 6905» .604 127 1783-6 | .2890 | .2506 | .0384 . .3318 | .608 128 1787-92 5 | 0689 | .0594 0095 » 04150 .054 i | 1 | For Nos. 125 and 126 the appearance of the escaping vein was the same as with Fig. 1. For No. 127 the horizontal length of the escaping vein was ; At the weir 59 .16 below weir .72, and after this distance, a constant narrowing in length. The same phenomenon occurred for No. 128. Lesbros made a number of experiments with other forms of approach, the weir being placed in some of them at the end of a canal of same width as length of weir. All these experiments were more or less complicated by other causes than those we are considering, and are of no value in assisting us to our final conclusions. The reader who desires to investigate them is referred to Lesbros’ original volume, Exeperiences Hydrauliques sur les lois de écoulement de PEau. Paris, 1850. Surrace Curve at Weir. Before comparing the various values of ¢ given for the foregoing selection of experiments, it will be well to discuss the form of the surface curve of the water as it approaches and passes the crest of the weir, and also to investigate the varying values of H—H,,, as given by Lesbros and others. Lesbros. With Fig. 10, Plate I., the surface on the central axial line of the small feeding canal—shown by vertical section normal to the plane of the weir, and bisecting it—had the following elevations above the crest of the weir; H, measured 11.48 above the crest, was .5089; this weir had both end contractions suppressed, with /=.6562. 108 WEIRS.—Surrace Curve. Leshios. TABLE XLII. Lesbros.—Central Surface Curve. Ga Lii I'ig. 10, Plate J. z 2 | Distance | Elevation | a | Distance | Elevation | + Distance ' Elevation | 4 up-stream | Surface 3 up-stream | Surface | °8 (horizontal): Surface & | from Weir. ‘above Crest A from er Crest.|| ™ below Weir. above Crest.’ 2 sae eM ee 3 a sae te male Fee, SE Mate Se pH M48 | 5089 |e | 98 | 4997 |) 0 4219 a 6.60 | 50836 «mm ! 820 4954 | vo 164 ae: | b | 6.40 | 5003 9 » | .656 | 4918 w 328 | 2900 c | 623 | 49st || o | 492 ° 4866 | « 492 | 1736 i 5.91 4990 pp.) 394 A810 y 656 | .0118 e | 5,25 4990 | q | 328 AT54 fz 820 |—.1660 fj} 459 5007) r .262 4688 | ig | 3.94 5010 | s 197 | 4620 | | h 3.61 5026 | t 131 . 4518 | i 2.62 5026 | w | 066 | 4400 fF | S 1.97 50230 i Ht 0 | £219 | | |e Ta) 5003. ¢ | | | ; i I The point 6 was in the plane of the mouth of the feeding canal, 6.4 feet above the welr. The irregularities in elevation from « to g seem to have been caused by the con- traction at entrance of the canal. The effect of the surface weir curve extended perceptibly to J, a distance of about 1 foot from the weir; and very slightly—if the above measurements can be relied upon for such small quantities—to the point /, at a distance of 2.6 from the crest. There was a notable loss of head, caused by contraction at the mouth of the feeding canal. For ; HT measured in still water in reservoir = .5089 iT’ 5 at 7, where surface curve begins = .£997 Apparent loss of head = .0092 This loss was partly due to head imparting velocity of approach, which can thus be computed: @ for above value of H was about .827, vide Experiment No. 2; hence a,=(.50+ 1.77) x .656 = 1.49, and pet =,55¢ with o=1, A =6 “a =.0063, head 1.49 2g absorbed in imparting velocity of approach. Therefore .0092—.0063 =.0029 = head lost by primary contraction ; this loss would, for A= .5089, diminish ¢ about 1 per cent..* We will call foregoing experiment No. 129, with h =(.4997 +.0063) =.5060 ; conse- quently c has a value of .655. * A small portion of this loss was due to “‘ frictional” losses as the water passed through the feeding canal. WEIRS.—Surrace Curve. Poneelet aad Leshros, 109 Poncelet cid Lesbros. For Fig. 1, Plate I, with a weir having full contraction, with /=.6562, the central surface curves were as follows, for T=.5915, and =.0951. TABLE XLII. Poncelet and Lesbros.—Central Surface Curves, (F=1.77 Fig. 1, Plate I. [ee Ea a ; | 43 | Distance | Elevation | Distance | Elevation ,; Distance | Elevation 3 up-stream Surface | 3 up-stream | Surface A (horizontal) Surface AY | from ae Ee oe AY | from Ponaee Crest. | & |below Weir. above ae | IE | 1148 5915Q)) g 394 ° 5840 |H,. 0 5394 la) 984 5915 | kh, 328 5807 | m, — .066 5220 | 5 | 353 | sore Lal 262 5758 fw | 181 4997 ie | 722 | 5899 ij 16s 5673 | o | 210 4659 dad 59 S879 k 098 Dd84 pj 295 AIST | e 525 | 5876 | l | .033 SATII g 394 3396 f | 459° | .5866 |, 0 ' 5394 + 4 sor ' | . | H | 1148 .0951 (2) g 787 .0866 | He | 0 0761 a 148 0951 | & + 591 | .0837 m 033 0682 | 3% 1.38 0945 i BOF 0810 on —-.066 0508 _@ 72s 0935 ji 295 0797 0 .098 0246 d@ 118 | 09232 , k 197 0787 | p BL. —.0115 » 1.08 ! .0906 | | 098 =, 0781 g 16 — 0542 | f | 984 0x92 | a 0 0761 - 295 297 The surface sections in plane of the weir for the foregoing heads were as follows ; 9915 5410 5394 5374 5682 5843 958 -0951 .0746 0761 0722 .0814 0948 H,, being mean elevation along weir; H,,' being in the centre of the weir; HH," being .06 from end; #7," being at end; H,,'" and H,,‘ being between end of weir and side of canal, #7," at a distance of .06 from weir, and H,,” at .26 from weir. It will be observed that the effect of the surface (axial) curve is sensible for a greater distance up-stream with small than with large values of H ; for (including two other measurements of this curve, the details of which we do not give) ; /f=.5915 Effect of curve is sensible a distance of .98 from crest. $= A81T 4, ‘ ‘ Je. TES ~ ag H=.2369 —,, : : = deri 2 H=.0951 ,, iy ; a 110 WEIRS.—Svurrace Curve. Poncelet and Lesbros. Another remarkable feature is that the surface elevation, with the head of .5915, between the weir and the side of the canal is .0043 greater than H. Leshios, With Fig. 4, Plate I, (bottom contraction suppressed), the form of the surface section in the plane of the weir varied with the head. With [7 above .42 there was a large descent from the sides of the weir, and a slight rise in the centre; with lower heads there was the same rapid dropping from the sides, then a slight rise, and then a very slight depression in the centre. The surface axial curve apparently extended back, towards the reservoir, a con- siderable distance from the crest. With Fig. 5 (contraction suppressed on bottom, and partially on one side), the surface section in the plane of the weir was highest on the end of the weir nearest the side of the canal, with a very marked depression for considerable heads at a point about two-thirds the length of the weir from this end. The axial curve, as in Fig. 4, appears to have been sensible a considerable distance up-stream from the crest. With Fig. 6 (contraction suppressed on bottom and partially on both sides) there was for considerable heads a very notable depression in the axial curve, after the water entered the feeding canal, indicating the loss by primary contraction. With H=.4649 the surface elevations above the crest were as follows : H 11.48 up-stream from crest ; in reservoir 3 ‘ ; 4649 D_ 6.40 #5 53 at mouth narrow canal . , +197 & 541 5 5 at lowest depression in canal . .2615 Ff 3.28 5 s highest point between Hand /#7,,' . .2986 fH} crest at centre. : ‘ ‘ ‘ é i . 2648 For this value of H, Lesbros found () =.6391; hencec=.5744, in c= ee lH2(29 H)* Calling this experiment No. 130, and calculating ¢ from head measured at point F as above (3.28 from crest) we have: = 2986 > t= 2986 6.787 4S.2351) B= = 2,718 5 2 hg = oF =.1148; h=H+,=.4134*; hence c=.685. This indicates a loss of head by primary contraction and friction on sides to the point F, of nearly 20 per cent. In the above instance the correction for h, reduces the value of the co-efficient of discharge about 40 per cent.. For; HT being .2986 C=1.116 hy 4154 c= .685 * Contraction being suppressed nearly completely on three sides, ht, =h.'. WEIRS.—Surrace Curve. Lesbros. 111 With this form of approach (Fig. 6) the surface section at mouth of canal—point D—declined from the sides and rose again at the centre ; in the plane of the weir, with H=.4649, H,} (centre) was .2648, and H,,™ (at ends) was .3478. Fteley and Stearns. Messrs. Fteley and Stearns determined, as shown by following table, the surface curves for a 5-foot suppressed weir, with varying inner depths (G) below crest. The measurements were made on a line 1.5 distant from side of canal, and hence 1. from central axial line. Single observations were taken, which were corrected slightly to form smoother curves by these authors; we give the original measurements, all made (except H) by point-gauge. TABLE XLIV. Fteley and Stearns. ee Curve at eee) Weir, 7=5.. Q constant. L¢. 3 | 3 4 | 5 | Distance from“ 7=3.56 | G=26 | @=17 | @=1 | G=5 | a | vg =.389 =.506 | v,=.705 v,=1.022 | », =1.529 | | h, = .0024 i =.0040 | h,=.0077 2h, =.01 62! 2, =.0363 | 2, | 6. up-stream=// 6143 | .6120 | 6059 «6©| «5904 | 5635 3.5 . | 6142 «6117s! ~=—«6060 | 5874 | 5635 3. - | 6135 6126 .6061 | 5903 | 5631 | 2 ‘ / 6084 | 6102 | 6039) «5894, 5621 | 1.5 : 6083 | 6066 | .6013 | 5882 | 5616 hy 2 6017 + 6002 5939 5833 «| «5596, J sa D847 5827 | .5788 5671 B47T4 | O cvestaw, 5226 52.26 5182 5097 5003 | ! 3 down-stream| .4251 4261 4212 4089 3914 i 38 , 385 3814 3786 3677 43739 HH, 0917 0894 0877. | .0807 0632 Castel. In the “ Mémoires de Académie des Sciences de Toulouse,” Tome IV., 1837, M. Castel in his Table VIIL., gives the form of 34 surface curves, with various values of 1, two widths of feeding canal, and G constant. The curves were determined by 10 vertical rods, pointed at their lower ends, attached to a horizontal bar, which appears to have been placed in the central line of the feeding canals. The results obtained by Castel show, without exception, that G and the width of the feeding canal being con- stant, the length of the surface curve diminishes as H and / diminish. The maximum length—horizontal ordinate—of the curve was 1.37, with G'=.56, H=.38,/=1.18, and 112 WEIRS.-——SurrFace Curve. Castel. width of canal = 1.18, the weir hence being suppressed ; the minimum length was .32, with G=.56, H=.20, 7=.10, and width of canal = 1.18. Herschel. Mr. Clemens Herschel, hydraulic engineer of the Holyoke Water Power Com- pany, has been kind enough to give us a section of the surface curve of the Connecticut River, as it passed over the Holyoke dam in 1883, in time of freshet. The form of the curve is shown by the following sketch. $$ S I OS) ___ bine of Abutment s'_ SURFACE CURVE : i : + 6 + = 6 a 2 uo 3 7 _ 2 D eo io Ss = te i nn t> io) Eee) ~ = > mf 1 : ‘ ® ‘ : : ‘ Povn TS aes. 10 é limber Dam abatt 30 tt. high The height of the surface of the sheet above the crest, H,,, was 5.2, with H=7.27 ; the horizontal length of the curve was about 50 feet. The dam is a long one, perhaps 1200 feet, and the pool formed by it extends a distance of some two miles above the dam. The section wax taken along the southerly abutment of the dam. Boileau. M. Boileau found that the horizontal length of the surface curve increased with H; “Traité de la mesure des eaux courants.” 1854. p. 55. H—H,,. For a weir .6562 long, with full contraction, H—#,, was determined to be as follows by Poncelet and Lesbros, and Lesbros, with forms of approach shown by Figs. 1, 2 and 3, Plate I. ; in these experiments H,, was the mean height in the plane of the weir. WEIRS. H—H,. Lesbros &e. 113 M—-If, iT IL, HH, a 6821 6273 .0548 .0803 29955 .5440 0515 .0865 B51 4849 .0502 0938 8724 3327 .0397 .1067 3625 3232 .0393 1084 3376 .2982 0394 1167 1985 1686 0299 1506 1788 .1509 .0279 1560 .1463 .1207 0256 .1750 .1070 0843 0227 a2 .0958 0755 .0203 .212 0771 0577 .0194 252 For weirs with full contraction H— 7, increases with the length of the weir, H being constant. For instance; with weir 1.97 long (Experiment No. 96) H=.349 and H—H,,=.0617 ; with weir .066 long (Experiments Nos. 133 and 134) H=.533 and H— Hf, .0026, also H=.267 and H—H,,=.0017. Castel’s experiments show the same results. For weirs with end contractions suppressed 7—H,, is much larger than for those with full contraction, as will be noticed by reference to Table XX VI. (Fig. 10, Plate 1), where with, Experiment No.1. i ‘ 3 H=.80 and H- lf,,=.1460 . a W251 , #2080 4 ete 2 « & ave, a3 In all probability with this form of weir H—AZ,, also increases with /, although as Z increases, losses of head by friction and adhesion on the sides diminish. By reference to Tables XXXVI., XXXVIII., and XL. (Figs. 4, 5 and 6) it will be observed that H—H,, is often a very large fraction of H, especially with small heads. For instance ; Experiment No. 117, Fig. 6 ‘ . M=.8366 and H— 7, =.3330 35 yy AQAS sy ay 5 . H=.0702 ,, H-H,=.0338 95 wy L009, 55 8 : . H=.0702 ,, H-H,=.0269 - gy Dil gy o# 3 H=.6732 ,, H-H,=.0580 ss sp LOL, sy. 95 : . H=.0528 ,, H-,,=.0223 The abnormal Value of H—Z,, in No. 101 (Fig. 4) is largely due to the friction and adhesion with such a small depth for the distance of 8.2 feet above the weir; hence when these retarding influences cease to be important, as in No. 97, where H is .67, Q 114 WEIRS. H-H,. Lesbros éc. H— Hf, should approach a value similar to that with a weir having full contraction— and which will be observed, is the case. For No. 117 (Fig. 6) the loss of head, shown by large value of H—H,, is chiefly attributable to the primary contraction at the mouth of the small canal, and to the head absorbed in imparting velocity ; in No. 124 the loss of head was chiefly caused by fric- tion and adhesion on the bottom surface. From the foregoing determinations we can pretty safely draw the con- clusion, that the horizontal length of the surface curve—from the plane of the weir to a point where the curve ceases to be sensible—increases directly with 7 and H; the experiments of Poncelet and Lesbros stand alone in showing that this length decreases as H increases. From the experiments of Fteley and Stearns, Table XLIV., with Q and / constant, and G varying from 3.56 to .50, it Gil fT land H being constant; this difference in length, caused by wide variation in G, is, however, not very marked in these experiments ; with the five given values of G, H could have been safely measured at a point about 3.5 feet from the weir. The Holyoke curve is of much interest, showing that with very high values of H and /, the length of the surface curve was 50 feet. This, in common with several of the other given curves, indicates that there is a summit or anticlinal point of slight height in the curve, soon after its origin in the feeding channel. H should probably be measured above, or up-stream from this wave summit. is probable that as increases, the length of the surface curve slightly increases, The measurement of H,, as well as H may be of some value in determining in a rough and indirect way, as was the case with the Lesbros experiments, the losses of head between the measuring point for Hand the crest. Owing to the mechanical difficulties in the way of the exact measurement of H,,, its given value will always be much less certain than that of H. As we have before remarked, we consider its determination as of no practical value in assisting in the exact reckoning of the discharge over weirs ; losses of head between the point for H and the crest, can be more accurately taken account of by other methods. Lesbros for the reduction of his weir experiments used the formula ‘ Q | LH, (29 [z-Ss])" thinking that this expression was analogous to the usual formula for orifices. His Cc deduced values of ¢’, it is almost unnecessary to observe, show very misleading results, WEIRS.—Errect or SUPPRESSION. 115 Errect or SUPPRESSION. We can arrive at some general conclusions as to the effect of suppression, or partial suppression of contraction, by a graphic comparison of all the quoted Lesbros experi- ments with /=.6562. On Plate VIII. are plotted these experiments, including those of Poncelet and Lesbros with Fig. 1; the total head, H, is shown by the vertical lines, and the co-effi- cient of discharge, ¢, by the horizontal lines. Our numbers are given on the plate for reference, as are also the numbers of the figures on Plate I., used as forms of approach for the nine series, or experimental curves. It will be seen by examination of Plate VIII. that the experimental curve for c formed by the series for Figs. 2 and 3, is slightly higher than that for Fig. 1. Lesbros attributed this difference to the effect of partial suppression on one side in Fig. 2, and on two sides in Fig. 3. Arguing from this assumed state of facts, he draws the con- clusion that in order to avoid a sensible partial suppression, the width of the feeding canal should be ten times greater than the length of the weir. He therefore considers that, so far as side suppression is concerned, there is entire parallelism between his weir with 7=.656 and canal 4.2 wide, and a weir with /= 1.97 and canal 12.1 wide ; the width of canal in each case being about six times the length of the weir, and the inner depth below the crest remaining constant at 1.77. He also assumes, by comparison of the curves for Figs. 1 and 2, that the increased discharge due to this partial suppression is more notable for small than for large heads, and hence in his table of co-efficients makes ce, with h=.03, 12 per cent. larger for Fig. 2 than for Fig. 1; and with h=.98, about 1 per cent. larger. These deductions are manifestly erroneous, as with a weir of great length and small value of h, the effect of even complete suppression at the ends would evidently have no sensible influence upon the value of c. With such a weir, full contraction or full suppression would only affect the particles of water forming into the escaping sheet or vein, for a distance from the end of the weir more or less in proportion to h. Hence with / very much greater than h, the effect of partial bottom suppression becomes the factor of importance. In fact Lesbros’ own experiments do not warrant the conclusion which he has accepted. For, if with Fig. 3, where each end of the weir was distant 1.77 from the side of the canal, there was an increased discharge compared with Fig. 1, where these distances were 5.71, it seems evident with Fig. 2, where these distances were respectively 1.77 and 5.71, the value of ¢ should be about half way between the values of c for Figs. 1 and 8. But, with h=.37, ¢ is the same for Figs. 2 and 3, and perceptibly higher than for Fig. 1. In the other experiment with Fig. 3, h being .1, ¢ is a trifle higher than for Fig. 2; with so small a head the chances of probable experi- mental error are larger than the discrepancy here shown ; and, it will be remembered, that the chances of error with h=.37 are much less, than where h=.1. We hence feel warranted in assuming that these discrepancies in the curves for 116 WEIRS.—EFFECT OF SUPPRESSION. Figs. 1, 2 and 8 are chiefly due to experimental errors, and that had the investigations been conducted with perfect skill, the curves would have been nearly identical.* The experiments of Lesbros in 1834 with Figs. 2 and 38, after his years of experience, appear to be entitled to greater weight than those made in 1828 (Poncelet and Lesbros), and we will therefore adopt the curve of Figs. 2 and 3, as representing with sufficient accuracy the values of c for a weir, with /=.66 and / from .1 to .6, having complete contraction. We will also assume that, with /=.66, h=.66, the sides of the feeding canal both distant 1.77 from ends of weir, and with an inner depth below crest of 1.77, the effect of partial suppression does not sensibly increase «; /.c., probably not over 3th of 1 per cent. . We are fully confirmed in this assumption by the Francis experiments (Nos. 112- 117 of Velocity of Approach). In No. 117 the inner depth below crest, or G', was 2.014, and 4 =.65, while in No. 116, inner depth was 4.6, and h=.62; the form of end con- tractions and / being identical. Using the very low correction for 7, applied by Mr. Francis, c for No. 117 was .6140, while for No. 116 ¢ was .6146; this shows that where G is three times h, there is no sensible increase produced by bottom partial suppression, even when the experimentation is conducted with much greater skill than was the case with Lesbros. By analogy, it is safe to suppose that with a distance between the end of the weir and the side of the canal also three times h, the effect of partial side sup- pression will not be appreciable. Complete Suppression. The curve for Figs. 2 and 3, and that for Fig. 6, Plate VIII, should represent the two extremes for comparing the effect of full contraction, and that of nearly complete suppression on the three sides of the weir. But, as: has before been shown in our dis- cussion of the Surface Curve, our values of ¢ for Fig. 6 do not by any means indicate the effect of suppression ; ¢ for h=.41 being .685, when 7 was measured in the feeding canal 3.3 up-stream from the weir, while on Plate VIII., with H measured in the reser- voir, c for the same head is .574. We might roughly deduce the proper values for « for Fig. 6, showing only the effect of suppression, but it will be preferable to take the curve for Fig. 10, where there was complete suppression on the two ends with full bottom contraction, as our standard for suppression; Experiments Nos. 1 and 2 should be corrected for losses by primary contraction ; Nos. 3, 4 and 5 were not largely affected by this loss, and hence need no correction. The upper heavy dotted symmetrical curve on Plate VIII., which follows the corrected experimental curve for Fig. 10, will be assumed as representing the value of ¢, * With large heads Lesbros states that the vein after its escape from the opening, was perceptibly diverted in a horizontal direction by the form of approach in Fig. 2, both for orifices and for weirs. This very delicate test shows, that for large values of h, this slight partial suppression had some effect. WEIRS.—EFFECT oF Suppression. Complete Suppression. 117 for a weir with full bottom contraction and end contractions suppressed, and with no retarding influences (losses of head) of much moment between the measuring point for Hand the crest. The lower heavy symmetrical dotted curve, on the other hand, represents c for a weir with full contraction ; the length of both weirs was the same, being .656. Assuming that the increased discharge produced by suppression is approximately in proportion to that part of the wetted perimeter suppressed, we have the expression, S Cy — Ce 6,11 toa) ore = “Ss Ce p c, being the co-efficient c, for a weir with all or a portion of its perimeter suppressed. c, being the co-efficient c, for the same weir with full contraction. S = length of suppression. p=l+2h=wetted perimeter. Comparing our two curves of reference we have the following values of z ; h=.656=1; c,=.652 and c,=.586; hence «=.168 h=.4; c,=.650 and c,=.594; hence «=.172 h=.2; c,=.656 and c,=.611; hence »=.194 AS ¢,=.676 and c,=.632 ; hence «=.30 For the last value of h, it must be remembered that our reference curves are much less reliable than for higher values of h. 8 Four experiments of Mr. Francis also afford us another opportunity for comparing the values of c for these two forms of approach, the length of both weirs being 10., viz. ; ( Experiment No. 49. Full contraction h=1.002, ¢=.6078 l 53 » 6. Contraction suppressed on both ends h= .985, ¢=.6232 f 9 » 52. Full contraction h= .801, c=.6110 t 3 » 8 Contraction suppressed on both ends h= .799, ¢e=.6233 Hence we have, (h=1.0, and x=.153 =10. . oe (h= .8, and «=.146 For Experiment No. 130, with Fig. 6, Plate VITI., with nearly complete suppres- sion on the three sides, it has been shown that with h=.41, c=.685; for a weirof the same length and head, ¢, is .594; hence x=.15. For Experiment No. 97, with Fig. 4, Plate VIII., it has been shown that the given value of ¢ is nearly normal, that is to say, not being considerably diminished by losses of head between the measuring point for H and the crest. In this instance, with complete bottom suppression, h=.67 and /=.656, c=c,=.607, while ¢, for same values of #2 and 7 would be about .586 ; hence x=.113. ; : ; ; S From the foregoing comparisons it appears to be reasonably safe to give a, in ¢,=¢, ( 1+ oe) a value of about .16 for heads from .3 to 1., when there is complete suppression on one or more sides of a weir. 118 WEIRS.—Errect or Suppression. Complete Suppression. Weisbach gives the following formula for weirs with end contractions suppressed ; d being depth in canal, or d=G@+H; - 2 2 CoCo O41 4.3693 (=) eon (1.041 + 3693 [4] ). C, a ¢ Hence z H 2 Q=e, 3 (1.081 +9698 [-z | ) (29 H)AUH. This formula is intended to take into account the effects, both of suppression and velocity of approach. With d very large in proportion to h, c, would be nearly 1.041 ¢,, and would represent the effect of suppression alone, as /, in this case would be insignificant. In this expression the ratio % does not vary with changes in / or H. We regard this formula as incorrect in principle ; it certainly is at variance with the experimental data which we have discussed. Our expression, c,=c¢, (1 +a 8 ), might be changed to c,=c, ( +z) in which A’ represents the p entire perimeter of the water, or A=2 7+2h. With this latter formula, however, « would have a much greater range than with the first. Partial Suppressioi. We have seen that with /=.66, if the sides of the feeding canal are distant 3 / or 3 h from the ends of the weir, the effect of partial suppression will be insensible. We have now to determine what effect will be produced with smaller distances between the weir and the respective sides of the canal. With a rectangular feeding canal, having vertical sides, with its axis normal to the plane of a vertical rectangular weir, we will call, as before, the distances from the ends of the weir to the respective sides of the canal, Z and L’, and the distance from the crest of the weir to the bottom of the canal, G. The least dimension of the weir, whether it be Z or h, will be called M, and the distance from any side of the weir to the respective side of the canal, where there is partial suppression, will be called N. The wetted perimeter will be termed » =/+2h, and the length of the side or sides on which there is partial suppression will be termed S’. The co-efficient ¢ for partial suppression will be called ¢,. Experiments Nos. 112-115 (Fig. 9, Plates I. and VIII.), where there was complete bottom contraction, and contraction nearly suppressed on both sides (Z and L’ both .066), should afford data by which the effect of a small value of N can be ascertained. Expe- riments Nos. 110, 111 and 116 of the same series (Table XX XIX.) will not be used: No. 110 was almost an orifice, and hence not a fair exponent for a weir; No. 111 was affected perceptibly by primary contraction at the mouth of the feeding canal ; and for No. 116 A was so small as to make c more unreliable than for the other experiments. The comparatively low values of H—H,, for Nos. 112-115 indicate that for them ¢ nearly represents the full effect of partial suppression. y Again, taking the formula c, =, (1 +a *) , 00 4 =e, (1 + o*) , we have with /=.656 ; P P WEIRS.—EFrecr or Suppression. Partial Suppression. 119 No. H=h c=¢, C, C, a a V2 508 635 “590 650 13 17 113 383 630 595 650 dl 7 114 329 631 598 651 cf 18 115 169 638 616 659 ll 20 The values of c, and ec, in the above statement are taken as before, from the two reference curves on Plate VIII. It will be observed that the differences between « and zx’ inerease as the heads diminish. This seems reasonable, as with a very minute head, even with NV as small as .066, it is apparent that there should be very nearly complete contraction, and hence that «’ should be nearly 0. The experimental curve for Fig. 8 on Plate VIII., when there was full contraction on the bottom and one side, and L = V=.066 on the other side, agrees substantially with the results shown by the series of experiments with Fig. 9, as above given. For, had there been complete suppression on one side, it is apparent that the experimental curve for Fig. 8 on Plate VIII. should have been about half way between our two reference curves on this plate, / being the same for the three weirs; the curve for Fig. 8 (Experi- ments Nos. 102-106) is, however, quite appreciably nearer the lower than the upper reference curve, thus indicating the effect of partial suppression. As illustrations of the effect of partial suppression, we have some experiments made by Mr. Francis, comparisons of which have been made in Chapter IV., Velocity of Approach, Experiments Nos. 112-117. For these experiments the length of the weir was constant at 10.; the width of the feeding canal was also constant at 14., there hence being nearly full contraction at the ends. Mr. Francis, it will be remembered, obtained the effective head h by the formula h =([H+h,]*—/,"")"", and ¢ by the formula Q=c 3 (2 gh)*lh. Weuseda larger value for h, making h = H+bh,, with b varying from 1.3 to 1.4. Taking the values of ¢ as given both by Francis and ourselves, we have the following values of a’; | Velocity of ¢ x | L Approach | G=W h No. Francis. | Smith. | Francis. ;| Smith. 112 4.6 00 .6089 .6078 0 0 113 2.0 } ‘ .6137 .6088 .010 .002 114 4.6 .6118 .6110 0 0 10.0 tsi f 115 2.0 .6143 .6107 .005 = 116 4.6 we .6146 .6138 0 0 117 2.0 } 6140 | .6116 = - For each of the three comparisons, there was full bottom contraction with the 120 WEIRS.—Errect oF Suppression. Partial Suppression. inner depth (G) of 4.6. These results indicate that with G=3h there is complete bottom contraction ; that with G'= 2.5 h the effect of partial suppression, if felt at all, is very slight ; and that with G=2 h, the effect of partial suppression only increases ¢, ths of one per cent., even when the very low correction for v, used by Mr. Francis is employed ; with our assumption of the effect of velocity of approach, the increased discharge caused by partial bottom suppression (G'=2h) only amounts to 3th of one per cent. . It is probable that partial bottom suppression of a weir with end contractions sup- pressed, would show larger values of x’ than those just given. From the foregoing comparisons we can roughly place the values of 2’, in c,=¢, 14 v= as follows, for heads from .3 to 1.0, viz. ; p ¥ 5 = av u 3 0 2 005 1 .025 - .06 0 16 Tt will be remembered that J/ is the least dimension of the weir, whether it be h or 1, and N is the distance from the side of a rectangular feeding canal to the respective side of a rectangular weir. It is evident that the ratio ca approximately measures the amount of contrac- tion : for, with /=1. and h=10., the distances Z and L/ need be only 3 /, or 3, in order to obtain full contraction at the ends; reversing the dimensions, and making 7=10 and h=1, Land I/ need be only 3 hf, or 3. In other words, with an opening of such unequal dimensions, the particles of water as they form into place to feed the escaping vein from a to b in the following sketches, have their direction changed by the 4 ib nt itu 5 HH rH rl if ol ul side or sides of the canal parallel to the line a b, and are not affected by the sides or side of the canal at right angles to the line a b. WEIRS.—Errect or Suppresston. Partial Suppression. 121 The distance from the point a to the bottom of the weir in one sketch, or from the points a and b in the other sketch to the respective end of the weir, can be assumed to be equal to the least dimension of the weir. In practice, for weirs / is nearly always much vreater than /, and hence partial bottom suppression is a more important consideration than partial end suppression, where G, L and L/ are equal. The experimental curve for Fig. 12 on Plate VIIL., illustrates the effect. of inclined sides (8= 45°) of the feeding canal, with Zand L’/ both .066, and full bottom contrac- tion ; it will be noticed with this form of approach there is more contraction than with Fig. 9—Nos. 110-116 on Plate VIII.—where / was the same as for Pig, 13, b= =.066, and sides of approach normal to the weir (8=0). The question of partial suppression is very intricate, and attempts to satisfactorily analyze the effect upon the discharge, due to various modifications of the forms of approach in the feeding canal, are of more interest as curious mathematical and experimental problems, than of practical importance. For, if it be desired to measure the flow of water over a weir with great accuracy, there should be either complete con- traction, or complete side suppression ; bottom partial suppression should, if possible, be avoided. A glance at Plate VIII. is sufficient to warn the experimenter of the danger of taking any other than these two simplest forms of approach. In the nine experimental curves drawn upon this plate, the value of « ranves from .481 to .711 with 7 and h con- stant, and the curves cross each other, in what at first sight appears to be a most contra- dictory manner. We have endeavoured to point out the causes of these irregularities, but without careful and laborious study it is hardly possible for the hydraulic engineer to fully comprehend such apparently anomalous results. Conclusivirs. We consider the following proposition as demonstrated, viz. ; To secure perfect contraction for a rectangular vertical weir, each side of the feeding canal must be distant from the adjoining parallel side of the weir, at least 3 times the least dimension of the weir. If this distance on all three sides be reduced to 2 times the least dimension of the weir, the discharge will be increased about 4 of one per cent. ; after the last limit, the discharge increases more and more rapidly as the distance diminishes to 0. This is a very different hypothesis from that adopted by Weisbach and others, of a fixed ratio between a and a,. For instance, by our proposition full contraction will result R 122 WEIRS.—-EFFECT OF SUPPRESSION. Conclusions. by having the following dimensions for the feeding canal, for three weirs, each having the same area of 12. (A) Let 2=12,4= 1; then (2+2[3h])x(h+2h)= 72=a,= 6a. (B) Let l= 4,h= 3; then (/+2 [31] )x (A443 h) =264=0,= 22a, (C) Let@= 1,4=12; then (/+2 [3 7] )x(A4+3 1) =105 =a, =8.7a. If our views are correct, the channel of approach must be as large as the given value of a, for each of the three weirs, in order to insure perfect contraction; we therefore have“ varying from 6 to 22 in order to produce this result. With the forms of a approach given in (A) and (B) there will be a notable velocity of approach, while with (C) the velocity of approach will be much smaller, Generally in practice with weirs 7 is much larger than h; hence making G=3h, and L and L' each = 2h, will insure almost perfect contraction. For weirs with full contraction on the sides, G can be reduced to 2 kh with hardly any appreciable increase of flow, but when end contractions are suppressed it will be well tohave G@ not less than 34. For shallow depths on a weir, such as .1 or .2, asa matter of precaution @ had better be made not less than 4 or 5 i, as some of Lesbros’ experiments appear to show that with such small heads the ratio between 4 and ( should be pretty large in oxder to secure complete contraction. Hydraulicians have nearly always confounded the effects of suppression and velocity of approach, by considering them together. As we have hitherto shown, there are three causes to be taken into account when the area «, is not very large in proportion to the area ca. They are: First.—The head due to velocity of approach, which increases the velocity, », at the weir. Seconp.—Suppression or partial suppression, which increases 1. Tuirp.—Losses of head by “friction” and adhesion, between the measuring point for H and the weir, which diminish +. These three causes are entirely distinct, and either one may be the factor of importance in determining the proper value to be given to cv. This can well be illustrated by comparing the lower reference curve (c,) on Plate VIII., with the values of ¢ for Experiments Nos. 130, 129 and 101, ? being constant at .656. In No. 130, with contraction suppressed on bottom and nearly suppressed on both sides, the head, H, was measured 3.28 from the crest, in the narrow canal shown by Fig. 6, Plate I. ; the deduced value of C for this experiment, in Q = C2 (2 oF )A 0 LE as 1.116, while, when the effect of v, is taken into account, ¢ is .685. The value of ¢,, TOF the same head is .593, showing an increased discharge due to suppression of about 15 per cent. Therefore in this instance the effect of v, was extraordinarily great; the effect of suppression, a smaller but still large percentage; while our third cause had but little effect. In No. 129, with contraction suppressed on both sides and complete on the bottom, the head H was measured .98 from the crest, in the narrow canal shown by Fig. 10, WEIRS.—ErFrect or Suppression. Conclusions. 123 Plate I.; the value of C was about .668, while « was .655: the value of efor same head is .590. Hence in this case the effect of suppression increased c, about 11 per cent., and the effect of », increased « about 2 per cent. . In No. 101, with contraction suppressed on bottom and full contraction on both sides, the head H was measured in the large reservoir—Fig. 4, Plate I. In this experiment the effect of », would have been hardly sensible, had J been measured in the canal. Butin this case « was .595, while ¢, for same head is somewhere near .660 ; this is doubtless due to our third cause, which much more than counterbalances the effect of suppression. In this instance adhesion of the water to the bottom of the canal, the depth in the canal being quite small, apparently was the chief retarding influence, as the assumption of a very high co-efficient for “ frictional ” losses in the canal, would only account for the smaller part of the total loss of head. It is hence apparent that the use of a formula like that of Weisbach,* which attempts to express the united effect of these three variables, will give entirely erroneous results, when the conditions are not identical with the experiments from which the numerical values in the expression have been deduced. DeETERMINATION oF vc, FoR WEIRS HAVING EITHER Futt Conrracrion orn Enp Conrraction SuppressepD, wird Futt Borrom Coyrraction ; 7 Nov LESS THAN .66. We will now endeavour to ascertain the laws governing the discharge over the two simplest forms of weirs—those with full contraction, and those with contraction sup- pressed at both ends and with full bottom contraction—.66 being the shortest length, and with heads from .1 to 2. . The co-efficients of discharge, ¢, will be denoted as follows ; c for weirs with full contraction : : ; » SG, ¢ ,, Suppressed weirs. ‘ ; ‘ ‘ « Se ¢ ,, weir of infinite length and finite depth ~ Se; The experimental data already given for these two forms of approach, can best be contrasted by putting the results in graphic form. This has been done on Plate VI. one of the co-ordinates representing /, and the other c. We have in all twelve series of experiments, each series forming one experimental curve. They are: Series I. c¢, Lesbros (Fig. 10) t= .66 Experiments Nos. 1-5. ys II. ,, Fteley and Stearns 5 t= 5. a 17-34 55 Ill. ,, J.B. Francis. 3 t=10. 5 6-8 IV. ,, Fteley and Stearns t=19 95 9-16 cs a \4 ay? * Weisbach for weirs with partial suppression, proposes cet =1.718 ( 3 ) » OF Cp=Ce (+ vail | ). c 124 WEIRS.—DETERMINATION OF ¢. Series Vc, Poncelet and Leshros (Fig. 1) } l= .66 Experiments Nos. 35-40 - VI. ,, Lesbros (Figs. 2 and 3) j i 41-46 is Vil. i, 4, (thick-edged) = 1.97 3 93-96 » WII. ,, Fteley and Stearns. : i= 2E8 a3 76, 80, 85 and 86 ‘3 IX. ,, Hamilton Smith, Jr. ‘ j= 2.6 35 57-68 = X. 4, Fteley and Stearns. ; i= 3.0 , 70, 72, 74, 79, 83 and 88 Py XI. ,, J.B. Francis. ‘ ‘ = 4.0 i 50 and 56 3 XIT,, 3 ; ‘ ‘ ?=10.0 3 47-49 and 51-55 Our number for each experiment is given on Plate VI. An inspection of these twelve experimental curves on Plate VI. shows: That all values of ¢, are larger than those of c,, for the same heads ; Also that c, increases as the length of the weir diminishes, and c,—with some minor exceptions—diminishes as the length of the weir diminishes. It is apparent that for a weir of infinite length, being finite and equal, there will be no difference in the values of «,, c;, and ¢,, as the effect of either suppression or con- traction will be confined to some finite horizontal distance from the ends of the weir. In confirmation of this proposition, we observe that as / increases, the curves for c¢, and ¢, approach each other ; those of the longest of each form—19-foot suppressed, and 10-foot with contraction—being quite close together with h=.6, and gradually diverg- ing to the maximum head of 1.6. We hence feel warranted in assuming that the curve for c; will be somewhere between these last two curves. We can now deduce the general proposition, That, compered with the curve of ¢; for a weir of tifirite length, with full contraction c, dauinishes as 1 diminishes, aud with coi- traction suppressed c, nereases as | dintiishes., This proposition applies where / is greater than .5; for smaller values of /, as we shall see hereafter, it does not strictly apply. There are minor discrepancies in some of our experimental curves, and hence it will be first necessary to determine what weight should be given to each series, and what experiments in the several series should be rejected, before we draw on Plate VI. our final theoretic curves, representing ¢, and c, for various values of /. We have already expressed the belief that the Lesbros curve with Figs. 2 and 3, more accurately represents c, for! =.66, than the curve of Poncelet and Lesbros. Hence we will transfer from Plate VIII. the theoretic curve for c,, /=.66, which was based upon results with Figs. 2 and 3. Series VII. is more or less uncertain, owmg to the thickness of the crest (.164). Judging from Lesbros’ remarks, Nos. 93 and 94 were almost free from the influence of this thick crest. Experiments Nos. 57 to 63 of the author, Series IX., will be accepted as deter- mining «, for /=2.6; the other experiments of this series will only be regarded as approximations. WEIRS.—DETERMINATION OF ec. 125 Series VITT. and X. of Fteley and Stearns, although indirectly determined, will have more weight than Series XI. of Francis, where two 4-foot weirs were separated by an interval of 2 feet, thus complicating form of contraction. Series XIT. of Francis will be accepted asx determining «, for /= 10. . Nos. 6 and 8 of Series IIT. of Francis will be considered a trifle too high, as these experiments were with a perfectly free discharge into the air, while in those of Fteley and Stearns, with which they are to be compared, the escaping vein was slightly con- fined at the ends of the weir. No. 7 will be assumed as too low, as in that experiment the escaping vein was more confined, than in Series IT. and IV. Experiments Nos. 20, 28, 29 and 32 of Series II. show abnormally large values of c, compared with the other experiments of the same series ; they will hence be dis- carded. Series I. of Lesbros (Fig. 10, Plate I.) is unreliable, owing to losses by primary contraction and “friction,” which we have before described. Hence we will not attempt to form a theoretic curve for ¢, with 7=.66. On Plate VIII. will be found this curve, which is approximately correct. It will be kept in mind that c, represents perfectly free discharge into the air, while c, is based upon the escaping vein being confined by prolongations of the sides of the canal, but which do not extend below the level of the crest ; this latter form is adopted as our standard, because Series II. and IV., upon which we chiefly depend for the establish- ment of c,, were made with this form of discharge. We will first determine the position of the curve for ¢;. With h=.6 c; is .616 or thereabouts, its position being closely determined by the close approximation of c, for 1=19., and ¢, for 7=10.. For the same head c, for /=2.6 is .604 or .605, and ¢, for 1=5. is .627 or .628; c, is therefore about half-way between c¢, for /=5. and c, for 1=2.6 ; hence we can assume that the + effect of suppression (both ends) is somewhat greater than the effect of full contraction, compared with ¢;. This assumption is fully confirmed by a careful contrast of the curves for ¢, and c, on Plate VI. With h = 1.6, placing c; at .614 corresponds with the above hypothesis. For heads from .1 to .25, ¢; should be very close to c, for /= 5. for the smaller value of h, and gradually diverging as the head increases. Having these points fixed, we can now easily draw on Plate VI. the central heavy line representing ¢; with h from .1 to 1.6. It will be observed, that with 4 from 1. to 1.6, ¢; is practically a straight line ; hence we can continue ¢; with / from 1.6 to 2. with same value of .614 with safety. Whether c; for heads above 2. can be assumed to continue constant at .614 is uncertain. We will allude to this point again in our com- parison of the analogies between orifices and weirs. For smaller values of /# than .1, the experimental data at hand cannot be consi- dered as reliable ; in fact c, for heads of less than .2 cannot be considered as being 126 WEIRS.—DETERMINATION OF ¢. accurately determined. Hence, it is a safe proposition to state that no wetr meusiremeit of water shonld be made with h less than .2, if accuracy ts essential. In drawing the curves for c, and c, we will be governed, first by the experimental data considered reliable, and next by the proposition, that compared with ¢,, the increased discharge produced by suppression and the diminished discharge due to contraction are approximately expressed by the following formule ; +) Y C=C; (1+ resi) Or C=C; ( tel): Co= Cj Pe OY t= ¢; [ag |, Ean ee “TE 2h)? no p In these expressions . and ¥ will vary so as to best suit the experimental data, a being in general rather larger than y. By reference to Plate VI. it will be seen that our theoretical curves agree quite closely with the most trustworthy observations, and that there are no striking dis- cordances. The data for c, with 1=2., and/=8, and for c, with J/=1. are insufficient; hence these curves are shown by dotted lines. and Some experiments of Mr. Francis, given on pp. 88-95 of ‘“ Lowell Hydraulic Experiments,” will be of service in testing the accuracy of our theoretic curves on Plate VI. All the experiments were with nearly full contraction, the contraction being less perfect for the larger values of 7. @ was constant for each series, but was not deter- mined. Taking value of ¢, from Plate VI. for the first experiment in each series, we can deduce (, and then ¢ for the other experiments of the same series, so as to make a com- parison with values of ¢, on Plate VI. WEIRS.—DETERMINATION OF c. 127 TABLE XLV. J.B. Francis.—Comparison of Values of H for Various Values of 1, (Q being Constant for each Series, (2 g)%=8.020 P ‘Diterence| Series. oo t i gy | Ne aE between oe co | candc, | > it 1 6.987 ) .3146 | | ' 6214 ) , | \ 4.096 7 | ; 6 3.500 | .5085 604. .609 ~ 005 TI. 14 | 6.987 | .3745 | 5.291 ; 6180 11 | 3.500 | .6049 | 601 606 ~.005 es 34 P | We 17 | 6.987 , .4681 eds 6141 ! 22 | 3.500 | .7580 595.602 = 007 | cee, tH | IV. / 32 | 6.987 | .6554 +} 12.087 6098 31 3.496 | 1.0656 | 588 | 595 —.007 ¥, 56 | 6.987 | 9554 ‘ 6047 | 55 | 5.487 1.1336 |"-21.096| .596 : 599 | -,003 BT 8.489 | .8361 \ | .608 | .609 ~.001 | ‘ Vi, | 70 8.489 | .3662 | 6193 | , 69 | 1.829 | 1.0627 (4 6.999] 581 587 ! ~ .006 | 71 + 5ABT | 4936 j 612 612 | 0 | i In the foregoing experiments with /=7, there was probably slight partial suppres- sion, which, if taken into account, would have somewhat increased the values of Q, and hence would have increased the deduced values of c. We can hence conclude that these experiments verify our curves on Plate VI. Tones Some experiments by General Ellis, p. 91 Transactions Am. Soc. of C.E., 1876, where the water first flowed over a short weir, and then over a much longer one, give the following results. @ is deduced from our values of ¢, for the long weir. 128 WEIRS.—DETERMINATION OF c. TABLE XLVI. LEilllis.—Orvifices used as Wetrs. (2 g)4 =8.020 Lone WEIR. : Suort WEIR. | | » 7 | Y val h Pe VI, Leakage. | L h | c ; 10, | 6085 | 6089 | .6133 15.580 126 15,454 : 2 1.8372 | .5803 » + 6070 6074 ° ,, 15.522 | 126 15.396 ,, 1.8290 | 5821 » 6100 | .6104 » | 16.637) .196 ]15.511 | ,, 1.8440 | 5793 5 | 26072 | .6076 » | 15.530] 126 15.404 , ,, | 1.8326 | 5807 | 6. 2849 | 2849 | 6262 | 3.0547 | .0637 2.9910 | 9 5793 | 6844 a 3778 | 8778 | 6171 | 4.5970} .0662 | 4.5308 || _,, 7849 | 6093; a 4800 | 4801 | .6130 | 6.5417 0702 | 64715 | ,, , 1.0079 5981 | | 4. 3930 | 3930 | .6144 | 3.2373 | .1593 || 3.0780 1.0001 1.0160 | 5621 The above values of « are contradictory, and show that there were imperfect methods of observation. This table is inserted as a eorreboration of our remarks on the experiments with orifices by the same authority, made at the same time as the foregoing weir determinations. Weis with One Mud Contraction Suppressed, We can now consider the effect of suppression at one end only. With this form of approach it is apparent that its co-efficient, ¢,, will be very nearly c;; for, in the equation, h hi = ( Teg! avi «x and y being nearly equal, ¢,’ will be practically the same as c;. In Table XXXIV. are given experiments by Fteley and Stearns, with contraction suppressed at one end: with /=4.0, Z was 1.0; with /=3.3, L was 1.7; hence with the longer weir there was more or less partial side suppression at the other end of the weir. Comparing the deduced values of « for these experiments, with values of ¢,; on Plate VI. we have; WEIRS.—DETERMINATION OF ©. 129 TABLE XLVIL Fteley and Stearns (Table XXXIV.). One End Contraction Suppressed. Comparison of ¢, and ¢, of Plate VI. ' eo ; | Differences | No. i h c Pla ri VL between ! | cand «,. ee en ee aie; | pg a oe ery ae —— | 69 401 | 176 | .6388 | 6379 | +,0009 71 : 269. 6285 | 6279 | +.0006 73 " 395 | 6212 6204 | +.0008 77 : 499 61776174 | +.0003 82 602 | 6177 6156) , +.0021 87 3 709 | 6156 6147 | +.0009 92 2 95 | 6196 » 6142 | +.0054 75 3.31 | 450 , 6178 , 6187 | —.00l4 | 78 , 569 | 6139 | 6160 | =0021 81 . 578; 6130 , .6159 | —.0029 84 i 88 - | aid | 6149 | —.0035 89 : 809 | .6105 | 6143 | —.0038 91 : 935} .6092 | 6142 | —.0050 | These differences are not excessive, especially when we remember that c was indirectly determined from our curve for c,, /=5. It is quite possible that in the determination of this curve for c, we were not enough influenced by Experiments Nos. 20, 28 and 29 (Plate VI.), and hence placed it slightly too low. If this be the case, ce for /= 3.3 in foregoing table would be practically identical with ¢,, while c for /= 4.0 would be somewhat higher than ¢,, on account of partial suppression at the other end of the weir; as before remarked in our discussion of Partial Suppression, the greater the head the more appreciable should be the effect from this cause. Final Conclusions. Whether or not the curves for ¢,, or those for ¢,, cross each other with increasing heads, the experimental data thus far given do not indicate, unless we accept Experiments Nos. 93 and 94 of Lesbros as being exact. Should they be so considered, it would show that the curve c, for /=2 crosses curve c, for /=2.6 at h=1.3, as will be noticed by reference to Plate VI. This question will again be considered from a theoretic point of view, when we trace the analogies between orifices and weirs. On Plate VII. will be found a clean copy of the theoretic curves on Plate VI. Tt will be observed that for considerable heads the curves for c, are reversed. For, s 130 WEIRS.—DETERMINATION OF c. Conclusions. if c, continues constant for heads above 1.6, as seems to be indicated by our diagram, then it is evident that the trend of the curves, both for ¢, and for c,, must be towards the straight line ¢;; otherwise with great depths, c, would rise to 1. or more, while ¢, would fall below 0. The following table has been compiled from Plate VII. In applying the co-efficients given, it must be remembered, that : The weir must have a horizontal crest, with vertical sides, with its plane at right angles to the line of the feeding canal. The inner face of the bottom and sides of the weir (lower end of the canal) must be vertical. The edges of the weir must be so thin, that the escaping vein will only come in contact with the inner corner lines. To insure accuracy, these edges should be formed by stiff metallic plates. The discharge must be free into the air, for full contraction. For suppressed weirs, the escaping vein should be confined by prolongations of the sides of the canal, but which must not extend lower than the Icyel of the crest; in case a suppressed weir has a perfectly free discharge, the following co-efficients are about 4 of 1 per cent. too low. Free access for air must always be provided under the escaping vein. HZ should for ordinary cases be measured for suppressed weirs at a point 6 feet up- stream from the crest, and the sides of the canal between /Z and the crest should be of smoothly planed plank. For full contraction H can be measured at any convenient point from 4 to 10 feet from the crest, and the smoothness of the sides is of no importance. When H and / are exceptionally great, the measuring point for H must be taken ata sufficient distance from the weir, to be above (wp-stream) the origin of the surface curve in the feeding reservoir. It will be remembered that at the Holyoke dam, with H=7.24 and / very great, the proper point for the measurement of H was about 60 feet above the dam or weir. Hf should always be corrected for velocity of approach; for this purpose Table XXV. should be used. When the water enters the feeding canal with a velocity greater than the mean velocity at the measuring point for H, screcus or racks should be employed to produce a more uniform velocity; vertical slats, beveled in the direction of the current, are the best; the openings should be so small as to cause a shght, but appreciable, loss of head as the water passes through the racks; the racks should be placed several feet above the measuring point for J. To obtain perfect contraction, each side of the feeding canal should be distant from the adjoining parallel side of the weir, at least three times the least dimension of the weir. But, as / is generally much larger than /), the distance from the weir end to the canal side, can be fixed at 2; the inner depth below the crest (G') can also be placed at WEIRS.—DETERMINATION OF c. Conclusions. 131 2h, without notable error for weirs with full contraction. These bottom and side distances (G, LZ and L’) had best be made not less than 1 foot. The surface of the water in the pool below the weir can be brought to the elevation of the crest, without forming any obstacle to free discharge, provided the vein retains its normal form; ample avenues for air to enter under the sheet must be provided, when the level of the lower water is nearly the same as that of the crest, otherwise atmospheric pressure will distort the vein from its normal position. The formule used are; (A) Mae 9 (2 yh)* lhe § (2g)* Uh. a a 3fy (B) Q=e 3 (2 yt (H+b%) =cR(2y)*1 [aso le zo Equation (B) is of high degree, but », = # can be easily obtained with sufficient ¢ c accuracy by one or two approximations. Conclusions. WEIRS.—DETERMINATION OF ¢. 132 | | ee ag l ot [ee fe poe pe, | ST [| [ARO GPO" LEO"; TE9° | 9e9" fp Ghee SP ae | lopg'|1E9" 9¢9°} 0g9" | azo" ! Ae) es Se | "FEO OFO FEO | GEN | GEO" eT | = “SFO ETI [SEO C9 | 869° | FEO GT) NFO |LEO EO" FeO 19E9" | BBO UT : “TL TE IGE9" O69") GEN | CED OT | sto" \tFO" Leg” 9° 869" | T60" | 169" 6 1 ae STD GED’ [CSO [TEN |LE9' | GEO 069° 8 9G9" [EFO |LE9 ve 169° | 0&9" | 2° $49" [OFO' |GE9N |LE9" |8c9' FEO | O69" | G19 | o | 129° 9° |FES9" |OE9" 069° | G19 G | OGD" [489° |6E9" 069° 169° | 069° | | FP | OG9" |9E9' |SE9° OED GG Coo | € | 199° [69° 99° |EE9 869° | 169° “eo | $69" |TFO [sso 99" 669" | 169° | e | 9¢9° ae GPO | 1FO" 189" | 269° | 969" ey 699° ESD OFO LTO Tro oe TED To} Ggs9 | ffl & 899° | LG9° Ga) g9i(i) z el 9 OL | cL y “Ma AA Jo yysueT = | ‘spuq y0q 4% pessorddng woroeIyW0g =v FID alg? FLO" FLY F19" FI9 TIO" FI Fo: F1L9" PE G19" 919° £19" Oc Geo 669° geo’ cro’ 99° TOF ‘ AL rau = CLILA °F¥[d WorZ 99°=7 205 *2) “pug vue 4v possoiddng uorovsyuo0g 10 ‘spug y0q ye possaiddng uoroeazUOH JO ‘UOTPOVALNOL [[QT SULAvY ALYZLo ‘sa1o \\ Loy! ayy hc) F9=0) wpnmiog uso WDavyosig fo quarliq-09 68u" 08 Gee" oxo" T6g° egy! cog" 869° 009 60 Goo R09" e119 619" Foo 0E9° | al | aN ihe 109° 09° |66G" r6W" 6 "209" F09" 009" 6G L6G" bso" HOD oe Taio abe ase 609" 909" 209" 6g" Fee" OBE” 609" 109" £09", 665" 968 Ieee" ‘O19 809° ]G09" LOD 260° FEE" O19" 909 [soo Gos" 969 I19" O19" 809° FO9" 109" 866" &19° LT9° G09" 909° |g09° 009° 19" era" |t19 9" 09" 709" 209 FI9"[€L9" 6 “909° FO9" G19 P19" |SL9° |I19° 809° os ee G19" e19" t 019 069" G19" |@19" £19" G9" FT9" G39 |FEO' 1FEO" Ge9' 1129" 1T29° 629" seg" Be "rea 99 CED FEO" FeO" WLI" Bed" Le 0° 19 BEI THEI ILE OI |OFI’ 6E9' peg" ug leu’ 09° lee leeg’ Gt pet Or} 4] sg | * { ‘ PUG euo qv oe s19" €eD" 669 Leg" 099° 9% CNG 6G" NY Egy" | Lae 965° | 06g" 109° | 964 809° 109" e19" | coo" S19" | T19 eo" | BLD" eco | ee @1 | 99 “ITAA JO IySua] =/ WOTpVAZUOH [NY ="9 “WHO JOT OL} 4B TWOTPOVAPMOL) [TOY Sesto ]pu UL Sutoq VOILE TIIATX A@TAVa = proyyT astayg WEIRS.—DETERMINATION OF ¢c. Conclusions. 133 For values of 4 and / intermediate to those given in the preceding table, ¢ can be obtained by interpolation, or more accurately by reference to Plate VII. Formule iu Ose. Lesbros considers the following equation for weirs, analogous with Q = Ca (2 qh)” for orifices (h for orifices being measured at the centre), G21. (29 Eoeak For his weir .656 long, with various forms of approach shown on Plate I., and with H=h, by this formula c’ had the following values ; e’ for Figs. 2 and 3 =.584 —.684 & ,, Fig, 4 =.601—.811 OF 5. ay, 8 =.622—.742 Os ax 6 =.718—.756 Gy, ae OS =.603 - .669 Giese. “nl, 3 =.638 —.718 Os ae NO) = .676 —.797 c 12 =.605 — .668 3? ? Boileau* gives for suppressed weirs the formula, Gee. pes = (aa HF H in which he attempts to take into account the effects of velocity of approach and partial A LH (29 H)4, suppression at the bottom. As he makes (1-57) =.417, and constant, we can re- duce his equation to our form, viz., ; G+H ; O= CEH PoP G25 2 (2 gy A ET i. 4 M. Boileau afterwards recognised the fact that (: — = is not constant,t and also that his new formula gave results considerably varying from his experimental results. He finally proposes the similar form,} with the measured values of H, and #, poce\” i aul. __j Hy A: 1 4% 7 | 1 ae ( TH and gives a table of corrections for his theoretical co-efficient as above, the factor of correction ranging from .96 to 1.07.§ We regard this formula as radically wrong in principle, and the experiments of M. Boileau, upon which this expression is more or less based, as inaccurate. Jaugeage des cours d’eau, Paris, 1850, par M. P. Boileau, p. 41. ‘«Traité de la mesure des eaux courantes.” Par P. Boileau. Paris 1854. Page 120. ‘¢ Traité de la mesure des eaux courantes,”’ p. 87. ‘* Traité de la mesure des eaux courantes,” p. 119. Cratt + * 134 The experiments of M. Boileau have been frequently quoted as authority, and it will not be out of place for us to here give our reasons for disregarding them. Taking from his “ Traité de la mesure des eaux courantes,” pp. 89-97, the four series, with end contrac- tions suppressed, with free discharge, applying our corrections for velocity of approach, and making (2 g)4 = WEIRS.—DETERMINATION OF ¢. 38,0227, the value assigned by M. Boileau, we have the following results ; Formule. I. No] @ | t | Hl uw|m’ | a | @ | a te 9547-1499 .08 .0001, 1600 | 1806 8088 2 | 944.2185 08 0001 2184 3023. .cO68 a 9449 | .2690!.12'.0003 .2693 .4308 “cos 4 3.639 || MOAT | +2986 .18 004.2940, 4978 | .6116 | 5 | ; 9547 .4306 24 .0012 .4408, .9209 | .6162 6 | 9547 |.5479'.32 .0021 | .5500 | 1.2816 |.ct53 7 | | 9449 .B774! .35 | 0025 | .6799 | 1.8795 G18. 8 [3 i) 9547 | 59TL| 87 | 0028 | 5909 1.4765 .6223 | - 4 — | No. G l | H te ha’ | oh | Q oR 1 f .2067 10.0002 2069 .8029 | .6325 2 | 2411.12 0003 .2414) .3811 | .6313 Ls | 2658 ' 14.0004 .2662) .4399 6204 4 ,.2789' 15 .0005 .2794, .4724 | .6286 eet) ea ye ee ee a a ! [,S070 25.0018 | 8988 .8040 | .6285 |g | | oe om .4381| .9318 | .6314 1 | | 5848.88 0030, .0878 | 1.2773 6364 pees |) 5840 43.0038, 5878 146116371 ~~ — iit . - : | No.| G bt tw iaet mrt # Q a 1 [+1870 .09 0002 .1872 2542 6190 2 a acetic 2432 STBL 6221 3 || | .2625 | .15 | .0005 | .2630) .4247 11200 4 2.690 | .9482 } 2904.17 .0006| .2910| 4932 , 6196 5 | 3396 | .22| .0010| 3406 6234 6184 6 "4008 | .27 | 0018 .4018 | .7988 | .6185 7 | 5512 | 48 | .0088 5550 | 1.8160 ; 6277 | _ IV. No. G ¢ , ty!) ae 0) bs re eer pe we ge g nly cg ak oe a 1 1631.10 acces LOL 540 2 1847 .11 .0003/.1850] .2884 | 5909 3 2139 .14| .0004 | .2143| .2932 11929 | 4 || 2854: 21 0010 | .2864 4697 [6044 ay ee | nee | 8182.25) .0013 8195) 5592 6106, 6 3543.29 .0018 | .3561| .6573 ° 6099 7 | 5578 ‘54.0061 .5639 | 1.3306 | .6196 8 | BT42 AT | 0067 | 5809 14032 6250 | WEIRS.—DETERMINATION OF c. Formule. 135 In all these experiments @ was determined by the flow through orifices: /Z appears to have been measured by placing a glass tube immediately above the weir, with its bottom near the floor of the feeding canal, the surface of the water in this tube being assumed as the height indicating W/; the pressure at the base of the weir is, as we believe M. Boileau was the first to point out, often in excess of the true value of //, but in the experiments we have selected, no serious errors ought to have arisen from this excess of pressure. Now it seems apparent that, comparing those experiments where the velocity of approach was but slight, our deduced values of ¢, should be nearly identical for the four weirs with equal heads, no matter whether or not our method of computing hi,’ be correct; that is to say, corrections for v, upon any theory must be so small as not to very largely change the values of ¢,. Contrasting the values of ¢,, with / less than 0.3, we have ; I; I. HI. IV. h c, h c, h o h Ce 15 .609 21 .632 19 619 .16 595 21 607 24 .631 24 622 18 591 2a .609 27 629 -26 621 .21 583 29 -612 .28 .629 29 .620 .29 604 We here see that with 4=.21, c, for the four series has the values of .607, .632, .620, and .583, all dimensions being practically identical, except G, which ranges from 3.6 to 2.0. The thoroughly reliable experiments of Mr. Francis, and Messrs. Fteley and Stearns clearly prove that with such a head, such variations in G will not appreciably affect the discharge, aside from the slight corrections necessary for the respective changes in v,. If increasing G increases c,, as Series II., III. and IV. indicate, then c, for Series I. should have a value of .64, instead of .61 as given. We attribute these discrepancies solely to errors in experimentation ; they are so large as to palpably demonstrate that the experiments have no value except as rough approximations. We have not selected for illustration M. Boileau’s experiments where he directly measured Q, as in these G was comparatively small. Critically examined they present almost equally untrustworthy results, as the ones just discussed. The Referees, Messrs. Simpson and Galton, for the Metropolitan Drainage, gave the following formula for weirs ;* Q'=5.5 (BP+.8 & vy. Q' being discharge in cubic feet per minute for each foot in length on the weir. d =head # in inches. v =velocity of approach in feet per second. Transforming this to our expression, with (29)*=8.02, Q=.718 3/ (29)* (H? + .067 H? v,°)%. The correction for H? is intended to represent effect of velocity of approach. This formula when used for the reduction of sixty-two experiments made under the auspices of these gentlemen, with / from 3. to 10. and h from .08 to .75, appears to agree very fairly with the measured values of @. These experiments must assuredly have been made under very peculiar conditions. Weisbach’s formule are as follows ;t * Copy of Letter to Lord John Manners, First Commissioner London, 1858. + Lehrbuch der Ingenieur. 136 WEIRS.—DETERMINATION OF ¢. Formule. Weirs with contraction; Q=«, (. + L718 [< J) 2 (29 HAL. s HH V"\ 1 Suppressed vee Q=c, ( 1.041 +.3693 Eada ) 2(29 WI4IH. These expressions take into account both velocity of approach, and partial suppres- sion or complete suppression ; they will give approximately correct results, except with unusual conditions. Mr. Francis proposes the following formule ; Weirs with both end contractions suppressed ; Q=3.33 [hh . < » SBe x» 3 : = 5,58 0-1 Ba. 5 » full contraction ; Q=3.383 (0-.2h) hh. Assuming (2 g)% = 8.020, and reducing to our form ; Weirs with both end contractions suppressed ; Q =.6228 2 (2gh)4 Ch. i 5 One 4, *3) r Q =.6228 3 (2 gh)4 (1-1 A)h. - » full contraction ; Q=.6228 2(2gh)4 (1-.2h) h. Mr. Francis considers that these expressions will apply when h does not exceed : : and with f# from .5 to 2.. Comparing the valves of c,in Q=c 3 (2gh)*th, deduced from these equations, with our values of ¢,, c; or ¢/, and ¢,, on Plate VII., we have for the limits prescribed by Mr. Francis ; Z he Francis. Smith. Suppressed weirs c= .623 c, = .614-.643 One end suppressed .602-.623 c,;=.614-.617 ( 1.5 5 581 c= .601 { 5 .610 611 5 i 598 601 | 1.6 583 591 Full contraction 4 j 5 617 615 10 1. 610 .608 ( 17 602 599 5 620 617 19. dis 616 611 [ L7 612 607 From the above comparison it will be seen that the greatest variation of the Francis formule from the values of ¢ as given by us, is in the case of weirs with full contraction, where the difference amounts in one instance to about 34 per cent. . Messrs. Fteley and Stearns give for suppressed weirs, Q = 3.31 0 1’ +.007 7, with h having a value of not less than .07. Assuming (2 g)* = 8.020, and reducing to our WEIRS.—DeEvTeERMINATION OF ¢. Formate. 137 form; Q=.619 2 (2gh)*7h+.0077. The correction of .007 J is intended to compensate for increased values of ¢ with small heads. We have shown with a suppressed weir having /=.66 and h=.8 (Experiment No. 1’ of Lesbros, Plate VIII.) that c,=.655 ; by the above formula « would be about .621, showing a variation of about 5 per cent. from the Lesbros result. The formule of both Francis, and Fteley and Stearns, are based upon the hypo- thesis that v,=c;; in our judgment the experimental data we have given®*™ prove beyond a doubt that c, is always greater than c,, for the same values of h. M. Graétff, in his “ Traité d’ Hydraulique,” expresses the opinion that the suppres- sion of contraction on the two ends of a weir has the effect of reversing the law of the co-efficients. That is to say, c, constantly increases with h, while c, diminishes as h increases ; the curves for c, and c, crossing each other. He also thinks that the length of the weir has very little effect upon the co-efficient. Expressions could be framed for ¢,, ¢; and ec, which would quite closely agree with the curves on Plate VII., but they would necessarily be complicated and therefore inconvenient for general use. The proper co-efficient for each of these three forms of approach can be obtained immediately from Table XLVIII. The formule of Mr. Francis are from their simplicity convenient for practical pur- poses, as they can readily be kept in memory. If modified as follows, they will give results sufficiently exact when great accuracy is not required ; Contraction suppressed on both ends; Q=3.29 G . jit. 9 ” » oneend; Q=3.29 1h’. Full contraction ; , : ; Q = 3.29 ('- O hk. The limits within which these expressions will apply with reasonable accuracy are, h from .5 to 2. ; 7 not less than 3 h. SHORT WEIRS. The smallest value of 7 for weirs, thus far considered, has been .656. For rect- angular orifices it has been shown that ¢ increases notably, as the least dimension of the orifice diminishes below this distance (.656), and it seems logical to suppose that there should be a similar increase in ¢ for very narrow weirs. The experiments of Lesbros and Castel prove that this supposition is correct. * The experiments of Castel, given in the following section of this chapter treating on Short Weirs, afford an additional proof of the correctness of our conclusions. T 138 SHORT WEIRS.—Experiments. Lesbros. Lesbros. Lesbros using his opening in a fixed copper plate .066 x 1.968 as a weir, with form of approach shown by Fig. 1, Plate I., obtained the following results. TABLE XLIX. Lesbros.—Short Weir ; Full Contraction, Free Discharge into Air. H measured 11.48 above Weir; hence H=h. (2 g)% = 8.0227 | No. | Lesbros’ Nos. H | HH, H—H, id Q C 131 1829-1830 | 1.9472 | 1.9390 | .0082 | .06562 .6090 .639 132, 1831-1833 9892 9849 .0043 of 2944 .650 133 1834-1836 .5331 .5305 .0026 sy .089 O1 | .652 | 134 1837-1839 2674 2657 .0017 rs _ 031 68 653 Castel, The only other data for short weirs are the experiments made by M. Castel at Toulouse, the results of which are published in the Mémoires de l’Académie des Sciences de Toulouse, Tome IV., 1837. The feeding canal employed was a rectangular horizontal box, 19.5 long and 2.428 wide ; this was supplied with water at its upper end by a pipe, with a valve attached, by which the amount of flow was regulated ; a metallic screen and fixed boards were used to prevent undulatory movements in the surface ; the weirs were made by copper plates, .007 thick, screwed to fixed frames. The head was measured by a point-gauge reading to .0001 metre, at the origin of the surface curve in the canal; the larger the value of v,, the less accurately was H measured, owing to greater oscillations of the surface in the canal caused by the increase in velocity. The discharge was ascertained by a measuring vessel holding 113 cubic feet; the times were determined to 1” by a stop-watch ; the minimum value of ¢ was 70”. There appear to have been frequently several determinations for a given experiment, although no details are given. Castel reduced his experiments by the formula (metrical) of ()=2.953 ¢ 1 H’*:; hence (2 9) was taken at 8.0232, which will be assumed as its value for our reductions. In the following table are given the experiments with normal width of canal, various lengths of weir, and (rf always constant ; it will be noticed that a good many of these experiments were with / greater than .66 ; our reasons for not having hitherto used these determinations will be given hereafter. SHORT WEIRS.—Exveriments. Custel. 139 TABLE L. Castel.—Rectangular Vertical Weirs. Feeding Canal 2.428 wide. G always 55%. (2g)? =8.0232 h=H+b oo and b constant at 1.4. Q=c 2 l(2g)4hi: . | | ‘i | ek. slic Mie Rage a o.| of Hay | Af ! h Q | ¢ No. 1 vi | vO, \ hed h W ¢ 135 |.0328 6526 02| 0 |.6526 .061 80.6681 166 ae 3287 05 ).0001 ».3288 1011 |.6122 136, (5899. 0 5x99 .05343..6720 167 2608! 04! 0 | 2608 |.071 38.6119 7, gy Leno 0 5249 .044.92 6730 [168 ,,-—«.1959! | «0 j.1959 Lose 44) 6118 138 4626 0 4626 Los 201.6748 169, 1624 0 |.1624 035 18 6137 139), 3947 0 | 3947 029526785 170, pe 0 |.1303 025 251.6134 | | 140 |.0653 7868. .05 | .0001 |.7869 1155 68.6386 171.3294 .7887/22/,0011 .7898 |7346 |.5940 141, 7218 04) 0 7218 1136 82 6394 ‘172, «7228 -.21|.0010 .7238 |.643.5 |.5931 142 6421 O 6421 11499 6100 173, 6519. 19) .0008 .6527 5506 | .5926 143, 5889, 0 | 5889 10118 6411 Hee) ae .5912' 17 |.0006 | .5918 |.475 7 |.5931 Lie ey OQ |.5217 OSL 44 G48 175, 5207 15.0005 |.5212 3920 5913 15,4623 0 “4693 070566429 176), 4551) .13/ 0004 |.4555 320.2 ‘5911 1460, 3911, 0 3911 .05509.6151 177.3934. 11/0003 .3937 |.256 4 5890 yg «aa 0 3255 .04203 6482 17x, 3297.09 |.0002 |.3299 197 1 os 148, 2608 0 |.2608 .03034 6521 179 ., .2618' .07|.0001 ion 139 8 oe 149.2008 0 | 2008 .02070 6587 180, 1991.05, .0001 |.1992 .093 13.5945 | | Isl, 1660! 04] 0 '.1660 |.071 131.5968 150 .0988 .6549, 06.0001 |.6550 17595 6283 182. 1401.0 1401 .055 gal 6044 151 ,,_—-:15925. .05|.0001 | 5926 15130 .6279 183 .o991/ | 0.0991 033 94.6176 152, 5236 0 1.5236 .125 69.6280 - | | | | 153, (1.4597 Q 4597 10333 6278 184 .6542 .6785 388 0032 .6817 '1.1647 5914 154, «| 3947, O [3947 .ON218 6274 189, OST 335.0024 5861 928 15911 155, 3294 0 3294 06265 6274 186, .5233, 300 0020 5253787 9, 5914 156 = ,,_—«(. 2615) 0 |.9615 .044 36.6280 isi 5 eis 263.0015 '.4628 | 649 8 589s 157 «L975 0 |.1975 .02910).6276 INS, .3921 .22 *.0011 |.3932 | 50x 9.5899 158°. 1697; 0 |.1627 .021 s3'.o294 189 . 326s 18 |.0007 [3975 BaT 4! 5007 | | | hoo! 2681 14 | 0004 1.2635 2808 5933 159 |.1637 17959’ .12/.0003 .7962 3825/6148 191 ., 1962 .10 .0003 .1965 1835 .6022 160 ,, (7396 .11|.0003 |.7399 3375 |.6143 ‘199 7 1690 08 0001 “1691 | 148 5] .6102 161, 6595 .10].0003 6597 2879 .6135 793 . 1299 .06 .0001 1300 101 4.6184 162, (5824 09.0002 .5826 2387 6129 194 0994.04 | 0 0994 .0GN 44 .6240 163. ,, |.5266; .08|.0001 1.5267 2049 eaews i | | | 164, 4564 07/0001 4565 1656 6133 195 .O849 4528 394) 0034 pe 96805963 165, 3963) .06 | .0001 3064 8ay joie le! “304 342.0025 | 3978 7811 5989 ' i; 140 SHORT WEIRS.—ExpeERiMents. Castel. TABLE L.—continued. No.| 1 ae ag i | ji | Q | é ‘No. | hE Vee es | BU, Gull se bev neu Jae og 197 | .9849 3264. 277 0017 |.3281 | .5937|.5996 213 '1.6483 1650 .21 |.0010 1660 3740 6272 198}, 2602.21 ,0010 1.2612 | .4931/.6016 214 —,, 1335.16 |.0006 1341 -.2738 | 6324 199 |, .1985|.16 |.0006 |.1991 | .2843/.6074 215 ,, 1027 12.0003 | 1030 | .1863 |.6392 200 | ,, |1663/.13 |.0003 |.1666 | .2202'.6147 . | ! 201 | ,, 1342.09 | .0001 |.1343 | .1602|.6180 216 1.9689 .3251'|.586 .0074 |.3325 1.2576 | .6229 202 | ,, sor 0001 }.1038 | .1109|.6296 217 YZ 2654 465 .0048 “2702 9281 | 6275 | | | : | 218, 1975. .325,.0022 1997 ) .5961 | .6343 203 |1.8117 .4068483,,0050 4118 1.1315 ,.6103 j19) | 1605 208 01 ee | pee em 20 | og AEB. AOD. DOSS | BABS | B20) .81B2 oon) | biaye AB | 000r |.1980 3083 ..6499 al | » 2641 .296',0020 |.2661 | .5905/.6131 52) | 1990! 14 0004 1024 2236, 6479 206, 1962 .21 |.0010 |.1972 3797 .6179 | | | mE 3s oo 0006 | .1597 | .2788 6226 p95 sana |aoea | 9: 009% 3141 1.3207 6298 208 | ., 1309] .13 |.0003 |.1912 , .2100}.6299 54 2612 .526| 0060 |.2672 11.0450 |.6337 He) a oe ae | L434" .6348 55, 1988 378 .0031 |.2019 | .6940|.6406 210 '1.6483 3192’ 471.0048 er 11,0087 |,6173 225 4, 144.298 0020 |.1664 5216 | .6436 ar) y tea" ave oan | sere | fees agi 228% ae PRES . .159 00 -119 19 .0001 .0001 .0001 .0001 .085 04 .070 35 .056 08 042 91 .029 81 .0004 .230 9 204 2 1759 149 8 12470 102 49 .081 30 .061 63 L044 07 2 028 54 | 3687 332.6 289 0 ALT 1.205 4 11665 .3950 |.133 3 3270 |.100 4 .0002 .0001 .0001 .0001 .0012 0011 .0009 .0007 .0006 .0004 .0003 .0002 .102 03 9 TABLE LI. Feeding Canal 1.184 wide. Castel. toa h=H+b es and b constant at 1.4. “9 i? a“ « 07 05 04 03 .0001 0 0 .36 | .0028 .0025 .0021 .0017 .0014 0011 .0007 -0004 .0003 .0001 .0035 .0027 .0024 | c fo, L df 6660 (258 | 1637 .2631 6665 259 1959 6679 260 |, |1657 6745 261 | ,, 1303 : 6460 |262 | .2582 .7530 6447 263 » -7195 6435 264 | 4, 6545 6437 265 | ,, |.5929 6149 200 5259 6467 267 ,, 4610 6480 268 3931 6543 269 3182 270] ,, | 265 6290 271°, 1946 6265. 6255 272 | 3012 1.7238 6249 273} . 1.6401 6245 274, 1.5928 6237 275 | ,, 5249) 6235 276! ,, 4597 6234 277} ,, 43940, 6240 278 3984 6263 279 | ,, 2628 230) ,, |1988 6218:281! ,, |.1637 .6189 }282 | ,, .1306 6163 1283 | ,, |.1004 6146 | 6134 [284 | .3294|.7815 6132 285 | ,, |.7231 6131 286 | ,, |.6601 .6130|287 | ,, 1.5869 .0018 0014 -21 | .0010 .0006 .0004 .0002 .0001 .0001 476, 0049 138 0042 400 .0035 355.0027 .0001 141 Q=c 21(29)% hls ia | a | 2632 .072 47| 6129 1960 |.046 55) .6126 1657 .036 20, .6130 1303 |.025 25) .6134 .7558 '.559 8 | 6169 7220 1.5202 6140 6566 14485 | 6104 5946 13846 | .6074 5273 |.3200 | .6052 4621 2617 | .6032 3938 2051 | .6010 13186 1488 | .5991 2628 11113 | .5983 1947 071 06) .5989 7373 607 4 | .6079 6428 5022 | .6049 5946 14450 | .6025 5267 |.3698 | .6005 4611 13016 | .5979 3950 12386 | .5966 3290 1.181 4 | 5968 .2632 129 6 | .5958 1990 .085 68 .5991 .1628 1.063 85, .6034 .1307 1046 44} .6101 1004 |.032 03) .6251 7864 |.7540 | .6137 7973 16650 | .6086 .6636 15771 | .6059 5896 |.4817 | .6039 142 SHORT WEIRS.—ExrPeriments. Castel. TABLE LI.—continwed. No. Z Hin : h! h Q | c No. 1 H iw, hf : hoe @ € | p 288 | .3294 1.5249 32 | 0021 | .5270 |.405 4 | .6015 306 | .6542 0965.09 0001 .0966 | 0663 .6309 289. ,, |4603|.27 |.0016 |.4619 .3311 |.5986 | | | 200, 3957 /.23 | 0012 | .3969 .2627 5963 /307 | .9849 |.4603 | .927) .0187 |.4790 1.1160 .6390 291 —,,_ |.3304/.19 |.0008 |.3312 .2001 |.5960 /308 —,,_-:(|3937}.769,.0129 |.4066 8667 | .6345 292, |.2625) 15 |.0005 |.2630 |1418 |.5967 |309) ,, |.3264|.618).0083 |.3347 6442 .6315 293 ~—,,_:(|.1959/.10 |.0002 |.1961 .091 65|.5990 /310; ., .2608).472 .0049 |.2657 4581 .6348 294 , |1650].08 |.0001 |.1651 .07159|.6057 |311 |, 1919 ].324/.0023 |.1942 | 2885 | .6399 295. ,, |.1319|.06 |.0001 |.1320 .051 81.6132 ai, -1568).25 |.0014 |.1582 2135 |.6441 296 ,, |0948'.04 0 .0948 03228 6275 313!) ., |.1309/.20 |.0009 |.1318 | .1632 .6473 | | i314 | , .0988|.14 |.0004 |.0992 | 10736521 297 6542/6037 |.756,.0124 |.6161 1.039 7) .6144 . ! 298 | ., |,5249].652).0093 |.5342 | .835 9/6119 /|315 sg 3780 .929) .01SS |.3968 1.0298 .6504 299 , |.4613|.567/.0070 |.4683 , .6837|.6097 316, 3333 791] .0136 | .3469 | 8349. 6450 300 ~—,,_:—«(.8898| 470) .0048 | .3946 | 527 6.6083 |317 |, .2638).594 .007 | .2715 | TTA .6443 301! ,, .8317].391/.0033 |.3350 .4125].6080 ;318 —,_—«(.1991 0 .0038 |.2029 | 3765, .6502 302, .2644!.80 |.0020 }.2664 .2932'.6093 319 —,,_ 1634) 3827/0023 |.1657 2790: .6529 303 ,,_ |.1946/.21 |.0010 |.1956 | .1N5 3} .6122 320, 1289 .24 |.0013 |.1302 , 1959 | .6582 304 ,,_:11591|.16 |.0006 }.1597 .137 9.6174 |321 —,,_: 1004.17 |.0006 |.1010 | .1348 .6628 305, , [1339/13 |.0004 |.1343 .107 1 92 : Of the foregoing experiments made by M. Castel, Nos. 135-261 and 315-321 have been plotted on Plate IX., with / and ¢ as co-ordinates. If his weirs were placed in the centre of each canal, so that in all cases J'= ZL, it is apparent that when L=3 / for either width of canal, / and / being constant, « should have similar values. We there- fore have two series of the Castel experiments for four values of J, with complete end contraction, viz. ; .033, .065, .099 and .164. By reference to Plate IX. it will be seen that while each separate series for these four weirs forms a curve of nearly perfect symmetry, the two curves often vary considerably with the largest heads; for instance, with 7=.065 and h=.79, « for No. 140 is .6386, while for No. 232 it is .6460, showing a variation of over 1 per cent.. This variation could not have been caused by erro- neous assumptions of the value of /,,’, as in both these experiments r, was quite small. These variations are perhaps due to experimental error, or perhaps J may not always have been the same as L’, for the narrow canal, so that there was in same cases partial suppression on one side. On Plate IX. have been drawn five curves in heavy solid lines, representing the most probable values of ¢ for lengths of weir from .329 to .033, all with full end con- traction. For the greatest length, /=.329, there was a slight partial bottom suppression SHORT WEIRS.—Experiments. Crusted, 143 for heads from .20 to .79; for the other four lengths there was complete bottom con- traction. In the determination of these curves, greater weight has been attached to the experiments with the broad canal, than to those with the narrow canal. The Lesbros Experiment No. 134, with /=.066, agrees closely with the curve for /=.065; Nos. 133 and 132 are about 13 per cent. too high. The series of experiments with the broad canal, where end contraction was partially suppressed, are represented by light dotted lines, except that for /= 2.23, when there was nearly complete end suppression, which is shown by a heavy dotted line. The series with 7=1.18 for the narrow canal, when there was complete end suppression, is also shown by a heavy dotted line. ne: , For all the Castel experiments }, in h,/ = os has a constant value of 1.4 in our reductions; this value is in all probability excessive for the larger values of v,, and consequently our deduced values of ¢ are probably in some cases too low, and especially so for the series with /=1.18, Experiments Nos. 315-321. For No. 315, where v, was the greatest, ¢ as calculated by us appears to be abnormally high; this would seem to show that / should have had a greater value than even 1.4 ; on the other hand the Castel experiments given in Chapter IV. (Velocity of Approach, Experiments Nos. 118 to 152) indicate that the smaller value of b of 141s too great. Taking them as a whole, the experiments given in Tables L. and LI. show that our correction for 1, is too small, while those in Table XXIV. show that it is too great; hence we can assume that the Castel experiments do not contradict our conclusions in regard to the effect of velocity of approach. Comparing the five heavy symmetrical curves on Plate [X., where there was prac- tically complete contraction on all three sides, it will be observed that as the length diminishes from .329, the co-efficient ¢ constantly increases for equal heads. With h=.8, c has the following values ; 1=.329 e=.o94 t=.164 e=.615 £=.099 ¢=.628 1=.065 c=.642 (=.033 c=.666 For heads diminishing from / =.3, with / constant, ¢ constantly increases ; for heads increasing from h =.3, ¢ for two of the curves slightly diminishes, and for the three other curves slightly increases. Taking into consideration the chances of experimental error, it cannot be said that these Castel experiments clearly prove, that with complete contraction, as /, increases above .3, ¢ does not remain constant. The irregular curve for /=.654 (Nos. 184-194) quite closely agrees with the curve for 1=.329; for this series (Nos. 184-194) there was a very slight partial suppression of contraction, but not enough to notably alter the form of the curve. We have seen 144 SHORT WEIRS.—Experiments. Castel. before that for weirs with full contraction, with fh constant, «, or ¢,, imcreases as / increases above .66; we can hence assume that the law in regard to the effect of / upon ¢, changes when / is about .5. Therefore, for weirs having full contraction, the co-efficicit of discharge for equal heads, tucreases as Lriuereases above .5, aid also inereases as L diminishes below .5. When h is a small fraction of /, the effect upon ¢, of variation in / will doubtless be small. In the curves on Plate IX., for the lengths .654, .985, 1.31, 1.65, 1.97 and 2.28, ¢ for equal heads constantly increases with the length ; this is partly due to the effect of changes in /, but more largely to the effect of partial suppression ; the weir with / = 2.23 being nearly suppressed on both ends. For the purpose of comparing the results of Castel with the data discussed in the first part of this chapter, there have been drawn on Plate IX. in light solid lines curves for c, with 7=.66, and for ¢, with /=1. and /=2,; these curves have been taken either directly, or by interpolation, from the curves on Plate VII. It will be seen that the curve for c, with /=.66, is for heads less than .5, higher than the irregular curve for Castel’s Nos. 186-194 with a weir of nearly the same length ; Castel’s experiments with this weir, agree more closely with Experiments Nos. 35-40 of Poncelet and Lesbros, Plate VIII., than with those of Lesbros from which this curve of ¢, was determined. The irregular curve for Castel’s Nos, 315-321, with complete end suppres- sion and /=1.18, agrees closely with the curve for c, with 7=1. taken from Plate VIL., except No. 315, which appears to be abnormally high. The irregular curve for Castel’s Nos. 222-227, with end contraction nearly suppressed and /= 2.23, is somewhat lower than the curve for c, with /=2. taken from Plate VII. ; had there been full end sup- pression for this series of Castel, the two curves would apparently have been nearly identical. Velocity of Approach Experiments Nos. 143 and 147 of Castel were with 7 = 2.43 and end contraction suppressed, with very nearly complete bottom contraction ; by comparing the values of « for these experiments with the curves for cv, on Plate VII., it will be seen that they agree very fairly ; for instance with /=2.4 and h=.26, c, by Plate VII. is .638, while for Experiment No. 143 (Velocity of Approach) with the same length and head, ¢ is .636. Nos. 148-152, in the same Table No. XXIV., do not, however, agree so well with Plate VII. ; the values of « in these latter experiments, when compared with the other experiments given in the same table, appear to be unreliable. From the preceding comparisons, we are warranted in the statement, that the experiments of Castel with weirs having a greater length than .50, agree as closely as could be expected with the deductions we have drawn from the data furnished by Lesbros, Francis, Fteley and Stearns, and ourselves. The experiments made by Castel with orifices having convergent mouth-pieces, and with weirs, form wonderfully smooth curves, when each series is plotted by itself. This symmetry for his weir experiments is WEIRS.—CastTeEt’s EXPERIMENTS. 145 somewhat more perfect, when // and C are used for co-ordinates, than when /, and c as deduced by us are the co-ordinates.* When, however, the various curves for different weirs are critically compared, they present discordances, not very great to be sure, but still considerably larger than one would expect with such apparently accurate experimentation. It may perhaps be uncharitable, but we cannot refrain from the con- jecture, that M. Castel devoted a good deal of time in his study in adjusting his experimental data, so that they would show a very high degree of accuracy when presented to the world. To one familiar with the very great physical ditticulties in the way of the exact determination of ( for weirs with small values of H, the exceedingly smooth curves shown by the Castel experiments, appear most suspicious. His experimental appli- ances were in every way inferior to those employed by Mr. Francis, and by Messrs. Fteley and Stearns ; hence the remarkable accuracy of M. Castel inferentially shows that these later experimenters did their work care- lessly ; this is an inference which we are by no means prepared to accept. We are frank to say that with the apparatus employed by Castel, with so small a measuring vessel that ¢ was often as short as 80”, and with heads as low as .1, we cannot conceive it possible that such smooth curves could have resulted, unless each experiment represented the mean of many separate determinations, and which does not appear to have been the case. It may be, that seemingly incongruous results were rejected, and only those were given which seemed to be harmonious ; such suppression is never justifiable ; an experiment made under normal conditions, which at first sight seems to be discordant, may upon closer study have an important bearing upon the problem to be solved. The perfect frankness of Messrs. Fteley and Stearns, who faithfully give a/d their results, affords an example which should be always followed. We are of the opinion that M. Castel, considering the appliances which were at his disposition, made his experiments with much skill, and that his results are fairly accurate, but that they bear such evidences of having been “doctored,” as to render them more or less open to doubt. Had it not have been for this opinion, we should have used many of his determinations in the first part of this chapter. SvuBMERGED WEIRS. A submerged or incomplete weir, is one where the surface of the water down- stream from the weir is higher than the crest. Dubuat proposed the following formula for such weirs, when there is no velocity of approach ; (A) Q=C1(2gh)* (HM, + ¥h). H,,=head measured up-stream in still water, above the crest. Hi, Sr gy ” down-stream ” ” ” h =H,—H,=etfective head producing the flow over the weir. This formula is deduced by the following reasoning : Dividing the section, / H,, at the weir into two horizontal sections /h and 1 H;, it can be assumed that the discharge through the submerged section / H, is represented by (B) Qi acl H, (2gh)*. Also that the discharge through the unsubmerged section // is represented by the usual weir formula, or, (C) Q" =o" 1LEh (2g h)%. * Castel in his reductions did not take into account the effect of v, . 146 SUBMERGED WEIRS. If it be assumed that c” and c’” are equal, by adding equations (B) and (C) together we have, (A) Q= 0+ Q’=c'l (2gh)* (M,+Fh). When there is an additional head due to velocity of approach at the measuring point for H, it should be added to h. Lesbros, very erroneously it seems to us, assumed that the entrance of the water from his reservoir into the feeding canal in Fig. 6, Plate I., and in a similar canal shown by his Fig. No. 19 which we have not given, should be considered as belonging to this class of weirs, and constructs his Table No. XXIV. from experiments similar to that from which we obtained the data for our No. 130. In this case (Lesbros’ No. 2015) he takes the head H measured in the still water of the reservoir as 7, (the canal being horizontal H,, is the same as H for the weir at the lower end of the canal) ; for 7, or what he assumes as the submerged portion, he takes the point # (p. 110) being the lowest depression in the canal, and .99 down-stream from its upper end. He hence has H,=.4649, H,=.2615, h=.2034; measured value of Q=.6391, and b=.7874, then by formula Y=c' 1 (2 9h)” H,,* he obtains c” =.4825. Strictly speaking there was in this case no weir, there being no dam at the head of the canal. The loss of head between the points H and # being caused, as we have before pointed out, by the velocity which the water in the canal had acquired at #, by contraction as the water entered the mouth of the canal at D, and slightly by friction on the sides between D and F# (vide p. 110). These experiments of Lesbros cannot therefore be considered of value in giving direct data for submerged weirs. They are especially alluded to here, because they have been quoted by Mr. John Neville, in his Hydraulic Tables, which is generally regarded as the standard text-book upon Hydraulics in the English language. Fteley and Stearns. The experiments by Messrs. Fteley and Stearns give us valuable data for this form of weir. The weir used by them had a length of 5 feet, end contractions suppressed, with inner depth below crest, G, of 3.17. The lower canal was about 8.7 wide. The head Hf, was measured 6 feet above the weir; the head H, was measured 6 feet below the weir. @ was determined by the flow over a sharp-crested weir, with free discharge, and then the level of the water in the lower canal was raised above the crest by stop- gates, the same volume of water passing over the now submerged weir. We have calculated @ for the following experiments by curve for ¢, with /=5. on Plate VII. Q 2 The correcting factor }, in h = H+ se) , is placed at 1.33. For corrections for velocity of approach for H, the same expression is adopted, hence h” = (1. +8 ) -H,, The value of C’ is then obtained by the formula of Dubuat, Q=C" 1 (2 g h’)* (+2 h”). Very likely our correction for v, with the submerged weir is not exactly logical * This is an incorrect formula for submerged weirs. SUBMERGED WEIRS.—Exrrriments. Fteley and Stearns. 147 but these corrections are not large, and any error from this cause will have but slight effect upon the values of C’. The velocity in the lower canal at the measuring point for I, is not given, although were it large it might possibly be an important factor; judging from the sketches of the apparatus, the section of the lower canal was large, and hence the effect of this lower velocity must have been slight. All these experiments were with the weir entirely submerged ; 7.¢., no air under the escaping sheet. The value of the co-efficient 2, given in the last column of the table, has been obtained by O=. 1 (2 gh”) (.915 H+ #h”). Fteley and Stearns. SUBMERGED WEIRS.—EXPERIMENTS. 148 i Ted exe O10e" LOTS LOTR” FFOO'| EE00": COP CETS” coo Sou" Ligh G99G"° GICL | LPO0'| S600") CLF) GOL | LEG’ ORIG) OGFL'| TE00"| EZOO" | ORE" GEFL | f 129" Osc" 0 619" c6u" OF Le" F19e"| 099°] GLO" | FZO0"| T6e) 8eCO | F69", £09 “ier 6eet'| o¢t9" &E00') FZ00"| Fk") 8609" 829 FIN CERF’ CELT) FEGG" ZEOO"| FZOO'| 968] CzEG™ | ceo" 98 CREE Tete ern 0Z00"| $100". E1k" £6TL"! “F291660" 06S", E60R" GFE EzO0" 4100") Lee” cOFS I CFO" Te9 ROLF GELO’ COTE! Ez00", 1100" Tee” 080g" | Gea exe eict" 98IF'| G9FG 1100" ITE91 EBS" OBLT') ROE) SORT TIO" let9" #8q" Ieee’ c61s'| 9TGt! 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See #1 | se! @ 9k Gz GEE | Ie FCG 6T €e 6 Bee ON saa oN ay7 ano passnd awnjo4 sums oy} pun ‘obsmyosiqT aang YR “Wa A, popsas0-dioygy waa0 Mozy Uosindmoj—'sulvag pup hapopir TIT WIAVi SUBMERGED WEIRS.—Exrerments. Francis. 149 Francis, Mr. J. B. Francis has published, in the Transactions of the Am. Soe. of C.E., September, 1884, an account of a number of experiments made with submerged weirs at Lowell, Massachusetts, in 1883. The feeding water wax drawn through a turbine water-wheel at rest, under a nearly constant head, from an upper canal into a lower canal. The amount of flow was regulated by the gates for this wheel ; the size of the opening presumably remained con- stant for each series of experiments ; the discharge into the lower canal was submerged. In this lower canal were placed two weirs, the feeding stream entering about midway between the weirs into the pool formed by them. The weirs were sharp crested, with end contractions suppressed, 11.22 and 10.98 feet long respectively, and having horizontal crests at the same elevation. The discharge was first gauged for each series by allowing the water to pass over the weirs with free discharge into the air; the water level in the canal below the two weirs was then raised a certain distance above the crests for each experiment. This diminished the head of the feeding stream or submerged jet, and hence decreased the flow; there were also fluctuations in the elevation of the surface in the upper canal; for each experiment the amount of head for the submerged feeding jet was determined. There were 5 series of experiments made, the gates regulating the supply not being changed during the experiments constituting one series. Now, calling 4 the area of the opening for the feeding supply, )” the head actuating the flow through this opening, being the observed difference in level between the surface of the water in the two canals, and Z the co-efficient of discharge for this particular opening, we have Q=Z A (2g Y)*%. The head Y¥ for any one series did not vary largely, and was never less than 12 feet; we have seen for orifices that with slight variations in such large heads the co-efficient of discharge is practically constant; we can hence assume that Z for each series was constant. Taking 42 experiments made under normal conditions where there was perfectly free discharge, from the given values of Z and H we can determine the most probable value of 7 .1 for each of the 5 series, Q being computed from the inter- polated curve for c, with /=11.1 on Plate VII. The depth of the water in the feeding lower canal below the crest of the weirs was about 5.8, so that the velocity of approach was never very great, and the effect of », can be neglected without serious error. The deduced values of ZA for each series do not vary much, showing that A was practically constant. These computations need not be given here; the results slightly vary from those indicated by Mr. Francis, who adopted a somewhat different method of compu- tation.* * Mr. Clemens Herschel in the Transactions of the Am. Soc. of C.E., May, 1885, has discussed these experi- ments of Mr. Francis. Mr. Herschel has calculated @ by the mean values of Z A, using the Francis formula for suppressed weirs, and making allowances for v,. 150 SUBMERGED WEIRS.—Experiments. Francis. We can now readily calculate Q for the experiments with submerged or “ drowned” discharge, as we have Z A and Y,in Q=Z A (2g Y)*. H,, was measured at a point 6 feet from each weir. H, was determined by the height of water in a vessel, which was connected with the back-water below each weir by a pipe having openings at its ends, distant 18 feet from the crests ; H, was hence the mean elevation of the back-water in the two lower pools. There was so much commo- tion in the water near the weirs, that it was necessary to measure H, at this consi- derable distance of 18 feet below the two crests. Using the same notation employed for the preceding experiments of Fteley and Stearns, and not attempting to make any corrections for velocity of approach, we have the following results. SUBMERGED WEIRS.—ExPeEriMENTts. Francis. —Submerged Weirs. 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 13.959 13.956 13.918 13.921 | 13.858 13.800. 13.719 | ‘13.706 13.545 13.901 13.903 13.938 13.949 13.945 13.943 13.930 13.903 13.860 13.785 13.663 13.136 13.128 13.133 13.123 13.063 13.043 13.016 13.147 13.060 13.265 End Contractions Suppressed and Full Bottom Contraction. i a we eS we =< = =F =i <1 oO “I Toe Fe F OM AT KH oO wm oO -~] | «32F 1.19 | .422 1.40 | .494 1.55 | .549 1.69 | .597 1.82 | .643 1.93 | .683 2.03 | .719 2.13 752 2.22 | .783 2.30 | .814 2.37 | .839 Vv 4.13 5.02 5.56 6.03 6.36 6.59 6.83 7.03 7.23 Sl Se) at soot ip a oS Ht 176 OPEN CONDUITS.—EXPERIMENTS. TABLE LI Xc. Darcy and Bazin.—Semicircular Conduits. ven (rs)4 Darey-Bazin. XXIV. A =pure cement. XXV, XXVI. sand (very fine). |A cement mixed with 4rd] A=partly planed plank. XXVIT. | A=small gravel (.03 to .07 dia.) fixed in cement. s=.0015 s=.0015 s=.0015 s=.0015 Top width (also diameter) | Top width (also diameter) | Top width (also diameter) Top width (also = 4.10 = 4.10 = 4.59 ; diameter) = 4.00 T= 63° | T=63° T =64° to 73° : T=54° No. d@ riv| « INoa| rie] a No dir |e! nm Nol 4 r fo ln 923 .59) 366 128.9 235; 61) 379 237 120.5 | 247) 63), -390 2.61, 107.8 260 | 454 2.17 780 224 83 503 3.72) 135.6 236 S88) 529 3.43) 122.0 248) 88 537 3.23 113.8 261) ... | 546 2.50 826 295 1,08 | 138.0 |237 1.09 635 3.87. 125.8 2491.07, 632 8.71 120.6 /262) ... | .619 2.69 820 2261.18 | .682 4.60! 143.7 2381.24, .706 4.30) 132.1 2501.24 .717/4.04| 123.0/263 ... | .681 2.93 840 227 1.34) .750 4.87, 145.1 ||239.1.41 | .787 |4. 51 131.3 2511.40 796 /4.95/ 193.2 [264 ... | .731 3.05 840 2281.47; .809 5.12, 147.1 2401.54; .839 eu 136.3 292153) 2564.51 | 195.8 265... TRL 3.22 458 22911.61| .867°5.29' 146.7 |241 1.69) .900 4.94/ 134.5 2531.68] .921 [4.64] 124.7 |266 ... | .826 3.33 849 S80 1.79 915 5.51) 148s 2421. 80} .941 5.20) 138.3 2541.79] .964 [4.87] 128.2 267 900 3.54 85° 931/1.83| .949'5 75| 152.5 msi 92) 983 5.38! 140.1 ,255'1.92 |1.015 [5.00 | 128.2 268 7 .968 3.73 85° 2321.94 | .992 5.91) 158.3 2441.98 1,006 5.48| 141.0 [2562.02 1.054 5.18 | 130.3 269 ... 1.012 3.95 88° 2332.05 1,029 6.06| 154.2 245), 041.022 5.58; 141.7 '257/2.14 1.096 /5.29| 130.4) | | 2342.08 1.034 6.11) 155.1 2462.09 1.038 5.66] 143.5 2582.24 1.129 nas) 12s j u | | | ‘9592.29 1.148 5.54 133.5 | | | 2 | i Series XXVIII. to XX XI. inclusive were made with a rectangular trough cut in four longitudinal solid timbers, nicely joined, and having a total length of 62 feet. Q was measured directly, with much accuracy. Series XXVIII. and XXIX. were made with the planed and very uniform wooden surface of the trough. 366.3 D=,2628 T=45 T = 39° — 43° 7 not given. T not given. No. 8 v n || No. s v n || No. 8 ” n | No. 8 v % eal ene | cacy 179 | .000 25] .167 |61.7 | 186|.000 71) .371 | 80.5 |} 193}.000 65) .403 | 62.0 | 199 .000 84, .633) 85.2 180|.000 71] .266 | 58.1 |187).001 80} .617 | 84.1 ||194}.00250| .823 | 64.5 | 200 .002 94/ 1.263 | 90.9 181|.001 83; .426 | 58.1 | 188] .006 51) 1.270 | 91.1 || 195|.007 25 | 1.463 | 67.3 | 201 | .007 23) 2.014 | 92.4 182|.0067 | .830 [59.1 | 189).014 41) 1.972 [95.1 | 196].016 10) 2.224 | 68.7 | 202 | .007 37 | 2.047 | 93.0 183} .015 25 11.250 |59.0 || 190/.030 18) 2.927 | 97.5 || 197.031 00 | 3.054 | 67.9 | 203 .015 57 | 2.835 | 88.6 184 | .032 40] 1.808 | 58.5 || 191) .039 66 | 3.392 |98.6 | 198 | .045 35 | 3.747 | 68.9 | 204 ' .029 38 4.095 93.2 185|.041 55) 2.077 | 59.4 || 192] .046 50 | 3.694 | 99.1 | 205 .044 73) 5.007 92.4 New Cast-Iron Pipes. XVI. XVII. XVIII. Vv = 366.1 D=.2687 Uv =365.7 D=.4495 U =365.4 D=.6168 T=5T° — 65° T=59° — 61° T not given. No s v n No. $s v n No. s v n 206 | .000 2 -289 78.8 219 | 000 24 489 94.1 229 | .000 27 .673 104.2 207 | .000 83 561 75.1 220 | .000 87 978 98.9 230 | .00175 | 1.63] 99.3 208 | .002 32 | 1.175 94.1 221 | .00209 | 1.601 104.5 231 | .003 68 | 2.487 104.4 209 | .005 31 |) 1.841 97.5 222 | 00475 | 2.503 108.4 232 | .00805 | 3.701 105.0 210 | .010 2 2.595 99.1 223) 01260 | 4.196 111.5 233 | .013 40 | 4.882 107.4 211 | .02255 | 3.888 99.9 | 294 | .02225 | 5.623 112.5 934 1 0225 6.342 107.7 212 | 03208 | 4.652 100.2 | 225 | 033.18 | 6.883 112.7 235 | 2038.1 82922 107.3 213 | .04041 | 5.154 98.9 226 | .03905 | 7.484 113.0 236 | .109 80 | 14.183 109.0 214 | 09547 | 8.048 100.5 | 227 | .09852 | 11.942 113.5 || 237 | .145 91 | 16.168 107.8 215 | .099 04 | 8.160 100.0 228 | .167 56 | 15.397 112.2 | 216 | .11978 | 8.924 99.5 ‘217 | .168 07 | 10.623 100.0 | 218 | .17072 | 10.712 100.0 | 228 PIPES.—ExPERIMENTS. Darcy. TABLE LXXVII. —continued. Old, Cleaned, and New Cast-Iron Pipes. Sig | se XXL | OCI. : } An old pipe. | No. XIX. cleaned. anole oe well A new pipe. U = 365.3 D=.7979 U = 365.3 D =.8028 | U =365.3 D=.9744 | l =365.3 D=1,6404 Tf not given. | T not given. T=69° —72° ! T not given. | No. | 8 | v n | No. | |__| | | ! 238.000 94 1.007 | 73.6 .000 52) 912) 89.3 | 954 00028 .800] 96.9 | 262 .000 451.380 101.6 239 | .002 02! 1.4835 73.9 .001 65 | 1.762 96.8 i 255 .0OL 19 1.765 103.7 | 263 “000 45)1.472 108.4 240 ).004 73) 2.320) 75.5 .004 98| 3.113) 98.5 | 256 | .002 69 2.700 |105.5 | 264.000 6 |1.559| 99.4 § vo 1 A ° w eS d 3 A ° s v v7 He om Heo wm mt ww tow Ww Ww to lo or oO 241|.011 50| 3.629) 75.8 | 249!.011 55| 4.659/96.8 | 257|.005 37, 3.789 /104.8 | 265 .001 2 2.602 117.3 242] 022.90 | 5.075 75.1 |, 250 .020 35] 6.247 | 97.7 ]258).011 05 | 5.420 104.5 | 266 .001 252.609 115.2 243'.032.00| 6.014 75.3 | 251 .027 36) 7.238 97.7 |259/.093 05| 7.841 [104.6 | 267 .0021 3.416 116.4 244|.041 05 | 6.801 75.2 | 252 .037 30) 8.438] 97.5 | 260].032 05 | 9.183 [103.9 | 268 .002 3 8.633 119.0 245 | 13981 |12.576 | 75.3 |) 253] .113 43 14.754) 97.8 y 261|.0407 10.368 104.1 | 269 .002 6 [3.674 112.5 | | i 270 ).002 5 '3.700 115.6 Lhen. Herr Otto Iben has collated in his “ Druckhohen-Verlust an geschlossenen cixeruen Rohrleitungen, Hanbury, 1880,” a large number of experiments of the flow through cast-iron pipes, made during the years 1874-1879 by engineers in charge of the water works of Hamburg, Stuttgart, and other German cities. We understand that Herr Iben simply acted as editor in compiling the data given to him, and hence is not responsible for the accuracy of the experimental work. The heads lost by the retardation of flow through the various pipes experi- mented upon, were in all cases determined by piezometers, either directly by vertical columns of water or quicksilver, or by Bourdon gauges. In the second general series of experiments made at Hamburg, the results of which will be summarized in the following table, the several elements were determined as follows : @ by absolute measurement of ¢ ; D, for the new pipes, by carefully measuring a few joints of each commercial size in use, by end diameters and also by quantity of water contained in the several joints; for the old pipes, although doubtless their diameters were considerably diminished by incrustations, D was assumed to be the same as for new pipes of the same commercial or founder's size; the loss of head, h, was generally determined by Bourdon gauges, the gauges being carefully compared for correction by PIPES.—Experiments. ben. 229 attaching them to a vertical stand-pipe before and after each series of experiments ; for Series X. open glass tubes were used for determining the pressures or piezometric heads, which permitted very exact readings. The new pipes™ had all been coated with tar, and hence probably had quite smooth interior surfaces. Old pipes, Series XI. and XIII, had also been coated with tar before they were laid in place. The following table gives a summary of the results deduced from these experiments. TABLE LXXVIIL Hamburg Water Mains, 1877-1879. v=n (rs)¥ New Cast-Iron Pipes, coated with Tar. (Nos. V. and V*. had been 3 months in use.) D=.335 0 D=.499 D=1,001 ' p=1.667 | } i=415, | 12397. |1=1073.| 1=930. | 1=930. ger. | gal DR | 1=580. 1= S514, L i * 41y Vv. ve VIII, | oc [2 Sa a . They 18 1 Uv qt vo nr v n Uv | % t 1% Vv nm Uv | nt 10 | 79 | 1.0:132. 20] 83] 2.311) 32:89; 16; 8 , 17 | 200 | O4 j128; 07 |105 17 92 [2.1] 97 3.3) 87 | 3.3] 85 44/92) 21) 97 | 18 | 149 | 09 108 1.6 | 110 21 | 90/28/97) 3.9) 88) 43}01 | 55 91 23/102 | 22} 181 | 13 |109| 1.9 }109 23 | 90/43/1051 48/92) 4.7] 01] | 26,112 | 26/192 ) 17 |110], 25 |109 2.8 ia 55/104 5.3| 87 | 5.1] 90 | 32) 108 | 29/129 | 22 us i. 5.3} 90 oe || E21 3.2 | 132 25 | 113) | _—_ 5.6| 91 , 4t |i | ae | ime | ao jae | ) i 60/92) 48) 125 | 38/193 | 36 |i | | | 6.5) 92] 4.5 )122 | £8 1/116 : 7.1| 90 6.0 |'149 | 6.1 114) i 8.0} 92 7.0 | 148 | 8.7 95 ! 7.2 | 129 | | | I : 7.9 | 129 : ! | a5 | 129 | : | | i 8.7 | 122 | | | | | | 10.5 | 132 | * Series V. and V*. were with a pipe, which had been three months in use. 230 PIPES.—Experiments. Jben. TABLE LXXVIII.—continued. Old Cast-Iron Pipes. (Nos. XI. and XIII. had been coated with Tar.) D =.335 D=.499 ! D=1.001 D =1.667 1=373. | 7=900. | 7=916. | 7=2149. | 7=7179. | 7=1808. 1=1736. (=781. = 4403. 19 13 19 2 14 15 22 22 25 years old.years old.tyears old.jyears old.tyears old. years old.| years old. years old. years old. Ii. VII. VIT*. XI. XIII. XIV. XV. XV". XXII. vin|vi|nitovoiowtv lw flv | ni ow n v n v n v n 0.3) 25 | 0.8) 26 | 0.8) 23 || 0.6) 74 | 0.7] 71 | 0.9) 67 0.8 50 0.4 42 1.6 64 0.7) 20 | 1.3) 3£ | 1.3) 26 ) 0.7) 77 | 1.2) 75 | 1.2] 61 1.1 47 0.6 43 2.2 60 1.0) 20 | 1.5) 34 | 1.5) 27 | 0.8; 81 | 1.6] 78 | 1.8] 58 1.5 47 1.5 44 it 60 1.6) 21 | 1.7) 34] 1.7, 97 | 1.2] 85 | 2.2] 79 | 2.3) 59 1.9 45 2.4 45 3.1 59 2.1} 21 | 1.8] 34 | 1.8) 27 | 1.2] 85 | 2.7] 80 | 2.5) 59 2.4 45 2.9 45 3.6 59 2.4; 21] 1.9] 34 | 1.9] 27 | 1.3) 89 | 3.2! 82 | 2.6) 58 2.6 47 3.5 46 4.0 58 1.6) 92 | 3.9} 80 3.1 46 4.4 45 4.3 58 2.4} 86 3.5 46 4.5 58 A glance at the foregoing results, especially for the new pipes, will show that the deduced values of m jump about in the most irregular manner, in one case n having the absurd value of 200, with D=1, and s=.000 29. The results obtained from the old pipes possess some value in indicating a very greatly reduced discharge from pipes unprotected by asphaltum coating, and which have been several years in use. Herr Iben gives 47 experiments made at Stuttgart, which have no more value than the Hamburg ones just summarized. He also quotes 8 experiments made at Bonn, which possibly are entitled to some weight. The Bonn pipe was a long force-main, supplied by pumps at the lower end; it was new, with no obstructions of consequence such as valves or sharp bends, and had been coated with a preparation of asphaltum. @ was determined by absolute measure- ments of g in a reservoir, and also by capacity of the plungers of the pumps; the two measurements agreed closely. Owing to the disturbed condition of the surface of the water in the reservoir, y was probably not very accurately determined. The head, h, was determined by manometers at the pumps (probably Bourdon gauges attached to the air chambers of the two pumps). There was a small amount of air in the pipe, which may have affected accuracy of results. PIPES.—Exprrments. Iben. 231 TABLE LXXIX. Bonn Water Works. New Cast-Iron Pipe, coated with Asphaltum. D=1.004 1=17 684 T=41° v=n(rs)2 Z ° © 3 h 8 nu No. Q | v h | 8 | n 1.241 1.568 21.3 | 00121 | 90.1 | 2717 | 1.230 1.553 21.3 | .001 21 | 89.3 1.675 2.116 32.8 | .00186 | 98.1 | 272’ | 1.665 2.104 34.4 | 00195 | 95.1 2.054 2.595 47.6 | .00269 | 99.9 || 273’ | 2.074 2.620 45.9 | .002 60 | 102.6 2.474 3.125 60.7 | .003 43 | 106.5 | 274% | 2.451 3.096 64.0 | .003 62 | 102.7 ~T Si ~T w bo bo bb “I Ww ~~] ie The experiments given by Herr [ben have been discussed at length by Herr Albert Frank, in Der Civilingenieur, Vol. XX VII, 1881, “Die Formeln tiber die Bewegung des Wassers in Réhren.” Lampe. In all probability the most accurate experiments of the flow through a long pipe, where the loss of head from friction, et cet., has been measured by piezometers, are those made in 1869-1871 by Prof. Dr. C. J. H. Lampe, and described by him in Der Civil- ingenieur, Vol. XIX., 1878, “ Untersuchungen itber die Bewegung des Wassers in Réhren.” We will therefore describe these experiments at some length. The pipe experimented upon was a cast-iron conduit, which conveyed by gravity a supply of spring water to the town of Danzig. It had a total length of 46 352 feet ;* the lower 9040 feet in length had a considerably steeper inclination than the upper portion, so there were two hydraulic-grade lines; the 4 experiments, however, were made upon the upper portion of the pipe. The pipe was laid in 1869; it was covered with 5 feet of earth; it had 3 curves, each of 10.3 feet radius, and a number of very easy curves as it followed the general contour of the ground. Its section was very uniform, and its diameter was almost exactly 16 Rhenish inches, or 1.373 English feet ; the constructing engineer states that the maximum deviation from this diameter was only .008 foot. The pipe was coated with a patent varnish, which did not appreciably diminish its section. Examination showed that from 1869 to 1871 the character of the inner surface had very slightly changed ; the only material adhering to the surface in 1871 could be readily removed by rubbing with the finger, and there being no signs of rust. The joints were 12 feet in length, united by lead and hemp packings. There were 26 air-cocks attached to the pipe along its course, at anticlinal points, the re- spective distances and elevations of which had been carefully determined. * Dr. Lampe gives most of his measurements in Rhenish feet; he assumes 1} Rh. ft.—.418 47 metre ; hence 1 Rh. ft.—1.029 72 English feet, which is the ratio we have used in reducing his data to English measurements. 232 PIPE.—EXPERIMENTS. Lampe. The mean velocity was ascertained by measuring the discharge, or y, in a masonry reservoir, situate at the outlet end of the pipe, which had an area of about 15 160 Rhenish square feet, and a depth of several feet, thus affording an excellent opportunity for the accurate determination of Q and »v. The pressures were determined by connecting a quicksilver manometer, first with one of the air-cocks and then with another ; these pressure determinations were not synchronous with the measurement of q, but in some instances several days apart. As in none of the 4 experiments was the pipe filled at its inlet, this lack of synchronism appears to form a dangerous source of error, the pipe being fed by the flow from springs, whose discharge must necessarily have been more or less irregular. Dr. Lampe, how- ever, states to us, in an explanatory letter, that he is satisfied no serious error could have arisen from this cause, there having been directly preceding or during these intervals no rains of consequence to notably affect the flow from the feeding springs; subsequent readings of the manometer, shortly after these intervals, also verified the constancy of flow. He considers that errors from this source will not change the given results more than 4 of one per cent. .* : Another source of error in these experiments, arises from the fact that the pipe had two hydraulic-grade lines. Although the several pressures were all determined above the anticlinal point at which these lines united, still this condition of the pipe was more or less unfavourable to extreme accuracy of observation, as the sucking or siphon action as the water passed over this summit could not have been perfectly regular, as doubtless the amount of air accumulating at this summit varied slightly from time to time; the consequent intermittent sucking action of the water as it flowed below this summit, hence may have appreciably affected the nearest piezometers. The very great length of that portion of the’ conduit experimented upon, however, makes it improbable that this last source of error could considerably affect the accuracy of the determination of the general value of s. In the following table we give the details of these piezometric measurements for Dy. Lampe’s first and third experiments, our Nos. 275 and 277, expressed in Rhenish feet. * There are several typographical errors in Dr, Lampe’s paper in Der Civilingenieur, especially as to dates, which he has been kind enough to correct for us. PIPES.—Experiments. Lane. 233 TABLE LXXX. Dr. Lampe.—Pievometric Determinations. Danziy Conduit Pipe. joe (Pipe Experiments Nos. 275 and 277. ) Elevations and distances given in Rhenish feet. Data for No. 275. October, 1869. So — — if I es || — Fi i ' Measured ' 5 | Elevation Head or Piezometric| Calculated "LossofHead Distance No. of — of Zero Pressure | Elevation Elevation rie | between | apart Between the Air-cock. ateach ‘ateach Air- at each at each - | the several _ of several several | Air-cock. cock,inFeet’ Air-cock. || Piezometer. | Air-cocks. | Air-cocks. | Air-cocks. | ' of Water. 22 — 69.38 | 29.33 | - 40.05 | ~ 30.76 =.29 501 | oe oer 21 — 92.97 | 49.91 | SOROG |) eee ||| ee 1148 | 6309.6 , .001 82 7 = 105.20 50.66 | —54.54 - 54.81 +.27 5.51 | 3107.4. .00177 Te> piles ie pee SUOe foun | 6388 | Line | oases 12 — 88.75 | 24.84 — 63.91 : - 63.11 —.80 4.08 2525.1 ° .001 62 10 - 108.65 | 40.66. —67.99 | -6808 +.04 | ome BINT > Gt ee 9 | - 9078 | 27.00 | -72,78 | sam | at) ate | goieg: | cota 8 | -100.62 | 23.67 | - 76.95 -17.37 +42 ee ee oe - 8 111.70 | 3958 { Rat | 82.14 +.02 | 5 is 25551 | 002 14 6 , — 102.93 15.33 —87.60 | ~87.12 — 48 a7] 1802.4 001 50 5 | — 95.64 5.33 | — 90.31 - 90.64 +.33 : | | | ; 402 50.26 amomee _sunihaae Data for No. 277. October, 1870. | 1 25 | — 60.46 | 11,33 « =48 48.75 Sen I age nae ‘Sania 24 - 65.25 1) 15.07 — -50.18 , 50.11 =07 | ser | S788 01 42 23 - 6853 14.68 53.85 | soe $09 0g 945.0 00112 22 - 69.38 | 1447 -5491 | 5524 | 4.38 ior! (aosy | .oo1a0 a j= 920% | 36.09 : -56.88 | -sr1s | +.30 641 | 4555.1 | 00141 19 | — 109.83 46.54 — 63.29 | -6344 | 4.15 16 | 605.6 001 25 18 = 108.10 | 44.05 ~ 64.05 — 64.28 +3 | 1s9 | 1igag 00157 17 - 105.20 ; 39.35 65.85 05.85 | +0 | 393 20955 | oold7 15 -120.47 | 5154 68.93 8.74 -.19 ai 580.2 | 00128 14 | -11813 | 4846: -69.67 ° -on | -13 | 53; aziz | 00169 13° | ~110.38 39.98 = 7040 * ofc $ =2e 473 | 3672.9 001 29 10, — 108.65 33.52 —75.13 -75.19 | +.06 es | AT85.5 001 43 8 | - 100,62 18.67 | “Ges | =8le “i | gad 5002.2 | 00129 -102.93 | 1454 | -8839 8868 a7 o79 | 18024 | .00155 - 95.64 4.46 | —91.18 — -9114 | -.08 | | | 19 : 42.04 | 308034 | .001 365 HH 234 PIPES.—EXpPERIMENTS. Lampe. The calculated piezometric elevations, given in smaller type in the preceding table, are for a straight line* uniting the assumed elevations in the extreme piezometers in each experiment, being the lines of most probable inclination, as calculated by Dr. Lampe; s for No. 275 is assumed to be .001 9504, and for No. 277 to be .001 3761. From the table it will be observed that the values of s between the several piezo- meters range for No. 275 between .001 50 and .003 36, and for No. 277 between .001 05 and .001 69. These variations are altogether too great to arise from possible variations in wand A; in fact, a critical comparison of the piezometric errors in the four experi- ments shows that these errors must be attributed chiefly to incorrect readings, or rather to erroneous indicated heights in the manometers. These errors, however, are not cumulative, as the maximum deviation of the indicated piezometric heads from the straight line of most probable inclination is, for No. 275 .82 foot, and for No. 277 .39 foot. We can hence conclude that, while for comparatively short lengths such as 1000 and 2000 feet, the piezometric heights are sometimes very inaccurate and mis- leading, still for the total lengths of 25 000 to 30000 feet they can be relied upon as not being far from the truth. As the readings at the various piezometers were not made at the same moment, that fact may in part account for the defective results. In the following table the final results of these four experiments are given in English measures; for No. 278 s is assumed to be .000 5936 instead of .000 5915 as stated by Dr. Lampe. For No. 275 Twas about 47°. TABLE LXXXI. Lampe.—Danzg Conduit Pipe. Cast-Iron, coated with Smooth Varnish, and free from Rust. D =1.373 v=n (rs)4 3 | oa t h 8 n October, 1869. 4.575 | 3,090 | 275 26 862 | 52.393 |.001 950 |119.4 : | 276. March, 1871. | 4.011 2.709 | 31719 | 51.692 001 630 | 114.6 277 October 8, 1870. “30r1 | 2.479 | 31719 | #3.650 001 376 | 114.1 278 » 1, 4 '2.334 1.577 | 25414 | 15.085 |000 5936] 110.5 . | Nirkivood, Mr. James P. Kirkwood} describes two experiments made at his instance, the first under the direction of Gen. George S. Greene, with a main in the City of New York and the second by a subordinate of Mr. Kirkwood, with a main belonging to the Water Works of Jersey City, U.S.A. Both pipes were cast-iron mains connecting reservoirs, * Strictly speaking this ‘‘ straight” line, is a curved ‘ level” line, nearly following the general curvature of the earth’s surface ; the distances doubtless being measured along the inclined line of the pipe. { The Brooklyn Water Works. Van Nostrand, N.Y. 1867. PIPES.—Experiments. Kivhavood. 235 and in both cases g was determined by the lowering of the surface of the water in the upper or feeding reservoir. The Croton (New York) main is said to have been “heavily tuberculated ” ; its diameter is given at 3 feet, being doubtless its original founder’s size; it had three quarter-circle curves, each of 90 feet radius ; losses of head due to primary contraction and to imparting velocity will be disregarded. The Jersey City main was laid in 1858; the date of the experiment is not given, but apparently was in 1860 or perhaps in 1859 ; its general diameter was 20 inches, but for 128 feet in length the diameter was 24 inches; this enlargement will be assumed to counterbalance losses of head due to contraction and to imparting velocity ; the water passing through the pipe came from the Passaic River. Six determinations are given, which probably formed one series, as the respective heads consecutively decrease from 30.262 to 26.325; the maximum value of 7 is 74.1, with h = 29.758 and v=1.51; the minimum value of 7 is 71.1, with h=27.302 and v=1.39. We will take the mean values of the six determinations, as given by Mr. Kirkwood. TABLE LXXXILI Kirkwood.—Croton and Jersey City Cast-Iron Mains. No. 279 much inerusted ; Condition of No. 280 not stated. ven (r s)% No. Name. | D W | u | / | h | s n | | } 279 Croton. Eg 21.204 | 3.000 | 11217 20.215 001 802} 81.6 280 Jersey City. 1.667 3.137 | 1.438 | 29715 28.128 ».000947) 72 y Uity ’ | . ! | | Mr. Kirkwood does not sufficiently describe the details of these experiments to enable one to judge of the degree of accuracy obtained in making them. If the Jersey City pipe had only been 1 or 2 years in use, as seems to have been the case, the very low value of 72.4 deduced for » seems altogether improbable. Rochester Main. Mr. J. Nelson Tubbs, Chief Engineer of the Rochester Water Works, New York, gives the following data in regard to the Hemlock Lake conduit-pipe bringing water by gravity to the town of Rochester.* The upper portion of the conduit is 3 feet in diameter, is 50 776 feet long, and has a fall in this distance of 27 feet below mean surface of lake ; this section is of riveted wrought-iron, 33; inch thick. The lower portion of the conduit is 2 feet in diameter, is 51 495 feet long, and has a fall of 116.72 feet to its outlet at the storage-reservoir ; of this section, 35772 feet is of cast-iron, and the remaining 15723 feet of riveted wrought-iron from ,3; to 1 inch in thickness. The flow through this pipe, presumably * Annual Report of the Executive Board, City of Rochester, N.Y., for the year 1876. Rochester, 1877. 236 PIPES.—EXPERIMENTS. Rochester Main. soon after it was laid, was measured by noting the rise of the water in the lower reservoir for a period of 8 hours. The banks of the reservoir were new, and it was hence thought that the measured volume, or g, was somewhat underestimated, owing to absorption in the fresh-made banks. The flow was found to be at the rate of 9 292 800 gallons in 24 hours, or, say Q = 14.378. Disregarding the small losses of head due to bends, contraction at entrance, and imparting velocity, we have a total head 27.00 + 116.72 = 143.72 ; assuming that 1 is a % constant in v=n (") for both sections of the pipe, the frictional head, h, for the lower section will be 7.701 times that for the upper section ; hence, TABLE LXXXIJIL _ oS, Nelson Tubbs.—Rochester Compound Pipe. v=n (7) . ; m being constant. | Wo. BR, @ | # 1 h | a 2.0034 50776 | 16.517 w ! | i302 |) 4.877 | 51495 | 127.203 nc | bore is Rosemary Pipe. In the Transactions of the Am. Soe. of C.E., January, 1885, Mr. F. P. Stearns has been kind enough to give at our request a description of some very carefully con- ducted experiments of the flow through a cast-iron conduit pipe, having a diameter of 4 feet and a length of about 1800 feet; this pipe forming a portion of the Sudbury conduit line for the water supply of the City of Boston, U.S.A., and being known as the Rosemary pipe. The pipe had been laid three years before these experiments ; it had originally been coated with a preparation of coal-tar and asphaltum, and at the time of these experiments its interior surface appeared to be very nearly in as good order as when the pipe was first laid. @ was determined by the flow over a weir, ten miles distant from the pipe, and an additional allowance was made for the infiltration into the conduit between the measuring weir and the }ipe; the co-efficients of discharge for this weir had been determined with great accuracy, so that for these experiments ( can be considered to be given with sufficient exactness. The head lost by friction, et cet., was determined by two piezometric columns of water placed near the extremities of the pipe; the piezometric tubes were small smooth brass tubes, laid in the conduit pipe, and resting immediately upon its bottom; the ends of these tubes were plugged, and holes drilled through their upper sides, near their ends; these holes were true and cylindrical, and at right angles to the axis of the conduit. The relative elevations of the zeros on the two piezometric scales were very PIPES.— EXPERIMENTS. Rosemary Pipe. 237 accurately determined. Much care was taken to prevent accumulations of air in the piezometric tubes. The pipe was of nearly uniform diameter, sv that the given mean area was within a small fraction of the truth. TABLE LXXXIV. EF. P. Stearns.—Rosemary Cast-Iron Pipe, coated with Asphalt and free from Rust. D=4.00 v=n (rs)% T =about 38° No. Q v l h 8 n 982 32.867 | 2.6155 ( 5557 |.0003180 | 146.7 283 46.972 | 3.738 rrazg 4) 243 [0007115 | 140.1 Id , 62.391 4,965 | mT" 1! 9133 |.001 921 142.1 285 77.852 | 6.195 | 3.230 |.001849 | 144.1 Tt will be noticed that 1 for No. 282 has an abnormally large value compared with the three succeeding experiments. No reason is known for this palpable error, which apparently amounts to at least 5 per cent. in the value of 1; it is probably due to the inherent defects of piezometric measurements in the determination of h, which in this case appears to be about 10 or 12 per cent. in error, if Nos. 283, 284 and 285 are assumed to be correct. Three experiments were made with the same pipe, the heads being measured by the difference in elevation of water at the two ends of the pipe; in @ and b the pipe was not full at the inlet end. Making no corrections for losses of head in imparting velocity, et cet., these experiments gave the following values for 1’. No. a ve = 2.497 wv = 130.2 No. 6 v = 3.121 n’ = 130.6 No. ¢ wo = 4,437 n’ = 136.1 Correcting No. ¢ for loss in imparting velocity, n would have a value of 148.5. In these three last experiments the attending circumstances were not favourable for great exactness. Hamilton Snith, Jun. The conduit pipe of the Spring Valley Mining Company, at Cherokee, Butte County, California, was laid in the year 1871, and at this time was the most daring work of the It is an inverted siphon, having a length of about 24 miles, an approximate diameter of 30 inches, and has a maximum depression of 887 feet below the hydraulic-grade line. It is made of double-riveted sheet-iron, which is subjected to a constant maximum tensile strain of 17 549 lbs. per square inch.* kind ever constructed. * This pipe is described at some length in Vol. VI. of the Trans. Am. Inst. of Mining Engineers, in a paper by Aug. J. Bowie, Jun., Esq., “Hydraulic Mining in California.” See also Vol. XIII, p. 35, Trans. Am. Soe. of C.E., February, 1884, paper by the author on ‘JV ater Power with High Pressirvs,” 238 PIPES.—Expreriments. Hamilton Smith, Jv. The length and head are taken from surveys made by engineers of the Spring Valley Co.; the author cannot state whether or not they were determined with proper accuracy. The given diameter of 2.43 was directly measured by the author at the only uncovered part of the pipe; it is probably slightly less than the mean diameter for the entire length. @ was determined by the superintendent of the company by the flow through standard orifices; three tests were made, under different circumstances, giving following results ; Ist 2 se wie Re .. GO = 51.54 2nd oes es si sat we QQ = 48.52 3rd wae se oa ae we Q@ = AQ85 Mean of Ist and 2nd tests “Ga a. @Y = 50.0 It is most likely that the adopted value of @ is somewhat too low, as in calculating the discharge through the orifices, no correction was made for partial suppression of bottom contraction. The pipe had been 5 years in use ; its interior surface at the time of the experiment was very smooth, with the exception of the rivet-heads, which for over half its length formed noteworthy obstructions. The pipe had originally been coated by a bath in boiling asphaltum and coal tar. TABLE LXXXV. Cherokee Wrought-Iron Conduit Pipe; 5 Years in Use. v=n (rs)4% | | Velocity of | No » D Q | v Z H | 8 ” Stones sent | | through Pipe. | | 2 ee ile 5 ; = | | 286 | 2.43 50.0 10.78 12798 | 150.0 | OLL 72 127.8 9.0 The inlet end of the pipe is funnel-shaped for some little distance, so that no cor- rection need be applied for contraction or imparting velocity. Stones, weighing as much as 25 Ibs. each, were sent through the pipe, witha velocity of 9.0; their velocity was doubtless much retarded by striking the rivet heads, which for over half the length of the pipe were of considerable size. In Chapter X. will be found a detailed account of 53 experiments made by the author at New Almaden, California. It will only be necessary here to summarize their results. PIPES.—EXpEeRiMENtTs. Hamilton Snrith, Jr. 239 TABLE LXXXVI. Hamilton Smith, Jun. 1877.—-Small Pipes of Various Kinds. ee eae n D h\¥ ; TH=57 to 68 WEE 9 gt “r= 4 (4 i) = (7 s)4 i | IL an IV. New wrought-iron, un- Same pipe as No. 1, with | Same pipe as No. 1, coated | First two joiuts pipe coated, no funnel, funnel-shaped mouth- with asphalt; with No. 1, coated with piece. | funnel. asphalt; with funnel. 7=60.172 D=.0878 | 7= 60.247 D=.0878 | 1=60.264 D=.0873 | /=16.685 D=.0876 No’ st | n No. 8 | v | a No.| s v nw ‘No. “8 mo et ae ee eee ee Suc Bs, oe Slee, . algteees ese, ee bs x 287 ae 99) 5.325 | 100.9 | 294.130 95 5.387 | 100.8 | 301 L130 64 5.443 101.9 | 305 '.230 92 6.882 | 96.8 288 ee 70 4.673 | 99.4 || 295.102 031 4.708 | 99.5 | 302 :108 38 4.761 100.2 306 .159 Bi 5.621 95.0 289 074 30 3.948 | 97.8 || 296 |.076 1513.999 | 97.8 | 303 052 19 3.224 95.5 307 .076 903.775 92.0 290 050 oy 3.175 | 95.5 |2971.051 57) 3.226 | 95.9 | 304 .026 93, 2.220 ' 91.6 | 308 .032 me 2.333 | 88.0 2911025 762.154 | 90.6 | 298|.026 412.199 | 91.3 | ! 7 | ! 292 012 27/ 1.421 | 86.6 | 299 L012 471.435 | 86.8 | 293 .007 50, .958 | 74.7 | 800.007 85) 1.052 | 80.1 | | . | | | \ | a Ste rte ea eas = _ Se 9) int, para reat Vv. VI. VIL. VIII. Old wrought-iron, un- New wrought-iron, un- | New glass. New glass. << ia se Bi coated. No funnel. | Funnel mouth-piece. No funnel. Funnel mouth-piece. 1=60.247 D=.0853 || l=60.127 D=.0523 | 1= 63.902 D=.0764 | 1=34.941 D=.0622 No.| s j v + m No} s 9 vw + mn» JNo} ss v n | No. veel ee ets WH Mies xe eaeecite es 309 133 12) 4.266 | 80.1 || 316 13322 3.878 92.9 |322 129 18 5.009 | 100.8 387 132: 310 105 10| 3.773 | 79.7 317 105 291 3.885 | 91.2 || 323 102.06 4.383 | 99.3 | 328 094 311 '.077 66| 3.211 | 78.9 1318 077 84.2.863 89.7 | 324.075 30/3.685 | 97.2 | 329.05 312 052 57/9.619 | 78.2 || 319 L052 99 2.295 | 87.2 ' 325 |,05077) 2.945 | 94.6 |/330/01 313 |.027 061.829 , 76.1 | 320 02749 1.578 83.2 || 326 1.025 01/ 1.955 | 89.5 | 314 |o12 56| 1.195 | 73.0 | 321 lo1s.12 1.029 | 78.6 | 315 .007 65 .910 | 71.2 | | | | | : a ‘ v7 251 4.373 | 96.3 585) 3.666 | 95.0 $26 2.652 | 91.3 7 97'1.398 | 83.6 240 PIPES.—ExperiMEnts. Hamilton Smith, J. TABLE LXXXVI.—continned. New glass. New wood (bored). No funnel. No funnel. 7=11.127 D=.0418 1= 62.05 D=.1052 No. | 8 v | n No. | s v n 331 | .230 88 4.439 | 90.4 335 | 13115 ! 3.986 | 67.9 (332.165 90 | 3.659 87.9 || 336 .103 06 i 3.519) 67.6 333 | 098 41 2.719 | 84.8 | 337 .07610 3.008) 67.2 334 .064 35 12.077 | 80.1 338 050 94 , 2.469) 67.5 | a7 02419 | 1.653) 65.5 | | Nos. 331 to 334, with the small glass pipe, Series IX., are the least reliable of the foregoing experiments, its mean section not having been as accurately determined as was the case with the other pipes. By plotting the several series, with v and n as co-ordinates, it will be found that very smooth curves are formed, No. 338 being the only experiment of the 53 which is notably irregular ; x for No. 338 should probably be 67.0, instead of 67.5, wide note in Table XCVII. These experiments can be con- sidered entirely reliable so far as the experimental data is concerned ; the danger of vw error in them is the doubt as to whether or not Soo correctly represents H—A. The following experiments were all with riveted sheet-iron pipes, and will be described in detail in Chapter X. The condition of surface was about the same for Nos. 340 to 355 inclusive, and can be considered quite smooth; the rivet heads for half the length of No. 356, Texas Creek pipe, formed noteworthy obstructions ; the curves in all these pipes somewhat retarded the flow, as did also irregularities at the joints. The danger of experimental error in these experiments, consists chiefly in the value of @, which was measured by the flow over weirs or through orifices ; in this regard No. 355, or the Humbug pipe, is the least trustworthy of the experiments. In all of them, except No. 356, h’ is a considerable fraction of H, and the deduced values of h may therefore possibly be appreciably in error. PIPES.—EX PERIMENTS, h=H- ” we TABLE LXXXVII. Hamalton Smith, Jr.—Riveted Sheet-Iron Pipes, Smooth Interior Surfaces, except Rivet Heads and Joints. 290° — f(D 4 ven (G ; =n(rs) T for Nos. 340 to 354 about 55°; for No. 356 from 50° to 60°. Hamilton Smith, Jr. 241 | | Maximum Velocity of Stones or Wooden ’ fol aie; o Q ; : a h y | 4 ” | Blocks sent through | | Pipe. 340 9105 | 6.525) 10.021 684.8 | 24.220) 1.562 ! 22.658] .033 09 | 115.5 9.42 341 911 5.644] 8.659 697.0 | 19.005) 1,166 | 17.839) .025 59 | 113.4 342 911 4,515] 6.927 713.9 | 12.850) .746 | 12.104) .016 95 | 111.5 5.79 343 911 3.972| 6.094 721.3 | 10.200, .578 | 9.622} .013 34 | 110.6 344 911 3.071] 4.712 730.6 | 6.555} .345 | 6.210) .00850 | 107.1 345 | 1.056 9.376) 10.706 684.9 | 24.510) 1.783 22.727) .083.18 | 114.4 346 | 1.056 7.572] 8.646 699.6 | 16.690) 1.163 | 15.527) .02219 | 113.0 7.4 (1) 347 | 1.056 6.097; 6.962 709.2 | 10.885) .754 | 10.131, .014 28 | 113.4 348 | 1.056 4.024) 4.595 718.4 | 5.130 .328 | 4.802) .006 68 | 109.4 349 | 1.230 (14.365) 12.090 684.4 | 24.390) 2.274 | 22.116 .032 31 | 121.3 11.85 350 | 1.230 12.587] 10.593 695.6 | 18.925 1.746 | 17.179) .024 70 | 121.6 9.58 351 | 1.230 10.054) 8.462 705.0 | 12.715} 1.114 | 11.601) .016 46 | 119.0 352 | 1.230 8.690) 7.314 710.7 9.550) .832 | 8.718) .012 27 | 119.1 353 | 1.230 8.128] 6.841 712.4 | 8.545) .728 | 7.817 .01097 | 117.8 354 | 1.229 5.199] 4.383 719.9 | 3.915) .299 | 3.616, .005 02 ; 111.6 355 | 2.154 45.92 | 12.605 | 1193.8 | 22.067| 2.471 | 19.596) 01641 | 134.1 11.24 356 | 1.416 31.721] 20.148 | 4438.7 [303.62|7.46 /296.16| .066 72 | 131.1 20.94 It is probable that for No. 349 is somewhat too low, and also that n for No. 350 is slightly too low; for Nos. 345 and 346 is known to be too low, owing to the stoppage of a stone in the pipe. @Q, and hence v and n, for No. 355 is probably too high, as will be seen by reference to Chapter X. Clarke. Mr. Eliot C. Clarke, in his account of the “Main Drainage Works of the City of Boston, Mass., Second Ed., 1886,” describes a number of experiments made as to the flow through the main sewage tunnel, lately built by the City of Boston, from the main-land, under Dorchester Bay, to Squantum Neck; the tunnel is placed at an Il 242 PIPES.—EXPERIMENTS. Clarke. average elevation of 142 feet below low tide. This tunnel has a circular section, with a diameter of 7.5 feet, the enclosing walls being of hard brick. It is an “inverted siphon,” with a total length of 7166 feet, through which the sewage flow from the city is all discharged ; the flow is caused by gravity; the sewage is pumped from a lower level to a reservoir at the inlet of this tunnel-pipe ; the elevation of the surface in this reservoir or pen-stock is sufficient to produce the flow by gravity through the tunnel, and through an additional conduit, to the discharging reservoirs on Moon Island. The pumps were started in January, 1884, and have been continuously in operation, except occasional stoppages of a few hours, up to the present time—February, 1886. The ordinary velocity through the tunnel is seldom greater than one foot, and often less than one-half foot. The tunnel can at any time be flushed by running the pumping machinery to its full capacity; the first flushing was done on June 12, 1884, which removed a considerable deposit of soft mud, horse manure, et cet.; since that date the tunnel has been flushed at regular intervals of about two weeks. The extra supply for this flushing is obtained by pumping salt water, which is added to the ordinary sewage flow. The following table shows the results of 7 experiments; Q for the first five was ascertained by the registered strokes of the several pump-pistons, with an allowance for “slip,” which was determined a few days before Experiment No. 361 by actual measure- ment in a reservoir ; for the last two experiments ( was obtained by measurement of g in a reservoir; these determinations of qg probably were within 1 per cent. of the truth: for the first 4 experiments a correction was made to the observed head, for loss of head at entrance; in the other 3 experiments the head was measured a short distance below the mouth of the tunnel, and no such correction was required. The tunnel has one quarter-turn of 9.75 radius, and one angle of 234°; the outlet end of the tunnel is divergent, which probably compensates for these two bends. TABLE LXAXVIII. Clarke.—Flaw through Dorchester Bay Tunnel-Pipe. 1=7166.; D=7.5, and r=1.875; (@ for Nos. 357 and 358 diminished considerably by sewage deposits. ) A=hard brick, moditied more or less by sewage-slime. No. h 8 Q v n Date. Character of Flow. 357 .520 | .0000726 | 41.06 929 80. | June 6, 1884 | Sewage. 358 566 | .0000790 | 44.08 .998 82. i, SOR ay ss 359 3.648 | .0005091 | 176.24 | 3.989 | 129. go bee Me About } sewage and } salt water. 360 .297 | 0000414 | 42.64 .965 | 109, un LB y- 5 Sewage. 361 4.165 | .0005812 | 173.56 | 3.929 | 119, | Oct.20, ,, | About } sewage and 3 salt water 362 3.975 | 0005547 | 167.78 | 3.798 | 118. Aug. 28, 1885 aa. Bw rm 363 3.680 | .000 5135 | 166.51 | 3.769 | 121. | Sep. 25, ,, oe mod 4 PIPES.—Experiments. Clarke. 243 The last three experiments—361, 362 and 363—are the most trustworthy of the series. Some experiments were also made of the flow through the conduit, which receives its supply from the outlet end of the Dorchester Bay tunnel ; this conduit is a square tight wooden flume or pipe 6.x 6., about one mile in length, and is made of planed plank placed lengthwise ; this, of course, gives a lower value of A than would have been the case with plank placed crosswise. The experimental section of the flume was 2486.5 feet in length, and straight. When the sewage in the discharging basins or reservoirs on Moon Island has a low level, the conduit is about one-half filled, but when the basins are nearly full the conduit is completely filled, and becomes a pipe. The ordinary velocity in the conduit is about 3 feet. From its bottom to the ordinary flow line, the sides are covered with a slimy deposit from 4 to 4 of an inch in thickness; above this line, on the sides and top, there is some slime, but not so much as_ below. This deposit is not removed by flushing, although when flushing is being done, the velocity at the lower end of the conduit is 7 feet. The following results were obtained in October, 1884, Q having been determined with approximate accuracy by the strokes of the pump-pistons; r v n Conduit or Pipe. Character of Flow. | | | 1.45 2.94 117. Conduit half-filled. Sewage. 1.41 2.87 117. 63 yi. BH 5 1.5 4.8 135. A full, and hencea pipe.| 1 sewage, and 3 salt water. | It would probably be dangerous to draw any general conclusions from the fore- going experiments, as in some of them A and « were not constant, and in others the value of @, and hence v, was only approximative. Other Experiments. Many other experiments given by French, German, English and American authorities have been carefully examined, but are regarded as too doubtful in point of accuracy to be of value. In many of them the inclination was measured by Bourdon gauges ; with ordinary values of s and /, such determinations of h are too inaccurate to give anything but very rough approximations of the true values of s. 244 PIPES. —h’. h’. 2 y) Whether or not the expression h’= = 3 correctly represents the losses of head due to contraction at the entrance and to imparting velocity to the water, can be deter- mined in two ways : First.—By having pipes with « and A constant, and with / and s variable. We will assume, and we think with entire safety,* that s should not be regarded as a factor directly influencing the value of n, but that v—which is the result of s—is the factor to be taken into consideration. Now, with two pipes having « and A constant, / considerably different, and H for each pipe of a value to make v identical, it is apparent qe 2H a % that if, by the use of the expression v =n ( “ 7 , m has similar values, the ex- pression is correct ; and conversely, if n differs beyond the limit of probable experimental errors the expression is inaccurate. Second.—By the use of piezometric columns along the course of a pipe, whereby / can be directly determined for a given part of the length; h being ascertained for a portion of the length, if the total head Hand the total length 7’ be known, the value of h’ can be easily calculated. Direct Method of Proof. The Bossut experiments, Table LX XII., afford fairly accurate and complete data in this regard. By reference to Plate XII., where they are plotted with » and nas co-ordinates, it will be seen that they present quite uniform experimental curves, the curve for the larger diameter of .178 being somewhat higher than for the one with D=.118. Now, in these experiments / varied from 32 to 192, and the several correc- tions of h/ applied to H are proportionately widely apart. Slight probable variations in a and A, aside from probable errors in H and Q, will fully account for the irregularities in these curves. Hence we may say that the Bossut experiments confirm the general accuracy of our expression /’ = a oF The three experiments of Bossut given in Table LXXIIL., when the inclination of the pipe proper was constant, also warrant this deduction. Plotting Experiments Nos. 36 to 52 inclusive of Dubuat on the same sheet, we see an experimental curve much more irregular than those of Bossut, No. 37 apparently differing 12 per cent. from the most probable curve for the series. So far as the effect of h’ is concerned, it will be observed that Nos. 38 and 48, where h’ is a large fraction of * The only experimental data, possessing any moderate claims to accuracy, which contradict this assumption, are the Darcy-Bazin series, Nos. XXVIII. to XXXI. inclusive, with very small conduits. We have shown in the preceding chapter that these seeming contradictions can be fully accounted for by probable experimental errors. PIPES.—h’. Direct Method of Proof. 245 H, agree quite closely with the curve formed by the other experimental points, when h’ was a much smaller fraction of H. Therefore it can be said that the Dubuat series proves that h’/= 2 v 290° 1s approxi- mately correct.* On the same sheet we will now plot Series I., II, III. and IV., of the New Almaden experiments made by the author, and given in Table LX XXVI. For I.; /=60.2; 4=new wrought-iron; end of pipe flush with inlet tank. [i 1] 66.2 a= 2 . ; funnel-shaped mouth-piece. IIL. ; 7=60.3 ; A=coating of asphalt ; a a 2 » LV.; 7=16.7, and A and mouth-piece as in Series IIT. The diameters varied from .0873 to .0878. The inlet end of the pipe for Series I. was a little rough, and o was assumed at .80; for the funnel-shaped mouth-piece used for the other series, 0 was placed at .98. The uncoated pipe, Series I. and IL, was new and free from rust; hence the varnish applied, in Series ITI. and IV., very slightly diminished the value of A. By reference to Plate XIT., it will be observed that each of these four experimental curves is almost perfectly symmetrical ; this proves that the experimental values of H and @ were accurately determined. Contrasting the curves for Series I. and II., where a was the same, they will be seen to be practically identical; this shows that the cor- rections for contraction for Series I. are the proper ones, or more exactly, that in ats ss the ise af thatactor ae: gives correct results. 29 @? 0” Comparing Series IT. and III, it will be seen that for the higher velocities the curve for III. is slightly the higher; this was to be expected, as for III. A hada slightly lower value. Comparing Series III. and IV., a notable variation will be observed, increasing with v, the value of n being 7 per cent. lower for IV. than for ITI., with v=5.5. In both series the effect of o was nearly eliminated, the value of h’ being nearly altogether vw f due to the head required to impart velocity, or 2g ; the relative correction of h’, or 7k was nearly three times greater for Series IV. than for Series III. This seems to prove 2 that =, does not sufficiently represent the loss of head absorbed in imparting velocity. 32 2) hi There appears to be no good reason to doubt the accuracy of the experiments con- stituting Series IV.t Of the experimental elements, 7, @ and H can be considered free from possible error of consequence. Owing to the smaller quantity of water con- tained in this pipe, a cannot be considered to have been quite as accurately ascertained * Nos. 53 to 59 of Dubuat are not used, as they present anomalous and contradictory results. + The accuracy of the experimental data for Series III. is absolutely proved, by the agreement of its curve with the curves for Series I. and II., and also by the curve for Series VII., with a glass pipe of nearly similar dimensions. 246 PIPES—h’. Direct Method of Proof. as in Series III.—the diameter and area of both pipes having been deduced from the weight of water in them. It is possible that A may have been higher for IV. than for III. Taking into consideration all these chances of variation or error, it is not probable that they could have caused the marked difference between the two curves. The relative corrections of h’, for the New Almaden Series VII., VIII. and IX., with glass pipes, were considerably different, as will be seen by reference to Table XCVII. Examining Plate XIII., where the curves for these series are plotted, it will be noticed that the curves indicate no serious error due to incorrect assumptions of the value of h’. The values of x are largest for the largest diameter, and least for the least diameter; the differences between the curves can be accounted for by 2 the variation in r. The relative differences of the correction aa ere however, for these three pipes considerably less than for Series III. and IV., and hence they are not entitled to as much weight, so far as the accurate determination of the true value of fh’ is concerned. Taking all the foregoing facts into consideration, it can be assumed that the New Almaden experiments indicate that the loss of head absorbed in imparting velocity 2 is somewhat greater than > g Unfortunately, these experiments were not reduced until a year or more after they were made, and it was then impracticable to repeat them, and to further investigate the proper value which should be given to 1’. Indirect Method of Proof. Darcy’s experiments should afford data from which the exact value of h’ can be deduced. It will be shown in the next section of this chapter, that a piezometric column should truly indicate the hydraulic head in a pipe or conduit, when the mouth of the orifice is exactly flush with the side of the conduit, and its axis normal with the axis of the conduit. M. Darcy states that his piezometric tubes were thus attached to his ex- perimental conduit pipes, and hence the heights of his piezometric columns should represent the true head at each of his 5 piezometers.* The head indicated by his piezometer No. 5 should represent the effective height of the water at the inlet. Supposing the regimen of flow to have been fully established at piezometer No. 3, the difference of height between columns Nos. 5 and 3 should indicate the losses of head due to contraction and impartation of velocity, and to * An account of the arrangement of M. Darcy’s piezometers, with a description of his methods of experimentation, will be found in the preceding section of this chapter. PIPES.—h’. Indirect Method of Proof. 247 “friction,” between No. 3 and the inlet end of the pipe. The loss due to friction can be segregated, by assuming it to be in the direct proportion of this length—No. 3 to the inlet end—to the length between Nos. 1 and 3; the loss of head, due to friction alone, between Nos. 1 and 3 is given by the respective heights of these two piezometric columns. This supposition involves the assumption that a and A are the same for the length 1 to 3 as they are for the length 3 to the inlet ; any variations in a and A are probably not large enough to produce serious error. Unfortunately M. Darcy does not give the height of the surface of the water in his outlet tank. Were this height known, we would have a reliable check upon the accuracy of his piezometric readings. We will select from the Darcy Series, Nos. I., II. and III., with small wrought- iron new pipes, and Nos. XVI., XVIT., XVIII, XIX. and XX., with cast-iron pipes of larger size. Nos. VII., VIII, [X. and X., with sheet-iron pipes covered with bitumen, present very discordant results, and will not be used; in fact, M. Darcy rejects Series IX. and X., as he conjectures that small pieces of bitumen may have interfered with the flow in the piezometers nearest the inlet. Piezometer No. 5 was attached to the inlet tank or cylinder. The positions of the other 4 piezometers, for the eight selected series, were as follows, the given dis- tances being in metres ; Distances in Metres. No. of Series. From Outlet End From From | From | From to Piezometer Piezometer Piezometer | Piezometer | Piezometer No. 4 No. 1. No. 1 to No. 2.’ No. 2 to No. 3.| No. 3 to No. 4. to Inlet End. L. 8.13 50. 50. 5.883 | .167 II. 8.35 50. 50. 4.805 30 III. 8.40 50. 50. 4.67 29 XVI. 6.59 50. 50. 4.716 2S XVIL 6.297 50. 50. 473 | 45 XVIII. 6.78 50. 50. 4.293 30 XIX. 6.912 50. 50. 4.17 276 XX. 6.912 50. 50. 4.17 276 The pipes all appear to have had a square and even connection with the inlet tank : hence o will in all cases be assumed to have a value of .82. In the following table : The first column gives our number of the experiment. The second column ; the mean velocity. 248 PIPES.—W’. Indirect Method of Proof. The third column ; the difference of height of piezometric columns Nos. 1 and 2. The fourth column ; the difference between piezometers Nos. 2 and 3. The fifth column; the difference between piezometers Nos. 1 and 3, which we assume to be the measure of the frictional loss of head. The sixth column; the difference between piezometers Nos. 3 and 4. The seventh column ; the difference between piezometers Nos. 4 and 5. The eighth column ; the difference between piezometers Nos. 3 and 5, being the total loss of head between No. 3 and the inlet end of the pipe. | The ninth column ; the frictional loss of head between No. 3 and the inlet, obtained by dividing the loss of head indicated in the fifth column, by the distance between piezometers Nos. 1 and 3—in all cases 100 metres—and multiplying the dividend by the distance from piezometer No. 3 to the inlet end of the particular pipe. The tenth column; the difference between the total loss of head between piezometer No. 3 and the inlet, and the frictional loss as given in the ninth column; this difference should represent the tiwe value of h’. The eleventh column; the theoretical value of h’, obtained by the expression h’= 2 290° The twelfth column; the difference, or lack of agreement, between the true and the theoretical values of h’. All these measures are given in metres. PIPES. —h’. Indirect Method of Proof. 249 TABLE LXXXIX. Values of hi’ deduced from the Darcy Experiments. 2 9=19.62 0=.82 All values given in metres, Series I.—New Wrought-Iron. D=.0122 U=11418 Maximum oscillation .005 at piezometer No. 5. 2 3 4 5 6 , 8 9 | 10 ll 12 | h’ v | 1—2 23 1—3 3—4 4—5 3—5 hy 2 Error. | True. Igo 65 .034 -043 042 .085 .004 .007 O11 .005 .006 + .006 66 072 | .092 .092 184 .009 004 .013 O11 .002 +.002 67 AT 154 .150 304 .018 .006 024 018 .006) .001 +.005 68 147 P26, ot 533 027 -005 032 032 .000) .002 — .002 69 169 , 378 376 154 .038 .006 044 .046 |—.002| .002 — .004 70 230 © 875 .784 1.659 .070 .006 076 .100 |;—.024; .004 | —.028 71 | .287 | 1.365 | 1.215 | 2580 | .120 |-.006 114 | 136 |=.042! .006 | —.o48 72 .343 ) 1.793 1.679 3.472 154 .009 163 || .210 |-.047] .009 — .056 73 392 , 9.253 2.146 4.399 -200 .010 210 .266 }-.056]} .012 — .068 T4 78 3.207 3.057 6.264 289 .010 299 | 379 1-.080| .017 — .097 75 | 573] 4398 | 4296 | @554 | 76 | .846 | 8.900 | 8.962 | 17.862 | .832 | .024 856 |/1.081 |-.225! .054 —.279 77 (|1.195 | 17.146 | 17.280 | 34.426 | 1.591 .102 1.693) 2.085 oe } -108 | — .498 eS | | 40.810 | 40.266 | 81.076 | | | | Series II.—New Wrought-Iron Pipe. D=.0266 UY =113.455 Maximum oscillation .04 at piezometer No. 78 .058 .018 O15 .033 0 -001 .001 | .002 ,—.001 0 : — .001 79 | 131 | 079 | 073 | 152 | .0055 | 0015 007 | 008 |-.001} 001 | ~.002 80 248 2015 22555 AST .020 | .008 028 | 10% .003 OOS — .002 81 | 368 | .520 | 495 | 1.015 | .046 | .018 | .059 | .052 007) .010 | -.003 g2 | 522 || .982 | 955 | 1.937 | .o87 | 027 | .114 | .099) 015] .021 | -.006 83 .667 1.583 1.543 3.126 140 | (049 189 3 .160 | 029 O34 — .005 84 .796 2.211 2137 4.348 -200 .067 267 | 222) 045 .048 | — .003 85 96] 3.184 3.132 6.316 .299 .093 392 22 070) .070 | O 86, 1.235 5.049 4.973 | 10.022 451 .216 667 |. .b12 155 116 | +.039 87 | 1.281 5.326 5.245 | 10.571 419 .231 .650 540 110| sf ~=.014 88 | 1.682 8.874 8.952 | 17.826 .768 oll 1.079 | 910 | .169 214 | —.045 89 | 1.998 || 12.790 | 12.811 | 25.601 | 1.155 -403 1.558 | 1.307 225] .803 | —.052 90 | 2.184 | 15.280 | 15.722 | 30.952 | 1.506 534 2.040 || 1.580 460) .362 | +.098 56.0975 | 56.2885 112.386 250 PIPES—h’. Indirect Method of Proof. TABLE LXXXIX.—continued. Series ITI].—New Wrought-Iron Pipe. D=.0395 U =113.36 Maximum oscillations .020 at piezometers Nos. 3, 4 and 5, i |e 3 4 | 5 |; 6 7 8 SP aby |) ae ae — = “ 1’ = | | 3 ; : No. ; @ « J-2 a 13: a defi 3—5 h, 2 | Error | | True. Bye | 98 | .063 | 012 | .010 | 022 «0 001; .001 |} .001 | 0 0 0 99 111 42" oB4 .078 .003 .002 | .005 .004 | .001 001 0) 100 .185 | .093 -089 182° .009 oe | oli | .009 | .002 .003 — .001 101 262 173 163, 336 O14 .010 oot | Gir | OOF -005 +.002 102 382 331 319 650 029 .016 045 .032 | .013 O11 +.002 103 559 | O54 632 1.286 061 .032 .093 .064 | .029 O24 +.005 104 788 | 1,217 | L1n2 2.389 | 117 .063 .180 -118 | .062 047 | 4.015 105) 915 1.59 Lass 3.123 - 083 93; | 165 .082 | 2063 | +,019 106 |1.095 | 2.913 | 2.155 4348 7 .213 | 120 © .883 | 216 | .117 | .091 | +.026 LO7 {1.920 | 6.277 | 6.038 | 12315 | 601 , .38t | .985 ] 611.374 | 279 | 4.095 | | 108 | 2.305 | 8.980 | 8.575 | 17.553 901 512 J.413 S71 542 403 +.139 | 109 | 2.597 | 11.427 ' 10.981 22.408 | 1.147 657 1.804 jitdd | .693 511, +.182 (32.011 31.679 64.690 | | | | Serics X V[.—New Cast-Iron. D=.0819 ¢’=111.586 Maximum oscillations .040 at piezometers Nos. 3, 4 and 5. = | \ 206 .088 | 010 010 | = .020 | 001 001 .002 001 | .001 | .001 0 207 71 : 045 | 058 O83; 005 005 .010 .004 | .006 | .002 _ +004 208 858 | 117 de 232 1 .007 014 021; .012 , .009 .010 | —.001 209 | .561 273 258 531 021 £035 056 |; .027 , .029 O24 | +.005 BO ereT | te | eo | we | 0d 07 11 || 051 | 059 | 047 | 4.012 211) | 1.185 1.145, i141) ° 2885 105 .130 | 235 113 | 122, .106 ! +.016 212 | 1.418 1.628 ! 158 + 3.208 115 2220 335 .160 | .175 152 +.023 913 | 1.571 | 2.0515, 199. 4.0415 145 28 | 425 202 7 223 eG | +.036 214 12453 | 4.721 4.826 9.547 i 288 .661 949 ATT) 472) 456 | +.016 215 | 2.487 5.034 4.870 9.904 | 446 .636 1.082 495 | 587 469 |) +.118 216 | 2.720 6.106 5S72 | 11.978 | .418 848 1.266 598 | .668 561 | +.107 ; | 217) | 3.238 8.503 8.304 | 16.807 | 665 LO%6 | 1741 | .840 ,901 795 | +.106 218 13.265, 8.646 | 8.426 | 17.072 | .641 | 1.106 | iar | 853 ' Soa! 808 > 4.086 a teal ee es I | | 38.7995 | 37.499 | 76.6985 | 4 PIPES. —h’. Indirect Method of Proof. 251 TABLE LXXXIX.—continated. Series X VIT.—New Cast-Iron. D= 137 =111.477 Maximum oscillations 0.30 at piezometers Nos. 3 and 5. 1 2 3 4 5 6 7 B 1 ® ) 304 1 12 ee |e -| —— a Ace Pa, a es Dyk ean I as eres | h’ No. | a ae ae oe ee ee a ee oe ae ee hy, ip Exror | | | Tries}. 5, iy) Wal De ees hy eiech | ee? lee 219 | 149 012 012 O24 001 003 . 00+ | 001 | .003 | .002 | +.001 220 ; .298 049 .038 .087 001 009 | .010 .004 | .006 .007 | —.001 221 5 1488 112 O97 .209 007 .020 | 027 O11 | .016 .018 — 002 222 763 | 251 224 AT5 O13 | 053 | 066 | .025 , .041 044 | —.003 225 | L279 .670 590 1.260 O47 155 .180 | .065 115 124 | —.009 934 |15TL | iis 1.045 | 2.295 | .080 240 | 320 | jis) 203! 29) || .618 225 | 2.098 1.758 1.560 3,318 130 345 | ATS 172 | 303 | 334 | —.031 226 | 2.281 2.065 1.840 3.905 A155 405 60 202. 48a8 | OOF — .036 227 | 3.640 | 5.162 4.690 9.852 B53 | 1.014 | 1.367 | .510 | AT | 1008 | —.147 228 | 4.693 8.787 7.969 | 16.756 £86 1.811 BOF .868 | 1.429) 1.669 | —.240 ; 20.046 | 18.065 | 38.111 | | Series X VIII.—New Cast-Iron. D=.188 Uv =111.373 Maximum oscillation .060 at piezometer No. 4. 229 205 O14 .013 O27 .005 .005 | .010 001 | .009 003. +.006 230 497 .085 090 175 | .010 Ol | .020 008 | .012 O19 ~ .007 231 758 188 .180 .368 | .017 O48 | .065 | 017 | .048 043 +.008 232 | 1.128 420 385 805 020 115 | 135 |, .0387 | .098 096 +.002 233 | 1.488 700 640 1.340 07 155 225 | .062 | .163 168 | — .005 234 | 1.933] 1.16 1.09 225 .10 25 38 103 | .277 283 | — .006 235 | 2.506] 1.955 1.855 3.81 ZL 46 65 | 175 | A475 4760 | —.001 236 | 4.323 } 5.704 5.276 | 10.980 675 1.338 2.013» 504 2.509 1.417 +.092 237 | 4,928] 7.593 7.068 | 14.591 .730 L774 2.504 I 670 1.834 L.s41 — .007 | 17.749 | 16.597 | 34.346 | | | 252 PIPES.—h’. Inilivect Method of Proof. TABLE LXXXIX.—continued. Series XIX.—Old Cast-Iron. D=.2432 Vv =111.358 Maximum oscillations .060 at piezometers Nos. 3, 4 and 5. 1 2 a 1 od 5 6 7 8 9 10 12 No. v 1—2 23 1—3 3—4 4-5 3—5 hg Error. True. 238 B07 | 049 | 045 094 | .005 010 .015 004 | .O11 +.004 239 452) 104. .098 | 202 .009 .018 O27 .009 | .018 +.003 240 707, 238 | 235 | 473 .020 .040 .060 021 | .039 +.001 241 | 1.106 585 565 1.150 .060 .085 145 051 | 094 — +.001 249 | 1.547 1.160 | 1.130 2.29 120 170 290 .102 ; .188 +.007 243 | 1.833 || 1.620 | 1.580 3.20 170 230 400 142 | .258 © + .003 244 | 2073 2.085 9.020 4.105 220 .300 520 82 | .338 +.013 245 | 3.833) 7.154 ° 6.827 ' 13.981 781 1.223 2,004 | .622 |1.582 + .268 | 12.995 , 12.500 | 25,495 | Serics XX.—No. XIX cleaned. D = 2447 ¢ =111 358 Maximum oscillations .12 at piezometers Nos. 3, 4 and 5. 246 278 027 025 052 .005 .005 01 .002 | .008 + .002 247 | 537 085 .080 .165 010 020 030 .007 | .023 +.001 248 | .949 253 245 498 03 065 .095 022 | .073 + .005 249 | 1.420 590 565 1.155 055 145 .20 .O51 | .149 — .004 250 | 1.904 1.035 1.00 2.035 12 255 B75 .090 | .285 +.010 251 | 2.206 1.385 1.35 2.735 .16 B82 48 122] .358 —.011 252 | 2.572 1.890 1.84 3.73 28 46 .68 166 | .514 +.013 253 | 4,497 5.838 | 5.505 | 11.343 701 1.428 2129 504 [1.625 +.092 11.103 | 10.610 | 21.713 An examination of the foregoing table shows: The heights indicated by piezometers Nos. 1, 2 and 8 are often discordant; sup- posing « and A to have different values in the two sections of the pipe (1-2 and 2-3) such variations would be constant for all the experiments of the particular series, and affect the readings of the three piezometric columns for each experiment the same way. Instead of this being the case, sometimes the indicated ioss of head between pilezometers Nos. 1 and 2 is greater, and sometimes less, than the indicated loss of head between Nos. 2 and 3. PIPES.—h’, Indirect Method of Proof. 253 The differences between the piezometric columns 3 and 4, and 4 and 5, also appear to be very contradictory. 2 Of the selected experiments, 44 indicate that ce is too small, while 37 indicate ‘ 2 that it is too great ; of the eight selected series, four generally indicate that a is toc Y oO small, and the other four that it 1s too great. Series I. is palpably in error, as the frictional loss of head between piezometer No. 3 and the inlet, is greater than the total 2 v =e cannot be too large a correction ; therefore, examining the + errors of the table alone, we see that they rarely form a considerable fraction of the “ true” value of 1.* indicated loss of head. It can be safely assumed’ that We can fairly deduce from the preceding observations, that the Darcy experiments 2, ts ee v : indicate that ban quite closely represents the true value of h’. Also, that our primary assumption, that his piezometers indicated the true head, is correct ; for were this not the case, and the piezometric columns Nos, 1 to 4 inclusive more or less lowered by the “sucking” action of the flowing stream—as M. Darcy surmises—their true heights would have been higher than the observed heights ; No. 5, being attached to the large feeding tank or cylinder, would not be nearly as much depressed ; hence the difference between the observed heights of Nos. 4 and 5 would be less than that given in our table ; = ae this would result in indicating that was in nearly all cases too large a correction. a 290 Such a conclusion, as before stated, we cannot admit. Conclusion. In the preceding data, the only proofs entitled to much weight, which show that 2 oe gives too small values for h’, are Series III. and IV. of the author's New Almaden experiments. These experiments were made with such care, that we feel disposed from them alone, to draw the final conclusion that, so far as loss of head 2 absorbed in the impartation of velocity is concerned, — does not sufficiently represent this loss. How much the expression is in error, we are not prepared to say, but pro- bably not very much. The head absorbed in the impartation of velocity to the flowing stream is in part, or in whole, represented by the vis-viva of the escaping jet from the outlet end of the * Not considering low velocities, where the probable errors of observation would fully account for any discre- pancy given. 254 PIPES. —h’. Conclusion. pipe. This vis-riva represents the force of each fillet of the jet, or, considering that the maximum velocity is in the axis of a circular pipe, and the minimum velocity by its walls, dividing the section into concentric rings, the mean of the sum of the squares of the velocities of the several rings will much more nearly indicate the true force, and will doubtless be slightly different from the square of the mean velocity. It is, however, not worth while to discuss this question from a theoretical point of view, as it can only be accurately and satisfactorily determined by experiments made with the proper care. A pipe, consisting of joints of drawn brass tubing, where « and A are almost exactly constant, should be employed. The inlet tank should be of rela- tively large dimensions ; a funnel-shaped mouth-piece should be attached, so that o will be nearly 1.. The surface height in the inlet tank should be carefully compared with the height of the centre of the outlet end, when the discharge is free, and with the surface of the water in the outlet tank, when the discharge is submerged; a sub- merged discharge will be required for low velocities. By varying the length of the pipe, the exact value of h’ can be readily deduced. To eliminate possible variations in a and A, for the short lengths the experiments should be repeated with different joints. Piezometers could also be attached to such a pipe, and the causes of their gcnerally erroneous indications be ascertained. Several series, with different diameters, should be made to determine whether or not variation in D affects h’. @ must be absolutely determined by measurement of g in a vessel of ample size. PigzoMeETERs. By attaching vertical tubes to an experimental conduit pipe, at points along its course after the regimen of flow has been established, theoretically the frictional loss of head, i, between two such attachments can be determined, by measuring the difference in elevation of the heights of the columns of water in the two tubes. These heights can be ascertained directly, by the use of vertical glass tubes ; and indirectly, either by columns of mercury in glass tubes, or by Bourdon pressure gauges. This method, at first sight, appears to be greatly preferable to the direct measure- ment of the total head H, where considerable corrections are sometimes necessary for the primary losses of head, due to contraction at the entrance, and to the impartation of velocity to the flowing stream. Induced by these seeming advantages of piezometric measurement, in late years nearly all experimenters have determined h by piezometric columns; the author being almost the only modern experimenter, who has preferred to deduce h from the observed value of £7. Dubuat assumed that the pressure on the interior walls of a pipe was diminished by the head generating the velocity of flow. Navier disputed this proposition, but, in general, writers upon Hydraulics during the last century have followed the theory of Dubuat. If this supposition be true, either in whole or in part, the indicated piezometric PIPES.—Pirzomerers. 255 heads, after the regimen of flow has been fully established, should indicate a less head than that due to the hydraulic head; in such case, however, the piezometric columns should indicate the hydraulic inclination for a pipe of uniform section, as the lowering effect upon each column would be identical. Darcy thought that the stream flowing through a pipe would have a “ sucking’ action as it passed by the mouth of a piezometric tube, and hence the height of the r column in the tube would be more or less lowered.* In the preceding section of this chapter, we have shown that such a deduction is hardly warranted by M. Darcy’s own experiments. Mr. Hiram F. Mills has lately very fully investigated the question of piezometric measurements applied to an open conduit, and has given a description of his experiments in the Proceedings of the American Academy of Arts and Sciences, Vol. VI., New Series, Boston, 1879. His experimental conduit was a straight trough of wood, of uniform section, 30 feet long, 1 foot deep, and 43,ths of a foot wide. This trouzh could be inclined at pleasure, so that velocities up to nearly 9 feet per second were obtained. At distances. of 2.5 feet apart, 9 openings, of various sizes and shapes, and in wood, brass and iron,. were pierced in each side of the conduit; each of these 18 openings was connected by a pipe with a small open tin reservoir or box, having a horizontal section of .9x.5. The height of the surface of the water in the centre of the conduit was observed, and simul- taneous observations made of the height of the surface in the two reservoirs opposite the central station. In all, some 6000 observations of surface heights were made. These experiments gave the following results : First.—When the mouth of the opening was flush with the side of the conduit, and the axis of the opening perpendicular (normal) to the side, the surface of the water in the piezometric reservoir had very nearly the same elevation as the surface of the water in the centre of the conduit. The maximum deviation of any single experiment is not given; the maximum mean deviation of any one series appears to have been +.0176, when the velocity, , in the conduit was 7.8. With high velocities, in general the water in the reservoir was slightly higher than in the conduit. Second.—When the mouth of the opening was flush with the side of the nas and the axis of the opening horizontally inclined with the line of the conduit, the water in the reservoir was somewhat higher than in the conduit, when the inclination was down stream; on the other hand, when the inclination was up-stream, the water in the reservoir was somewhat lower than in the conduit. Third.—When a pipe, projecting into the conduit, was attached to one of these inclined orifices, there was a much larger difference of elevation between the water in * Recherches expérimentales, relatives au mouvement de l’eau, dans les tuyaux. Paris, 1857, p. 217. 256 PIPES.—Piezometers. A/ills. the reservoir and in the conduit, than when the mouth of the orifice was flush with the side of the conduit. Fourth When a projecting pipe with a square end was attached in line with an orifice, whose axis was perpendicular to the side, the water in the reservoir was con- siderably lower than in the conduit. With a projecting pipe, the difference in elevation between the conduit and a reservoir, amounted to.44, with =8.0; this appears to have been the greatest difference observed.* The following general conclusion can be drawn from Mr. Mills’ experiments ; When the axis of an orifice is normal to the side of a straight conduit, and the mouth of the orifice absolutely flush with the side of the conduit, the piezometric column will truly indicate the surface elevation of the water in the conduit; but, especially with high velocities, a very slight deviation from the given conditions will cause notable errors in the piezometric columns. If the mouth of an orifice is placed so that the current of the stream impinges upon the section of the mouth, the piezometric column will be higher than the surface in the conduit ; on the other hand, if the line of the current forms an acute angle with the axis of the orifice, the piezometric column will be lower than the surface in the conduit. There is every reason to believe that the foregoing propositions will equally apply to piezometers attached to a pipe under pressure. The sucking or lowering effect of a current passing by a projecting pipe, has long been utilized by the steamboatmen of the Mississippi River, who, strange to relate, discharge the water from their coal barges by boring holes in the bottom! This rather surprising feat is accomplished by inserting a spoon-shaped iron, which projects below the bottom, with the outer side of the spoon pointing up stream; when the empty coal barges—large flat-bottomed boats—are being towed up stream, the sucking action of the current keeps the barge free from water. Naturally, when the boat is not in motion, the holes are plugged. These experiments of Mr. Mills simply prove that the summit of the liquid column n a piezometric tube, properly attached to a conduit, is in the true line of hydraulic inclination ; for the axial surface line in his open conduit was the true “ hydraulic-grade line.” * In this experiment air was drawn in through the piezometric pipe, so this difference of .44 probably does not fully represent the full ‘‘sucking” effect of the current. Mr. Mills states that in other similar experiments, the details of which are not given, the difference between the surface of the piezometric column and the hydraulic-grade 2 i ter than 7. line was greater than om PIPES.—PIEezometers. 257 The following sketch illustrates the effect of attaching piezometers to a pipe under pressure. ok a AA a a K a re B BY Bila" b eS wp Let the horizontal pipe B b, of uniform section and uniform condition of interior surface, be attached to the large reservoir R, by a trumpet-shaped mouth-piece, so that the co-efficient of contraction will be nearly unity ; we can hence neglect the primary loss of head due to contraction at the entrance. To the pipe attach the vertical tubes, A’ B’, A" B’, and A” B”, open at the top and connected with the pipe in the manner prescribed by Mr. Mills. Let the reservoir be kept constantly full with the liquid, and assume A a as the head required to impart velocity to the stream flowing through the pipe. Assume that uniform motion is fully acquired at the point B’. The liquid in the three tubes will then rise to the points a’, a’, and a”, all in the straight line ab.* The line aa’a’a’’b will then represent the hydraulic pressure line, or the hydraulic-grade line, and the pressure against the interior walls of the pipe at the point B’ will be represented by the line a’b’, at B” by the line a” b”, and at the end of the pipe by 0. This will be absolutely true in regard to pressure, if we assume that the threads of liquid move through the pipe in lines parallel to its axis. Suppose we now attach to one of the tubes, A” B”, a curved tube, shown in the sketch by dotted lines, having its open mouth at c normal to the axis of the pipe ; conceive that this extension of the tube has no thickness, so that it will not disturb the normal flow through the pipe. The liquid in the vertical tube will then rise to the point A”, in the straight line A D, parallel to the line ab. Strictly speaking, if ¢ is in the axis of the pipe, the liquid will rise higher than A”, as the velocity at c will be greater than the velocity due to the head A a, which represents the mean velocity head for the entire section of the pipe. Now if we make H= A B=total head; h’=A a=velocity head; h=a B=H-—A’ =“ effective” or “frictional” head, which is absorbed in overcoming all sorts of resist- ance as the liquid flows through the pipe; 7=B b=total length of pipe; /’=length from B to any piezometric tube; h,=piezometric head in any tube. We have shown 2 before that h’ is pretty closely represented bya, , v being the mean velocity in the pipe. re * The point b in a circular pipe will be in the centre of pressure, which will be slightly different from the centre of the section, but this refinement can be neglected.. ‘ : LL 958 PIPES.—PIEZoMETERS. Al’ v 9 2 [' Therefore, h, =h— T3 and approximately, h, = (zz = = (z-3) —_ br) (iF) : The Darcy experiments given in the last section of this chapter, have been reduced in accordance with the foregoing expression, allowance being made for effect of primary contraction. Jacobson has made some interesting experiments with piezometers attached to a trumpet-shaped mouth piece, which are described by Dr. Lampe in Der Civilingenieur, Vol. XIX. Mr. Mills has shown, that when proper care has been observed, piezometric attach- ments will accurately indicate the surface height in an open conduit, and one would naturally suppose that the same method would give equally satisfactory results, when applied to pipes. Entirely the reverse of this is true, for not a single series of pipe experiments thus far published, where 4 was determined by piezometers, gives thoroughly satisfactory results. Of all such experiments, we regard the Lampe series as the most reliable ; this accuracy resulted from the very great length of Dr. Lampe’s experimental pipe, so that the piezometric errors were not relatively large enough to vitiate the final deductions ; but, if in that pipe such considerable lengths as 2000 or even 3000 feet had been experi- mented upon, the deduced heads would often have been entirely untrustworthy.* M. Darcy appears to have used great care in his pipe experiments, and it is quite unlikely that there were any errors of consequence in his given values of J, D, a, Q and v, as he used very proper methods for the determination of these elements. It will be shown in the next section of this chapter, that his work, when carefully analyzed, is most unsatisfactory and contradictory, if one considers the time and cost involved in the execution of his labors. It seems probable, therefore, that his errors chiefly resulted from erroneous values of h and s. Messrs. Fteley and Stearns have shown the highest skill and accuracy as hydraulic experimenters; they were familiar with Mr. Mills’ paper, and fully realized the necessity of observing the proper precautions in the connection of piezometric tubes to the experimental conduit. In their experiments with the Rosemary 4-foot pipe, Nos. 282-285, one of the four differs considerably from the other three, and this divergence is so considerable that we can only attribute it to misleading piezometric heights. t It is not worth while discussing other defective data, where errors may be attributed to the use of a rough instrument like the Bourdon gauge, or where the bad results may have been due to errors in other elements than h. Comparing the results obtained by Darcy, et cet., with those obtained by Couplet, Bossut and ourselves, we see for the latter vastly smoother curves, and curves which * A description of Dr. Lampe’s methods of experimentation, with a tabulated statement of his piezometric errors, has already been given. Pipe experiments, Nos. 275-278. + It may be remarked that the experiments with the Rosemary pipe were made by the assistants of those gentle- men, neither of them being at the time able to personally superintend the execution of the experiments. PIPES.—PIczometers. 259 rarely contradict each other. Taking for instance, the author’s New Almaden series, where @ was ascertained by absolute measurement of g, the curves are in all cases almost perfectly symmetrical, and the general harmony of the curves for the different series proves the substantial accuracy of the work. With proper care, with i deduced from H, we are satisfied for velocities above 2 feet and diameters not less than 2 inches, that experiments can readily be made, which will show no comparative errors in nx of over 4 of one per cent. as a maximum. We are hence of the opinion that the discrepancies presented by the experiments of Darcy, Lampe and Stearns, must be attributed in a great measure to false piezometric heights, and not to unknown causes proceeding from slight changes in the water, such as we have conjectured affect the discharge through very small orifices, or for orifices and weirs with very low heads. Piezometric errors can be attributed : First.—To imperfect connections of the piezometric tubes with the conduit ; the effect of error from this cause should be constant, that is, always + or always —. Second.—Accumulation of air in the tubes; this seems difficult to avoid, although rmauch care has been observed by Darcy and others to prevent the presence of air. Third.—Obstruction of the small tubes, by sediment of various kinds; a leaf clinging to the mouth of one of these tubes would considerably affect the height of the column. Fourth When small glass tubes are used to directly show the heights of the columns, the uncertain or irregular amount of capillarity may be a source of error ; this, however, will not be the case when a mercurial column of proper size is employed. In addition to these chances of error, there are probably others, which can only be ascertained by more careful investigation than has thus far been devoted to the subject. In all ordinary cases, where h’ is a small fraction of H, we feel assured that experi- ments with pipes, when h is deduced from H, are incomparably more reliable than when / is obtained by piezometers. We have shown that the expression h=H— cannot be very greatly in error, and any possible error from this source will in 2g 07 general have no notable effect upon the deduced value of n.* Concxusions. Variation in n, caused by Changes in A, D aid v. D Taking the simple expression, v = n (7's)* =n . s) , we will now proceed to see what conclusions can be drawn from the foregoing experimental data, so far as the 2 * Error by the use of h=H—- 3 Z 2 qe will be confined to a fraction—probably quite a small one—of ai , while with 2 . . vy . piezometers errors appear to be in some cases even greater than “gai when » is not large. g 260 PIPES.—Conciusions. Variations in n. relative effects of D, v and A upon the co-efficient 1 are concerned. In this discussion it will be more convenient for pipes to use D than. For open conduits r will be used, the ratio between these quantities being constant for circular pipes, a¢., 4 to 1. Of the 363 selected experiments, we consider the New Almaden series—Nos. 287- 339—much the most reliable. The pipes employed were of small dimensions to be sure, but the mean areas were carefully obtained, and @ was ascertained by absolute measure- ment of g. The only considerable danger of error in these series is in h; we have shown that h as deduced is probably slightly too great, and hence n probably too low, but errors from this cause will not largely affect the deduced values of n, except for Series IV., when A’ was a large fraction of H. Such possible errors for the other 9 series would hardly be appreciable for the smaller velocities, and as the lengths for 7 of the series are nearly identical, any slight errors in h will affect the several curves in a nearly similar manner. From these New Almaden results, we will endeavour to deduce some general principles, which will enable us to more intelligently discuss the remaining data. On Plate XIII. are plotted the New Almaden series, Nos. V. to X. inclusive, with v and n as co-ordinates ; the values of 7 for the pipe used for Series III. are shown by a symmetrical curve which has been drawn from the data given on Plate XII., where the experiments constituting Series I., II. and III., have been plotted; the experimental points for these three series are very nearly in the same curve. The seven curves on Plate XIII. represent the experiments made with seven distinct pipes; these pipes can be described as follows : A nearly Constant and very Low. Series III. New and smooth wrought-iron, covered with an asphaltum varnish, and very smooth ; D=.087; /=60.8; 0=.98. Series VII. New glass; almost perfectly clean; D=.076; 1=63.9; 0=.97; composed of 12 joints. Series VIII. New glass; a few spots of dirt adhering on inner walls, but not enough to appreciably retard flow; D=.062; 1=34.9; 0=.82; composed of 6 joints. Series IX. New glass; almost perfectly clean; D=.042; J=11.1; 0=.82; composed of 2 jomts. The area of this pipe less accurately obtained than for the others. Series VI. New wrought-iron, free from rust, with no varnish, hence A is very slightly higher than for the four preceding series ; D=.052; 1=60.1 > 0 =.825. Figher Values of A. Series V. Composed of 4 joints of old pipe, of which the interior surface was covered with a thin hard scale, with occasional small nodules; and 1 joint of new PIPES.—Concxusions. Variations in x. 261 varnished pipe. Total 7= 60.2; of which 52.7 was rough, and 7.5 smooth. D=.085; 0=.98. Series X. Red-wood, bored by usual pipe-auger; surface as usual for such auger holes in moderately soft wood ; D=.105; 1=62.0 ; 0=.80. By reference to Plate XIII. it will be observed that the experimental curves are nearly perfectly symmetrical ; the only exceptions being Experiment No. 311, where n appears to be a very little low, and No. 338 where » appears to be fully 1 per cent. too high.* This symmetry proves that for each series H and @ were correctly deter- mined, ee Examining the 7 experimental curves, we see the following results due to changes inv, DandA: Effect of Variation 7 v.—In all cases n increases with v; very rapidly for velocities below 2 feet, and more and more slowly as v further increases; the trend of all the curves as v diminishes is manifestly towards the axis of the co-ordinates, so we can roughly assume that the curves have a common origin at this axis. Hence, with an inpinitesimal velocity, n will be infinitesimal, no matter what values D and A may have. With A very nearly constant and D variable—Series IX., VI., VIII., and III. or VII.—as D increases the curves become somewhat more inclined for velocities above 2 feet. Hence, it is probable that the effect of v upon n, increases with D. With D very nearly constant, and A variable—Series III. and V.—as A increases the effect of v upon n diminishes. This is shown still more forcibly by the curve for Series X., with D about one-fourth larger than for Series III. ; for this curve, as v increases above 2.5 it has comparatively but little effect upon n. Hence, with not very large values of A, and such small values of D, an increase in v above 2.5 will have but little effect upon n. Also the lower the value of A, the more will an increase in v uncrease n. Effect of Variation in D,.—With v and A constant—Series IX., VI., VIII, and ITT. or VII.—n notably increases with D; with v= 4, for D=.042 has a value of 89, and for D=.076 a value of 98; with the same velocity and D=.062, » has a value of 95.5. Hence we can generally assume that n, v and A being constant, always increases with D. Effect of Variation in A.—With v and D constant—Series V. and III.—the smoother pipe shows much the larger value for n. For Series X., with a bored wooden pipe, D=.105; for a similar pipe with D=.087, or the same as for Series III., the values of n would be lower, and would be not over 62 for v=2, and 64 for v=5. With the same values of v, n for Series III. has the respective values of 90 and 100. Hence * The value of n for this experiment should probably be 67.0 instead of 67.5 as plotted. Vide note in Table XCVIl. 262 PIPES.—Concuusions. Variateons in n. an increase in A, with v and D constant, greatly diminishes n, and in these experiments for v above 1, A is undoubtedly much the most important of the three variables. The curves for Series III. and VII. almost absolutely coincide; the respective diameters were .087 and .076; this agreement of the two curves shows that the lower value of A for the glass pipe compensated for its smaller diameter. We will now critically examine our other experimental data, taking them in the order in which they have been given, for the purpose of determining which of them shall be selected as authority. Couplet. Versailles Pipe.—The experiments of Couplet were the first ever made with pipes of considerable length; they seem to us much the most reliable experiments extant, with very low velocities. DD was constant for the 6 experiments selected— Nos. 1-6—and therefore any probable error in D will not notably affect our curve— certainly not its form. 7 seems to have been measured with sufficient care; t was quite short, being only {3 seconds for No. 6; this experiment, however, was repeated with similar results ; as the times were taken to } seconds, it is probable that any error in ¢ will not affect n more than 3 per cent. ; now, as the values of increase from 66 to 82, with a limited range of 7, such an error will not vitiate the general accuracy of the curve. Plotting the curve, with v and n as co-ordinates, we see it is symmetrical, n rapidly increasing with v; these 6 experiments will be accepted as authority. Bossut.—The experiments of Bossut will be found plotted on Plate XII. By reference to that sheet, it will be seen that the curves for Series I. and II., with tin pipes having the respective diameters of .118 and .178, are fairly symmetrical, that for the larger diameter being somewhat the higher.* The curve for Series IV., with a lead pipe having a diameter of .089, is on the other hand a little higher than that for the tin pipe of .118 diameter. The point for Series III.—tin pipe, D=.118—is considerably lower than the curve for Series I. with the same diameter. Comparing the curves for Series I. and IT., with that for the New Almaden Series III.—D =.087, it will be seen that the three curves fairly agree in form, but that the curves for the Bossut Series I. and II. are a little too high, making allowance for differences in diameters ; the Bossut Series III. on the other hand is a little too low. Dubuat.—On the same sheet will be found the curve of Dubuat’s experiments with a tin pipe having a diameter of .089. This curve is almost identical with that of Bossut, Series I. We can fairly deduce from these experiments of Bossut and Dubuat, that our first proposition in regard to the effect of v is correct. As to the effect of variation in D, the experiments are conflicting. As to the effect of A, that quantity was nearly constant. * When we speak of an experimental curve or point being ‘‘ higher” or ‘‘lower,” it means that the ordinate representing the value of » is greater or less. PIPES.—Concwusions, Discussion of Experimental Datu. 263 Scotch Authoritics.—For Experiments Nos. 60-64, the great danger of error consists doubtless in the values of @, and hence v. For two experiments with the Colinton pipe—Nos. 61 and 62—Mr. Leslie states that @ was the mean of quite a number of observations, the variation between maximum and minimum being in both cases about 6 per cent.. How these measurements were made does not appear. Some weight will be attached to the Colinton series; the experiments, however, are of not much value, as nothing is known in regard to 4; they indicate that for this pipe, increasing v from 5.3 to 14.5, increased n from 96 to 115. The Loch Katrine experiment, No. 64, was with D=4, and A evidently quite low. Probably H, a and / are given with reasonable accuracy ; how @ was determined Mr. Gale does not state. This experiment is a very important one, as there have thus far been published the results of experiments with only one other iron pipe of the same size. These experiments—Nos. 282-285 of Mr. Stearns—with about the same velocity give na value of 140, as against 112 given for the Loch Katrine pipe. Evidently, one or the other of these results is much in error; their respective claims or chances of accuracy will be weighed hereafter. Darcy.—¥Examining the Darcy Series I., II. and III., with new wrought-iron; for Series I. we see a very ragged experimental curve, n for No. 67 apparently being as much as 20 per cent. in error, if Nos. 65, 66 and 70-77 are assumed as correct. The curves for Series IT. and III. are much less contradictory. They indicate a rapid increase in n with D; n, for velocities of 3 feet, being about 66 for D=.040, 85 for D=.087, and 94 for D=.130. They generally indicate an increase of n with v, although contradic- tory in this respect. Series IV., V. and VI. with new lead pipes, having the respective diameters of .046, .089 and .134, give curves practically identical, n being near 81 with v=1, and 100 with »=5. This would seem to show that with lead pipes 1 does not increase with D, but largely with v. Series VII., VIII, [X. and X. were with sheet iron pipes lined with bitumen, having the respective diameters of .088, .271, .643 and .935. These 4 curves are fairly symmetrical ; that for No, VIII. is considerably higher than the one for No. VII.; the curves for Nos. IX. and X. are nearly identical, and a little higher than the one for No. VIII. ; for the curves IX. and X., n is 108 with v=2, and 131 with v=11. They indicate that does not increase with values of D above .6. Series XJ. was with a new glass pipe, having D=.163. This curve is perfectly symmetrical, and from v=1.5 to v=5, closely agrees with the curve for the New Almaden glass pipe, with D=.076. Series XII. and XIII. were with a cast-iron pipe, with D about .12. The pipe was an old one with deposits on its inner surface ; in this condition 7 is nearly constant at 59; the pipe was then cleaned, its mean area again determined, and n is 80 for v=.37, 264 PIPES.—Conciusions. Discussion of Experimental Data. and 99 for v=3.7. The flow through this pipe was hence increased 1.6 times, with the same inclination, by the removal of the deposits. Series XIV. and XV. were with an old cast-iron pipe, with D about .26, and was first experimented upon in its bad condition, and then again after being cleaned. The results were similar to those for the preceding series; with the same inclination the flow was increased about one-third. Series XVI, XVII., XVIII. and XXII. were with new cast-iron pipes, having the respective diameters of .27, .45, .62 and 1.64. For velocities above 4, n for Series XVI. has a pretty constant value of 100; for Series XVII. a constant value of 113; and for Series XVIII. a constant value of 108. The curve for Series XXII. is exceedingly rough, and the experiments for this series are palpably worthless. The results for the first 3 series are contradictory, so far as D 1s concerned. Series XIX. and XX. were with an old cast-iron pipe with D=.80, afterwards cleaned; » for Series XIX. has a pretty constant value of 75, and for Series XX. a pretty constant value of 97. Series XXI. was with an old cast-iron pipe, with D=.97, very well cleaned, and n, with v from 1.8 to 10.4, has a constant value of 104. Summing up these results : Effect of Variation in v.—They generally indicate that with smooth pipes m increases with v, very rapidly with v less than 1, and more and more slowly as » increases from 1 to 16. There are, however, frequent contradictions to this conclusion for velocities less than 1. With old cast-iron pipes covered with deposits, n is nearly constant, with values of v from 1 to 12.5. In none of the series, excepting No. XXII., which is manifestly defective, is there any noteworthy lowering of n for high velocities. Effect of Variation in D.—In the majority of cases n increases with D. The most notable exceptions are with the lead pipes. Liffect of Variation in A .—They show, without exception, that the lower the zine of A, the greater is 7. Contrasting Series XIV. with a cast-iron pipe in bad order, and Series VIII. with a pipe covered with bitumen, both pipes having about the same diameter, for velocities from 2 to 4 feet, the value of n for the smooth pipe ranges from 105 to 111, while for the rough pipe n is about 68. We hence see that the Darcy series can only be considered entirely conclusive in regard to the great effect of changes in A. In other respects they are so frequently contradictory, that they can be of no service to us in the determination of our final curves for n. In the greater number of cases they confirm the general accuracy of the propositions deduced from the New Almaden series. As heretofore stated, we regard PIPES.—Concusions. Discussion of Experimental Data. 265 the very uneven results shown by the Darcy series to be largely due to the false indications of piezometric columns. We have devoted a good deal of space to M. Darcy’s pipe experiments, because they are the most elaborate investigations of the kind which have ever been executed, and because they are generally considered to be the highest authority. M. Darcy deserves very great credit for his demonstration that the condition of the wetted surface has such an important, and often controlling effect upon the flow. This disproof of Dubuat’s fallacious proposition is the most brilliant and valuable discovery in the science of Hydraulics during the present century. Iben.—The Bonn experiments, Nos. 271-274, with D=1, and A presumably low, show an increase in x from 89 to 106, with v from 1.6 to 3.1. The rough method employed for obtaining h, renders these experiments more or less unreliable. Lampe. Danzig Pipe. The dangers of error in these experiments, Nos. 275-278, have already been discussed. These experiments will be accepted as authority. Kirkwood.—We regard these experiments as of very doubtful authenticity, and they will not be used. Rochester Main.—This experiment appears to be entitled to considerable weight, as gq was absolutely measured. It is a pity that Mr. Tubbs has not more fully described the methods followed in obtaining his experimental data. Stearns. Rosemary Pipe.—These experiments, Nos. 282-285, are of much im- portance. We consider that in them there is no danger of considerable experimental error, except in the piezometric determinations of h. As the results of these experi- ments are flatly contradicted by the Loch Katrine experiment, No. 64, we will here- after discuss their relative chances of error, using as a guide the open conduit experi- ments of Darcy and Bazin, and Fteley and Stearns. Smith.—No. 286 of the author, with the Cherokee pipe, can only be regarded as an approximation, as there is too much uncertainty as to the proper values of the experimental elements. The given value of 1 may be in error either way 10 per cent., the chances perhaps being that 7 is a little too high. The North Bloomfield series, Nos. 340-354, are entitled to a good deal of weight. The chief uncertainty in them arises from the indirect method of obtaining Q by weir measurement. The chances of error in the individual experiments will be fully stated in Chapter X. These experiments show that increases with D, and with v, in the same manner as shown by the New Almaden series. For them, A was appreciably higher than for the New Almaden smooth pipes, and probably higher than for the Danzig and Rosemary pipes ; the joints were roughly made, and the rivet heads also slightly added to the roughness. The curves in these pipes also slightly retarded the flow. M M Eure 266 PIPES.—Concivusions. Discussion of Experimental Data. No. 355 for the Humbug pipe, with D=2.15, is more unreliable than the North Bloomfield series, on account of uncertainty about Q. The chances are that x is placed a little too high. A for this experiment was about the same as for Nos. 340-354. No. 356, with the Texas Creek pipe, D=1.42, is of great value, being by far the most authentic experiment on record, with a very high velocity, and a pipe of consider- able size. The chances of error in this experiment are less than for the North Bloomfield series. A can be considered about the same, or perhaps a trifle higher, than for Nos. 340-355, and about the same as for No. 286. The curves for this Texas Creek pipe were sharp enough to appreciably retard the flow, possibly diminishing m one or two per cent... Darcy-Bazin.—The only experiments we have given with iron pipes having diameters larger than 2.4, are No. 64 with the Loch Katrine pipe by Mz. Gale, and Nos. 282-285 with the Rosemary pipe, by Mr. Stearns; each of these pipes was 4 feet in diameter, and the value of A appears to have been not very different. To determine which of these determinations is trustworthy, we will use the Darcy-Bazin Series XXIV., XXV. and XXVI., with semi-circular open conduits, each having a diameter of about 4 feet, and with A,a plaster of pure cement for No. XXIV., a plaster of cement mixed with one-third sand for No. XXV., and planks partly planed for No. XXVI. In all these experiments / was directly measured, and Q obtained with a fair degree of accuracy. The general accuracy of these Darcy-Bazin experiments has been con- firmed by the open conduit experiments of Messrs. Fteley and Stearns. Some authorities, notably Humphreys and Abbot, have insisted that the air, even in calm weather, considerably retards the flow in open channels. This assumption has been proved by M. Bazin to be incorrect. In any event it seems clear, that comparing values of 7 for full circular pipes, with values of n for semi-circular open conduits, 7, v and A being identical, n cannot be higher for the conduit than for the pipe. We are therefore on the safe side in using the Darcy-Bazin data.* Fteley and Stearns.—Open Conduit Experiments, Nos. 444 and 445, were made with a section of the Sudbury conduit, where the brick-work was covered with a thin plaster of pure Portland cement; for these experiments the wetted surface was probably smoother than for the Darcy-Bazin Series XXV. (cement and sand), but on the other hand the semi-circular form of the latter conduit was more favourable for flow, than the oval form of the Sudbury conduit; hence the values of n for these two conduits should be about the same. From the given data, we will now select those experiments with low values of A, which we regard as most trustworthy ; they are as follows: * The Darcy-Bazin Open Conduit (Series LI. and LII.) experiments showed that in a wooden rectangular pipe 7, in v==n (rs)%, had about the same value, as for a rectangular open conduit with the same values of 4, v and r. PIPES.—Concuusions. Discussion of Experimental Duta. 267 Name of Pipe. Authority. Bee ‘ Material. bh a D v New Almaden Author 331-334 Glass 0 042 | 2.1—4.4 5 5 316-321 Wrought-iron 0.2 052 | 1.0—3.9 $4 m 327-330 Glass 0 .062 1.4—4.4 322-326 : 0 076 2.0—5.0 301-304 Wrought-iron 0.1 087 = | 2.25.4 " 293 i 0.2 088 | .96 299-300 0.2 | .088 |1.05—1.4 Versailles Couplet 1-6 Lead (hy 44h 18—.47 North Bloomfield Author 340-344 Riveted sheet-iron 1.5 91 4.7—10.0 Bonn Tben 271-274 Cast-iron coated i: 1.00 1.6—3.1 North Bloomfield Author 345-348 Riveted sheet-iron 1.5 1.06 4.6—10.7 " ‘ 349-354 ? 3 15 1.23 |44—121 Colinton Leslie 61-63 Cast-iron (2) 1.33 |5.3—14.5 Danzig Lampe 275-278 Cast-iron coated I, 1.37 1.6—3.1 Texas Creek Author 356 Riveted sheet-iron 1.6 1.49 20.1 Humbug is 355 - ia La 2.15 12.6 Cherokee 3 286 5 se 1.6 2.43 (2) 10.8 Rochester Tubbs 281 { pease aa } 1.20 i. Loch Katrine Gale 64 Cast-iron coated 11 4.0 3.5 Rosemary Stearns 983.985 | 33 4 L. 40 '2.6—6.2 Conduit Nos. e . : 223-234 Plaster of pure cement si .37—1.03, 3.0—6.1 ae }\ Darey-Bazin 235-246 | Plaster of cement and sand 1. 38—1.04) 2.9 —5.7 247-259 Partly planed plank .39—1.15) 2.6—5.5 Sudbury-eonduit [Filey Stearns { 444 Plaster of pure cement | fie (| 2.05 2.7 445 : 7 ti 1g6 2.5 The foregoing experiments have been plotted on Plate XIV., with v and 7 as co- ordinates. It will be seen by an examination of this diagram, that the positions of the experimental curves and points, with a few exceptions, are in harmony with the con- clusions drawn from the New Almaden results. That is to say; with v constant, 2 steadily increases with D; with D constant, for small velocities rapidly increases with v, and this increase in n becomes less and less as the velocities become greater. The exceptions to these general principles or laws are as follows : No. 282 of the Rosemary pipe, which indicates that n increases with low velocities. 268 PIPES.—Conctiusions. Discussion of Experimental Data. This experiment stands alone in this respect, so it should have but little weight as against the mass of other testimony. No. 64, with the Loch Katrine pipe, which indicates that n for D = 4.0 is less than n for the Danzig pipe with D=1.37, and only slightly greater than n for the North Bloomfield pipe with D = 1.23, the value of v in all three cases being the same. With the same velocity and diameter (v=3.5 and D=4.0), for the Rosemary pipe 1 1s about 140., as against 112. for the Loch Katrine pipe. Looking at the three curves for the Darey-Bazin semi-circular conduits, where 7 is a variable, we see that with 7= 1.0, or D=4.0; For the surface of pure cement ... as a sa n=154. with v=6.0 i sas > 9) cement with one-third sand ... is n=141. with v=5.4 Bay fs »» 9) partly planed plank. . oe “8 n=128. with v=5.0 A plaster of pure cement is doubtless a smoother surface than that of a cast-iron pipe covered with a coating of asphalt or tar; probably a plaster of cement and sand has about the same character of surface as such a pipe. For the same velocity of 5.4 the Rosemary curve shows a value of n of 143, which closely agrees with its value for the conduit plastered with cement and sand. We hence see that the Loch Katrine experiment is directly contradicted by the Rosemary and Darcy-Bazin experiments, and indirectly by the Danzig and North Bloomfield curves; we are therefore justified in rejecting it.* The Colinton curve, Nos. 61-63 with D = 1.33, is considerably lower than the North Bloomfield curve with nearly the same diameter. Without looking for other reasons, we can fairly attribute this to roughness of surface in the Colinton pipe. For Nos. 345 and 346 of the North Bloomfield series, x appears to be a little too low, compared with the other experiments of the three curves; we have before stated that owing to the stoppage of a stone in this pipe, 7 for these two experiments has too small a value. No. 286 for the Cherokee pipe and No. 355 for the Humbug pipe are somewhat contradictory ; neither of these experiments ranks high in point of accuracy. Rejecting these exceptional experimental results, we can now proceed to draw curves on the diagram, representing the values of i: for various diameters, the inner surface being in all cases quite smooth ; these curves are shown on Plate XIV. by sym- metrical, solid, fine lines. A for the four New Almaden experimental curves for small pipes, was exceedingly low ; in practice such smoothness will rarely be attainable. Hence for the diameters of .05 and .10 the curves for will be drawn somewhat lower than is indicated as their proper position by the experimental curves ; they will also be made flatter, with velocities above 3., than the experimental curves, owing to the supposed effect of increasing A, * Possibly A for this Loch Katrine pipe may have had a higher value than is indicated by the remarks made by Mr. Gale in regard to the condition of its inner surface. PIPES.—Concuiusions. Proper Values of n. 269 The North Bloomfield curve for D= 1.23, is on the other hand for a riveted pipe, carelessly laid with poor joints, and having two rather sharp angles; we will assume the curve for this pipe to nearly represent the curve of 1 for D=1.0. The datum point for the curve of n for D=4.0 will be Open Conduit Experiment No. 244, with the Darey-Bazin semi-circular conduit, covered with a plaster of cement and sand. The form of this curve is in harmony with the North Bloomfield and New Almaden experimental curves, following the law that, A being constant, the effect of v upon 1 increases with D; that is to say, the larger the diameter, the more the curve for n should be inclined. The curves of » with D between +. and 1. have been interpolated from these two curves; the curves with D from 5. to 8. have been drawn, so that they are harmonious with the other curves. By reference to Plate XIV. it will beseen that the various curves of 1 agree very fairly with the experimental data considered reliable. The curves closely approximate to the given values of r for the Darcy-Bazin conduit (cement and sand), and those for the Sudbury conduit. The Rosemary curve (Nos. 283-285) is about 2 per cent. too high ; the Rochester compound pipe shows nx about 4 per cent. too high; the Danzig curve is slightly too high ; the experimental point for the Humbug pipe is slightly too low, and that for the Cherokee pipe about 9 per cent. too low. Extending the curve for D=1.5 from v=15 to v=20, it will be noticed that it agrees almost perfectly with the experimental point for the Texas Creek pipe. A for this pipe was somewhat higher than the standard assumed for the curves of n; hence it is probable that the curve for D=1.5 should have been more steeply inclined, or, in other words, that we have somewhat underestimated the effect of v in increas- ing 7”. In the following table are given the values of » corresponding to the curves on Plate XIV., with D from .05 to 8., and v from 1. to 15. for diameters up to 4... If it be required to find » and v, in our equation =n (7's)”, 7 and s being known, they can readily be obtained by two or more approximations. For instance, let D=4. (hence r=1.) and s=.0001; assume n to be 100.; then v=1.. Looking at the table or the diagram, we see that with D=4. and v=1., » should be 123.; taking n as 125., we have v=1.25. Again looking at the diagram we see that 125. is about the proper value for n with D=4, and v=1.25. It must be kept in mind that the total head H must be corrected for losses of head in imparting velocity and contraction at the entrance ; this should be done by h= H— Igo For short pipes one or more additional approximations may be needed in order to obtain f, and hence s= f before the proper value of 2 is determined. 270 PIPES.—Conciusions. Proper Values of n. The given values of n can, in our judgment, be used with entire safety for computing the flow of reasonably clean water, either through well-made cast-iron pipes, or through riveted sheet-iron or steel pipes, where the rivet heads do not form quite a notable portion of the area. The pipes must be properly coated with a varnish of asphaltum and coal tar, or some other preparation equally good; the joints must be smoothly united, and any curves must be well rounded. These remarks apply to diameters from 1. to 8... For diameters less than 1. the given values of n are probably somewhat too high for either cast or riveted pipe; they are suitable for ordinary lap-welded pipe, which has also been coated. These values of 7 will also apply to open semi-circular conduits, plastered with a mortar of cement and sand ; for a hard smooth surface such as pure cement, n will be several per cent. larger. For open conduits of rectangular section n will be several per cent. smaller than our given values. For values of D or 7 larger than those which are given, for the same degree of smoothness » will continually increase with D. For a riveted sheet-iron or steel pipe, with its inner surface properly coated, with D=20. and v=say 5., 7 will probably have the very great value of 180, or perhaps even a higher value. For velocities less than 1. the proper value of 7 is more or less uncertain ; it can be approximately determined by reference to the curves on Plate XIV. For values of D and v, intermediate to those given in the table, 7 can best be obtained by interpolation on Plate XIV. ; it must be remembered that A for the two least diameters, has a smaller value than for the other diameters. 4 PIPES.—Conctusions. Proper Valwes of n. 271 TABLE XC. Values of Co-efficient n, in v=n (18 )% for Circular Pipes, or Semi-circular open Conduits, having quite Smooth Interior Surfaces, and no Sharp Bends. (A for D= .05 and 1. Var ,, D=1. to8 DIAMETERS. D=4r. Velocity | ae ee a dere Re ee ° 05 | -10 | a, ee) a | p35) $f | BY T é ee a ea ae 1@) § .. 80.0} 96.1 | 102.8 | 108.8 | 112.7 | 116.7 | 120.2 | 123.0 } 127.8 | 131.8 | 134.8 | 137.5 2 77.8 88.9 | 104.0 | 110.9 | 116.2 | 120.3 | 123.8 127.0. 129.9 134.3 | 138.0 | 141.0 | 143.3 3 824 93.7 | 108.7 |115.6 §120.8 |124.8 |128.3 ,131.4 ,1342 | 138.6 | 142.3 | 145.4 | 147.6 4 85.6 | 97.0 |112.0 | 118.9 | 124.0 128.1 | 131.5 ea ae 141.9 | 145.5 | 148.6 | 151.0 5 87.6 | 99.3 | 114.4 [121.3 | 126.5 | 130.6 [134.1 [137.1 1140.0 | 144.7 | 148.1 | 151.2 | 153.6 6 89.1 101.0 |116.3 | 123.2 |128.6 | 132.6 | 136.3 | 139.4 | 142.3 [146.9 | 150.5 |153.5- : 90.0 | 102.4 |118.0 | 125.0 | 130.4 | 134.6 |138.2 |141.5 [144.5 |149.0 | 152.7 | 8 90.6 103.3 |119.3 1264 132.0 136.3 140.0 143.3 146.3 |151.0 154.9 | 9 90.7 | 104.0 | 120.4 | 127.7 |133.3 |137.7 | 141.6 | 145.0 |148.1 | 152.8 | 156.7 10 90.8 | 104.5 |121.4 /128.8 | 134.5 | 139.0 142.9 | 146.4 |149.7 | 154.6 11 90.9 } 104.7 | 122.0 |129.7 | 135.6 | 140.2 |1442 | 147.7 | 151.0 | | 12 | 91.0 |104.8 {129.5 | 130.4 |136.4 |141.1 ) 145.2 148.8 | 152.2 | | 13 91.0 | 105.0 | 122.9 | 131.0 | 137.1 | 141.9 [146.1 149.8 | 153.2 14 | 91.0 | 105.0 /123.2 | 131.5 | 137.6 | 142.5 1146.7 | 150.5 | 154.0 | 3 15 | 91.0 | 105.0 |123.6 | 131.8 | 1380) 142.9 | 147.2 151.1 154.6 20 (2) 123.9 | 132.9 | | : | No attempt will be made to give the values of n for pipes having rough inner surfaces ; the degree of roughness cannot be properly estimated, and any computation of the flow must necessarily to a great extent be guess-work. When it is required to compute the flow through such pipes, it must be remembered that for ordinary diameters, an increase in v above 2. or 3. has but little effect upon n. Also that A isa function of D; that is to say, a degree of roughness which would greatly lower n for small diameters, would have but little effect for great diameters. There is no authentic evidence proving for pipes, that with large values of A, n decreases with an increase in v. Some of the Darcy-Bazin experiments with open conduits, where A was exceedingly high in relation to r, seem to prove that for velocities above 2. does diminish as v increases; the Mississippi experiments of Humphreys and Abbot apparently show that this is true for rivers. If it be true for conduits and rivers, it is doubtless true for pipes. Some experiments made by Professor W. C. Unwin* show that the resistance encountered by a rotating disc, with a rough surface, is measured by a higher power than the square of the * Proceedings, Institution of Civil Engineers, London, 1884-85. These experiments are more fully described at the close of this chapter. 272 PIPES.—Conciusions. Proper Values of n. velocity ; this may indicate that in our Chezy formula, with high values of A, decreases with an increase in v. An iron pipe, unprotected by any surface coating, will soon become incrusted either by a hard scale, where water impregnated with lime flows through it, or by tubercular excrescences where soft water is used. These incrustations have the effect of not only diminishing the area, but also of increasing A; the flow is hence diminished in a double way. Some of the Darcy experiments, and those of Iben given in Table LXXVIIL,, illustrate the greatly diminished flow due to these two causes. The author has had a large experience with riveted sheet-iron pipes in California, and has found no difficulty in protecting them both from rust and the formation of tubercles.* The Danzig and Rosemary cast-iron pipes had both been laid several years before the given experiments were made with them the high values of m for these experiments prove the efficiency of the interior coating. Formue. The Dubuat formula for pipes and streams is, in old French inches, v= 297 (r4—.1 ( )’ —.3 ( xk 77. 2s “*— 1), in which Z is the hyp. log.. Transforming this cs =a (;+16 s into English feet we have, (A) pc LE O28). hd (7% —.0298). ee) 8 8 Prony proposed for pipes, in metrical measures, + g Ds=.000 17 1+.003 4160” Transforming this into English feet, we have, (B) v= (.006 65+ 9421 7s)*¥—.0816. Prony suggests as a simpler form, but which must not be used for very low velocities, v= 26.79 (Ds)” metric, or, (C) w= 97,05 (rs8)*. Eytelwein proposed (English feet), DO (D) v= 47.9 (sr) Mr. Neville gives, (E) v= 140 (rv ae —I11 (7 s) “Io, Darcy’s formula for new cast-iron pipes was in metrical measures, Rs = (.000 507 + —E) v’, which in English feet becomes, 5 % (F) eee \ sooo yea ee * The reader is referred to a paper by the author, published in the Transactions of the Am. Soc. of C.E., Feb- ruary, 1884, for detailed information in regard to such pipes. PIPES.—Formu.a in USE. 273 Weisbach gives, - 1D a (G) ate 29h) in which the co-efficient ¢ is greater for low than for high velocities. Weisbach recog- nised the fact that ¢ diminishes as D increases, and also that ¢ increases as the inner surface becomes rougher. For general use he makes ;=.014 so400° ee metric) ; hence we have in English measures, 1 % (HH) v = ( ——__—________rs2q)\. .003 5975 +e Hagen’s formula* in metrical measures was, (I) v= Spas * it 023 577 + .000 115 19 —.000 004191 7,4+.000 000 092 29 2 | | oo In the foregoing expression T, is the temperature in degrees of the Celsius scale. Dr. Lampe proposes in metrical measure : 2, (a) s=.000061 341 <, +.000793 32-” ; and also suggests, D? D > 1.802 (6) s=000 7555 as Transforming ()) into English measures we have, (J) pala Pe a, In the Kutter formula, given in Chapter VII., his co-efficient, «, of rugosity for a plaster of cement and sand is placed at .011; with this co-efficient his formula becomes in English measures, y 206.3 a ee 002 1) ra)? O11 ve D’Aubuisson, Saint-Venant, Lévy, Dupuit, Grashof, Gauckler, Boileau, Frank, Boussinesq, and Reynolds, have also proposed formule for the flow in pipes and streams. The formule of Bazin for open channels have been given in Chapter VII. Fanning, in his treatise on “ Water Supply Engineering,” gives tables, showing vary- 1+ ( 41.66 + % ing values of the co-efficient ¢, in v= fe 2g7' 3 , in which he recognises the effect of » . changes in A, v and 7, upon ¢ * Druckhéhen-Verlust in geschlossenen eisernen Rohrleitungen. Otto Iben, Hamburg, 1880. NN 274 PIPES.—ForMUL# IN USE. In the following table are compared, with widely varying values of v7 and D, the resulting velocities given by some of the preceding formule. The values of v as deduced from our diagram on Plate XIV., are first given, and then the values computed by the several formule, in the order of their agreement with our results. TABLE XCI. Velocities in Circular Smooth Pipes, as deduced from Formule of Prony, et cet.—r and s being known. Prony | D ‘ Pinte SIV Lampe | Kutter | Darcy (Weisbach Dubuat .005 a 59 . 94 | 1.01 1.09 1.01 | 98 | .10 15 6.21 5.48 3.24 5.14 6.78 | 5.94 ) 5.86 9 7.43 Af 10.46 9.45 5.29 8.39 11.50 9.71 | 9.62 | 12.96 { .000 4 95 1.01 1.03 1.09 89 97 39 | 86 1.0 025 9.56 10.03 8.50 8.64 8.94 7.67 '° 7.59 | 9.29 wl 19.6 © 21.6 17.0 17.3 18.7 15.3 153 ° 4 f .000 07 1.03 1.01 1.09 94 we 81 73 | 66 4.0 .001 2 4.84 4.87 4.87 3.90 3.63 3.36 3.28 3.28 OL 15.5 15.8 14.1 11.3 11.5 9.71 9.63 | 11.08 ea f .000 025 97 2-3-1403 80 9 69 61 BL ~ {| 0007 5.83 5.84 5.81 4,23 3.96 3.63 3.55 3.46 ai .000 008 99 (1) 92 1.20 12 52 61 4 .36 .000 155 5.0 (2) 4.77 | 4.83 3.16 2.85 2.70 2.62 230 | aos _ An examination of the preceding table shows that the formule of Dubuat, Prony, Weisbach, and Darcy give results, especially for large diameters, widely varying from the experimental values indicated by the curves on Plate XIV. By giving n a constant value of 120, we would obtain considerably better results for these comparisons, than those deduced from any one of these five expressions. Darcy’s formula gives very slightly better results than that of Weisbach. The Kutter formula, for D less than 1., gives very much less than the true velocities ; it agrees very well with our experiments for D from 3. to 8., except for very slight inclinations, when it gives results much too high. The Lampe formula agrees quite closely with our results, and can be used without very serious error; aside from its much closer approach to the truth, it is much preferable to the Kutter expres- sion, on account of its greater simplicity. PIPES.—ForMuULz IN USE. 275 From the experimental data, contained in this chapter and the preceding one, we ye: ; 1 Tees could frame formule, giving the co-efficient NV and the exponents ~ and;,* in v= Nr!» sm, constant values for each assumed value of 4; such expressions would give results approximating the truth. We prefer, however, to use the simplest form; the data we have given will enable the student to make a fair guess as to the value of 1, where A is high in proportion to 7; the assumption of such values of A must neces- sarily be a matter of judgment. Professor Osborne Reynolds has published in the Transactions of the Royal Society, 1883, an account of experiments with glass and lead pipes, the diameters ranging from } inch to 1 inch. With single joints of glass pipes up to 1 inch in diameter, by the aid of coloring matter in the water, he found that up to a certain limit of velocity for each diameter, the water moved in parallel threads or rings; the instant this limit was passed eddies formed, and parallelism disappeared. He made a large number of experiments with two lead pipes, having the respective diameters of .0415 and .0202, with a very wide range of pressures; the “frictional” loss of head was determined by two piezometric tubes placed 5 feet apart. The experiments with the larger diameter showed that with the least pressure {s=.000 80 and v=.11) m had a value of 39; with increasing pressures 7 increased to 91 with v=.74, and then dropped immediately to 86 with about the same velocity ; this point Professor Reynolds considers to be that at which parallel movement ceases and eddying movement commences. The pressures were steadily increased beyond this limit, until s became 3.2; with this “inclination” v was 23. and n was 126; the increment of n with the increase in s was, however, quite irregular. With the smaller diameter the least pressure was equivalent to s=.0086, when v=.26, and »=39; the limit at which parallel movement ceased was about s=.0516 and v=1.45, when » was 90. Above this limit ~ was 92; then m decreased to 80 with v=2.1, and with increasing pressures increased to 110, with s=3.9 and v=15.; as with the larger pipe, the increment of n with s was quite irregular. So long as parallel motion continued, these experiments agree roughly with the Poiseuille formula of v= 52 500 (1+.033 68 7, + .000 221 772) s D®, the velocity increasing directly with the inclination and the square of the diameter. They indicate that as D increases, the limit of parallel motion diminishes ; that is to say, with a considerable diameter, such as 1 foot, eddying movement will commence with a very small value of v. Professor Reynolds thinks this critical velocity is very nearly in the inverse ratio of the diameter; hence the critical velocity being .74 for D=.04, it should be about .03 for D=1.0. For low velocities the experiments of Jacobson and Reynolds confirm the general accuracy of the form of our curves on Plate XIV. ; for very high velocities the experiments of Reynolds indicate that continually increases with v, the pressure increasing with the 1.722 power of v, while we are of the opinion that becomes nearly constant. Professor Reynolds appears to think that the effect of temperature is by far more notable when the motion is parallel, than when it is eddying ; this agrees with our conjectures as to the effect of 7. One of the Reynolds series indicates that the curve for , formed by using n and v as co-ordinates (as we have plotted our experimental data), has an abrupt break the instant parallel motion ceases.t Although this may be quite possible for a short smooth pipe, we do not regard it as definitely proved by Professor Reynolds ; his methods for determining s and v can only be regarded as approximations ; especially for the determination of s his piezometric measurements may have been quite misleading. The only trustworthy method of obtain- ing s and v for very low velocities is as follows: The pipe or pipes employed should be of lead, having a length of at the very least 100 feet for a diameter of one-half inch, and a still greater length for larger diameters. The discharge should be submerged, and the outlet and inlet tanks should be placed side by side; the respective heights of the surface in these tanks should be observed by hook-gauges attached to * Eytelwein, we believe, first proposed the use of ‘‘ fractional” exponents. + It may be remarked that with the Darcy experiments, the irregularities in the curves for are generally greatest when the velocity was less than .7. 276 PIPES.—ForMULz IN USE. the same vertical post, so that their zeros can be quickly and accurately compared ; changes in inclination should be obtained by successively lowering the outlet tank. A trumpet-shaped mouth-piece should be attached to the inlet end, by which means the effect of contraction at the entrance can be nearly elimi- nated. The head absorbed in imparting velocity can be obtained with sufficient accuracy by a9 Q should be determined by the measurement of qg in vessels of proper size; an iron pail in the form of a cone trun- cated near its apex, such as we used for our Greenpoint experiments, is the most convenient kind of vessel for such a purpose. D should be obtained by the weight of water contained in the pipe. Care must be observed to prevent accumulations of air in the pipe, and the curve or curves in the pipe should be smooth and gradual. By such a method the value of n for a range of velocities from .01 to 2. can be obtained with the utmost precision. With such a pipe we hazard the guess that the curve for m will be almost perfectly symmetrical. The question of parallel movement in pipes as contrasted with eddying movement, has but little practical interest, as with conduit-pipes of ordinary size parallel movement in all probability ceases at a very low limit of velocity. In fact with such a pipe composed of many joints, each of which forms more or less of an obstruction, it is difficult to conceive of parallel motion, unless at almost infinitesimal velocities. Professor W. C. Unwin* has lately made some very interesting experiments in regard to the friction of discs having various conditions of surface, rotating in water having various temperattres ; one experiment was also made with a polished disc rotating in a syrup formed by dissolving sugar in water. These experi- ments showed that the friction of a polished disc diminished 18 per cent. by raising 7’ from 41° to 130°; the amount of friction increased rapidly as the surface of the disc was made rougher; for the smoother surfaces the resistance varied as the 1.85th power of the velocity ; and for the rougher surfaces with a power of the velocity ranging from 1.9 to 2.1. How close the analogy is between the friction of such rotating dises, or the friction of vessels moving through the water, and the movement of water through conduits, we are hardly prepared to say. * Proceedings Institution of Civil Engineers, 1884-85. The same experiments to which we have before referred. 277 CHAPTER IX. EXPERIMENTS WITH ORIFICES AND WEIRS. Cairornia, 1874-1876. Soon after the discovery of gold in California in 1848, associations or incorporated companies were formed for the purpose of building ditches and storage reservoirs for the supply of water to the placer mines. The amount of capital invested in these hydraulic works aggregated many millions of dollars, and a single company often sold water to hundreds of mining claims. The cost of water was by far the most important item in the miners’ bill of costs, and hence it became necessary to have a standard of measure not only accurate, but also so simple that the amount of discharge could be readily com- puted by the common miner. This was accomplished by the discharge of the stream of water sold to each customer through a rectangular, square edged, vertical orifice, with free discharge into the air, and having a constant head. In different parts of the State the standard opening varied ; the width varying from 2 to 4 inches, and the head above the top of the opening from 4 to 7 inches. Each square inch of the opening was called ‘‘a miners’ inch ;” hence in a locality where the standard opening was 2 inches wide, if the miner wished a flow of 50 miners’ inches the orifice was 25 inches long, and if only 10 inches was needed the length was reduced to 5 inches. This method is analogous to the pouce d’eau used in Southern France, and was probably first introduced or suggested in California by some French or Mexican miner ;* the simplicity of this mode of measurement combined with a sufficient degree of accuracy, soon brought it into general use on the Pacific Coast, wherever water was sold for mining, irrigation, et cet. . The standard, which had been in use since 1852 or 1853 in the mining districts supplied by the Eureka-Lake, Bloomfield and Milton water companies in Nevada County, California, was an opening 50 inches long, 2 inches wide, with constant head above opening of 6 inches; the flow from this was called 100 miners’ inches. If less water was needed, the opening was reduced in length by false pieces ; if more water was *Mr. John Dunn adopted this system of measurement at Nevada City, California, in January, 1851; so far as we can learn, this was its first application in the State, for the purpose of gauging or selling water. 278 MINERS’ INCH OF WATER.—Ca.Lirornta. needed, as many openings were used as was necessary ; for instance, if 350 miners’ inches was needed, three standard openings and one-half, or 25 inches in length, of another opening were employed. Generally the miners bought water for 10 hours per diem at an agreed price per inch; for example, a miner using 350 miners’ inches, for 10 hours each day, at the rate of 15 cents per inch, paid the water company 52.50 dollars per diem, and received the amount of water which would flow through orifices, having an aggregate length of 175 inches, a width of 2 inches, with a head of 6 inches above the top of the opening, during a consecutive period of 10 hours. When water was used for the whole 24 hours of the day, the flow was termed ‘a miners’ 24 hour inch,” and of course meant 2.4 times the amount of discharge of ‘a miners’ 10 hour inch.” . The shallow placer mines became in time exhausted, and the water from these ditches was used upon a much larger scale in the deep placers, where the gravel, stones and earth were washed away by great streams of water discharging from pipes under high heads. The old system of water measurement became objectionable on account of the large size of the measuring tanks necessary to afford sufficient space for the openings, and a new standard or module was adopted, being the discharge from an orifice, 12 inches wide, 123 inches long, with a constant head of 6 inches above the top of the opening; the discharge from this orifice was roughly assumed to equal that from two of the old standards, or 200 miners’ inches. It was desirable to know the exact discharge through each of these modules, and the author made at Columbia Hill in Nevada County, California, the experiments we are about to describe, the cost of the measuring apparatus being defrayed by the three companies before spoken of. APPARATUS AT CozLumpBra Hitt. A vertical section of the wooden canal, measuring tank, et cet., is shown by Fig. 7, Plate XV., and the horizontal plan, showing form of approach, et cet., by Fig. 8, Plate XV. The feeding reservoir had an area of several acres, and was supplied by the flow from a higher reservoir with a nearly constant stream during each experiment. There was some little wind during the experiments, which caused more or less fluctuation in the surface of the approach to the orifices or weir. The hook-gauge, reading to.001 inch, was placed 7.6 feet from the weir, as shown by the plan, in a wooden box, having a number of small holes bored through its outer side from top to bottom, so that the interior surface level quite speedily responded to fluctuations in the channel of approach; the openings in the gauge box were, however, not large enough to permit the interior surface to be much disturbed by local wave movements. The gauge was read at regular intervals of from 10 to 30 seconds, depending upon the total length of time of the experiment; means were deduced from these several readings, for orifices by the 4 power of each observation, and for the weir by the 3 power EXPERIMENTS WITH ORIFICES AND WEIRS. 1874-6—Apparatus. 279 of each observation. This method was thought to be preferable to that sometimes followed of having one small avenue of communication between the feeding canal and the gauge box. The inner depth below crest, G, for the weir was 3.8 feet; this was also very nearly the distance below the bottom of the several orifices experimented upon. The feeding canal was 7.3 feet wide, at the dam where the various openings were placed. The water, after its escape from the opening in use, passed into the lower wooden canal or flume, dropped down 2 feet at the step A, and discharged at the end of the flume C, until the commencement of an experiment. Gratings or racks were placed as shown by the section. There was an over-fall on each side of the flume, each 16.5 feet long, with the crests level and at equal elevations; these crests were 1.3 feet above the bottom of the flume. The stop-gate, B, was a sliding wooden gate, which could be put in place in about the space of two seconds. The measuring tank was a rectangular box, made of narrow, well seasoned, planed pine plank, with an inner horizontal section of about 9 x 17.3, and a depth of 8.5. It was held by heavy outer timber frames well braced, and firmly tied each way by iron cross-rods. In spite of these precautions the tank became a trifle larger when it was filled than when it was empty; this was determined by several measurements on the outer sides from fixed points, and amounted to an average increased size of 2.14 cubic feet in 1874, and in 1876 to .86 cubic foot. The leakage of the tank was ascertained by filling it with water, and noting by a hook-gauge the fall in its surface per minute. In 1874 this leakage amounted to an average of .078 cubic foot per minute while the tank was being filled; in 1876 this leakage amounted to an average of only .03 cubic foot per minute. Low-water mark was fixed by two upright sharp steel points placed in opposite corners of the tank, and whose summits were at precisely equal elevations; before each experiment the water surface was adjusted by means of a draw-off cock with great exactness to these summits. Immediately above these points two wooden boxes were placed, open at the bottom, and having sharp steel hooks placed near the top of the tank ; the points of these hooks were in a vertical line above the lower points, and the height of the water in the tank was the distance between the lower point and the upper hook. The horizontal section of the tank was very carefully measured in 1874 and again in 1876, by 8 sections one way and 5 sections the other way. The tank had been very carefully made, and the greatest difference between any two measure- ments for the length-wise sections was only .022. The tank capacity, when filled, in 1874 was, 9.0319 x 17.3494 x 8.1075 =1270.48 cubic feet; less spaces occupied by tie-rods and boxes=2.31 cubic feet; making net capacity 1268.12 cubic feet. In 1876 capacity, when filled, was 9.0442 x 17.3620 x 8.1100 = 1273.47 ; less tie-rods, et cet., = 2.31 ; making net capacity 1271.14 cubic feet. 280 EXPERIMENTS WITH ORIFICES AND WEIRS. 1874-76.—Apparatus. A second hook was placed between the lower point and upper hook, so that when desired, measurements of volume could be made with either the upper or lower section of the tank, When the plate or frame forming the weir or opening at the dam, D, was in place, and the water in the reservoir brought to the desired level, the false-gates in the feeding canal were removed and the water passed through the opening, discharging at ©. Low- water level was then accurately adjusted in the measuring tank, and the gate at B was closed ; in a few seconds the incoming water filled the flume from A to B, and the time was taken when the water began to spill over the over-falls into the tank; the end of the experiment was the moment of time the water reached the fixed hooks near the top of the tank. An observer was stationed at each of these hooks; the times were determined to 4th second. The discharge was hence the net capacity of the tank, plus the amountof water above the over-fall level contained in the flume from Ato B.* The height of the surface above the over-fall was measured at several points for each ex- periment ; in Experiments (Weirs) Nos. 66-68 there was more or less uncertainty as to the exact mean height, because of the boiling of the water; any errors from this source even with No. 68 (Weirs), when @ was largest, could not have exceeded 2 or 3 cubic feet. The boxes enclosing the hovuks kept the surface of the interior water reasonably quiet, so that the time of filling to the hooks could be gotten very exactly. As the boxes were open at the bottom, the water level at the hooks fairly indicated the mean surface level in the tank. An abstract of the notes for Experiment No. 68 (Weirs) will show the methods of observation followed : Date, November 8th, 1876, The total head, 4, was 1.7327 taken from the mean of 7 readings, varying from 1.7296 minimum to 1.7362 maximum. Time, 71.7 seconds. Discharge. Capacity of tank ... abs ies sie ss 1271.1 Above over-fall in flume... ae 4g ee + 42.0 Leak of tank As a ie cnn st + 0 » 9» gateat B we en 0 a3 + 0.1 » 5, damat D ake rere — 0.1 Splashing over sides of tank... as ses + 0.3 1313.4 1313.4 =a7-= 18.318 cubic feet per second. Water approaching weir with surface velocity of .68 ft., from reservoir to beginning of surface curve at weir. The small leakages, as just noted, were generally directly measured; they were * The comparatively small quantity of water in the air, between the over-falls and the surface in the measuring tank, was not fully taken into account ; this error was nearly balanced by the depression formed by the surface curves approaching the over-falls. EXPERIMENTS WITH ORIFICES AND WEIRS. 1874-76.—Apparatus. 281 largest in Weir Experiment No. 66, when they amounted in the aggregate to 2.2 cubic feet. In all the experiments with orifices the velocity of approach was too inconsiderable to be worthy of attention, and hence H=h. For Weir Experiments 64-68, effect of velocity of approach was doubtless slightly sensible for Nos. 64 and 65, and for Nos. 67 and 68 may have increased the discharge } of one per cent.; as these last named experiments can only be considered as approximations, owing to the small capacity of the measuring vessel compared with @, no attempt will be made to correct them for v,. The standard of measure was a ‘“ New York” level rod, which agreed closely with the standard in the U. 8. Coast Survey Office in San Francisco. EXXPERIMENTS WITH ORIFICES. Columbia Hill. Small Heads. The discharge through the old module, 50 inches long and 2 inches wide, was determined by three experiments with an opening cut as true as possible in a pine plank, 3 inches thick, with the lower side or crest 1 inch broad, and then beveled at an angle of 45°. The inner corners were square with the inner face, and the escaping vein only came in contact with these corners. The length was 4.169, being .0023 in excess of the standard; the width for Experiments (~) and ()) was as nearly 2 inches as the means of measurement at hand—the broad end of a carpenter’s square corresponding to the standard—permitted its determination; the swelling of the wood reduced the width of the opening a trifle for Experiment (c), as shown by the gauge fitting more tightly in the opening than it did after Experiments « and b. This diminution in size will be neglected. The time, ¢, is the mean of the determinations by the two observers at the measuring tank. TABLE XCILI. Determination of Value of Miners’ Inch. Eureka Lake Standard, 50 Inches long, 2 Inches wide, 6-Inch Head above Top of Opening. Full Contraction. (2 9)4=8.018 Temperature of water about 65°. | | Orifies Size of Opening. | | Total Discharge. - a i ee ae lle i we Tank. | Flume. | Leaks, | Total. | (a) | 4.169 | .16667{ .5827 | 488.7 |1268.12] 11.41 | 1.36 |1980.89 | 2.621 |.6163 | 6169 (6) i me .5855 487.7 | 1268.12 11.41 1.97 1281.50 | 2.628 '.6165 |.6170 (c) ” ” 5888 488.0 | 1268.12 11.41 1.30 1280.83 2.625 | 6140 6146 Means | |.6156 |.6161 1 4.1667 | .166 67 | .5833 | 2.618 | .6156 | .6161 00 282 EXPERIMENTS WITH ORIFICES. 1874-76—Small Heads. In the preceding table Q=C (2gh)¥lw; and Q=c 21 (29)% (H,"—H;"); H,= At+Ss H=h—-> . Itis assumed that the value of c for module—No. 1—will be the mean of (a), (b) and (c), notwithstanding the slight variations in h. The lower value of ¢ for No. (c) than for Nos. (a) and (b), was doubtless chiefly caused by the slight diminishment of size before alluded to. The chief danger of error in No. 1 is in the value of w; probably in this case the limit of error for the value of ¢ is not far from z4,>. The discharge through the new and larger module, called the Bloomfield standard, was determined by two experiments. An opening with sides made as nearly square as was possible, was cut through a plank .12 thick, fastened to an outer plank of the same thickness having a still larger opening. The discharge was perfectly free into the air, the escaping vein only coming in contact with the inner corners. For No. (d) the length was 1.063, and the width 1.0005; No. (c) was made the day following, and the dimensions were then, /= 1.0655 and w=1.0018. The co-efficients and discharge for the exact module with /=1.0625, w=1., H,= .5, H,=1.5, and h=1. are obtained in the same manner as for the Eureka Lake standard (No. 1). TABLE XCITI. Determination of North Bloomfield Standard, called 200 Miners’ Inches, 123 Inches long, 12 Inches wide, 6-Inch Head above Top of Opening. Full Contraction. (2 9)4%=8.018 Temperature of water about 50°. Orit Size of Opening. eee ; Total Discharge. | 3 2 . t w Tank. | Flume. | Leaks. | Total. | (d) | 1.0630 | 1.0005 | 1.0035 255.1 | 1271.1 17.8 0.7 1289.6 || 5.055] .5918] .5983 (e) | 1.0655 | 1.0018 | 1.0029 253.7 | 1271.1 16.7 0.7 1288.5 | 5.079} .5926| .5992 Means | 5922) 5988 ® | 1.0689 | 1. i | | 5.045 | .5922) .5988 | In the foregoing experiments there was a surface current opposite the hook- gauge of .105; an allowance for this velocity of approach would not change the last decimal in the above value of ¢ one point ; 2.., about rodo5- Two indirect measurements by weirs were made by the author, of the discharge through a number of these Bloomfield modules placed side by side, the intervals separating them being about 6 inches long. In both cases there seemed to be a sensible increase in the discharge ; that is to say 10 adjoining openings discharged more than 10 times the amount a single opening would discharge. These experiments were not made with sufficient care to absolutely determine this point, and hence will not be given here. D’Aubuisson states that with three orifices placed side by side the co-efficient ¢ was larger than when only one orifice was used. On the other hand, Francis’s experiments with two 4-foot weirs, separated by an EXPERIMENTS WITH ORIFICES. 1874-76.—Small Heads. 283 interval of 2 feet—Weir Experiments Nos. 50 and 56—show a slightly lower value of ¢, than would probably have been the case with one weir of the same length having full contraction, as will be seen by an examination of Plate VI. Bazin’s experiments with several adjoining orifices indicate a considerable increase in the co-efficient, when the number of openings was increased.* The 4 following experiments were made with orifices nearly circular cut through thin sheet-iron plates; these plates were firmly bolted to an outer plank having a somewhat larger opening, so that there was no warping on account of the thinness of the iron. The edges were made as nearly square as was possible by dressing with a fine file. ite cone Thickness of Diameters. | | , “Maximum. | Minimum. | Mean. | 3 .0090 2542 | «95950 | S583 4 0085 419500 A1TT | A185 5 | 0085 6637 | 6615 6627 6 | .0060 1.0110 | 1.0070 | 1.0089 The mean diameters were deduced from 6 or 7 measurements for each orifice and were thought to be very near the truth. The same care was taken in making these experiments, as was practised with the preceding experiments, but they cannot be considered as reliable, as they were single determinations. TABLE XCIV. Flow through Circular Orifices, with free Discharge into Air. Full Contraction. (2 9 )¥=8.018 Temperature of water 50° to 55°. 5 Total Discharge. f | ! H=h | | t | qo Le Hi ce | | Tank. | Flume. | Leaks. Total. Size. ; | | 1 Orifice No. Mean Mean Diameter) Area D a : i 1 ea a a | a ee ee — 3 2533 | .05039| 1.3315 1832.4 | 516.0 2.9 0.9 519.8 2837 | .608 poet 4 4185 | .1376 | 1.2045 | 1749.0 1971.1 5.4 Les eee 7305. 6035, .6041 5 6627 | .8449 | 1.0887 || 750.4 | 1971.1 | 8.3 0.6 1280.0 /1.706 5913 5929 6 1.0089 | .7994 | 1.0457 | 333.9 | 1271.1 | 13.0 0.3 | 1284.4 lone 5869 5913 | The co-efficient Cis obtained by G@=C (2 gh)’ a. The co-efficient ¢ is obtained from Table IV., giving the relative values of C and c for circular orifices. * Recherches Hydrauliques. Darcy and Bazin, p. 61. 284 EXPERIMENTS WITH ORIFICES. 1876.—Small Heads. A small whirlpool or vortex kept forming and re-forming on the inner side in No. 6. In all the other experiments with orifices, the surface of the water above the opening was quiet, or free from any apparent whirling movement. The values of ¢ for these 6 experiments will be found plotted on Plate III. Weir Experiments.—Colwinlia Hill. The weir experimented upon was a plate of boiler iron, .026 thick, with the crest .008 wide. The opening was 1.754 deep; the length on top was 2.584 and on the bottom 2.587. The edges were cut square with the inner face, and were made as true as could be done with a fine flat file. These experiments were conducted with the same care used in making those with orifices. The inner depth, G, below crest was 3.8, and the width of the feeding canal at the weir was 7.3. Hence for Experiments Nos. 57 to 64 there was practically complete contraction, while for Nos. 65 to 68 there was a slight partial suppression, but hardly enough to warrant corrections being applied. As before stated, velocity of approach will not be taken into account, although its effect was doubtless appreciable for Nos. 64-68. For No. 68 the central surface current, from the reservoir to where the surface curve at the weir became sensible, was .68. TABLE XCV. Sharp-Crested Weir, Free Discharge into Atv, Very nearly Complete Contraction. (2 g)4=8.0177 Temperature of water 50° to 60°. Wer | Mean | Total Discharge. Wa. A Length. t QO; C l Tank. | Flume. | Leaks. Total. DT 5659 2.586 358.4 (aa | [235 | 0.2 1283.8 3,582 .6087 58 .6163 2.586 | 318.7 T2a1 14.4 0.2 1285.7 : 4.034 .6032 59 .6470 2.586 296.8 1211 16.0 | 0.3 1287.4 4.338 .6030 60 .6703 2.586 | 281.9 1271.1 16.1 0.2 1287.4 4.567 .6020 61 O72 2.586 | 260.3 1271.1 17.1 0.3 1288.5 4,950 .6021 62 1.0681 2,586 144.2 127 h.1 24,3 0.6 1296.0 8.988 . .5890 63 1.1063 2.586 137.3 1271.1 26.8 | 0.3 1298.2 | 9.455 878 64 1.2033 2.586 120.7 1271.1 30.1 | 0.3 1301.5 | 10.783 .5910 65 1.3257 2,585 106.6 1271.1 33.5 0.3 1304.9 i 12.241 5804 66 1.5391 2.585 83.8 1271.1 35.2 ee 1308.5 f 15.614 | .5918 67 | 17195 | 2.585 || 72.1 | 1971.1 40.4 0.3 131L8 18.194 | .5840 68 1.7327 2.585 Te (27 1.1 42.0 0.3 1313.4 18.318 .5812 There was some little question as to whether termination of No. 62 was very exactly determined or not, and hence No. 63 was made with nearly the same head, which showed that no serious error had been made in No. 62. EXPERIMENTS WITH WEIRS. 1876.—Columbia Hill. 285 Experiments Nos. 57-63 can be considered as reliable, and in these C' is practically identical with c. Nos. 66-68 can only be considered as approximations for the reasons before given. C is obtained by Q= C3 (29 M)*41 4H. All the foregoing experiments were made at Columbia Hill, California, in lat. 39° at an elevation of 2900 feet above sea level. The apparatus employed was well suited for the exact measurement of Q up to a maximum of 9 or 10, and it was the intention of the author to have made a large number of additional experiments with weirs from .866 to 2. in length, and with square and round orifices of various sizes with heads up to 3 feet. Unfortunately, professional engagements prevented the completion of the contemplated investigations. EXPERIMENTS WITH ORIFICES. North Bloomfield and French Corral. Great Heads. In July, 1874, the author made a series of experiments at North Bloomfield, California, and also one at French Corral in the same State, for the purpose of de- termining the effective duty of water-wheels, known as hurdy-gurdies, which are almost exclusively used for water-motors in California.* Incidentally measurements were made of the discharge through tapered nozzles of various forms, some of them having divergent adjutages, and through annular square-edged openings, where contraction was more or less perfect. The measurements of discharge and of the mean diameters were not made with the highest degree of accuracy, so that the following results may be in error as much as 3, or possibly 4 per cent.. They have sufficient value, however, to warrant their insertion here, as the heads employed were exceptionally great. The several nozzles and rings were placed at the end of a discharge-pipe, which was fed from a long main. The level of the water in the pen-stock at the head of the main was known, as was also the elevation of the nozzle. More or less water was being drawn from the main while the experiments were in progress, and hence a portion of the total head was absorbed in friction through the pipe. This consequent loss of head for Nos. (Orifices) 7-17 was determined by a new Bourdon gauge, placed at the upper end of the discharge-pipe at a known elevation.t The readings from this gauge were corrected by closing the openings in the main, and noting what difference there was between the pressure shown by the gauge, and that due to the head ; these corrections varying from 1.6 to 3.2 lbs. were then applied to the gauge readings. By this method the actual or effective heads at the nozzles were determined with comparative certainty to within 2 feet. As the least head employed was 312 feet, errors from this source would not amount to more than 3J,. The discharge was measured over a sharp-edged rectangular weir, with very true * A detailed description of several forms of the hurdy-gurdy will be found in the Transactions of Am. Soc, of C.E., February, 1884. + This gauge was placed at an elevation about 5 feet higher than the lower end of the discharge-pipe. 286 EXPERIMENTS WITH ORIFICES. 1874.—Great Heads. sides, made of + inch boiler iron, edges beveled on the outer side, and .8660 long. The water escaping from the nozzle was conducted by a square flume to the weir, the current being checked by a rack and cross boards; the weir was placed at the end of the canal, and there was nearly complete contraction. The height on the weir was read by a hook- gauge, reading to .001 inch, placed sufficiently far from the weir to be out of the influence of the surface curve. In Nos. 15, 17, 7 and 9, the discharge is thought to have been less accurately measured than in the other experiments, as the water in the feeding canal for the weir for these Nos.—15, et. cet.—was more turbulent than in the succeeding experiments, where the surface in the canal was quite quiet. The forms of the nozzles used are shown by Figs. 1, 2, 3, 4, 5 and 6, Plate XV. The discharging nozzles A, B, C and D all screwed with a very nice fit into the larger nozzle HI, which screwed into the lower end of a long discharge-pipe of smooth sheet-iron, having a regular taper, with a maximum inner diameter of about .6. The larger nozzle, H I, was of hard cast-iron, first accurately turned and then smoothly polished; its sides were very nearly straight—its inner form being nearly the frustrum of a cone—having a convergence of about 8° 25’. The discharging nozzles A, B, C and D were also of polished cast-iron, very smooth, and with sides slightly curved. The rings E, F and G were of thin saw-plate steel, held in place by annular screw- caps. The pipes k L and MN, to which these rings were attached, were of smooth cast-iron, but not turned. The sections of the large nozzle H I and of the discharging nozzles and rings, were as nearly circular as they could be made by a skilful mechanic. They were kept perfectly free from rust by rubbing with oiled cloths, the oil being wiped off by a clean cloth at the beginning of each experiment. The surfaces were hence probably smoother than would have been the case had the nozzles and rings been made of brass or any other comparatively soft material. The diameters of the nozzles A, B, C and D were measured at their smallest sections by calipers, and then determined by a finely graduated scale; the holes through the rings were cylindrical in form, and were measured directly by the same scale. This method of obtaining diameters of such small size was not sufficiently accurate, especially with A and E, where a slight error of measurement would sensibly affect the value of the co-efficient of discharge. After C with its divergent mouth-piece or adjutage had been experimented upon, it was cut off close to its minimum section at , and again used as C’. The temperature of the water was not taken ; it was not far from 60°-65°.. In the reductions a cubic foot of water is assumed to weigh 62.4 lbs... The discharge over the measuring weir has been re-calculated from the original notes, the co-efficient ’ (in this case c,) being taken from Plate VII. The original notes are not at hand for No. 18; the total head for this experiment, however, has been corrected properly for head lost by friction in pipe. 287 1874.—Great Heads. EXPERIMENTS WITH ORIFICES. TAOX GTAViL : i | : 179° F'06 f2eL | Ve “LOST || Fo°8¢ OT 960° e6sl seg Tt (puey : ye jou rine 9 81 | | | | | e99° LIFL Cele | L9G | FLO Geel TPES — 6'8EE GE9 G00" L¥80°° LRGG" GZ09" | OOS E" a A LT | g99° STF : Mele | E96 | SLO Peel UFée | 6'8EE &€9 G00" LFS8O" | 8069" | FE09" | 60EE" a J | Ot g19 PEF | EOE . HEE | FTE O'SET TPE | 68EE 664 600° 4660" | FEte | LEO | OG6T ss a | gt | | SRury | FIOT Fel grote €€6 | SOE Ayer LFEE = GREE €6T 800° L101 | LELTT) G16G° | F89g _ a TI I10'T OcTL ,GLTE | OTE) Lele GEL LTEE | COREE €G1 800° ATOV (OFLTT 16S | ggg - a él c00'T OLFT | ONE | FE LT&e Cert GEE | FbEC L16 400° 8980° | OFL8 | Gee | Toe . O él L00°T OFT LEGE | 99 W Lee OGFI PPS | e OEE LT6 GOO" 8980° | E148 | eu6g | 1F9F 7 0 Il 900°T VeFL | VOTE | Vee | CTE — GCFET THEE GREE L16 G00" 8980 | PRPS | ge6G | RGGE | i 0 OL 936° LUFT | Cele | L9G | F LOE eel LES |G NES F219 400° ogs0" | 9c6L' | 9960 | Toet | a ad 6 F00'T Gol “ eTte | Fe | L60€ GTEL Lhe | 6 Ree F19 G00" 080° | OOI8’ | $964 | olfr | * a 8 OFO'T OTTT i eece | 9OTL ] GLTE 9 LET LPEe | 6 SEE | S16 G00" Teg" | 9TEE | FOLD | L6Ee" . O99N" Vv 4 . a f y |peogt) “peeHL | ‘sqry ut | -qoog ur 29 o=e ou "PeaT] [Jo sso] poyeorpuy eamnssarg | peo ‘pee | 7 2 a @ ara |\yen| 3 ‘eBreyo xy 5) ri is : gil, : eee Se aqreTg ‘a[ZZON| “ON -siqq yo | “Aq10010,/ BA (poqoert0g) a8ney-eansserg ion uae quatorga | [Roar | oe ars -09 “OanL | “peoy{ oatoayy Jo woryeururdeqeq, hice ae “Ia AA Sutansvoyl slo's = 44 &) “pig, ‘Aine wiusofyng ‘paywong yosr ‘sbuay pun sajzzoary ybnosy) abinyoseg 288 EXPERIMENTS WITH ORIFICES. 1874—Great Heads. The foregoing experiments were originally calculated by a modification of the formula of Mr. Francis for weirs,* which gives considerably lower values for Q with small heads, than does the interpolated curve for /=.866 taken from Plate VII. We now make the co-efficient of discharge for nozzle A, c=1.040, and for nozzle C’,c=1.006. It seems altogether improbable that ¢ for either of these nozzles, or in fact for any of the nozzles, should be above .998 ; no probable error in the measurement of the diameter of A would account for this difference of 4 per cent.. It may be that the curve for c, with 7=.66, which has been drawn on Plate VII. from Lesbros’ experi- ments, and from which the preceding values of c, have been deduced, is too high, and that the lower values of c,, as shown by the experiments of Poncelet and Lesbros (Weir Nos. 35-40) and of Castel (Weir Nos. 187-194), are nearer the truth than the determi- nations of Lesbros. Had these lower values of c. been used in the determination of Q, the final results in the foregoing table would, apparently, have been much more satis- factory. It is to be regretted that in these experiments Q was not obtained by absolute -measurement. Any indirect calculation of @ is always more or less unsatisfactory, unless made over a weir, or through an orifice, whose co-efficients of discharge have been ascertained under precisely similar circumstances by measurement in a vessel of proper size. Despite their imperfections the foregoing results are sufficiently accurate to show : First.—With great heads and smooth converging mouth-pieces, the co-efficient of discharge, c, is nearly 1. Secoud.—With great heads, short divergent adjutages have but little effect upon the discharge. Third.—With great heads and small orifices, with complete contraction ¢ will be about .60, and the size of the feeding channel probably need not be as great in propor- tion to the orifice, in order to avoid partial suppression, as is the case with small heads. It may be incidentally observed that with equal quantities of water, the discharge from the rings gave a slightly higher useful effect upon the water-wheels than resulted when the nozzles were employed. Experiment No. 18 was made at French Corral, California, at an elevation of about 1600 feet above sea-level ; all the others were at North Bloomfield, the elevation of the orifices being 2950 feet above sea-level. The greatest danger of error in Experiments Nos. 1, 3 and 4 with orifices, undoubtedly consists in incorrect assumptions of the value of «. For Nos. 7-18 (Orifices) danger of error perhaps about equally rests in incorrect values of a and of Q. The linear measurements for these areas were made with considerable care, by a finely divided ivory * The original computation is given on page 18 Transactions Am. Soc. of C.E., 1884. EXPERIMENTS WITH ORIFICES. 1874—Great Hedls. 289 scale ; this scale was compared with, and corrected by, a long box-wood rod, which in turn had been compared with a steel scale in the U.S. Coast Survey Office. This was a vicious method, as subsequent experience has fully shown us. Errors, with the use of several scales roughly compared, may very likely be all in one way ; this is especially true of personal error. The only proper method in obtaining such small linear dimensions, where great accuracy is required to avoid notable comparative error, is by direct comparison of the orifice with the final standard. It has been shown that the co-efficient ¢ for No. 3 is apparently 1 per cent. too great, by comparing ¢ for No. 3 with the Holyoke experiments. Supposing this error to be entirely due to an incorrect value of a for No. 3, it would involve an error of .0012 in the measurement of the mean diameter. This is a considerable quantity, but we now can conceive it to be a possible error, as we have of late seen more than one instance where errors of several comparisons have all been in one direction. Very likely the steel scale, which was not very finely divided, in the U.S. Coast Survey Office, may have been slightly different from the standard of Professor Rogers.* With such large orifices, as those employed for Experiments Nos. 2, 5 and 6, the danger of comparative error in incorrect measurements of the dimensions of the orifice, is, of course, very much less than for the smaller orifices. * A slight error in the reference standard would not affect the Columbia Hill experiments, nor in fact those at North Bloomfield. Because the measurement of g was made by the same standard. 290 CHAPTER X. EXPERIMENTS OF FLOW THROUGH PIPES. Tue following experiments were made by the author several years ago, the final results of which have been published in the Transactions of the American Society of Civil Engineers, 1883. A detailed description of the methods of observation employed will here be given, to enable other hydraulicians to determine what weight should be given to the several experiments. New ALMADEN EXPERIMENTS. In the month of April, 1877, 53 experiments were made at New Almaden, Calli- fornia, with small pipes of wood, old wrought-iron, new wrought-iron, and glass, with velocities from .9 to 6.9. The object in view was to determine the general laws governing the discharge through pipes, with widely varying conditions of the inner surface, and with varying velocities. It was thought that very exact measurements with small diameters would give more reliable data, than rougher experiments with large diameters, where it was impracticable to determine the discharge by absolute measurement, on account of the considerable cost of building measuring vessels of proper size. The apparatus employed at New Almaden is shown by Figs. 1, 2 and 3, on Plate XVI., the discharge in all the experiments being perfectly free into the air. The measuring vessel was a rectangular wooden tank, of planed, tongued and grooved red-wood plank, held by outside wooden frames securely bolted together, so that there was no perceptible enlargement when the tank was filled. A cock was placed at the bottom, so that the water could be drawn off, and accurately adjusted to the low-water mark. The measuring points for height were as follows: At the bottom a vertical sharp steel point, A, to the summit of which the water was brought with great precision at the beginning of each experiment. A sharply pointed steel hook, B, distant vertically 1.726 above the datum point. A similar hook, C, near the top of the tank, and distant vertically about 2.15 above B. EXPERIMENTS WITH PIPES—New Atmaven. Appuratus. 291 The tank could therefore be divided into two sections. The capacity of the lower section (1) was as follows, each dimension being the mean of 12 measurements ; (L) April 12, 1877... .. 1.9712 x 1.9918 = 3.9263 ae eB ye Ay a .. 19739 x 1.9955 =3,9389 Mean ... Se say wet 3.932 x 1,.726=6.787 Grooves in corners, .022 x .0l1x1lL7x4= + .002 Space in water-cock, .17 dia. x .44 =.010, one half filled + .005 Capacity (L) = Ais a a ot ... 6.794 cubic feet The discrepancies between the measurements of April 12th and 16th, were partly due to slight changes in the tank, and partly because the sections were not taken at exactly similar points. The capacity of the upper section (U) was determined in the same way, and was as follows ; (U) April12, 1877... 00... 9814 x 1.9701 = 3.9036 si EG ype ee et L9777 x 1.9722 =3.9004 Mean ... 3.902 square feet. From April 10th to 14th the height of (U) was 2.157, and from April 15th to 17th 2.153 ; there was a + correction for grooves in corners of .028x.011x2.2x 4=.003 cubic foot, hence ; April 10-14... w+ (3.902 x 2.157) + .003 = 8.420 cubic feet » 15-17 0... (3.902 x 9.153) 4.003=8.404 ,, ,, The total capacity (T) of the tank was therefore ; (T) April 10-14 ve es 6.7944 8.420 =15.214 cubic feet » 15-17 fs ob 6.794 + 8.404 = 15,198 ” 19 There was a slight leakage from the tank, which was carefully determined twice each day, and a table of corrections prepared, which it is not necessary to give here. The maximum leakage was on April 10th, as follows ; When (T) was used .0027 cubic foot per minute ” (L) ” ” 0011 ” ” ” The minimum leakage was on April 17th, as follows ; When (T) was used .0010 cubic foot per minute ” (L) ” ” 0003 When everything was in readiness for an experiment, a tin trough, which carried the escaping water from the end of the pipe over and beyond the tank, was quickly removed, and the discharge allowed to enter the tank. In order to prevent commotion at the surface, when the water approached the measuring hook, an assistant caught the jet in a tin pipe of 4 inches diameter, the lower end of which was held some distance 292 EXPERIMENTS WITH PIPES.—NeEw ALMADEN. Apparatus. below the surface ; by this means the exact moment the surface level reached the hook could be very closely determined. For the small pipes the discharge was sometimes measured by two buckets, one of which (B) held 94.27 lbs., the weight being ascertained by a Fairbank’s platform bullion scale, sensitive to 3th oz. This scale was also checked by a similar scale; they were both known to be exact for a weight of 763 Ibs. (flask of quicksilver). The tempera- ture of the water was from 57° to 68° Fahr.; hence weight of 1 cubic foot = 62.35 lbs., and capacity of (B) = 1.512 cubic feet. The other bucket (b) held 29.60 lbs. =.475 cubic foot. This bucket measurement was quite imperfect compared with the tank, but the limit of possible error with them was not over 1 per cent., with the chances of error (mean of three measurements) probably under one-half of 1 per cent... This was deter- mined by using first the tank, and then the bucket, for the same experiment; for instance in No. 316, mean of two tank measurements gave @=.008 31, while by bucket @ =.008 35. The times were determined by a large stop-watch (T. S. Negus), which had a + rate of 359; hence a minus correction of .0004, 7.2, t’x.9996, had to be applied; the beats were 5 to a second, hence 4th second was the limit of accuracy for time. The water was fed to the pen-stock through a 24-inch hose, connected with a hydrant having a high and nearly constant head ; hence a very regular supply could be obtained. The disturbing effect of the water as it came from the hose was checked by the cross plank shown on the sketch. A waste-way or weir was placed on the side of the pen-stock, and a small quantity of water allowed to pass over it in all the experiments. An engineer’s level was set at a point equidistant from the pen-stock and the end of the pipe, and the difference in elevation determined before and after each experiment, the level rod being held first on the weir, and then on a sharp nick or bench-mark cut on the extreme end of each pipe. The elevations from the centre of the pipe to these points were measured, as also the height of the surface of the water in the pen-stock above the over-fall; these last measurements being taken first at the commencement, and then at the ending of each experiment. The total head, H, hence was always the difference in elevation between the surface of the water in the pen-stock, and the centre of the discharge end of the pipe. Two distinct measurements were made for each experiment when the tank was used, and three when the buckets were used. It would occupy too much space to give all these figures in detail; a copy of the notes for Experiment No. 287 will show the methods of observation. EXPERIMENTS WITH PIPES—New AtmapEen. Apparatus. 293 Head. | Water Point | Tine oxcupied ae Size in Cubic Feet Se No. oe all above Surface above | Total oe ee of Measuring Vessel.) Feet per sa i of above Centre | Head. | rome | Second. : Over-Fall. | of Pipe. | B4it 8446 031 7 4 39.8" 9 59.3"! (T) | 287} 196 _.200 | .034 J | 1232.6" 10" 51.8” | 8.248 8.246 © 7 52.8" 7 52.5” 16a 8247} «032s | «051 «(8330/4736 Ss Leakage | Rate =i 8x .0020 +.016 : 472.4" 15.230 052 24 ey Whenever there was any appreciable difference in the times, as was the case in two or three experiments, the measurements were repeated. The various inclinations were obtained by lifting the pipe, which rested on temporary brackets, the pen-stock not being moved. The glass pipe rested on planks, laid lengthwise. The pipes were all straight, except the glass pipes which were slightly curved, as hereafter noted. The standard of measure was a ‘‘ New York” level rod, which agreed very closely with the standard scale in the office of the U.S. Coast Survey in San Francisco, A steel tape was used in measuring the lengths of the various pipes, which (with average temperature during the experiments and a constant strain) read .018 too long in 60 feet; thus making a constant minus correction of .0003, ve., i’ x .9997, for each foot in length. The temperature of the water was from 57° to 68° Fahr. The water was almost perfectly limpid. The diameters of the pipes—except for the 4 inch glass pipe, and 14 inch wooden pipe—were computed from the weight of the water contained in them. This was determined by filling each joint, and weighing the total amount of water contained by the several joints constituting one pipe; the length being the sum of the lengths of the joints, and hence being somewhat shorter than the length of the pipe in place, on account of the spaces between the ends of the joints. Description of Pipes. Series .—A lap-welded iron, gas or water pipe, about 1 inch in diameter, which had never been used, and was almost entirely free from rust. Its inner surface was quite smooth, except immediately at the ends of the joints, where there was a little roughness caused by the cutting of the threads for the outer screw coupling. The ends 294 EXPERIMENTS WITH PIPES.—New Aumapen. Deseription of Pipes. were a trifle smaller than the interior section of the pipe. Just at the weld the dia- meter was slightly smaller, than for the other axis. All the iron pipes used, were connected by the usual outer screw coupling, the joints not butting squarely together, but having intervals from .01 to .04 in length ; no account was taken of these trifling enlargements. The diameters of the two ends of each joint were taken at a point about .1 from the end. The upper end of the pipe was just flush with the side of the pen-stock ; the co- efficient of contraction at the entrance was assumed at .800, on account of the slight roughness at the end. The length, /, in place was (60.190—.018) 60.172; the sum of the lengths of the five joints was (59.964—.018) 59.946. The arithmetical mean of the end diameters was .086 85, and by weight of water contained by the several joints the mean diameter was .0878 (vide Series ITI., for method of determination), thus showing that the central parts of the joints were appreciably larger than the ends.* Serivs I.—After the completion of the seven experiments with Series I., a funnel was screwed on the upper end of the pipe, projecting into the pen-stock as shown by Fig. 1, Plate XVI. This mouth-piece was 1. long; outer diameter .542 and inner diameter .086; it was of cast-iron not polished, but quite smooth; its form is shown by Fig. 3, Plate XVI. The length of the pipe was measured from the beginning of the taper of the funnel (the funnel proper not beiny included in the length, /), and was (corrected) 60.247. The diameter was .0878, being unchanged from Series I. The co-efficient of contraction was assumed at .98. ; Series I[[.—The pipe was then taken apart, and each joint immersed in a boiling bath of asphaltum and coal tar, thus giving the interior a very smooth and glassy surface, but which slightly diminished its diameter. The measurements to obtain mean diameter were as follows; Direct measurement. | Measurement by Weight. 086 25 Sum of lengths of tive joints ee oe 5O964 .086 42 Error of tape —~ ve he we os. 20I8 .O86 25 “59.946 O86 42 wees -086 50 | Sum of the weight of water in five joints... 22.42 Ibs. L087 33 Let, 086 25 | m=l|bs. of water in 1 cubic foot ... we = 62.55 086 08 Piength of pipe filled ... ae we = 59.946 086 00 D = required diameter. ates ee eee w= weight of water contained in pipein lbs. =22.42 Mean =.086 39 Hence, Da {_™ \ #5.087 30 | (= 7a * The maximum end diameter was .0875, and the minimum .0865 ; the mean, computed from end sections, was 086 85. EXPERIMENTS WITH PIPES.—New Atmapen. Description of Pipes. 295 The accuracy of the scales and of m (weight of cubic foot of water) had been’ verified by comparisons of the buckets (B) and (b) with the tank. The mean diameter was assumed at .0873, thus giving a slight value to the direct measurement. The length, J, of the pipe in place was, Lower end of pipe to funnel oo bi — 60.157 Funnel, from lower end to beginning of flow... 125 — 60.282 Error of tape... ow bah aes —.018 i = 60.264 Co-efficient of contraction at entrance was placed at .98. Series [V.—The first two joints of the pipe used in Series III., with the funnel mouth-piece. The mean diameter, as deduced by weight of water, was .0876; /, including cylindrical portion of funnel, was (corrected) 16.685 ; co-efficient of contraction, .98. Series V.—This pipe consisted of 4 joints of old lap-welded pipe, 52.7 feet long, and 1 joint at the upper end, for connection with the funnel, of new pipe, 7.4 feet long. The old pipe had been in use for 2 years, constantly filled with water, which had but little velocity. The water was from a clear spring, evidently containing lime. The interior surface was covered with a thin hard scale, with occasional small lumps, and was rather rough. It had never been coated with an asphaltum preparation; it was when new, doubtless of very nearly the same diameter as the pipe used in Series I. The short length of new pipe was smooth. The diameter was substantially the same, .0853, both by end measurements, and by weight of water ; the diameter of the old pipe was from .0846 to .0850, and of the new joint, .0871. The length, including cylindrical portion of the funnel, was (cor- rected) 60.247. Co-efficient of contraction at entrance was assumed at .98, as in Series IJ.-IV. Series VI-—A new 8-inch lap-welded pipe, free from rust, but not coated with asphalt, consisting of 5 joints. The upper end of the pipe was flush with the side of the pen-stock, and was dressed smooth, with sharp clear-cut edges ; .825 was adopted as the co-efficient of contraction. The mean diameter was, by end measurements .0515, and by weight of water .0524; .0523 was adopted. The measured diameters varied from .0508 to .0521. The length (corrected) was 60.127. Series VII.—A glass pipe, formed of 12 joints as follows ; 296 EXPERIMENTS WITH PIPES.—New AtmaDEN. Description of Pipes. Length. Diameters. 6.199 { .0875 = entrance. The largest diameter was .0875 ss » smallest ,, » 0719 6.342 { 7 The greatest difference between diameters at a 689% Joint was (.0844-.0825) .0019; generally they 4.100 { ‘O817 were pretty evenly matched. acne f 0822 The weight of water contained in the 12 joints 062 1 “0800 was by two trials 18,22 and 18,22. 4.607 { ae Hence D= /—” * 076 32 : T ee ao1t [ ott ~ as The measured length of the pipe in place (cor- 6.499 { ‘0719 rected) was 63.902; hence showing that the {0735 intervals between the joints were very small. 4.644 : : \ .0765 mye .0760 nee { 0746 0754 6.008 { “ors7 0746 4918 { “o746 f 0744 ai38 \ .0729 =discharge. Tape $e) «07751 = mean. 63.872 A diameter of .0764 was adopted, and /= 63.902. A funnel mouth-piece shown by Fig. 2, Plate XVI., having outer diameter of .422 and inner diameter of .0875, was fitted accurately to entrance end of pipe, the joint being very nearly absolutely exact ; the co-efficient of contraction was assumed to be .97. Series VIII—A glass pipe, formed of 6 joints as follows ; Length. Diameters. 6.070 { .0717 =entrance. The largest diameter was .0717 -0708 » smallest ,, 20D78 4.410 { 0679 The greatest differences between diameters at a se joint was (.0675 — .0633) .0042. 6.050 i ae The weight of the water contained in the 6 joints ‘aes was by two trials, 6.61 Ibs. and 6.59 lbs.; mean Pave { (0608 = 6.60 lbs. 6.145 { “per Hence *=( me — in : Im B08: | ean : = -0596 = discharge. The measured length of the pipe in place (cor- 34.943 06304 =mean. rected) was 34.941, ° Tape — .009 34,934 EXPERIMENTS WITH PIPES.—New Atmapen. Description of Pipes. 297 The adopted diameter was .0622, with 7/= 34.941. The end of the pipe was flush with the side of the pen-stock ; hence co-efficient of contraction =.82. Serics IX.—A glass pipe, formed of 2 joints, as follows ; Length. Diameters. .0450 = entrance. The water was not weighed in this instance 5.400 0495 5 ? f .0404 | .0£04 = discharge. the scales not being accurate enough for such 5.730 small quantities. As with the other glass pipes the mean diameters were smaller than the mean 11.130 .0421 =mean. of the end diameters, it was assumed that the Tape — .003 same would be true of this series. 1L.127 The diameter adopted was .0418. The measured length in place (corrected) was 11.127. The end of the pipe was flush with the side of the pen-stock, and hence .82 was again assumed as value of the co-efficient of contraction. Series VII, VIII. and [X.—The glass pipes were all of “ lead glass,” of German or French manufacture. Their sections were not perfectly circular. Some of the pipes were curved ; the one having the sharpest curve, being a joint of Series VII., which in a length of 64 feet, had an ordinate of .035. Hence it was not possible to lay the pipes in a perfectly straight line; the general direction, however, was very nearly straight, and the curvature was so slight as not to deserve special consideration. The ends of each joint were ground smooth; the joints were butted together, and a water-tight connection made by placing over each connection a piece of tightly fitting rubber tubing. This tubing made a perfectly water-tight joint, except in one or two cases where a few drops of water leaked, but the leakage was too inconsiderable to deserve notice. The inaccurate fits at the joints, amounting in one instance in Series VIII. to a difference in diameters of .0042, doubtless somewhat added to the friction. These irregularities of surface, however, made no eddies or cross currents perceptible to the eye, the water flowing through perfectly limpid, and showing no visible signs of commotion. The pipes were used for the first time; their interior surfaces were smooth, with no scratches. One or two of the joints in Series VIII. had a trifling amount of dirt adhering to the inner surface in spots; not enough, however, to seriously increase the resistance. All the joints for Series VII. and IX. were almost perfectly clean. Series .—A pipe of 8 joints, made of heart red-wood, and bored by the usual pipe-auger. The interior surface was of the usual smoothness of such holes. The pipes were new, and had not been covered with any artificial coating. The connections were made by driving one joint into the other, an iron outer band preventing the wood from splitting. QQ 298 EXPERIMENTS WITH PIPES.—New Atmapen. Description of Pipes. The diameters were quite uniform, varying from .1042 to .1062; the mean was 1052, which was adopted as the correct size. The length (corrected) was 62.05. Reductions. The value of (2 g)”% was computed by the formula given by D’Aubuisson, in metrical measures, = 9.8051 (1. +.002 84 cosin 2 2) ( = ‘) ay? ‘ r= 6 366 407 (1. +.001 64 cosin 2 /). J=latitude ; r=radius terrestrial spheroid ; e = elevation above sea. Reducing this to English measures, = 32.169 54 (1 —.002 84 cosin 2 2) ( z= ‘) z v= 20 887 510 (14.001 64 cosin 2 /). e=400, and 7=37° 10° ; hence (2 g)4 =8.0179. In the following table, showing the results of these 53 experiments, /, D, y, t and HZ are experimental values. t Pe the value of o of .980 for Series II., III., 'V. and V., may possibly be a trifle too high, but in any event it would be .94, and this smaller value, if adopted, would not considerably a= : D Q=-1.. v= . ois the assumed co-efficient of contraction at entrance ; change 1. his deduced by following formula ; /)’ = 2 QV as ae or, tw=o (2gh')*; hence 9 g wee d ( g ) ’ — ; e ;3 h’ being the assumed head required to give velocity to the water, taking go into account contraction at entrance, and / being effective head absorbed in overcoming friction and adhesion, as the water passes through the pipe. The final co-efficient n is obtained by the Chezy formula, v=n (7 s)* ; 7 being Sc a se de h hydraulic mean radius = 1 D, and s being sin of inclination = — ; hence / Q=% 7 ae . 8927 1 re ss 299 Redwetions. EXPERIMENTS WITH PIPES.—New ALMADEN. ‘Jouun; YT ‘paqyeoo ‘adid aaoqe jo squtol “TOTIUN YT “T soLiag sev ‘odid jo pus aSreyostp [Ty ayinh you pIp 6B “ON UT JoqeAy ‘SyIeUIOy ‘Jauun; YIM Apeydse yy pezeoo *T sottag se adid eueg 6001 ayinb you pIp OOS “ON UT TaqzVAy | ‘ou ‘payvooun ‘Mau ‘WOT YOUT T I omy asi | 0'S6 £99 | ele . fue 139° Sypsog 896 | cage | 494) 086 |oco'r sgs'9 9°16 || ec9'T joso| “ |¢orT | osee G56 GFIe | sory; eles cae oso9 j2zge) L699 | TOLT TIT soutag | 6 IOL]} esses |ost] ose’ |ege's || etre | ‘adil jo pus oreo iy | 9g || exe eto. TG GOT vor | tee [seo] “ ey) ae | e16 | Test |s10, “ G99 T | 661% | ge | sore ‘oot; “ gtee — 9ee'e v6 | eect lecc) “tte F , 666'E odd ourg gg | seT'9 love} “| 90¢°9 | sort CTT seteg | SOOT] LFeL [OLR] O86 [ATES ' LRee i ate Ick | Ge eee | Ree? gor 9 ges 6FO, LR), ler 906 8 ocet eit] «6 of egart | Fats lees | acoe cre, lctee | care ami ei6 | Ltr ou “| 0ge'F _ free | #66 6c0'9 Teg“ 0649 | &L9'°F ST sariag | 6001 | 1792 [6397 008" | oge’s | wee'g ou x 4 0 H a “WOTOe.14 “peo -m0r) jo “peo “£qrO0079 A aATPOHA quoromyg | [eIOL | ay IL ‘LIST wopmupy mayr *x(8 -1) TIAOX ATAVE RR EEO" SP TF0° 66 10° 06 610" OG NEO’ 8 EEO" L€ 900° 69 800° T€ €10° €49 610° 16 oO" OG NGO" 19 G80" OX £O0° 09 800° F0 €10° G6 610° 06 €60' 66 860" FG GEO 0 “‘puoseg aed as.ryp SI SFP G'99¢ Tg e'ORd Pred €'L94 Oe Less S01 F082 e669 greg oF GIs OT6s (@6re GFLIT F'E6L 6°LE9 L8Eq PCLP tm 2 "T@889 A. TE OF poarmbeyy ouhy gocat |! F06'S1 | ! L6L°9 9836'S 626'S1 L3E°ST a F08"9 10x" OFs'ST | Ges'aT SEs" 0ee'¢I GLE cos'9 | 089 gocer J | 0gz°T CFEGT gea'GI b Suiins | oT, JO Syrordeg » 986 G00" “F40 900° OSE | “[oss0 460 900° 160 900° 3 GOO" 00° G00" FSO 900° FCO 900° FG0 900° FO 900° FLO 900° FG0 900° #0 900° F0 900° FG0 900° F20 900° FSO 900° F40 900° FS0 900° DvD “VOU UV9TY w=a Ud “w fo UOyourwUNnLnjag— yf? “yjnugy MOPUDTT EL80° &L80° CLSO° EL80° “19qVOUT eI UBITT G89'9T G89'9T F909 9609 F909 #9609 LEG 09 LFe09 L¥G 09 LPG09 L¥G'09 L¥G 09 LTE09 eLT09 GL 09 eL109 ELT09 6LT09 6LT09 GLT09 2 ‘yySue'T 90€ GOE FOE £06 G0e 106 So Da ol SoD Le ol oy — D Ol EXPERIMENTS WITH PIPES.—New AtmapEN. Reductions. 300 ‘JeuUN YYIA ‘ssvps your F “66 ceu'9 “ITA shes | e-ooT! Gucg ‘| | ‘grat | Gye GES | EuO'T GIR OLE | LGR OR9F e16 | 1ee'9 “jeuuny ou peqvevooun MOU yout g | ‘TA soueg | 66 OL0'X | STL | or oe, || sgn | VOL | deat ; (eee ' eel | gre 94809 “UOT 0 “499. rn i sO pey soa iG G ? gall) donee -un “aoat plo “yey }°ZG | 68 | 629° ! LGs | EEED ‘A SOLIOg | 108 | 00'S | 0°88 Gey | | ‘panurypwoa—" { J sarag | 0°66 esol | wu Y | “syIValay f 4 ‘pve OATPONT ico. 20 19" i. ver) Jv | C90G a ‘Fie gee E10 a CO’ - ‘Foo. he LOT i 1ge “OE ORG xXO" ce | Ike O86 | MU oD “WOTPOBIY -uo0g jo POLI A 09 “cc Orso ERLE 60.070" JoLo8 GOOEY EEO" | ELN GEOT | Te G00" foust | stat | 6e goo | 906 —- G6E'G | £6 F00" LOSE CONS | GT 900° E6C0 — GNe'e | LB 100° bee's | 8£8°8 | ce x00" plr O16’ | 0% 400" ost | S6T'L | €8 900° F890 | est | oF OTO gues || 619° | 46 10° 9F8'F | Ile | CE RTO’ '€96°9 | e1re | 96 160° ales °° 99c'F | 8E FEO" | £69" GEES | 90 10° | | ‘trot ll gre | gt eco | \ Wf a | 0 | ‘puossg peep, ' tod [qe], ey asaeyo | “SI ‘panuyuo— TTAOX WIAVL CRGh L | ¥84 700°} t920° | cOG'¢ ae99 leec'eL | F9q F001 1910 | ZO6'E9 |, | : | GFIS | GLb '8t1 GOO"; £690" | LE1'09 { i : SOL | Gir StL 600" | § | 261-09 g90¢ [SIs "SFT G00 ecco" | 1C1'09 were jsie NFL GOO" E40" | LET'09 el0e siet “spt s00" weco’ | #eTO9 OINT cIgT 7 peer eee i aes SFI B00'| ¢eg0" | Le1'09 BRIX | 008'9 | | | e1¢ ‘oh eet eoo"| eeno’ | Pred CIR iGtL $00" gogo’ | LFG'09 6089 GIL G00°| ESR0" | LFE'09 | 662'9 ULL GOO" | eg80° | LFG°09 '862°9 YI4G00°! $480" | 4FG"09 laeest |g12 G00" egs0" | 4¥6'09 “€GEUL 1 G712,900° GRO", AFE'09 I (SIF 9619 | he ccs ; Daas a 120 900°] 9480° | 989°9T FGROL) 9TE'ST? | | i ie 120 900°| 9180" | ¥g9°9T 699 |6oesr). SF aXe 7 b v a 2 ‘JassaA | ‘[ASsa A. : Tha 09 «|: Surans ‘vaIy eee aa poatnhay! -eapy jo) uvepe | UC | WUT auty, | Ayiovdug uUvoy GI& TIE ele clé T1¢ Ole 60€ 806 301 MONS, Reduct EXPERIMENTS WITH PIPES.—New ALMADEN. ‘ It | | "1 07 O'L9 JO ONTeA B OATS prno ‘poydope yr ‘tory ‘UTIL =2 epeut per} s9r}0 '8q} $601 Jo ‘ontea 4soTpeuts | se udyey SBM 9 SOUET {87894 ' aoe mpenien So ddet tees ga9 | TO9T [990] “ sour eel + tetto! BaF 9649 GOD | Ege qysys {10a v sum o1949 gee “ON AT || Gi19 LOL | 8Fl i 606'¢ | 697% | 9F 10 GOL ,CL@ EL - G06 [See GLO GCL P | 066" i STOF | 800°E | G1 9t0° LI8G | OTST | 269 800° ccOT’ COCO | Lee ‘jeuuny ou ‘Meu ‘poom yout FT 9-19 G6E°9 | 10c | 9699 | GISE | 6G OO) TL6F (SNe el | 669 800°. 60" GOED | 9EE Sx sereg | 6°19 SETS |98| O08 +rEeu's | DOE | GOFEO | 6'SEF : 406°G1 669 800° | ESOT’ GOED | Gee ! | ( | : T08 ! gtL) | 00T i: 9TR | LLOG | GRTOO | FOOT | GLE GLE 100° RIFO': LELIL | Fee Stk | g6oT | TAT e / 9961 | 6126 | €1 600° Get oar GLE 100°|81FO = LET IL | €€€ ‘Jeuuny ou ‘sseps yout ¥ 618 ; OFS T | 608° 7 | Gols 6E9e | CO G00 | LEG 1 alr GLE 100° |81FO" LeTIT | GEE "XT semeg | 7°06 | 699° |9SF] Ocs’ | GE0'e 6eFF § 60900°, OSL GL ‘e216 100° | 81FO' | LETIT | Tg sp w 105 “odid Jo pus odtreyotp ges | seg [SFO] “elo NOEL | Ge FOO’ «STILT | GLF 6£0 £00") cc90" | TFE'FE | 08s 19 70U PIP O&E “ON UT TOqe AA e'16 968°. | S9T ‘ G40 G49 | 90800'| GETR 8619 6€0 00°, GE90° | IF6'FE | 6ZE ‘jauung ou ‘sses your= | OGG | GREE | TIE fs 099" 999% FLITO | 6609 | 462°9 6£0 €00°| GZ90" | Lt6 FE | cE ‘TILA seteg | $796 OEO'F | GFF) OC38° | ELOG ele | 66 E10 | MITE | L6L°9 6€0 €00° | G690° | IFG6'FE | Lee | G68 86G°T | €90° = 199'T Gu6'T | 96 800°} O'6&L | 6619 TSG F00'| F9L0° | G06'ED | 9GE 976 | PHOS [EFT] “ 186 | SFES | OGETO’ | 9'60G | 2649 | $89 F00'| F940° | GOB'EO | ace “‘panurjuoo— JT A sarwagy |) C16 GIS | Pes} 046° 960'S G39 | 689I0°| G06 | SESSL | FS F00'| F9L0° | cO6'E9 | FZE # q M a Pe a 0 a b » a 7 “WOT}OVIY ‘puoossg | ‘TassaA | ‘Tossa A. . ; “pea -wog yo | ‘:peayt tod Tha 03 | Sutms agarge | UPPPE Eo SyIvuLey peo “£49079 A. 3 aay wens 5 vq | ‘yysueT | on aAtpayy querIgg | TeqOJ,- esreyo jpeambey) voy] JO | wea | 5 -09 “sIq eury, | Ayroedeg WN | ‘panurquoo— TTAOX ATAVE 302 EXPERIMENTS WITH PIPES.—New Autmaven. Reductions. Of the foregoing experiments, those of Series IX (short }-inch glass pipe) are the least reliable. The others are believed to be very near the truth ; of course for the low velocities, where @ was measured by (b) having only a capacity of .475, the deter- minations of 7 are not quite as trustworthy, as when larger measuring vessels were used. The method of ascertaining D, by the weight of water contained, gives far more accurately the mean areas for small pipes, than any practicable process of direct mea- surement. The accuracy of the assumed values of o for Series I. and IT. can easily be proved by a comparison of the values of n for the two series ; there being for Series I. full contraction, and for Series II. hardly any contraction. The indicated results from these experiments have been fully analyzed in Chapter VIII. The facilities for making these investigations were very kindly afforded by Mr. J. B. Randol, Manager of the Quicksilver Company, to whom the author here desires to express his obligations. North BLooMFIELD Pires. The following experiments were made at North Bloomfield, California, in October, 1876, in behalf of the North Bloomfield Gravel Mining Company, of which the author was then the General Manager. Three pipes were experimented upon, of about 15, 15 and 11 inches diameter respectively. They were made of sheet-iron, from .0054 to .0091 in thickness, single riveted. They had been several years (about 5 years) in use, and hence the rivet heads were somewhat smoother than when the pipes were first made ; no deductions for these heads will be made in computing areas. The pipes were made in riveted joints, each about 20 feet long, put together stove-pipe fashion, one end hence being slightly larger than the other. The pipes had been originally coated by immersion in a boiling bath of asphaltum and coal tar ; at the time of these experiments the interior surfaces were quite smooth, and almost absolutely free from rust. The three pipes were laid side by side across a sharp ravine, the outlet tank being about 25 feet lower than the pen-stock (inlet tank); the discharge was conducted by a ditch some 600 feet long to a weir, over which the discharge for each experiment was measured. The co-efficient, «, for the same weir had been previously determined by absolute measurement, as we have already described. Figs. 4, 5 and 6 on Plate XVI. show the profile of the pipe-line, a plan of the pipe and ditch, and upon a larger scale a sketch showing position of the weir, hook-gauge, et cet. . The pipes had each a funnel-shaped mouth-piece, shown by Figs. 10, 11 and 12, Plate XVI. These funnels were included in the measured lengths, 7, but the diameters were obtained by measurements of the pipe proper; the co-efficient of contraction at entrance was assumed at 1., and it was thought that this consequent neglect of the EXPERIMENTS WITH PIPES.—Norra BiLoomFieLp. Deseriplion. 303 effect of contraction would be practically compensated for, by neglecting to consider the increased diameters of the mouth-pieces. Any errors produced by this hypothesis would be inconsiderable. The flow through each pipe was first determined when they were in their normal positions—Experiments Nos. 340, 345 and 349; the total head, H, for Nos. 345 and 349 being the difference in elevation between the surface of the water in the pen-stock, and the surface in the outlet tank. In order to obtain lower velocities pieces of similar pipe were attached to the ends in the outlet tank, thus extending the several pipes on the general line of their inclination. The discharge was hence free into the air, and IT was assumed as the difference in elevation between the surface in the pen-stock, and the centre of the discharge end of the pipe; in Experiment No. 344 the escaping water did not quite fill the end of the pipe (top of jet being .04 below upper edge of pipe in the plane of the discharge end), and in this case H was assumed as the difference between surface in pen-stock and centre of the escaping jet (about .02 lower than centre of pipe end). The difference in elevation between the pen-stock and the outlet tank was ascer- tained by two careful level-lines. The surface of the water in the pen-stock was read by an assistant from a gauge-rod ; his readings were to the nearest hundredth of a foot for Nos. 340-351, and for Nos. 352-354 to the nearest two-hundredth (.005). There was a maximum variation in the surface level at the pen-stock of .02 in Nos. 340, 345 and 347; in the other experiments there was no perceptible variation. In No. 349 there were a few small eddies—hardly vortices—slightly disturbing the surface just above the entrance of the pipe; in all the other experiments the surface in the pen- stock was practically quiet. In No. 349 the submerged jet of escaping water struck with much force the opposite side of the outlet tank, some water rising close to this side as much as .5 above the general surface level in the tank ; in No. 345 the submerged jet made less commotion. In both these cases the height of the surfaces in the outlet tank was measured on the north side of the jet in comparatively still water. In all the other experiments—except No. 344 as before stated—the top of the escaping jets rose higher than the top of the pipes, owing to their inclined position. The limit of error in the determination of H could not have been over .02 for any one of these experiments, except Nos. 345 and 349, where, owing to the disturbed con- dition of the surface due to the submerged discharge, H might possibly be .05 or .06 in error. The mean diameters and areas were obtained by measuring the circumference of each pipe with a steel tape at 14 points nearly equidistant ; the inner diameters were then computed by an allowance for the thickness of the iron, which for the mains varied from .0054 to .0069. No allowance, however, was made for the thickness of the asphaltum coating, which, had it been taken into account, would have reduced the stated mean diameters perhaps .001. 304 EXPERIMENTS WITH PIPES.—NortH BioomFIELp. Description. The maximum variations in the inner diameters shown by these measurements were for the mains (ic. pipes in place, not including funnel mouth-pieces) as follows ; Maximum. Minimum. Mean. 11 inch pipe bss 923 .889 .9105 13 5 .. 1061 1.049 1.056 15 5 .. 1,240 1.221 1.230 The above mean diameters were slightly changed by variations in the sizes of the joints used in extending the pipes at the discharge end. The diameters can fairly be considered as being correct within .0035. The lengths were carefully measured ; the values given of 7, should be within .2 of the truth. The standard of measure was the same as that used for the pipe experiments at New Almaden, and for the measurements with the 2.6 weir; agreeing closely with the U.S. Coast Survey standard. The course of the ditch feeding the pipes was at right angles to their line, and at the junction of the pen-stock the width of the ditch was 10.5 feet; there was hence no approaching velocity of any moment at the entrance, to be taken into account. During the experiments the mouths of the two pipes not in use were tightly closed, so that the flow was entirely confined to the pipe being experimented upon. Measurement of Q. By reference to Figs. 4 and 5, Plate XVI., it will be seen that the arrangements for the measurement of Y at the weir, were somewhat defective. The surface of the water, from the outlet tank to the hook-gauge at the weir, had a considerable inclination in the ditch; for instance in Experiment No. 349, the surface at the tank was 77.90, while at the gauge it was (75.364 1.46) 76.82, thus having a fall of 1.08 in about 600 feet horizontal, and giving a considerable initial velocity to the stream, which is perhaps not fully taken into account by the mean velocity, v,, where it was measured at the section ab, Fig. 4. As there were no screens across the ditch, this initial velocity was not at all checked. Unfortunately the surface velocity from E to H was not measured ; this would have much more satisfactorily shown the effect of the approach- ing velocity, than its computation from v, at the section a b as hereafter given. The position of the hook-gauge was also objectionable, being placed on the down- stream side of the weir in comparatively dead water, where the surface was in all probability slightly elevated by the force of the current, as the stream passed by towards the weir. The hook-gauge, reading to .001 inch, was placed about .7 north-west from the side of the flume in a vertical box, extending some distance below the crest of the weir ; small holes were bored through the sides of this box from top to bottom, so that the level of the interior water very speedily became the same as that in the exterior canal. EXPERIMENTS WITH PIPES.—NortH BioomFietp. Measurement of (. 305 For each experiment the water was allowed to run for some 20 minutes, until a nearly perfect regimen was established in the ditch, and then the hook-gauge was read every minute for a period of from 10 to 30 minutes, until the readings became practically constant. The mean of these last readings for a period of from 6 to 13 minutes was taken as the correct value of H at the weir. The largest fluctuation—difference between maximum and minimum—in the final gauge readings was in No. 347, being in that experiment .0014 (.8118-—.8132); the average fluctuation in these readings for the 15 experiments was .0007. The measured value of H, aside from the defects of position, can therefore be con- sidered as having been very accurately ascertained. There was but little wind at the time, which accounts for the steadiness of the water. There was a slight leakage at the stop-gate A, and also under the weir; also a very slight leakage from the pipes. This was in part measured, and in part estimated ; by reference to the following table it will be observed that the total leakage was an incon- siderable fraction of Q. The mean velocity of approach, v,, as measured at the section ab, Fig. 4, ? 2 . Plate XVI., has been reduced by the formula h= H+ b a b having a constant value of 1.4. | The inner depth below crest was 3.08, so contraction can be considered as complete, except with No. 349, where there was probably a slightly increased discharge, due to partial bottom suppression. Hence for the following reduction, ¢ will be considered as c,, and its value will be taken from curve on Plate VII., for /= 2.6. 306 EXPERIMENTS WITH PIPES—Norru Broomrtetp. Measurement of Q. TABLE XCVIII. Hamilton Smith, Jr.—Determination of Volumes of Water passing through Pipes. 15 North Bloomfield Experiments Measured over Sharp-Edged Iron Weir ; Full Contraction. 4 = 3.08 (2 g)4=8.018 Q =c8 (2g h)4Ih No. wl e@ a ae l ae aa 0 Leakage. Q \ | | & a at hk Get dese 340 M186 3200228508 2.586. 5969 | 6.475 050 6.595 341 7702 28 0017 719 | 2.586 .5989 5.614 030 5.644 342 6604 | 24 | 0013 | 6617 | 2.586 6018 4.478 037 4.515 343 6062 22 | 0011 | .6073 . 2.586| 6034 | 3.947 » .095 3.972 344 5092 | 18; 0007 5099» 2.586) 6065 | 3.053 018 3.071 I | | | | | 345 , 1.0913 | 42 0038 | 1.0951 2.586 5908 (9.359 O17 9.376 346 9414 | 38 | .0031 | .9445 2.586 5944 | 7.542 030 7.572 347 8124 | 30) .0020 | Slit | 2,586] 5977 6.072 025 6.097 348 | 6116 9.22, .001l | 6127 | — 6032) 3,999 025 4,024 | 349 | 1.4611 | 58 | .0073 | 1.4684 , 2.585 5828 | 14.339 036 14.365 350 | 1.3344 | 55 | .0066 | 1.3410 | 2.585] 5855 | 12.563 024 12.587 351 | 1.1442 | .45 | .0044 | 1.1486 | 2.586 .5897 | 10.034 020 10.054 352 | 1.0349 | 41 | 0037 | 1.0386 | 2.586, .5923 | 8.666 024 8.690 353 9889 | .38 | .0031 9920 2.586 593 810 02k 8198 354 7292 27 0016 7308 2.586, 6000 5asl | 018 5.199 | | Stones weighing from 2 to 18 lbs. were sent through the pipes, during Experiments Nos. 340, 342, 349, 350, 346 and 345, and the time of passage determined by a stop- watch beating } second.* After No, 347 had been made with the 13-inch pipe, two stones were put in the entrance, but were caught by a projecting edge of a joint, and remained lodged in the pipe during Experiments Nos. 346 and 345. Some time after these investigations the pipe was taken apart, and the stones were found lodged as stated. These obstructions doubtless retarded the flow in Nos. 346 and 345. * The moment of the escape of such a stone from the lower end of a pipe, can be very well determined by sound. EXPERIMENTS WITH PIPES.—Norrn Bioomrreip, Velocity of Stones. 307 ! Times. Weight of | Specifi | Pipe No | a Stone in Granity | Description of Stone. | Maximum. | Minimum. Pounds. | as, aie see | = : 11-inch aq J bo 4G. 2 . Oval quartz pebble. { ; 72.7 17 2.3 Rough edged. Volcanic tufa. | 5 5 sa 126.5 3 8 2.5 Round quartz boulder. 7 A 123.3 8 2.1 | Rough edged. Volcanic tufa. 1 | 5 5 | , | 15-inch a | 63.3 6 2.3 Smooth edged. Volcanic. } om | 87.6 18 2.3 Nearly round. 3 a6 | 76.1 | — 5 2.3. ' Rough edged. a | fy ; 72.6 10 aE Black quartz boulder, nearly round. , I | 13-inch 346 | ide | 94. 8 er This experiment unreliable, as after Ex- | periment No. 347 for flow had been made (No. 347 preceding No. 346 in ' point of time), two stones had lodged | in pipe. The water was nearly clear for Nos. 340 to 343; it was muddy for Nos. 344 and 354, and more or less discolored for the other experiments. The temperature of the water was not observed. It did not, however, vary much, and must have been not far from 55° Fahr. The final reductions of these 15 experiments will be found in the following table. As before stated, the co-efficient, 0, of contraction at the entrance will be assumed as 1.; hence the effective head, h, is obtained by, /’= oF andh=H-—h’. As heretofore, / (2 9)%=8.018. No corrections will be made either for the rivet-heads, or for the two main angles in the pipe line, which were respectively 9° and 11°. Probably the projections or irregularities at the “stove-pipe” joints retarded the flow more than the rivet-heads and curves combined ; this, however, can only be a matter of conjecture. Reductions. EXPERIMENTS WITH PIPES.—NorrtH BLOOMFIELD. 308 ro Poth ty eas EN re ie ~ - © ane rat —=, el | 1 O'sIT O9TIL 919¢ 666° OGP COL | Exe't | cer’ €98T 1 66 T | 66TL | T'G0L | SPL PoE) Ustt | SLIT | Lisl NGL GPFas GOO TG |OT9'COT | LF89 | 8ceUs8 | 688lT Ose lL | Fels eee | SPL | EUs FOI T6ll RTL'8 OGG 000°66 | OG4°ZOT | FIL |o60's essl'T OS@ 1 | L'°01L 6°69 SPL | CGE | SGI | O6IL TOO TT FILL (l@1Lcer] gos’6s | g0°COL | G9t'R | FE0°OT) ERRI'T O€eT | O0°SOL | 6069 , SFL | [SE ! SEs 8u'6 »-O9TEL GLULI OT LT GEG6SL! G8EES |) TE'COL | €69°OL L8eGT| esstl O€e' LT 19469 |8'089 | STL | O€€ 93 88°TT | Eel Yl ce FLEE = 068 FG O6LL | 6ECOL | 060°EL COE FT 68st O€@T |Ft89 |9'699 | SFI | 6FE | S601 F601L CONT soe" O€1°S eYL6 | GOCOL | GOSTF | FEOF | SELLS” 9G0'T |PSTL F90L. “EL | STE LSI FELL) LELOr FGL G8g'0OL} SU8'16 ' FLEOL | G969 | 1L60°9 | Baz’ 9G0'T |G 604 5169 Ocl | LEE Fell | jr OELL) LEGEL COLL (069°9T ehG3 GFCOL 9F9'8 l@1g°h | ggg" 9G0'T 197669 | 9289 | OGL | OFE 6 FIT PPIL| LELCe SslLT |OLG'E OL LL, LEGOL | 9OLOT 9L6°6 | SGL8" 9G0'T | 6789 |6CL9 | OCT | GE | | USOT | TLOL! O69 Cte" Gogg GFLG6 | O€ GOL | GIL | 1TL0'E | 8IS9° 116° 90EL SceL, BL PPE OIIT | XOIL! 669'6 slg 00601 TLC6 | PECOT | F60°9 ‘E16 | SIg9" I16° Giles Gels SL &Pt& 6 TIT 614 TIL) FOTseL 9FL° OG8'GL FF'6S | GEOL | 166°9 ‘ote 81g9° 116° 6E1s | t'902 | 8s GPE Sel PEI! 6ES LI 99LT [GO0'6L) SI@E8 |) GE SOL | 649°8 | FFO'E |) SIE9 116° 0°69 | E689 | eh IPE ssl GP '6 GGIL}| 899°GG C9GT |0GG'F% SLs! OF GOT | TGOOT | Gcg°9 | 11g9° GOT6” 8789 | OLL9 | BL OFE | ‘pu | ‘00! _ wv | u " e hp ie ee te. Se |» a 2 | wey jew a ‘sodig | aa | : ! “UdATS 9104} ysnoryy _ & -_ 2 | eer «-20g] TUS SENOS “AqI0019 | a Suny “suexy, Jo Aqro0ja A. “proxy Suyaedarr ieees -fyojo , THAOX “Bory dogourerqy “a3 | Va IAL aATOOPAA [UL peqsosqy esol WPOWPA) oqey,) uvepl | weep qeuey | peoH "QLOT ‘0290790 ‘pjayiuoorg YI0AT ‘XIOX WIAVIL "(8 t) WHA we VW fo WOYMULUWLAVgT— "IC ‘YU WORVUDET EXPERIMENTS WITH PIPES.—Norrs Broomriznp. Reductions. 309 These 15 experiments were reduced in 1876, and the results published in the Trans. Am. Soc. of C.E., 1883; the values of m thus deduced are given in the last column of the preceding table. They have been here again reduced from the original notes, in accordance with the views expressed in our chapters on Velocity of Approach and Weirs. By comparison it will be observed that our present values of x appreciably vary from those first given. These discrepancies nearly altogether result from a much smaller value being now attached to the effect of velocity of approach at the measuring weir, than was thought by the author in 1876 to be a proper correction for that cause. He was then a believer in the Bernoulli theorem “ that no force is lost,’ but from a careful study of the experimental data which has before been fully discussed, has largely changed his views in this respect. Owing to the initial velocity in the ditch feeding the weir, it is probable that in these later computations we have underestimated h,’, but this is perhaps sufficiently compensated for by the faulty position of the hook-gauge. The diameters, as we have before explained, are probably very slightly underestimated. Owing to slight partial bottom suppression at the measuring weir, which has been disregarded, the chances are that n for No. 349, and perhaps for No. 350, is placed a little too low. The results here given, with the exceptions of Nos. 345 and 346, can be regarded, however, as being nearly as accurately determined, as is practicable with large pipes, where @ is obtained by indirect measurement. In point of accuracy this series cannot be considered as comparable with the preceding experiments with small pipes made at New Almaden. The foregoing remarks well illustrate the uncertainties attending the weir measure- ment of water, unless the investigations are conducted by one thoroughly familiar with the minor laws governing the flow over weirs. It is hardly to be expected that an engineer, busied with important works of construction, can devote sufficient time to. master a subject, which has perplexed the minds of such great physicists as Bernoulli and Dubuat. Hvumsvue Piree.—26 inches diameter. The main water supply for the North-Bloomfield mine is conducted across Humbug Cajion by two sheet-iron riveted pipes, each 26 inches in diameter, and having a united / delivery capacity of about 90 cubic feet per second. In 1873 the discharge through one of these pipes was determined by the flow through apertures under a constant head, but under conditions not favorable for very exact measurement. It was not practicable to measure Q directly. The length, mean diameter, and head ({) were obtained with accuracy, the methods employed being the same as with the experiments just described. 310 EXPERIMENTS WITH PIPES.—Humsvue PIPE. The pipe had been coated with a preparation of asphaltum and coal tar when first laid in 1868, and its interior surface was very smooth at the time of the experiment. The thickness of the iron was .0054; as it was single riveted with light rivets, the rivet heads formed no obstruction worthy of consideration. Neither were the bends in the pipe sharp enough to perceptibly retard the flow. The stove-pipe connections were similar to those employed for the three North Bloomfield pipes. The discharge was submerged in the outlet tank. The course of the ditch into which the water escaped, was at right angles to the line of the pipe, as was also the case with the feeding ditch at the pen-stock ; hence there was no disturbing element at either end from velocities in the ditches. The pipe had a short funnel-shaped mouth-piece, which is mecluded in the given length ; co-efficient of contraction is hence assumed at 1., and h = H— 5, : g TABLE C. Hamilton Smith, Jr., 1873.—North Bloomfield 26-Lneh Sheet-Iron Pipe. Determination of n, inven (rs)\4=n (HP), | ; | Maximum Head | | | | Velocity N | Taal Mean Mean a Boy Total | oe | Effective | of Stones O | HenSs- | Diameter. | Area. co Nae Head. | . Head. y passing | | ; Imparting peaaeL | | Velocity neue | | ne Pipe. l D a | | HT | h' h , 355 | 1193.8 2154 . 3.643 45.92 12.605} 22.067 2.471 19.596 : 1a? Ae | | 4 Cubical wooden blocks were also sent through the pipe, and occupied a very slightly larger space of time in passing through, than did stones weighing 10 to 15 lbs. The element of uncertainty in this experiment is the value of @. It is thought that the quantity given should be within 4 per cent. of the truth, with the chances that it is stated somewhat in excess of its real value. The reason for this supposition is, that while the velocity of stones passing through the pipe was only 11.24, the calculated mean velocity of the water was 12.60, which appears to be too great a difference. In other experiments these differences were as follows : EXPERIMENTS WITH PIPES.—Hovumpua PIpe. 311 a i Ve ee of | & Diameter Mean Experinent. of Pipe. : Srey Velocity | Bieler of Water. SS femmes lates == No. 340 1 I 9.42 10.02 91 yy Beet L&R 6.93 \ 12300 350 i) GBS 10.59 »» B55 2.15 11,34 12.60 a ee: * Ae 20.94 90.14 | | The inclination of the Bloomfield pipe, No. 355, was not quite as steep as that of Nos. 340 and 349, and not nearly as steep as No. 356; nor were the rivet heads in it any larger than those in Nos. 340 and 349, while they were much smaller than those in the Texas Creek pipe, No. 356. It may be also observed that the velocity of the stones in No. 355, was slightly greater going down the incline to the synclinal part of the pipe, than in ascending the hill on the opposite side. Texas Creek Pipze.—17 inches diameter. The experiment with this pipe is most interesting, owing to the great head of 300 feet, and consequent velocity of over 20 feet per second; we will therefore describe it in considerable detail. The pipe experimented upon was laid across the Big Cation branch of the South Yuba river, Nevada County, California, in 1878; the following described measurements were made in 1878 and 1879. The elevation of the pen-stock is about 5500 feet above sea-level. The reader is referred to Trans. of Am. Soe. of C.E., February, 1884, for a profile of the pipe line, and for mechanical details which are not here pertinent. The pipe is made of wrought-iron sheets, double riveted on the long seam, in joints of about 20 feet in length; these joints for a linear distance of 1350 feet are put together stove-pipe fashion, and for the remaining 3090 feet by an inner sleeve, with an outer band, the spaces between the pipe and the band being filled with lead. The templates used in the shop where it was built, show the following dimensions ;* * The thicknesses of the iron as given here, are slightly different from those stated in the paper in the Trans. Am. Soe. of C.E. 312 EXPERIMENTS WITH PIPES.—TeExas CREEK PIPE. 1350 linear feet, iron .0069 thick, internal diameter = 1.4156 220 —,, x gy HOOLD” 55 re i = 1.4146 240 ,, 43: oy. 0091. 5, ‘3 3 = 1.4175 250, » 97, 0104, a ‘5 = 1.4137 320, wag SOLTE y, $5 4 = 1.4243 610 ,, xy 97 90182 4, 4 = 1.4233 1460) a oe BOISB ye ‘* - = 1.4297 Diameter by arithmetical mean =1.4188 4440 feet total length net oy ae 14196 The mean of a large number of end measurements, made after the pipe was delivered, was 1.4166, and is probably more accurate than the geometrical mean of 1.4196 obtained from the templates, as this latter size was probably a trifle diminished by the draw of the rivets. The pipe had an unusually heavy coating of asphaltum and coal tar, which diminished its internal diameter fully .001. Hence 1.416 can be assumed with safety as representing with much exactness the mean diameter, D. The rivet heads for the larger thicknesses of the iron formed noteworthy obstruc- tions for, say, one-half the length. The curves were made comparatively easy, but with the very high velocity of 20, doubtless somewhat retarded the flow. Neither of these two retarding influences will, however, be taken into account in our computations, as there are practically no data of value to enable one to even approximately estimate their effects. Comparing the condition of the interior surface of the Texas Creek pipe with those used in Experiments Nos. 340-354, it can be considered as being appreciably rougher. There was a short funnel-shaped mouth-piece at the entrance, which is included in the stated length of the pipe; the co-efficient of contraction for such an entrance would be in the neighborhood of .90; as the length of the funnel is insignificant compared with the total length, this co-efficient, 0, will be placed at .92, and the increased dia- meter of the funnel will be neglected. The discharge was submerged, the pipe terminating at the bottom of the outlet tank. Neither the velocity of the water entering the pen-stock, nor the velocity of the water escaping from the outlet tank, practically had any effect upon the flow through the pipe. The total head, H, was the difference in level between the surface in the two tanks. It was determined by the mean of two lines of levels, and, as stated (303.62), is within .35 of the truth. The length of the pipe as laid was twice measured, and 7= 4438.7 as given, cannot be in error more than 1.5. The standard of measure was the same as that used in all the preceding experi- ments—U.S. Coast Survey. This pipe line is across a very precipitous mountain cation, as is shown by the EXPERIMENTS WITH PIPES.—Texas CREEK PIPE. 313 profile before referred to, which accounts for the rather large limits of error as above given. The temperature of the water during the several trials varied from about 60° to about 50° Fahr.. The water was froma clear mountain stream, and was perfectly limpid. The volume of water passing through the pipe was measured in May, 1879, first by the flow through orifices 1 x 1.0625, and then over a weir 5.48 long. These measure- ments were made at the lower end of a wooden box or canal, situate just above the upper mouth of the pipe. The canal was horizontal, 16 feet long, 134 feet wide, with its bottom 1.83 feet below crest of weir (G7). The water entered the upper end of this canal with considerable force, which in these first two experiments was only partially checked by a grating or rack placed across the upper end of the canal. These two experiments, after allowances for wastage at the pen-stock and a small loss by leakage, gave the following results ; Q by weir measurement = 31.69 Q by orifices ‘ = 30.90 Neither of these experiments, however, was regarded as reliable owing to the large initial velocity in the canal before spoken of, and for which in both cases a considerable correction was applied to H. There was much more suppression of contraction at the bottom with the orifices than with the weir, which doubtless accounts for the smaller value of Q deduced from the flow through the orifices. After greater precautions had been taken to check the primary current ia the canal, the experiment was repeated on June Ist, 1879, the total head, 7, being almost exactly the same as it was in the former trials; we will use this last determination for our data. The weir was 5.500 long, vertical, with square corners; the crest was a plank 1 inch wide, planed true, and with perfectly square corners. The discharge was hence free into the air, the escaping vein only coming in contact with the inner corners. The inner depth below crest, G', was 1.83. The weir was placed in the centre of the canal, with each end distant 4 feet from the side of the canal, so that there was almost perfect end contraction. As H was 1.50, with G=1.83, there was an appreciable bottom sup- pression, which we will assume added 14 per cent. to the discharge, this correction being in accordance with the experimental data for partial suppression we have given in the first portion of this volume. The velocity of the surface in the central line of the canal opposite the hook-gauge was very nearly 1.00, while the mean velocity, v,, at the gauge was 736 ; we will adapt the former velocity with the factor of correction, )=1.; hence ,/= 5 = 0155." Phe 4 2 * With v,=.736, and b=1.4, h'.=b 577 O18, SS 3814 EXPERIMENTS WITH PIPES.—TeExas CREEK PIPE. primary or initial velocity as the water entered the canal was not entirely checked, as is Indicated by the considerable difference between the surface and the mean velocities. This value of h’,, .0155, is probably a little too low. The measured head, H, was 1.5050 ; the water in the little gauge-bux, in which the hook-gauge was placed, was quite steady, the fluctuations in level being minute. The co-efficient c, (full contraction) on Plate VII. for /= 5.5 and =1.50, is .5932; adding 14 per cent. to this for partial bottom suppression, for this weirc=.6011. Then, with Q’=e2 (29 4)*1h, and (2 g)*=8.017, we have; HT* ha’ h Z c a 1.5050 -0155 1.5205 5.500 .6011 33.129 Of this volume there was wasted at the pen-stock the flow over a weir with full contraction, and with very little velocity of approach, as follows ; H=h t c., Plate VII. Q A17 1.50 605 1.3806 There were also losses by leakage at the pen-stock and by small leakages from the pipe, aggregating about .102=Q”. Hence Q’—((" + Q”) = Q= 31.721, being the volume passing through the pipe. Hence we have by the formula, if "% ven a a ; l TABLE CI. Hamilton Smith, Jr., and H. C. Perkins, 1879.—Texas Creek 1i-Inch Shet-Iron Pipe. | oa Wee, (targeted (laeee = Mean Mean otal | Losses o ffective cay ae No. ' Length. Diameter. : Area, Head. | Head. ' Head. a elocity of | Gs. é t | Wooden I D a Q ao) ae | N | ho om Blocks. | i pera Be ee, ecm ell = ; a 356 | 4438.7 1.416 ee 31.721 20.143 | 303.62 | 7.46 | 296.16 | 131.1 20.94 Six blocks of wood, each 4 inches x 6 inches x 64 inches, with corners rounded, and loaded so they would just sink—specific gravity 1.05—were sent through the pipe; the time of passage varied from 211 seconds to 2193 seconds ; the mean of the two least times was 212 seconds, showing a velocity of 20.94. Two similar blocks, unloaded— specific gravity of, say, .8—passed through the pipe in 213 seconds and 216 seconds respectively. A small round stone weighing 4 lb. came through in 231 seconds, and another weighing 3 lbs. in 232 seconds, thus having a velocity of 19.2. * With a trifle less water passing over the weir, and with its length 5.480 instead of 5.500, the measured head, H, was in the first experiment in May, 1879,4.4850. This shows the effect of the initial velocity spoken of. EXPERIMENTS WITH PIPES.—Texas Creek PIre. 315 The above experimental value of x* compares favourably in point of accuracy with Experiments Nos. 340-354. * The same value for n is given in the Transactions of the American Society of Civil Engineers, although the values of h.' and c for the weirs are somewhat different, one discrepancy balancing the other. The computations as here given have been made from the original notes, in conformity with the views expressed in the first portion of this volume. 316 CHAPTER XI. EXPERIMENTS WITH ORIFICES, 1884-5. HOLYOKE. Betne especially desirous of accurately determining what differences there were between the co-efficients of efflux for submerged vertical orifices, and those for the same orifices with a free discharge, the author during the summer of 1884 made a large number of experiments at Holyoke, Massachusetts. The experiments with a free discharge, were repeated in 1885. The general arrangement of the apparatus employed is shown by Fig. 1, Plate X VIT. | The supply of water was admitted to the pressure tank, A, by an iron pipe, with a stop valve just above the tank; this pipe received its supply from an upper reservoir, not shown by the sketch, where the surface of the water was maintained at a nearly constant level. The orifices to be experimented upon, were pierced in brass plates of 4 inch thick- ness; those for the two round and two square orifices, were .71 square, the orifice proper being slightly divergent, with a thickness of ;4, inch, as shown by Fig. 2, Plate XVII. The plates were made of this considerable thickness, in order to secure entire rigidity, so that the form of the orifice could not be changed by any possible unequal strains, due either to the wood screws by which the plate was fastened to the side of tank A, or to changes in the plank to which it was screwed. A rubber gasket, between the plate and the wood, formed a water-tight joint. There were 5 brass plates used; two circular, having the respective diameters of about .05 and .1; two square, having the respective sides of about .05 and .1; and one oblong, being a rectangle having a length of about .3 and a width of .05. A plate of boiler iron, with a round orifice having very nearly the same diameter as the .1 round orifice in the brass plate, was used for a few experiments in 1885, to determine whether or not there was any difference of discharge for orifices in different metals. The holes for the round orifices when examined by a microscope of high power proved to be almost absolutely perfect ; the edges of the rectangular openings were not so perfect. || The sizes of the orifices were determined with great care by Prof. W. A. Rogers, of Cambridge, Mass. The means of 3 sets of observations gave the following results, with 7'=62° for the 6 plates ; EXPERIMENTS WITH ORIFICES.—Ho yoke. Apparatus. 317 Round .05 Brass Diameter .049 802 Area .001 9480 rm l i si .099 994 5 .007 8531 ‘5 al Iron ‘ .099 999 “3 .007 8538 Square .05 Brass .049 756 x .049 800 #5 .002 4778 si wl y .100 100 x .100 089 < .010 019 Oblong .3x.05 ,, .299 736 x .049 892 5 .014 954 The measurements of the brass plates were made after the completion of the series of experiments made in 1884 ; after the experiments in 1885, the orifice in the brass plate, D=.10, was carefully remeasured and found to be .099 9984, or a trifle larger than the original determination. This enlargement may have been due to wear, or possibly to a little rough usage in June, 1885, by the mechanic who used this plate as a standard in making the orifice of nearly the same diameter in the iron plate. The apparatus for exact measurement, employed by Prof. Rogers, is exceedingly perfect, and the foregoing sizes are doubtless within a very small fraction of the truth ; the bronze yard of Prof. Rogers, which had twice been compared with the Imperial yard in London, was his standard of reference. /! The standard of measurement for capacity and heads was a graduated box-wood rod, whose slight errors had alse been determined by Prof. Rogers. The tank B had two discharge orifices, at a and b; when the discharge was sub- merged, the orifice at b was closed, and the water flowing through the experimental orifice, O, escaped at a; when the discharge was free into the air, the water escaped at b from the lower tank. These tanks A and B were very solidly built, with the bottom 5 inches thick, and sides 3 inches thick. With the foregoing arrangement it was essential for accuracy that there should be no leak from A into B, and no leak from B. It was thought that these conditions had been almost perfectly complied with. This arrangement was used in the year 1884. The measuring tank © was made of pine, 3 inches thick, with outer clamping frames of wood, held together by iron rods. Its inner section was nearly 2.5 x 2.5, with a height of 4.5. The inner surface was smoothly planed, and the joints of the plank very neatly united. In this tank were placed a steel point at B, and three fixed steel hooks at M, U and T. Just before the commencement of an experiment, the surface of the water was accurately adjusted to either B, M or U, depending upon which vertical section of the tank was to be used for g. The water escaping from the tank B, was then introduced by a bent tin pipe into the measuring tank ©, and the time taken, by either one or two stop-watches, the instant the assistant caught the jet at a or b in the tin pipe; when the water in the measuring tank had risen to nearly the summit of the desired hook,—either M, U or T—the tin pipe was quickly removed, and the time again taken. In 1884 the difference between the surface of the water and the standard hook, was measured by a small hook-gauge, and the capacity computed from this measurement; in 1885 the much more accurate method was employed of determining the capacity between the surface of the water and the summit of the fixed hook, by using a wooden rod, .25 x.25 and 2.5 long, graduated so that each division represented 318 EXPERIMENTS WITH ORIFICES.—Hotyoxe. Apparatus. .001 cubic foot ; this rod was depressed in a vertical position until the surface of the water rose just to the summit of the fixed hook ; the depth of the submerged portion of the rod instantly gave the correction to be applied.* By waiting until the water in the tank wax quiescent, these adjustments could be made to within .001 of a cubic foot. The tank © was measured 5 times, by means of 7 equidistant horizontal sections, with 8 measurements to each section.}/ These measurements gave the following results, for the entire tank, B—T; Aug. 12, 1884 Tank nearly new 27.286 ay lg 39 oa » 24 weeks in use ces ee a i 97.306 Sept. 3, ,, st » 34 4 . : 27.303 May 4, 1885 oe » during the caus ates had ‘Bean comewhus enlarged by action of ice wi oe 27.333 ee NDE 53 oes At termination of May, 1885, series.. Hy 27.331 The enlargement from August 12-27, 1884, was due to momen of the wood. The tank was so firmly made, that there was no difference appreciable between its size when empty, and when full. The tank was also measured on May 4th, 1885, by an iron vessel, which will be described hereafter, whose capacity had been determined to be by weight of water .9310, and by capacity of cylindrical tubes .9312; the mean of these measurements being .9311 with T=46°. By this vessel the capacity of the tank—B to T—was 27.345t. As- suming the rod measurement of 27.333 to be correct, the capacity of the iron vessel was .9307, which was assumed to be its size when used for g at Holyoke in 1885. The contents of the several vertical sections or divisions of the tank were determined, by assuming that the total capacity, or B—T, was 27.333; the results of the flow through the orifices, with nearly uniform heads, filling first one section and then another, were compared, and the sub-division of the total capacity thus made ; B—M =16.968 M—Ue= 4.881 \ 10.365=M_—T U—T= 5.484 B—T = 27.333 Computed from the rod measurements these capacities were as follows ; B—M=16.975 M—U= 4.875 U—T= 5.483 B—T = 27.333 * In 3 or 4 experiments the water was above the hook ; in these cases the correction was ascertained by a hook- gauge. + Tank C adjusted at B, and 29 pails of water poured in, the pail being held by the handle when filled ; the surface of the water was then .0555 below hook T. The horizontal area of the tank at this point is 6.1837 ; hence, 29x .9311—27.002 .0555 x 6.1837= .343 (T of water 48°) 27.345—=capacity tank C, from B to T. EXPERIMENTS WITH ORIFICES.—Hotyoxe. Apparatus. 319 Errors in direct measurement of the small sections were necessarily much larger, proportionately, than for the whole tank. Hence the indirect method of segregation was preferred ; very likely 4.879 would have more accurately represented the capacity of M—U, than 4.881, the quantity adopted ; this difference would represent a com- parative error of z 755 In the value of M—U, 5,55 in M—T, and 4,455 in B—M. The vertical elevations B—M et cet. remained constant during the experiments of 1885, showing that the tank remained unchanged in its vertical plane. // The leakage of the measuring tank in 1884 was quite appreciable, ranging from .0025 cubic foot, to .000 05 cubic foot per minute. In 1885 the tank when filled to T, lost a maximum of only .0026 cubic foot per hour; as the longest experiment in this year was only about 30 minutes, this loss was not appreciable, being probably due entirely to evaporation. We therefore had in 1885, a measuring vessel without leakage and of unchanging form, a desideratum rarely obtained. The stop-watches used beat to } seconds. In 1885 only one watch was used, whose rate when compared with a good second pendulum clock, varied from +.0019 to +.0009; part of this maximum variation was doubtless due to personal error; a uniform + rate of .001 25 was assumd as a correction for May, 1885, being the mean of a large number of determinations, with times varying from 60 to 1800 seconds. For June, 1885, a constant + rate of .001 was adopted. When the experiments made in 1884 had been computed, the results presented were slightly inharmonious. The curves for each orifice were symmetrical, but they did not agree comparatively close enough, one with the other. Determined to know the reason of these discrepancies, in 1885 we made a new series of experiments, all with free discharge, using every possible precaution to prevent error. These new series of experiments revealed the fact, that in 1884 the slight known leak from the tank B was large enough to affect the results; this leakage was so small as not to notably —hardly appreciably—affect the co-efficients from the three larger orifices with the larger heads, but it became quite notable with the two smaller orifices with small heads. The tanks A and B rested upon a plank floor, which was entirely open below ; during 1884 the leakage from both A and B seemed to be only a few occasional drops falling from this floor; doubtless much the larger portion of the leakage from B was absorbed by evaporation. // The possibilities of error for the series of 1885 will be critically discussed hereafter. Details of the 1885 series will only be given; we will hereafter state the general results of the 1884 series. Free Discuarce, 1885. For heads of less than 1., the experimental plates were placed at the lower end of the tank B at 0’; the centre of each orifice being .75 above the floor and distant 1.5 320 HOLYOKE ORIFICES.—FreEeE Discuarce, 1885. Methods. from each side of the tank B. A hook-gauge was placed at f, and the elevation of its zero-point determined at least three times each day by a delicate spirit level, extending from the bottom of the orifice to the summit of the hook. The maximum variations of these determinations were very slight, as the wood of the tank B was thoroughly damp. This gauge was read to about .0002. The water came to the orifices through the racks g and g’ very evenly ; the per- fection of the jets escaping at o’ proved that the supply was evenly distributed, this being a wonderfully delicate test. The surface of the water at f was so steady, that no enclosing box was used for the hook-gauge. The iron pail was sometimes used for g for the .05 round orifice with small heads. When this vessel was employed, it was quickly shoved upon a horizontal plank, into position to catch the jet; this instant was the beginning of ft. When nearly full the pail was held by the handle about half an inch above the plank, and the time ended when the pail was brimming full. We may remark that in all these experiments, both of 1884 and 1885, the quantity q received in a measuring vessel was as nearly as possible the true quantity discharged from the orifice during the observed time, 1.* For heads above 1.7, the experimental plates were placed at 0, on the side of the tank A, and, as at 0’, with their centres .75 above the floor, and 1.5 from the sides of the tank. The heads for this tank were measured by the hook-gauge at c, which was read to .0005; the elevation of the zero point of this gauge was determined, by first measuring the height from the bottom of the orifice to the fixed hook d, which was exactly vertical above the centre of the orifices; the water in the tank was then brought to the level of the hook d, and a reading taken of the gauge at c; these measurements were generally repeated three times each day. The variations of this zero-point were appreciable, being as much as .003 during the first period of twelve days, when nearly all the experiments were executed, and being probably due to unequal swelling of the wood mounting of the gauge, and the tank. The possible error from this source could not be over .0015, if indeed as much, for any one experiment ; such a possible error for H over 1.7, would be barely appreciable. When an orifice was placed at 0, the lower end or side of the tank B was removed, and the water was conducted from 0 to the end of B through a small wooden square pipe or trough, which was perfectly tight, not a single drop escaping from it. From the end of this wooden pipe, the water was at will directed into the measuring tank by the bent tin pipe before described. The use of the screen in the tank A, enclosing the iron feeding pipe, did not result ina perfectly uniform supply of water to the larger orifices, when the highest heads were employed. This was indicated by the slight imperfections of the .1 round jet. * Excepting the leakage from the tank B in 1884, which will be discussed hereafter, and slight losses with the .3x .05 orifice, when dividing inner plates were attached. HOLYOKE ORIFICES.—Free DiscHarce, 1885. Methods. 321 Experiments were made, without any screen around the feeding pipe, to see whether or not this uneven supply of water had any appreciable effect upon the discharge. The experiments were made at an elevation of 90 feet above sea level, in latitude 43° 12’; hence (2 g)* = 8.0210, vide Table IT. In the following table, the exact time of the beginning of each experiment is given, in order to chronologically trace variations inc. That is to say, month, day, hour and minute. Hours from 8 to 12 are a.m., and those from 1 to 6 are p.m. The given temperature, 7, is that of the water. During the experiments made in May, the temperature of the air did not differ more than 8° from that of the water. During the experiments made in June the temperature of the air varied only 3° from that of the water. The corrected time, or t, is only given; the noted time, ¢’, was always y}> larger than t in May, and 5,459 larger than ¢ in June. HT is the mean head during each experiment from the centre of the orifice. O is the maximum variation or oscillation of the head during each experiment. This varia- tion was largely due to a slight leakage of air into the feeding pipe; this leakage caused a variation in the amount of vacuum in the iron pipe between the valve and its lower end, sometimes causing a slightly irregular flow; after the cause of this trouble was discovered, and the pipe made perfectly tight, there was but little difficulty in maintaining a nearly perfectly uniform head for each experiment. The hook-gauge was read either every 30, or every 60 seconds, depending upon the length of the experi- ment. is the arithmetical mean of the gauge readings; where O was largest the means of the } power of the various readings were also calculated, but they in no case differed appreciably from the arithmetical means. The co-efficient c, representing the arithmetical mean of several experiments with nearly identical heads, has been calculated from C’ by Tables III. and IV. The experiments were commenced on May 6th, and concluded on June 6th, 1885. TT 322 EXPERIMENTS WITH ORIFICES. TABLE CIL Holyoke, Massachusetts, May and June, 1885.—Flow through Vertical Orifices, with (2 g) 4=8.0210 Q@=C a (2g H)¥ Brass. .05 round. : | q | H No Date, T | t } fv Q | 1885. | I weasel, | CORPO Tg | oo 0 May 9, 2-35 212.2. .9307 9307 | 004386 | .1849 | .0010
p> o° oO 190 | .6611 6454 .6478 21 | g4go |-8472 || 200. | .6475 6475 6477 6474 6479 .6482 ly Ww .6478 || .203 | 6481 6441 6436 23.6438 |.6436 || 240 |.6438 6433 6432 6329 24 6340 .6334 || .283 | .6336 6444 6442 25 6452 6451 || .282 | .6453 .6468 324 EXPERIMENTS WITH ORIFICES. 26 27 28 29 30 31 33 2-18 2-21 2-25 2-29 8-51 8-55 8-59 11-47 11-51 11-54 11-57 8-44 8-48 8-51 8-55 5-01 5-21 11-31 11-34 11-37 11-40 11-10 11-15 11-18 11-21 11-24 9-08 9-49 10-04 49° 49° 49° 51° 52° 49° 49° 49° TABLE CII.—continued. 173.7 173.4 173.3 162.6 162.5 162.6 162.7 161.1 160.9 161.5 161.1 862.0 861.3 143.3 143.2 142.7 143.2 130.2 129.7 129.8 129.8 129.7 2050.0 585.3 658.8 Vessel. 9307 5.484 ” 9307 ” ” ” 16.968 4.881 5.484 Correc- tion. q 9307 9307 .9307 9307 9307 9307 9307 .9307 9307 9307 9307 9307 9307 9307 —.043 | 5.441 —.054 | 5.430 .9307 9307 .9307 .9307 9307 9307 9307 9307 9307 —.065 |16.903 —.061 | 4.820 —.094 | 5.390 .9807 @ 005 364 || .005 374 .005 377 .005 374 .005 358 .005 367 .005 370 005 724 .005 727 .005 724 .005 720 005 777 005 784: .005 763 .005 777 .006 312 .006 304 .006 495 .006 499 .006 522 .006 499 .007 148 .007 176 .007 170 .007 170 .007 176 .008 245 .008 235 .008 182 H 2823 2823 2825 2825 2830 2830 2830 .3355 8351 .8350 3345 .3378 3359 8353 3356 4024 3995 4349 4365 4373 4875 .5359 5359 5370 5365 5855 7201 7186 7220 .0005 .0005 .0017 .0006 .0001 .0001 .0091 .0011 .0010 .0010 .0005 | .0010 | .0005 .0055 | 0010 HOLYOKE.—FReEE DiscHarcE, 1885. Reductions. 325 TABLE CII.—continued. lw =I Lo os) 30 31 32 33 6461 6473 6474 .6470 6446 6457 .6461 6324 .6332 .6329 .6330 .6362 6387 .6369 .6382 .6368 6384 .6303 .6296 .6312 .6289 .6249 6274 6262 -6265 .6276 .6219 6217 .6162 REMARKS. Mean. H c 6470 || 282 |.6472 | 6455 || .283 |.6457 | | | 6329 |) 335 |.6330 | I 6375 | 336 |.6376 | 6376 || 401 | .6377 | i | 6300 | 437 |.6301 | 6265 | .536 | .6265 6199 | .720 |.6199 | Brass. .05 round.—continied. For this determination expiration of ¢ was not taken very exactly. — .042 | 326 EXPERIMENTS WITH ORIFICES. TABLE CII. —continued. Correc- No. Date. T t Vessel. a q Q H O May 9, 10-23 524.3 | 4.881 | —.048] 4.833 | .009218 || .9132 | .0035 34 » ae FORE T 50" 5883 | 5.484] —.080| 5.404 | .009186 | .9122 | .0020 oe LOST 592.5 . ~.063 | 5.421 || 009149 || .9059 | .0015 cela: ae 1814.6 | 16.968} —.038 | 16.930 |} .009330 || .9289 | .0010 egy | ERE 1816.7 —.022 | 16.946 | .009328 || .9288 | .0012 cap. la 521.3 | 4.881 | —.018| 4.863 || .009328 || .9303 | .0015 35 52° i: ee. “ALS 519.3 . ~.034 | 4.847 , .009334 | .9284 | .0022 oa dees 585.1 5.484 | —.032 | 5.452 | .009318 | .9292 | .0008 coe, SS 584.8 " ~.080 | 5.454 || .009326 | .9278 0 oe 1s ee 432.7 | 5.484 | —.037 | 5.447 || .012588 || 1.7383 | .0030 36 < @ 206 | 431.4 5 ~.049 | 5.435 || .012599 |} 1.7376 | .0030 gp 23S 431.2 3 —.049 | 5.485 || 012604 | 1.7423 0 ‘sg » B, Bil], 659.2 || 10.365 | —.047 10.318 |) .015 652 || 2.7231 | .0100 a 4 OR 659.2 ‘ —.026 | 10.339 | .015 684 | 2.7349 | .0040 » 13, 2-25 576.2 | 10.365} —.049| 10.316 | .017 904 || 3.5790 | .0080 38 > » 240) 54° 576.2 : —.050 | 10.315 || .017 902 || 3.5668 | .0050 cae «| 578.5 » | —.031 | 10.334 |) .017 864 || 3.5641 | .0040 a 13, 1485 506.1 | 10.365} ~.037 | 10.328 || .020407 || 4.6581 | .0100 39 cp | 508.7 . -.046 | 10.319 | .020285 || 4.6152 | .0140 a. | EO 506.8 » | 7044 10,321! .020365 || 4.6276 | .0010 Brass. .1 round. —— ——4 : — a =e | | May 12, 8-31 722.2 1 10.365 , —.031 , 10.884 | .014 309 | 1296 | .0025 40 s « B47) 20 724.5 » | =.027° 10.338 ' .014 269 | 1291 | .0007 ll cde see SE | 726.9 | 4 ~.003 | 10.362 014255» .1288 | .0011 | | i: | | | | 4l(y, 12, 9-42 | 51° |} 507.3 | 10385 ~.044 } 10.321 | .020 345 2644 | .0090 | | | | ! | | eZ | » I, 248) | 3946 | 10.365 | ~ 033 , 10.332 026183 | .4563 | .0030 | 2.58 | | 393.5 | | 10.323, .026234 | .4583 | .o010 | | \ HOLYOKE.—FRree Discuarce. Reductions, 1885. 327 TABLE CII.— continued. No. Mean. c REMARKS. 3d 35 36 37 38 39 6174 6155 6152 6196 6195 .6190 .6200 6187 6197 6111 6117 6111 .6071 .6070 6057 .6067 .6056 6051 | 6043 6059 .6160 6194 .6113 .6070 6060 6051 910 929 3.57 4.63 6160 | .6194 .6113 |} Brass. .O5 round.—continued. For this series, adjustments and corrections for g, were made with the utmost precision. .6070 6060 | .6051 | | | D=.099 994 a=.007 8531 40 41(?) 42 .6310 6305 6306 6281 6154 6152 6307 6281 6153 | 129 264 457 .6155 ' Jet enlarges and diminishes three times in falling 7 feet. 6337. The section of the jet is uniform; the swell looking in front, being just opposite the stricture looking sideways. 6288 j Head for this experiment jumping about badly. A little manifestation of alternate swelling and contraction of the jet, as above noted. Jet perfect at orifice. 328 EXPERIMENTS WITH ORIFICES. TABLE CII.—continued. | i | Correc- No. Date. | T | t Vessel. pie q Q iT O =a Hs | | | May ll, 2-16 | 329.1 | 10.365 | —.044 | 10.321 | .031361 || .6613 | .0005 43 o oy 226 | 51° 329.6 i. —~.042 | 10.323 |} .031320 || .6596 | .0005 yoo BBE ‘| 329.6 _ —.042 | 10.323 |) .031320 || .6618 | .0020 i | i gg, tle, SLA8 a6 | 748.0 , 27.333 | —.065 | 27.268 , .036454 || .8993 | .0025 [oe = 208 750.2 ‘ ~.036 | 27.297 || .036386 || .8998 | .0022 y dd 48 542.6 | 27,333 | —.065 | 27.268 | .050254 | 1.7460 | .0070 45 yoy £85 49° 545.6 —.027 | 27.306 | .050048 || 1.7283 , .0010 i ap, see! 545.3 2 — 032 | 27.301 || .050066 | 1.7300 0 ie , 6 257 ms 402.9 || 97.333 | —.064 } 27.269 | .06768 | 3.1806 | .0030 ie 9. ele 403.4 7 ~.047 | 27.286 | 06764 || 3.1776 .0005 | » 6, 2:09 336.6 | 27.333 | 4.002 | 27.335 | .081 21 4.5969 | .0015 47 | yo Ded 49° 335.0 " —.113 | 27.220 ) .08125 || 4.6019 | .0025 | oe ae 336.3 —.035 | 27.298 || .08117 || 4.5948 | .0020 | gm @ & Be) oe 335.9 | 27.333 | -.030 | 27.303 | 08128 | 4.6023 | .0090 i oy ee Ss 336.6 ‘ 0 27.333 | .08120 || 4.6019 | .0005 | ! Je bigs. “Ge. OBES 19° , 402.2 | 27.333 | —.035 | 27.298 | .067 87 | 3.1985 | .0010 | yoy 9-29 $02.2 | ” ~.031 } 27.302 | .06788 © 3.2007 | .0005 I | a » 5, 219 ie | 401.5 27.333 | —.037 | 27.296 | .06799 || 3.1946 | .0010 | yyy, 2-87 402.1 | ‘3 ~.006 | 27.327 .067 96 3.1941 | .0010 | Sl 4 3 646 48° 400.7 | 27.333 | -.033 | 27.300 | .06813 || 3.2213 | .0050 | ; » By 10-06 336.0 | 27.333 | —.031 | 27.302 | .og126 || 4.5968 | .o025 » wy (10-24 336.5 2 0 27.333 | 08123 | 4.5965 | .0015 52 ow om «LOO age 335.6 - ~.062 | 27.271 | 08126 || 4.5971 | .0015 So. ag. OR 337.8 = +.064 | 27.397 | .08110 |} 4.5933 } .0040 me ee ots 337.0 % -.012 | 27.321 | .081 07 | 4.5906 | .0210 HOLYOKE.—FRreEE Discuarce, 1885. Reductions. 329 Mean. iT 43 44 45 46 47 48 49 50 51 52 rn eS ee lo Lo 2 0 6112 .6103 .6090 .6038 6044 6048 6013 .6013 .6012 .6015 .6010 .6025 6024 6039 6037 6026 .6017 .6015 6017 .6008 .6007 .6119 .6096 .6042 6025 6013 6012 .6024 .6038 6013 661 .900 | 1.73 3.18 4.60 4.60 3.20 3.19 3.22 4.59 .6096 6042 6013 | | | .6012 6024 6038 6013 TABLE CII.—continaed. REMARKS. Brass. .1 round.-—-continued. ' Perfect jet for the entire fall of 7 feet. Perfect jet as above. ¢ Jet very nearly perfect for 2.5 feet; perfect at orifice. 6025 | Jet very nearly perfect, and sometimes apparently perfect, from orifice to where it strikes trough. Jet seems to be perfectly true for .2 from orifice ; then it sometimes begins to twist a little, amd at other times is perfect for a distance of 1.5 from orifice. For Nos. 48 and 49, no rack or protection of any kind around discharge end of iron feeding pipe. Jets for both heads are exceedingly ragged and twisting, being very far from smooth even immediately at orifice. For No. 50, one vertical plank, .93 high, placed across tank, A, just in front of feeding pipe ; open on top. Jet ragged and twisting even at orifice; not apparently as bad a jet, however, as for Nos. 48 and 49. 3 For No. 51, two vertical planks, 1.86 high, placed across tank, A, just in front of feeding pipe ; open on top. Jet ragged and twisting. For Nos. 52 and 53, four vertical planks, 3.92 high, placed in front of feeding pipe, open on top. Jet twists a good deal, and is perceptibly not absolutely perfect near the orifice ; very much better, however, than in Nos. 48 to 51, but not as good as No. 47. UU 330 EXPERIMENTS WITH ORIFICES. TABLE CII.—continued. No. Date. | F t ! Vessel. oe q Q iH O | May 5, 1121 | 403.7 | 27.333 0 | 27.333 | .o6771 || 3.1883 | .0340 se 1. 6 [lee ) ee | aon . _.013 | 27.320 | .06772 || 3.1900 | .0020 ie 8 | 403.5 : _.003 | 27.330 || .06773 || 3.1875 | 0 | | ba | 4, 5 taf 4B 337.4 | 27.333 | —.012) 27.321 || 08098 || 4.5973 | .0020 | | 5d) («CO AL as? |) 336.1 | 97.333 | — ant 97.316 | 08127 || 4.6190 | .0140 June 6, 11-43 | 5243 | 27.833 | —.040 | 27.298] .052056 | 1.8767 | 0 56 | » » %I20l 62° 5249 _.055 | 27.278 | .051968 || 1.8767| 0 gap ADEE | 526.3 7 _.041 | 27.292 | .051856 | 1.8637 ge Vie Bie OE ae | 501.4 | 27.333 | -.040 | 27.293 | .054434 | 2.0487 | 0 og BS | | 500.7 i -.074' 27.259! 054442 | 2.0487] 0 | oy 8 807 337.6 | 27.333; -.088 27.295 .08085 | 4.5583 | .0010 58 | 4 » B22 | 62.5" 337.5 | 3 ~ 054! 27.279 | .08083 || 4.55171 0 4a. di ey | 337.6 > ~.054 ee 08080 | 4.5477 0 Iron. .1 round. Ih June 6 16-08 532.4 27.333 | -.058 27.275 | .051930 | 1.8156] 0 5 | ay 2080 | Bie | wary ‘ ~.106 | 27.227 051 207 | 1.8222 | .0030 ~~ 10-51 532.5 P ~.060 | 27.973 .051217 | 17966] 0 | 8 433 531.8 | 27.383 | -.043 | 27.290 | .051316 | 1.8070 | .0040 ai, — Asi] ee 531.6 “ ~.077 97.256 051 972 | 1.8028 | .0010 Be es TS 533.4 ’ —.060 | 27.273 .051 130 | 1.7936 | 0 ' 4 6 (9-08 428.9 | 27.383 | -.037 | 27.296 | .063 64 | 2.8046 | 0 a +, « 984) #1 430.0 2 +.057 07.390 06370 28076] 0 p> 9-39 428.7 —.045 | 27.288 06365 , 2.8076] 0 | , 6 821 332.4 | 27.333 | —.051| 27.282| 08208 , 4.6766] 0 62 | , 4 $36 | 605° |} 3322 . —.067 | 27.266 | 08208 | 46766] 0 oe. BS) 332.2 a ~.053 | 27.280 || .082 12 | 4.6766 | 0 Mean. HOLYOKE.—FREE Discuarce, 1885. Reductions. 331 53 54 55 56 57 58 5996 6004 .6033 .6022 .6030 .6038 6038 .6012 6014 6015 6028 .6038 .6014 1.87 bw 05 4.55 .5996 6004 .6028 .6038 .6014 | TABLE CII.—continued. REMARKS. Brass. .1 round.—continued. Jet apparently twisting a little more than in No. 52. Orifice well oiled with sperm oil, before No. 54. After No. 55, orifice found to be pretty nearly free from oil. ie oe Nos. 56 to 58 with normal conditions every way. | ) D=.099 999 a=.007 8538 59 60 61 62 6035 6022 .6066 .6060 6062 -6060 6033 6035 -6030 6025 6025 6028 6041 | 6061 6033 .6026 1.80 2.81 4.68 6041 6061 £6035 .6026 Jet nearly perfect ; perhaps twisting a trifle more than with brass plate, D=.10, with same head. ’ Jet twisting more than for Nos. 59 and 60. Jet twisting a good deal, after distance of .2 from orifice ; perhaps a little more than with brass plate, D =.10, and same head. 3382 EXPERIMENTS WITH ORIFICES. TABLE OII.—continwed. Brass. .05 x .05 = es san cy ee eg I Correc- | | No Date. | T | t ; Vessel. ee. q Q | vae O | ! dn Se _ ieee | ‘May 9, 3-80 ge14 ' 4.981 | ~.027} 4.854] 007124 | .3117 | .0100 7), = eae) 760.7 5.484 | -.057 | 5.427 || .007 134 | 3150 | .0030 a. (@; ag 560.4 | 4,881 | -.062| 4.819 | .008599 | 4619 | .0070 ane e401 | 5.494] -.060| 5.424) 008474 , 4528 | .0035 i : (8 £49 i798 | 4,881] -.046| 4835) 010077 | .6504 | .0040 ae) a. eee] 536.7 | 5.484 | —.071| 5.413 | 010086 |) .6518| .0010 ! | » 9 «B19 413.2 || 4.881 | —.060) 4.821 || 011668 | .8863 | .0080 | BB ese ca. |, SRE | ae) BE _ 054 | 5.430 || .011 585 || 8734 | .0025 non BAd 469.3 | 053] 5.431 || 011573 | .8706 | .0005 | erm, 14 868 | 52" |) 648.2 | 10.865] -.082} 10.383] 015941 | 17017 | 0190 , 14 959 630.7 | 10.365 | -.052| 10.313 |) 016352 | 1.7887 | .0040 68 |, » 1015 | 52° || 631.0 , | ~.042 | 10.323 |] .016 360 | 1.7919 | 0110 yoy 10-384 628.6 , | 046} 10.319} .o16416 | 1.7921 | 0 | = Hon 14, 10-85 504.7 | 10.365 | -.044 | 10.321 || .020450 | 2.8171] 0 2. Gun gt 505.3 | =.052| 10.313} 020410 | 2.8121] 0 | yg tae | 330.2 | 16.968) -.006| 16.962 | .020431 | 2.8121] 0 om. 188 831.7 , | -.021 | 16.947 |) 020376 | 2.8021) 9 » i 148 | 440.8 || 10.365 | —.038 | 10.327 | .023428 | 3.7045 | .0020 70 | , 4 201] sie | 443.9 . | -.009 | 10.356 | .023330 | 3.6941 | 0 J» » 2 442.7 , | =.028 | 10.337 | 023.350 | 3.7012 | .oo10 | » 14, 233 650.3 | 16.968} -.015 | 16.958 || 026070 || 4.6298 | .0020 1 |, 948 | Bie |) 396.4 | 10.365) -.031 | 10.834 | .026070 | 4.6266) 0 la on 300 396.0 | , | -.046 | 10.319 | .026058 | 4.6256] 0 | ! | HOLYOKE.—F ree Discnarce, 1885. Reductions. 3383 TABLE CII.—continued. Mean. -049 756 x .049 800 = .002 4778 = area. 63 64 65 66 67 ”) 68 69 70 71 6420 .6396 .6366 .6340 6287 6286 .6236 6237 .6240 .6149 .6152 .6149 .6170 .6130 .6124 .6130 6125 6124 .6107 6107 .6096 .6098 .6096 .6408 .6353 6238 .6157 .6127 .6113 .6097 313 457 651 877 1.70 1.79 2.81 3.70 4.63 .6410 6354 6149 .6157 .6127 .6113 6097 REMARKS, Jet very regular, with diamond-shaped facets, soon after leaving orifice ; not cruciform. Jet as above. Jet as above. Jet as above. Head jumping about for this experiment. Jet very nearly perfect, and cruciform for .6 from orifice. For this series, adjustments and corrections for g made with great precision. Jet as in No. 68. Jet very nearly perfect ; cross continues for 1.2 from orifice. Jet very nearly perfect ; cross continues for 1.5 from orifice. 334 EXPERIMENTS WITH ORIFICES. TABLE CII.—continwed. Brass .lx.1 No. Date. if t | Vessel. | ee | | Q H 0 10nN, May ll, 11-09 484.6 | 10.365 | —.042' 10.323 | 021302 | .1790 | .0050 72 | 4, 4 1292} soc | 480.9 | ‘4 ~.032 eo 021487 | .1814 | .0005 s 4: tie 479.6 | ~.067 | 10.298 || 021472 | .1814 0 | : » 11, 1023 792.5 | 27.883 | -.030 | 27.803 || .034 452 ! 4804 | 0005 el ae eae: || 8 795.1 |, ~.020 | 27.313 | .034 352 . 4789 0 = tke eat | 671.3 | 27.383 | -.032 | 27.301 | .040669 | .6762 | .0035 Tce a: Grete) ee 670.5 |, ~.026 | 27.307 | 040726 6774 + .0005 «tty. “sae 5723 | 97.333/ 0 27.383 | 047 760 | 9371 | .0035 7 | oy gp 808'| Bor | ree] ~.048 | 27.985 047801 | .9412 0035 oy 9-28 | 215.7 | 10.365) -.o41 | 10.324 | .o47 863 || .9399 | 0 1 | i » 14, 544] 51° | 426.7 | 27.333 | -.030 | 27.303 | .06399 | 1.7155 0 78 | 5 25, B38.) ae 4) Bia ~.004 | 27.329 | 06394 | 1.7095 0 < » Be 496.9 |, _.031 | 27.302 .06395 | 1.7095 0 | » 1, 9-05 | 337.5 | 97.383 | -.036 | 27.297 | 08088 1 2.7455 0 | yn p B20) ae | sire r ~.005 | 27.328 | 08090 2.7455 0 < ge O84 , ear a : ~.007 | 27.326 | .08092 || 2.7455 0 | | | » 15, 9-49 | 989.8 ; 27.988) —.034| 27.299] .09420 | s.r 0 7’ | 4 os ©68T0-08 | 4g” «|| aaa Z ~.075 | 27.958 + 09412 |] 3.7375 0 =: ee, POT | 289.8 ‘ —.050 | 27.283 | 09414 | 3.7375 0 | ' ! » 18, 10-30 961.7 27.333 | -.002 | 27.331 | 10444 |) 4.5865 0 791 m= 10-43 |) so | 261s : -.035 | 27.298 |] 10439 || 4.5865 0 gy 10-55 261.7 : ~.030 } 27.303 |] 10433 |] 4.5865 0 Brass 3 long x .05 wide | a [ne May 12, 2-44 | 687.6 || 27.333 | -.042 | 27.291 | .039690 | .2609 | .0001 so | , » 304) 52° | esas : ~.051 | 27.282 || .039.625 | .2608 | .0003 < Bee | 687.2 ; ~.060 | 27.273 | .o39687 | .2611 ° .0015 HOLYOKE.—Fnree DiscHarce, 1885. Reductions. 335 TABLE CII.—continued. -100 100 x .100 089 = .010 019 = area. | Mean.|| ¢ REMARKS. 265 79 | 6278 6272 | .181 |.6292 Jet very short. No facets as with .05 x .05 with small heads. : No signs of any interior vortex. 6273 1 i 6185 73) eya7 | .6181 | .480 | .6184 || Jet considerably rippled. | 6154 | 74 6157 6155 | .677 .6157 | Jet perfect, except a little rippling. .6139 75 |.6131 |.6138 | .939 | .6139 | Jet very nearly perfect ; sometimes perfect. Shape cruciform. i 6143 | 6079 76 |.6086 ;.6084 | 1.71 |.6084 | Jet very nearly perfect ; a little uneven 1. from orifice. 6087 6074 77 =| .6076 |.6076 | 2.75 | .6076 || Jet as above. 6077 | | | 6063 | 78 |.6058 .6060 | 3.74 | .6060 | Jet more perfect than with preceding head. .6060 | oe | Jet al bsolutely perf fect than with her head et almost absolutely perfect ; more perfect than with any other hea 79 |.6066 |.6065 || 4.59 | .6065 Sertite acibee. : 6062 | a _ ee oe mas -299 736 x .049 892 = .014 954 = area. 6478 : so |.6469 |.6474 | .261 |.6476 Jet nearly perfect ; close to orifice, entirely free from ripple. .6475 No signs of any interior vortex above orifice. EXPERIMENTS WITH ORIFICES. TABLE CII.—continwed. No. Date. T | t | Vessel. es | @ vel 0 pee | ei May 12, 207 |... 540.2 27.333 | +.065 | 27.398 / 050718 |} .4421 | .0007 2 ae. “206- | 536.7 ' ~.087 | 27.246 | .050766 | .4425 | .0010 | | , 12, 12-20 | 444.0 97.333 | —.056 | on.a17 | 061435 | .6568 | .0098 82 es (ee ae 440.6 |, ~.038 27.295 061950 | .6713 | .0015 ow a BL | 440.8 | : ~ .064 [nae | 061862 | .6674 | .0015 cae 11-30 | «877.6 27,333 | —.046 "97.287 07226, .9172 | .0065 88 | » » U46! srr | 3794 j 5 | =0Or , Breee | 07197 || .9156 | .0025 12-01 | BS ow _.032 | 27.301 | .07213 | .9178 | .0010 —* | | | | | | », 15, 4:25 9721 97,3388 | -.015 27.318 10040 |, 18240 0 84 ~ » 438) $2 | 2788 | f +.008 | 97.341 | 10022 | 1.8120 0 i. ana |, ~.007 ) 27.326 10021 | 1.81200 ; 7 | 85 ~ 1 889.) Be 293.7 , 27.833 | +.006 | 27.339 | qaaa1 87180 | 0 : } 1 6 ny B AOE ; 219.7 | 27.333 | +.057 | 27.390 | 12467 | 2.8270 ; .0020 « ap 308 218.8 z 017 27.316 | 12484 | 2.8300 | 0 ~ &. Sete 190.2 27.333 | —.056 | 27.277 14341 3.7475 | .0030 87 |. 430 | 49° || 1998 4 |, | 125 97.208 14335 | 3.7500 | 0 oy aD 190.6 | ‘ 0 | 27.333 | 143 40 | 3.7423 ' .0020 ‘ | | | , 15, 321 169.8 27.333 | —.095 | 97.238 .160 41 4.6980 | 0 88 gigs oe 52° || 170.1 I Ps —.068 | 27.265 | .160 29 4.6970 0 oy BOB] 49° 190.6 5 ~.077 | 27.256 | .14300 | 3.7390 0 a |S | 190.9 | ‘ ~.045 | 27.288 | 14294 || 3.7340 0 HOLYOKE.—FreeE Discuarce, 1885. Reductions. 337 TABLE CII.—continued. H c REMARKS. 81 83 84 85 86 87 88 Brass. .3 long x .05 wide.— continued. .442 |.6361 | Jet as for No. 80. .6359 .6362 6320 6304) .6312 | .665 .6312 | Jet a little rippled. 6313 . 6291 6271) .6280 | .917 | .6280 || Jet regular, but not quite perfect 6277 6197 .6207 | 6203) 1.82 |.6203 | Jet fairly true. 6206 .6180 | .6180 | 2.72 | .6180 .6182 wie 6184 2.83 | .6184 || Jet twisting a little. 6176 : 6171! .6176| 3.75 |.6176 , Jet twists a little. 6180 .6170 6166 | 6168] 4.70 | .6168 || Jet fairly true; twists a little. .6169 With vertical brass sheets of various thicknesses, placed aeross centre of orifice ; the sheet being normal to the plane of the orifice, and projecting into feeding reservoir. 90 6197 | Brass sheet .74 x .75, and .0008 thick ; hence area of orifice = (.299 736 al ee — .0008) x .049 892 =.014 914 for Nos. 89 and 90. res Jets somewhat truer than for experiments with this orifice without the | sheet ; the form of the jet apparently unchanged .618 | 1 .6182 .6182 | 3.73 | .6182 338 EXPERIMENTS WITH ORIFICES. TABLE CII.—continued. - | Correc- | No. Date. T t = Vessel. | “peo q ! Q Bi 0 | 1 ei May 8, 8835 ae 189.4 | 27.333 | —.089 | 27.244 | 143 84 | 3.8107 .0110 ee es 8-46 189.6 5 —.007 | 27.826 144 12 3.8230 0 99 » 8, 8-59 ig , 290.1 27.333 | —.016 | 27.317 12411 2,8226 .0005 a se a 9-11 220.6 | $4 0 97.333 .123 90 9.8229 .0005 lI 93 » 15, 2-21 5 | 933.3 |) 27.338 | —.047 | 27.286 | .116 96 2.7620 0 se ay 2 33 233.4 i (2) () | 2.7668 .0010 F i { Ls. | | i i | | | ] | — | | (a June 5, 9-43 319.4 |) 27.333 : —.068 | 27.265 | .087 28 | 1.8120 94 A a 9-56 64° 312.7 7 » ~ 083 | 27.250 || .087 14 1.8010 » 37~—«CL 0-10 312.5 ” | _.086 | 27.247 .087 19 1.8010 | » 9, 10-32 254.7 | 27.333 | —.080 | 27.253 | .107 00 2.7120 0 95 yoy 10-45 63° 255.4 . 023 | 27.310 | .106 93 2.7120 0 yyy =~ «L 0-58 255.5 is —.024 | 27.309 10688 | 2.7120 0 i » 5, =L1-18 195.4 || 27.833 | —.078 | 27.255 | .139 48 4.6600 0 96 yng ~~ LAB 63° 194.8 4 —.096 | 27.237 .139 82 4.6720 0 yy ~~ L187 195.4 ‘4 —.052 | 27.281 .139 62 4.6700 0 . HOLYOKE.—FRreeE Discuarce, 1885. Reductions. 339 TABLE CII.—continued. No. C | Mean. H c REMARKS. Brass. .3 long x .05 wide. 91 Hol el 6161 | 3.82 | .6161] Same sheet, and same area as Nos. 89 and 90. Sheet, however, pro- .6162 jects through orifice (plate }” thick) and extends .085 in length on outer side of plate, thus dividing contracted vein. A very small loss of water in these experiments, by dropping from this projec- 6175 aren & tion, which is not caught by wooden pipe or trough. 92 6164 6170), 2.82) .6170)) The jets follow along the dividing sheet their forms unchanged. 6246 | Brass sheet .67 x.64, and .0182 thick; hence area of orifice= F 9 7 9 93 6246 | 2.76 | 6246 (299 736 —.0182) x .049 892 =.014 046. This sheet does not pro- , ject through orifice. A perfect fit was not made for this sheet | against the plate, and hence a little water comes out in spray under- neath the edges of the sheet, which was chiefly lost, z.e., not going _ into measuring tank. The jet unites after its escape from the orifice; perhaps not perfectly, but nearly so. The correction for g for the second determination was not noted ; it | ' -was not yery different from the first. { | | 6242 | 94 | 6251) 6249; 1.80 | .6249 | 6255 | | Brass sheet, .67 x.67, and .04017 thick; hence area of orifice= : ' (.299 736 — .040 17) x .049 892 =.012 9503. This sheet does not | project through orifice, and fits perfectly to the face of the brass 6255 plate. : ecll For Experiments Nos. 94 to 96 there was a very slight loss of water Se | et by spray, which did not go into measuring tank ; this loss was 6248 | probably greater for No. 96 than for the other two experiments. . _ The escaping jets for these experiments united at a distance of about | ' 05 from the plane of the orifice ; from this point the jet had the .6220 same appearance as when no sheet was employed—Nos. 84 to 88. 96 | 6227] .6222 , 4.67 | .6222 .6220 340 HOLYOKE ORIFICES.—F ree Discuarce, 1885. Reductions. The results obtained in May, 1885, under normal conditions for the five brass plates, have been plotted on Plate III., with values of ¢ as ordinates, and values of HT as abscissee. From these experimental points, a curve of the most probable value of c, with T from 48° to 54°, has been drawn for each one of the five orifices. Except for the values of H below 1. for the .05 round orifice, and Nos. 41 and 67 which were known to be unreliable, these curves do not as a maximum differ more than .000 75, or sath part, from the given values of c, and rarely more than .0005, or aath part. The symmetry and harmony of these curves almost conclusively demonstrate the general accuracy of the experimental data. In Chapter ITI. we have fully discussed these results. Chances of Error and Variation. The methods employed in 1885 can be considered perfect, except in the measure- ment of t; this could have been more accurately obtained by an automatic chrono- graph, with the beginning and ending of each experiment graphically registered by means of electric currents. The chances of error and variation in detail were as follows : Orifices.—The dimensions of the orifices were measured by Professor Rogers with the greatest care. The given areas of the .1 round and .1 x.1 orifices are doubtless free from appreciable error ; the areas of the .05 round and .05 x .05 orifices may perhaps be slightly appreciably in error. Such errors will of course be constant. The edges of the orifices both in 1884 and in 1885 were always free from the slightest tarnish ; before each series of experiments they were wiped perfectly free from any trace of oil or grease, except m two experiments with the .1 round orifice, where the edges of the orifice were purposely wet with oil. Professor Rogers found by experimentation with the .1 round orifice, that its diameter increased with increased temperature pretty closely with the general co-efficient of expansion for brass. The temperature of the water for the May, 1885, series only varied 6°, so that enlargement by expansion was not appreciable. Measuring Vessels—The standard of capacity was the value of 27.333, for the measuring tank from B to T. We have already described the several measurements of this tank. We consider this value as having been determined certainly within sath part of the truth. Any error in this value will be constant for the complete series of experiments in 1885. The segregation of the capacities of the several vertical sections of the measuring tank may be slightly in error, especially for the section M-U, which may possibly be in error .004, or auth part. Such errors will only appreciably affect values of c for the two smallest orifices. For the three other orifices the entire tank was nearly always employed. The errors of adjustment to the lower datum point or hook were very slight. HOLYOKE ORIFICES.—Free Discuarae, 1885. Chunces of Error. 341 These adjustments could be made with wonderful precision, and for the point B with rapidity, as the water at this point soon became quiescent. The errors of correction for capacity between the surface of the water and the upper datum hook, could also be obtained within a limit of about .001 cubic foot, by waiting until the water became almost absolutely quiescent. This required considerable time, and when the entire tank was used, this correction was made generally with a limit of accuracy of about .003 ; such an error would not be appreciable. When the smaller sections of the tank were used, greater care was observed in making this correction. As before stated, the losses from the tank when filled, only amounted to about .003 cubic foot per hour. Its form also remained unchanged. Taking all the chances of comparative error in obtaining Y by the measuring tank, we may assume that for the smaller sections, an aggregation of errors all in one way might amount to on ; for the entire tank such errors could not exceed so. When @ for the .05 orifice was obtained by the iron pail, with the capacity of .9307, there might be a possible error of say g, or .0019; the capacity of this pail was known certainly to .0005; any error would chiefly arise from incorrect observation of the instant when the pail was filled. Time.—Errors in ¢ arose from an uneven rate for the watch and from personal error. Probably the rate did not vary beyond the limits of +.001 and 4.0015; +.001 25 was the rate adopted; error in rate hence probably did not exceed yqth. Personal error had a probable limit of about .25” counting errors both at beginning and ending; for short times, such an error would be quite appreciable, being -3 = 3, for No. 32. Head.—When the plates were placed at o/ for heads below 1 foot, the united errors of zero determination for the hook-gauge, error in the gauge-rod, and errors of observation probably did not exceed .0012, and could not possibly have exceeded .002 ; the readings were taken by an assistant who had had long experience in such work, and his determinations were frequently verified by the author. The surface of the water was nearly absolutely quiescent, and the position of the hook could be readily fixed within a limit of .0008 or less; when the height of the water varied during an experi- ee in the few cases when these oscillations of the surface were irregular, a special note to that effect is given in Table CII. When the pressure tank A was used, united errors of zero-point, rod and observation probably did not exceed .002, and could not possibly have exceeded .004. Errors in H would of course be greatest with the smallest heads ; for heads above 2.5 feet they could have had hardly any appreciable effect upon c. The given value of H represents the mean of a number of observations, which probably nearly eliminated errors of observation ; errors of zero-point and rod would be nearly constant for the ment the error from this source was roughly in the proportion of 342 HOLYOKE ORIFICES.—FReEE Discuarce, 1885. Chances of Error. experiments from which the mean value of c was deduced. Hence for small heads the mean errors in H for such a series were probably less than .001, and we can hardly conceive that they could have exceeded .0015. A mean error in H of .0015 for H=.3 would affect ¢ about zth part, and for H=.9 about jth part. Variation in Temperature.-—The range of T in May, 1885, was only 6°; such a variation we surmise did not appreciably affect c for the .1 round and .1 x.1 orifices, unless possibly for Nos. 40 and 72, when H was quite small. For the .05 round and .05 x .05 orifices with heads less than 1., we conjecture that even this slight variation in T appreciably affected c. An increase in 7’ having the effect of diminishing c. Impurities in the Water.—The water used in all these experiments was drawn from the Connecticut River, above the City of Holyoke. There are many manufacturing establishments on the river above this point, and the water doubtless contains small quantities of grease or fatty matter. The water of the Connecticut can hardly be called muddy, even in times of flood ; its color is dark, in common with most of the streams of New England, probably due to vegetable acids. In May, 1885, the river was at a pretty high stage, and the water was less limpid than in June, 1885, when the river was at a lower stage. The action of the water had only a very slight tarnishing effect upon the faces of the brass plates, and none whatever upon the edges of the orifices ; this latter result was doubtless due to the fact that these edges were frequently rubbed with oil, when not in use. Barometric Variation.—The following table gives the height of the barometer, reduced to freezing point and sea level, as observed at the U. S. Signal Station, at the City of Springfield, about 12 miles distant from Holyoke. Barometric Elevations. National Armory, Springfield, Massachusetts. Reduced to Sea Level. (Station 215 feet above Sea.) | Date, 7A 1885 i stale Date, 1885 9 PM. i | 7 AM. | 2 P.M. 2 PM. | 9 P.M. | May 6 | 30.191 | 30.167 | 30.188 | May 11 | 30.104 | 30.133 | 30.202 » 7 | 30.199 | 30.1171 30.076 ,, 12 30.296 | 30.299 | 30.336 , 8 | 30.099) 30.038 | 30 099 » 13 30.341 ' 30.293 ; 30.330 » 9 | 30.010 | 29.858) 29.887 4, 14 30.291 | 30.138 | 30.080 » 10 | 29.887 | 29.866 | 29.891), 15 30.067 30.108 30.337 | SUBMERGED AND Free Discuarae. 1884. From August 13th to September 3rd, 1884, 185 experiments were made with the 5 orifices in brass plates, of which 114 were with free discharge, and 71 with submerged discharge. As before stated there was during these experiments a slight leakage from the HOLYOKE ORIFICES.—SubMERGED AND FREE Di1SCHARGE, 1884. 343 tank B. For free discharge the amount of this leakage can be determined closely, by a comparison of the results of 1884 with those of 1885 which are free from error of this kind. For submerged discharge the leakage must have been somewhat greater than for the free discharge, as the water in the tank B then stood at a considerably higher level. We will assume a constant leakage for each of the 10 series of experiments ; that for the 5 series with free discharge being assumed a quantity which will cause the results of 1884 to agree pretty closely with those of 1885; for the submerged discharge we will assume that the leakage was increased nearly one-third. The 1884 experiments were made in the following order; x is the assumed rate of leakage per second for each series; 7’ is the observed temperature of the water. Date, 1884 Series. x ae August 13-14 .10 round, free .000 10 72.5° 5 14-15 a » submerged | .000 15 » 16-18 .05 x .05 9 -000 13 73° » 18-21 3 free .000 08 80° a 82 .05 round, ,, .000 18 82° | ) 22-25 = » submerged | .000 20 | 82° and 72° | yg «~BBepe 2 | 10.10 " .000 17 | 68.5° and 71° | September 2-3 » free .000 12 70.5° is 3 sox OD 45 » .000 15 | a 3 : 3 submerged .000 20 In the following table are given the results of the 1884 experiments, corrected by the foregoing values of x H, @ and c’ are the mean values of the head, flow and co- efficient c, as determined in 1884; these means generally represent the results of Zaaal 3 experiments ; c” is obtained by O +c’=c", being equivalent toQ+a=c" a (2g A)” ; hence c” represents the true co-efficient for the 1884 series. c¢ is the co-efficient, as determined in 1885, and has been obtained by the curves on Plate IIJ.—most probable curve for c for each of the 5 orifices. For the submerged orifices, the value of c for free discharge is also given, to illustrate effect of submergence. 344 HOLYOKE ORIFICES.—SuBpMERGED AND FREE DISCHARGE, 1884. TABLE CIII. Holyoke Experiments with Orifices in Brass Plates, 1884, Discharge Free and Submerged. Corrected for losses by leahage, by experiments of 1885 ; this correction, or x, constant for each of the 10 series. .05 Round. Free Discharge. Submerged Discharge. x=.000 18 z=.000 20 | ‘ 1885 rae Q e ce 4 oe No. h | Q | c & c= free. 4.64 .0202 .5990 6044 6050 | 97 4.08 | .0188 | 5953 .6016 .6054 3.76 -0181 -5990 .6050 6056 98 2.16 0137 | 5954 .6041 .6090 2.80 .O157 5998 .6067 .6069 99 437.0062 5990 .6183 6294 1.97 .0132 -6008 .6090 .6099 1.02 .0096 .6094 6208 .6168 | 412: .0062 .6159 .6338 .6305 | | .1 Round. ~ Free Discharge. Submerged Discharge. x=.000 10 a=.000 15 ag isss | om, , | 1885 iT Gos c ¢ : { aes h Q | c c sie epee 4 64 .0814 5998 -6005 6014 | 100 3.97 0750 | 5980 -5992 -6018 4.14 .0768 .6001 .6009 6017 101 3.57 -O711 5974 5987 .6020 3.94 .0750 .6002 .6010 .6018 102 2.99) 0650 5975 .5989 .6025 3.64 .0721 .6008 -6016 -6020 | 103 2.58 0605 .5982 5997 .6028 3.14 | .0671 .6012 6021 6024 | 104 2.00 0533 5989 .6006 -6035 2.64 .0616 6022 .6032 6027 | 105 1.51 -0462 5987 .6006 .6051 1.94 .0529 .6037 6048 .6036 | 106 985 0375 .6001 .6025 -6084 1.40 .0451 6054 -6067 6055 . 107 648 0304 5997 6027 .6122 1.02 .0384 -6055 .6071 .6080 108 .250 0189 6000 .6048 .6229 .502 | .0273 6132 6154 .6148 315 | .0216 6121 .6149 .6200 HOLYOKE ORIFICES.—Submercep anp Fre Discuarce, 1884. 345 05x05 Free Discharge. Submerged Discharge. x= .000 08 x= .000 13 | NA Ly c= HH @ | oe! e” BO || seg! Q ¢ * me ¢ | c = free, 4.65 | .0261 6083 .6102 6097 109 | 4.06 0242 6036 .6068 6103 3.94 .0240 .6082 .6102 -6105 LO. 2 Berd 0179 .6048 .6092 .6140 3.36 0232 .6090 .6112 6113 || 111 .350 | .0072 | 6091 .6201 .6390 2.55 , .0194 6105 .6130 -6129 | 1.39 | Ol44 6154 .6188 6178 597 | .0079 6335 .6399 6372 | | ~~ Shel Free Discharge. Submerged Discharge. 2 =.000 12 xz=,000 17 ] ron | - OT ad gil ” 1885 - i a 1885 ved Q ce € | : No h ( | c c " ieaiines, 4.65 .1050 .6059 .6066 . .6061 112 3.95 .0963 6037 6048 6062 3.89 .0960 | .6061 .6069 | 6063 115 3.11 .0856 .6040 6052 6070 3.21 0872 | .6061 6069 .6069 114 2.32 0737 .6026 6040 6079 2.70 0802: .6074 .6083 6074 115 1.52 0598 .6038 6055 | 6095 | 1 2.21 0725 6077 .6087 | .6080 116 | T71 0425 .6029 6053 © .6149 1.68 .0633 6076 6088 .6089 117 | 410 | .0312 .6058 6091 | 6205 1.15 .0524 .6079 .6093 6117 — 118 | 207 | .0222 .6070 6117 6277 i 774 | .0430 -6091 6108 | .6148 538, .0360 .6114 .6134 .6179 | 3874 | .0301 .6131 .6155 | 6214 3x05 | Free Discharge. Submerged Discharge. x= .000 15 z= .000 20 | \ . » | TSK 7 Lg _ | 1885 eA Q c ¢ | : | No. h W 3 ¢ | OT cae tres 4.60 1583 .6156 .6162 .6170 119 2.77 1232 6178 1 .6188 | .6180 fF 3.80 144] .6163 .6169 | .6175 120 1.63 .0949 6194 | 6207 | 6212 3.00 .1280 .6169 6176 ' 6179 ° 121 614 .0582 .6198 6219 .6321 216 | .1090 | .6184 | .6192 | 6192 | | | 1.34 .0862 6223 6234 | 6232) .830 .0681 6235 6249 6285 557 .0562 6275 6292 | 6333 | YY 346 HOLYOKE ORIFICES.—SurMERGED AND FREE DiscHarcE, 1884. Revarks. The preceding comparison of the results of 1884 and 1885, with free discharge, shows a pretty fair accordance, except for heads less than 1 foot. With the .1x.1 and .3 x .05 orifices, with heads below 1, the corrected co-efficients for 1884 appear to be notably lower than those for 1885. Possibly this may be in part due to the much higher temperature of the water in 1884. These lower heads were, however, less accurately determined in 1884 than in 1885, and the lower the head the greater the danger of error in incorrect assumptions of the value of the leakage w. Submerged Discharge. The height of water in the tank B, above the centre of the submerged orifices, varied from .57 to .73. The head in the tank B was determined by a hook-gauge at e. The zeros of this gauge and the one in the tank A at c, were compared by filling the tanks A and B to about the height of the hook e, as shown in the sketch; the supply of water was then shut off, and simultaneous readings made of the two gauges. The gauge at e was enclosed in a wooden box, with small orifices pierced in its bottom and sides. There was some commotion in the surface of the water in the tank B, with the three largest orifices, with heads above 2 feet, but not enough to notably affect the general surface elevation in B. For submerged discharge, with heads above 1., with the three largest orifices, the chances of error in c” (including leakage) are probably not over ;3,; for the two smallest orifices, with the same heads, the probable errors in c” are not over z},. The smaller the head, and the smaller the orifice, the greater is the danger of error in c”. The 25 determinations for submerged discharge are plotted on Plate V. The heavy solid curved line represents the most probable curve for c, for the orifice with free discharge, as shown on Plate III. The dotted curve below the curve for free discharge, is the most probable curve for c, for submerged discharge. The experimental curves or lines for the 5 series appear to be fairly symmetrical, except No. 99 for the .05 round orifice ; this experimental point, judging from the other experimental curves, seems to be about .003 too high; the dangers of error in this experiment are greater than in any of the others. By reference to Plate V., it will be noticed that the experimental submerged curve for the .3x.05 orifice, is slightly higher, for h above 2, than the curve for free discharge. This was doubtless caused by the divergent sides of this orifice, which were .021 in length, being the full thickness of the brass plate ;* when the orifice was submerged these divergent sides doubtless notably increased the flow. These conpara- tively long divergent sides for one orifice, were purposely used, to determine their effect. The short divergent sides for the .05 round and .05 x .05._ orifices, probably * In the other four plates, the length of the divergent sides of the orifices was only .0069, HOLYOKE ORIFICES.—Supmercep Discuarce, 1884 Remarks. 347 appreciably increased the submerged flow; this increase was probably hardly ap- preciable for the .1 round and .1 x.1 orifices, when the length of the divergent sides was only “ths of the diameter or side of the orifice. The results for submerged discharge, as indicated by Plate V., have been already discussed in Chapter III. In the foregoing table the discrepancies between the results of 1884 and 1885 have been attributed almost entirely to the leakage from the tank B in 1884. It was known at the time that there was a slight loss from this source, but some of the corrections given seem to us to be improbably large. The temperature of the water in 1884 ranged from 68.5° to 82°, while in 1885 it was from 48° to 54°. We conjecture that an increase in 7’ of 25° or 30° will very slightly, but still appreciably, diminish the co-eflicient c, even for the .1 round and .1 x..1 orifices with the largest heads. By ‘appreciably ” we mean a change in ¢ of .0005. This surmise, however, is not substantiated by a comparison of Nos. 47 and 58 with the .1 round orifice, where a variation in 7’ of 13.5° does not appear to have affected ¢; this can hardly be considered as conclusive, as such a variation in 7, according to our conjecture, would only change ¢ .0002 or .0003, a quantity barely appreciable. For the three other orifices we feel confident that for heads less than 1 foot, the higher temperature in 1884 notably diminished ce. We hence regard it probable that the given corrections for /eakaye are excessive. On Plate III. have also been plotted, three experimental values of c, for the Greenpoint circular orifice of about .02 diameter, and the six experimental values of ¢ obtained by the author in his California experiments with orifices; all of these being with free discharge, and full contraction. In conclusion it may be observed, that for all the Holyoke experiments H is always equal to 1. That is to say, the area of the feeding canal or tank, a,, was always so very much larger than the area of the orifice, a, that the velocity of approach, », , was almost inappreciable, and the head, h,, imparting this velocity was absolutely inappreciable. The Holyoke Water Power Company constructed at its own expense the tanks employed at Holyoke, and also furnished without charge the services of all the needed assistants. This was done at the instance of its hydraulic engineer, Mr. Clemens Herschel. We here desire to express our appreciation of the company’s liberal aid in behalf of the science of Hydraulics, and also our personal obligations to Mr. Herschel, for his valuable advice and co-operation during these investigations. GREENPOINT. The experiments at Greenpoint were all made with a circular orifice, having a diameter of about .02, pierced through a hardened steel plug. The inner face was true ; the orifice proper had divergent sides .016 in length. The diameter of the orifice was measured by Professor W. A. Rogers, of Cambridge, Mass. ; the mean of several deter- minations was .02015, with 7’=50°. The orifice was almost absolutely circular, with 348 EXPERIMENTS WITH ORIFICES—Greenpornt. Appuratus. nearly true edges; this orifice was, however, not quite as perfect as the circular orifices in brass plates, used at Holyoke ; under high magnifying powers, in some places on its edge, there could be seen a very slight rounding of the corner.* The steel plug, through which the orifice was pierced, was .042 in diameter, and was very firmly “bushed” in the centre of a rectangular cast-iron plate, .04 thick, and 64.69. On the outer side of the orifice an iron plug, fitted with a handle, could be screwed to the iron plate; a soft leather washer secured a perfectly tight joint, when the plug was thus attached. The feeding or pressure tank employed was a solid cast-iron vessel, 3.65 high. The upper 8 feet of this vessel was cylindrical ; this tube was carefully bored, and had a nearly circular and uniform section with a mean diameter of .50451, with T about 50°, The lower portion of the vessel was a parallelopipedon, having an inner base of .52 x .58, and an inner altitude of .58. The iron plate before described, formed one of the vertical sides of the parallelopipedon ; it was attached to the vessel by many screws, a rubber gasket forming a tight joint. Near the bottom of the vessel, and on the opposite side to the plate, a hole was pierced; a short pipe with a right bend was screwed in this hole, and to the pipe was attached a glass tube, having an inner diameter of about .0315, with a nearly uniform circular section ; the axis of the glass tube was parallel with the axis of the iron vessel (tank) ; a tight connection for the glass tube was made by putting rubber rings around it at the bottom, which were tightly pressed by an iron “ follower.” During all the experiments made with the iron tank, there was no leakage whatever, all the joints being perfectly tight. The iron tunk was firmly fixed in an exactly vertical position, and a vertical wooden rod, with a graduated scale drawn on white paper glued to the face of the rod, firmly placed immediately behind the glass tube. Horizontal “sight-bars” were placed in front of the tube, so that the height of the liquid column in the tube could be read at several points, with avery fair degree of accuracy. Alllinear measurements were carefully reduced to the standard of Professor Rogers, being the same standard employed at Holyoke. As in some of the experiments 7’ had a wide range, it became important to determine the effect of changes in temperature upon the diameter of the orifice. This was determined by Professor Rogers, with a maximum variation in T of 86°. He found that the diameter increased with 7, but with a somewhat irregular movement ; this was probably in part due to the composite character of the plate, In one ex- periment the increase in 7’ of 76°, resulted in an increase in D of 54, for each 120 000 degree, being a larger rate than the usually accepted co-efficient of expansion for either * The orifices in the five brass plates, and the one used at Greenpoint, were made by Messrs. Buff and Berger, of Boston, who are skilful workmen. The circular orifices were entirely satisfactory ; the rectangular orifices were not as perfect as was desired. It may be remarked that it is exceedingly difficult to make perfect rectangular openings ; they can only be finished up properly, by the aid of a microscope with pretty high power. EXPERIMENTS WITH ORIFICES.—Grernpont. A pprcratus. 349 cast-iron or steel. These determinations proved that no notable error would ensue, by the assumption that with an increase in 7, the diameter of the orifice would increase in the same proportion as the diameter of the cylinder of the iron tank; therefore the increase in the areas would also be directly proportional. The measuring vessel employed was a bucket or pail made of best quality “ galvanized” sheet-iron, having the form of a truncated cone, with a bottom diameter of 1.25, a top diameter of .5,* and an altitude of about 1.45. A curved handle was attached to the pail; when supported by this handle the capacity of the pail was increased from .0006 to .0007, although the bottom was stiffened by light cross-bars ; the amount of this enlargement was determined by filling the pail exactly to the rim when it rested on its bottom; the pail was then lifted by the handle, and the dropping of the surface level noted; this was about .0033, showing the gain in capacity to be .000 66. The capacity of the pail was determined in two ways: Frst-—By filling the iron tank and its attached glass tube with water; the screw plug closing the orifice was then removed,t and in a few seconds the pail was quickly placed into position to catch the jet ; at this instant the height of the column of water in the glass tube was noted ; when the water had fallen in the iron cylinder to near the bottom, the pail was quickly removed, and the height of the column in the glass tube again noted. The pail not being filled, the experiment was repeated, the concluding instant being the moment when the pail, supported by its handle, was brimming full. The mean of three determinations, with T from 46° to 49°, showed that the capacity of the pail, when supported by its handle, was equivalent to 4.640 in length of the sum of the mean areas of the iron cylinder and the glass tube. The sum of these areas was .200 69; the capacity of the pail, held by the handle, was hence .9312. This method was not entirely satisfactory, owing to the varying amount of capillarity in the glass tube, which will be commented upon hereafter. Mr. Ross E. Browne has found with very small orifices, that when a plug is removed quickly, the jet is at first abnormally long, quite an interval of time being required for it to diminish to a permanent length. In our experiments we could hardly notice such a phenomenon ; the jet with us for all liquids became constant, certainly inside of a period of 2 seconds. Second.—By ascertaining on very accurate scales the weight of water contained in the pail, the pail resting on its bottom. The gold bullion scales in the U.S. Assay Office, in New York, determined the weights, with various temperatures, showing the following results. The given weight is the net weight of the water, with the pail brimming full ; * The top diameter should not have been more than .35 or .4; this smaller top size would have permitted more accurate determinations of the instant when the pail was full. t It is almost unnecessary to observe that the female thread into which this plug screwed had a much larger diameter than the orifice. 350 EXPERIMENTS WITH ORIFICES.—Grzenpoint. Apparatus. the weight of a cubic foot of water at the given temperature, has been taken from Table I. ; the capacity of the pail is deduced from these values ; 7’ was determined by a standard thermometer. Peerene Weight of a Capacay Cubic Foot or Pail. Troy. 0984 226-231 w = .0656 232-236 Ww = .0328 175 -179, 180-184, and 237-243 Co-ethictents of discharge (©) as ordinates. Heads, in feet (Weh) as abscisse. Plate IL. 6. .700 —| .690 . 680 _| .670 Seca Es ee Pl 060 650 . O40 - 630 G20 . 610 ' 164 59 _ 163 ISB © 210 6 209 162 | w- 6562 34 | 157 = 7 208 o - \ 5 206 6 6 -———_# 207 205 204 161 . 600 2il2 ~ 45160 1650 © 213 “ait 0 3); 4. oO. .590 6. Julius Bien & Co Tith Plate Ill O ; c 1 22 3 4 $5 6 ol iB 9 lou 12 138 14 15 16 17 18 19 2 21 22 23 24 25 26 27 28 29 3 31 3.2 33 34 385 36 37 38 39 4 41 42 43 44 45 46 47 48 49 O 21 22 23 24 25 26 27 28 29 3 31 312 39:34 35 36 37 38 39 4 41 42 43 44 45 46 47 48 49 5 | ’ ] T L a - : 660 | — \ 655 | \ 655 = = By as EXPERIMENTS r 19 Vv : ~ = = WITH L | VERTICAL ORIFICES. | = > ml wPT j ns TT 650 } aa FREE DISCHARGE AND FULL CONTRACTION, ESCAPING JET ONLY ee TOUCHING INNER EDGES. “ 22 | Spi gies sy s = 3 a\80 HDLYOKE AND GREENPOINT 1885. | | Hamilton Smith jr. - 645 [Be J Te 40°- 54°, = 645 | = ORDINATES = CO -EFFICIENT OF DISCHARGE, OR c. 6 i IN 9 005 010 018 020 ‘ ozs eh ‘ \L\ iz ap pe fg pe a ep oO zh ss |__| \ } << _ - +}. = ABSCISSA = HEAD FROM CENTER OF OPENING, OR H-h. -— 4 640 is ge ° 2 2 3 4 5 6 7 8 0 1 \ ‘ fee pe ep ey a: } ‘| 4 4 = 640 a | | I t 8 c r | 5 ~. The heavy horizontal short lines above and below each experimental point aa | \ en = ~ madicate the range of experunental valites. = t— “at 635 + T The eaperimental points marked are tor the Qolumbia Hull eaperunents 635 | ab ae " | | ies een ees | made by H. Smith prin 187# and 1876’. atts | | | | spo \ \ ies = | | ai a2 —~ | + = 630 | \ | 3 MA PI 123} | | i \ \ ~ | 630 I i ry 4 oe : : 7 | ae : : 83 . | == = | as : : | wordt Some oN = | | | | cs | hh al 32 \ | 7 122 7 7 secie oo _ M3 ji | 625 | h ot = z - 625 I \ \ a | aad | | Sates Lo | 2 : LES” [ [ina = [ | = _ bere _ | Se = A = [ft - L = \ IT | el | | 620 ; \| a ~ | ae ; i - 620 - id L \ D33 035 i les i aaa 7 =a] T 7 ao ow 4 SS - ze = i eH es ee ee = ee salle i es | | ~J i | L = on _— = —- + + —-— —-+j | ~ i ae y 615 = Ge S [a 615 — <= S855 | I x = Se | ic ai : | 4 2 > Pe eran ee —— f {| ! 1 i | i, | 610 at a 1 oe 7 610 oy eo SSS Se ee de = oS eh ee ea alte = P fee alae Hie coos al 2a =e ee olor 605 - 45 paper 605 iin a oi pose ll SSS Se eee ee ———_ se ls tae ihe ae ea E ? rep. Ste + ile — : is gee = = = = a deat = pee a = a ae | |__| 4} | — | —-_}. -—_}--_,- —~s=— — 4 fp] —_|}——_—_+—___— ———}— 600 et 600 aa — = => > ca Shee le = = SS T = - ——— - = Aa Noa esi ee eee Fas —|-—- +--+} = = —= = = = = sees jo} stale ——— — — aaah — — —- —— iis apes eens | s - i 595 Mises Oe eae = - gabe | eal y oe —__|—___| - f —j—_____} = = = { } _| | cs | | 590 \ Julius Bien & Co Lith Plate VI. A o a 4 5 6 1 8 9 1 1. 1.2 13 14 1.5 1.6 1.7 18 19 2 -710 2 ———_ _ ss = aes 4 .710 | DEPTH OF WATER ON WEIR (h IN FEET. | 700 4 .700 DIAGRAM .690 SHOWING VALUE OF CO-EFFICIENT c,IN FORMULA Q=c 4 (2 gh) fih aa ane FOR SHARP-CRE STED HORIZONTAL RECTANGULAR WEIRS, THE WATER ESCAPING FREELY, OR NEARLY FREELY, INTO THE AIR 325 Deduced from WEIR EXPERIMENTS .680 - or re Poncelet,& Lesbros, Lesbros, Francis, Fteley & Stearns, and Smith. o The heary curved line indicates assumed value of C tera wer of infinite S length, with till bottom contraction. 670 344 ~ AU experimental values above this line are for weirs having end contractions suppressed . 670 ; , , below» noon noon v4 till or nearly Hall, contraction | haere ee aot. .660 +o Dee ai : Bie a he ee be coee Seen pee ee Sas Se .650 650 | ae ef . a | ee sere | Ae ——— ae | : cae | — ; | ieee ——— E Se ree oe S| = eee ren as) .640 640 eS] Cae a ie = [ee] a eT = ee eee [a Lee 10! a -630 Ps ———— 630 | eet) ' a ames 15° | Sar [eae Se | 9! EK 2 ee 10 | }——— ul ¢ 9 eee es 620 eGet Fteley and Stearns 19° EIR OF INFINI|TE LENGTH | -610 -610 600 -600 | | | 990 590 = -580 BRO 18 19 2 Julius Bien & Co Lith, -710 -700 -690 -680 .670 - 660 -640 -630 -620 -610 -600 -590 1 6 ah 8 Ll L2 13 14 L5 1.6 1.7 18 19 Plate VU. 2 (h IN FEET: -710 | | 9 DEPTH °F WATER O p were | | DIAGRAM SHOWING VALUE OF CO-EFFICIENT c,IN FORMULA Q=c # (2 gh) Fih FOR SHARP-CRESTED HORIZONTAL RECTANGULAR WEIRS, THE WATER ESCAPING FREELY, OR NEARLY FREELY, INTO THE AIR Explanations. The central heavy curved line shows value of Ctor a weir of intinite length, with depths trom O.1 to 2 feet The lower curved lines show value of C tor weirs with fill contractions trom 19 feet to .06 foot in length. The upper curved lines show value of Cfor weirs from 19 to 2 feet in length with end contractions suppressed; with escaping water contined below the weir by projections of the sides of the feeding flume, but which projections do not extend below level. of crest of wair. (In case the water discharges with entire freedom, C will have aslightly larger value.) The sides of the feeding flume to be of smoothly planed. plank. The head (X.) ts shown by the vertical lines; tt should be measured ata point 6 feet above suppressed wews and corrected tor velocity of approad. The inner depth of water below crest of weir to be at least 3N, and never less than 1 foot. With weirs with (ull contraction, sides of feeding carval should be distant trom ends of wetr at least 2h,and never less than 1 foot. -700 -690 670 -660 640 620 -610 600 1.590 ew tn dk: LT 18 19 -580 Ww 120 110 100 90 80 70 60 50 40 30 3 4 5 6 7 8 9 on T ‘BB ©.92 73 “63 . VELOCITIES IN| FEET 66 ; I—E 57 59 «83 :70 ere 7 ——- s oT 80 “ are [eres tates eee Ks ~~ See es ; _|,60 "60 BB 43 SET 1 = Series VI VII VIII. 45 ; " - - 2- XI XI XIV : x .@? ,e? 406 a 3 " 3 = + XV XVI XVI 5°" a9 | gai ha «9S ees = ; a AT St .23 99,2. ti Se =| ae 7 79 Curve for r =.70 (a aa aT .73 S FF z ud oO SS 5 ee oe ee Le ul ' Oo ° [_ Curve for ,, 7 1 FIG.2 2.05 Ree One 444 Sri a SUDBURY CONDUIT y:%s 38 : Ze FTELEY -STEARNS EXPERIMENTS. 26 2.33 | RECTANGULAR OPEN CONDUITS. DARCY -BAZIN EXPERIMENTS --—_——_- ILLUSTRATING EFFECTS OF CHANGES IN v AND r. The value of v ts written opposite each experunentat point Ordinates = Co-cffictent 1 tv Vv -n( rs)? Abscesse = Velocity wm feet. { Ordinates Qiic Taglar ntetoanae SGRIES | we lee gees seo l 2 3 4 1 2 3 110 100 90 80 70 150 140 130 120 Julius Bien & Co. Lith Plate NU : 2. 3. 5 6. he 120 7 - 1@ 120 | | 277152 \ VELOCITIES IN|FEET or" | | | .-- | | - | | sae? ae el dae | lo + —__——-+___—_——--+; lo | 31-33 | BOSS|UT © | Tin \Pipes D118 | | \ 100 SSS 100. a Fk z Wl oO L Ww W ' ° oO 90 * 90 EXPERIMENTS WITH PIPES BY BOSSUT, DUBUAT AND HAMILTON SMITH J® BO ' 80 ; AOSCUSSH Velocities tn feet t Ordinates — Co - efficient W,tn i Vv nm(rs)?,;, 8s beng corrected . for contractton and tmpartatton ' of veloctty’ 1 "283 2 S I pl bscisse pe hE SE hah fh ee ocales ‘ ‘: i | : lOrdinates 912.9 1 3 8 78 9 10 i 1 1 I 70 + 70 i | 1 ! | 4 36 | | | - | | 2 3. 4. 5. 6, 7. Julius Bion &CoLith 150 — 149 - B 130 Cherokee D 2.4: “286 120 Loch Katrine 2 D=4.0 64 110 105 100 90 80 75 + z | j / | | 70 7 7 | — | | | | | | | | | | | | \ | \ | | { | | | | VELOCITIES IN FEET. 65 > 2 : . ? ; ne 2. 3. 4. 3. 6. Ts 8. 9. 1o, Ths 155 150 145 14) 0356 130 9 5 19 Ss N 5 Al 13: 105 100 95 90 3 80 75 Plate XIV. 19. ape a Seis ee pa ae U LN3ldls43-090 1.42 18. ITs 16, Se | =D. 20 15. | / ; 4r 3 =i i a mn (rs)2 Couplet et cet. 25 Mb 1. to 8. 14. The dotted trreqular lines nin v for D = 15 10 DIAGRAM A for D=.05 and .1 quite smooth interior surfaces, A and no sharp bends. D -\2.5 13. M 5 L Abscisse - velocity in feet or v. Ordinates =co-efficient n Showing values of havymé Based upon experiments of shown by heavy lines. represent concat expertments of Darcy and Bazin, and Fleley and Stearns. For circular Pipes, with diameters from .05' to 8’ 2.15 Hum bug D. Il. 65 Julius Rien & Co. Lith Texas Creek D. i} | | | | T | | | | | } | | | | | | | | | } | | | | { | | | | | | | | | | | | | | | | | | 19 14. il. 12. VELOCITIES IN FEET. 10, Plate XVI. SKETCH —— or i Fee ee DITCH and POOL and WEIR Fig. - ’ 7 NFA ke 2.5 8 ‘ Where water (lowing through ' ELEVATIONS 7 », 6 ' OF Scale 24 pipes was measured i BOTTOM or CANAL Scale an t! Same datunt as tor hig. 6 o 5 10 S fxs - 3 B= Fig. 4 ‘| Cm fog ; Wes S E< f See | Pa / sation Fg c / or ys beio= 25, 0% = L¢ ein Flow poten’ H- 73.08 3 am Bottom of Wetr= 75.360 ' ow 4 = Seale for Figs. 2 and 3 12 eee ae | 7 Hoop eile D oS ELLE DEAD WATER f ———— [e} HOOK GAUGE pe SSG ] WEIR a ae “ | / S 7 rr Bi . NORTH BLOOMFIELD PIPE EXPERIMENTS, FIGS. 4-12. se i STOP GATE ‘ | /¢ SKETCH fe fe or Z ¥ DITCH and PIPE and LINE | | Scale 2¢00 INLET FOR PIPES Figs. 7-12 Q ae Scale 72 i oO 1 2 3 a 5 6 - 8 9 10 FEET i a ce a ee PLAN i z Fis. 5 : SS \peN stock = Fig.7 Fig.9 { ss “ELEVATION 103.000 PIPES \t sae — S iT x { z <—€ FEEDING FLUME PEN-STOCK 100! [3 oe ELEVATION END VIEW . FUNNELS = ce Fig.8 Figo) uray 5 PEN-STOCK ELEVATION Fig 6. Suet Shstss oc isecccesse cnt 1 gees | Horizontal zac o } (Vertical 960 wae ewe pe mapa woe weapon 4-- PLAN OUTER DIAME'TERS Julius Rien & Co Lith, Plate TI. 2849 ol i Oe AS 44 tS AG 47 de 45 } q ' Wee) OR ah cen | ome — a 860 a as > 660 Sah doh eters: ath ats | e I 2 655 VDIe TYR 355 28 EXPERIMENTS | r err 2 WITH | i ERTICAL ORIFIVES. yon |e. dee te 7 ‘y8 y very y AD sa Brea at seit 5 fg Sree en AND FULD. CONTRACTION, ES CADING JIT ONLY eau 5 TOUCHING INNER EDGES, 7 ILYOWKE AND GREENPOINT 1885 flamiton Smith je . Tes (PS al © r % 6-15 me 6415 ANTES = CO-BKEFFICIENT OF DISCHARGE, OR c. aC) ous .020 025 feactee eet fe be tate oe de hee ted AL= HEAD FROM CENTER OF OPENING, OR Heh. B10 VSL es ces ee os0 cis ee | O26 Pao ow 4 short Lines above and belin each eaperumental pond SS eaptinenttial valiies 635 snes marked are for the Columba Hell eryurunents Waa wm ISTH and ISTH 630 ood 625 625 620 620 615 615 610 — : = : 610 605 1 Pa TES eee em 605 600 GOO y ae ADDS ti | i 5 a | 500 oe o a4 3a 38 A SF AG SG 1 at 42 8 ae 1s 26 17 ap ao 5 Julius Bien & Co Lith Plate IV. -620 610 - 600 L 2 > T Plow through a circular orifice, with 1 -.o2015, of WATER ,QUICKSILVIER AND TICK OIL. ne Au with tree discharge tito atr, and tull contraction. Greenpowtl Barperinverds lbs t - = Hamilton Sietth Aft = | Ne Abscrssce = heads trom center of UOritie, in feel. x ‘i ; NY Ordinates ~ values oC Cin YW Coa (eglhy % £740) bee ns eee L NL Bahcet a eee, A ra ~ _~ ii .130 Bale melee ata Sees) ee eco ae a 2, 730 Pg Sees 4-------- he 162-195 -— ----~-------4------------- b----K : cue I ¥. ol | \ . “oricas,, } Ree Seek i eo| eae Sain cere di Soe g Oith Pah Sates es ais __|.720 f 1 : i | | 7190 L——-——_-- Ses ives 2a Sis ES See, Be =3 = Let eS -710 i 1 .700 | pected), soesns palin scatter | eee : : 2 ' SSMS, Sa ledee sees e200 ! | | 1 ODO! eae EE. agents, Soka Ae nGi Ga a aes (a. aeSeecee gap 4 1 1 i | | | | | | | : | 680 25 ba kets sg pe esas | 5680 | \ { | 670 |. 4 DS ie esha ee end OO . 660 sierae ea ei sae Sins ob tee ee a BOD -| .620 6.10 .600 Julius Kien & Co Cith, 630 -610 630 620 610 -600 620 610 630 620 te 3. Plate V. HEADS (h) IN FEET TMAMILTON SMITH J® SUBMERGED ORIFICES HOLYOKE 1884 0. = 05. Curve of free ascharge 103 et Ft ial Lok fy 7 te evens ere ee eee 30 X 05 120 a = ———_——— a. 4, 630 620 -610 -630 620 .610 610 -600 620 -610 630 oO te o Tuhus Rien & Co Lith ‘| 2 S 2 ° ° ° ° Ss ° fo] o ° ° oS zt 2 a oO oa oO Ve] + a 0 a: e 8 S © 3 8 © 8 S 6 © 8 2 zNI = eek ASS See aes i] co i oy i I oO Me te ae Nm hg lle a S eran 2s hs , = —4 a a +} qq ; 2 soit) ne nee 5 fe=)2 2 a et S : 2 ee ee | PA io ear \ j i ‘ : 1 : 4 | ; ! \ : | f i ‘ 4 | | \ : : \ PF . 7 tay . 7 F 7 7 : a ™ | “| 4 S 3 2 2 a 2 q s oa a oil eee Com-mutalemite ee a So ee ae a we = - os | ‘i . PS ! : ‘ © / I \ \ i | | ‘ \ | | \ \ ' 0 ’ i 4 : ~ | \ \ \ \ | ee \ \ \ ! a { | \ | \ \ + | = | aoe i et | | - : i ! 1 “J i ' = i INFINITE LENGTH L700 -600 590 670 660 650 640 630 620 610 es a Julius Bien & Co Lith 710 .690 680 -650 -6-40 .610 -600 -580 Pe oe TO ae ! ! berm rt Nal oO a = (A 74 Leo? Be . lot. 3.33 . 4c lo 34Y 4D sur EG % 37 7 ee HUY, da Pron Siete We 3-76 Plate VIL. i. we ae 1.6 17 18 1a i eat | i So Se = 4.710 H | | | | | | | | | i i | | | | | 7 i pee SS 4.700 ; | | ow ag j vet “3 ~ G90 i | t | See ear reaps sofa = BBO ' ; | | 1 {; i I , —--— Se Ba \ bs ! | | | pieaiet ge a ee + ~ .660 v ae ij Be Ni Sage deneteer - me Pope pe seaaia ae 650 4 nee |e oe eT | Lae bh ee ry | eee Lea | 640 oa eee | ees 10! { ———— aT 630 a a ee eee | asp aaa PP coe le oe 3 la was Se : | lee geese 620 | | 1 gee a .610 | a 7 : a Se 19 ‘ ee at | ee [tae eal is? HL. ieee ee a | a ee 600 q | 1 ee | —— [ree + [| 7 i eel He ac -. | i | [ee | 5 ; soe OO r [ \ [Po ee ONS ae pe P| | el = | | Ses Pesca ie, eae) | i We : -/ 580 3 14 LS 16 ay 18 19 zZ Pilate VIO. iL 2 3 A ‘5 6 4 3 9 Tis i a as ne . ees po PLS oe ee =f , at ee eye HEADS IN FEET H-h, : L. Qs Le Ss 7 rn | re | | \ | ! ! 700 | \ | ' | 700 -_ ane | 700 \ - | | | | pee ld. \ SNe reek Seis: Saeko \ : Pe collet SE SS ‘ Fig \ \ | B30 | \ \ | a. 4 ae ee Giese creme beep | eters Sata ee ea eee \ \ | \ ‘ are ae | Fgsaqws ‘ eae eo t : a ——— ov} . % | rea lk se | Wn teen lie ‘ Ne | . eee Hy % NO aS SS =o i Thos = : é Fig.t2 128 \ 4 Pen ES to i230, ee eee Cp “ R oe ae Se : Se oS : .650 5 ~ Se 5 o Ss figpre.¢ S 6 pres * 99 See : G7 : | ie ee | ASS SS _ oe ee oat ug i2l nth, fo a) , eo eSeeeates oc oe 8 ‘ pee. spat cstakg. . Sess ee ene ‘ PP | ' : NN7 : (fen { ee | | , : a i \ eres | : ! | ( eae ee os ; Si | 550 ' DIAG RAN : Sata memtcencir ame! “eal Jom hal aah + ciara te ee —--— set etnies = getline es $23 showing ’ : n 3 Le VALUE OF c,inQ-c3 (2 6h) 2th phen ; Bagh aga ae eee | Deduced {tom LESBROS EXPERIMENTS with weir .6562 ft.loné and various forms of approach. Halways oh. (except Nos 129 aud 10). 1900 Typer heary dolled line ts curve for weer with end contractions suppressed, and full boltom contraction . Lowery ‘ se i a 3 x complete contractton . Figs 12 eteet. retir to particular form of approach used, as shown by Plate 1, pe ee —— es, Beep ey See ec el 1 2 3 4 5 B 7 8 9 I Julius Bien & Co Lith 710 .700 -690 .680 -650 640 -630 -620 -610 -600 -590 -580 Plate [X J 2 3 4 2 GO at 8 9 L | | 1 HEADS (hi IN FEET ' : | 1 i i ; \ , | | : = | = e as saeionsl ee | SHORT WEIRS : ‘ L - ca : CO-EPFICIENT ¢,in Q-e 3 (2 eh) 21h, Bes = from | er ys ” . , 7 Y Trt ot ‘ CASTELS EXPERIMENTS i with feeding canals 2.428 and 1.184 wide, 1 from .033 to 2 232, G = 558 1-033 Gj 139 - h-head corrected for vy os Tt ened ™ 1 138 f 231 ki 137 ‘ 9 136 i ' 9 | 3 steele OAS Ss ‘ eee ne entail Pe 135 | : 3 a ee | | ‘ © 7 2 "135-13 229 i pa pad Bee tes {20-201 2 : T= 1188 Re? ‘ \ : ! eet r2 fe ses ss 0820 149% | Xe Ie ge ON i 1=2,23Q : ; Oo | : ‘ Lesbros 1 =.066 [. 5) 132 L= 1.972216 j u 232! \ 233 oe ' aa @) | : 1140 - ON , W=/. Plate Vil 2 Poe = ih RS los2 “239 (1 = 165218 Q220 : I 142 aa ~COeeStStC ot || SS. oe =. p——— 140 | an eT: ? = 13 2088 \ = / 2. Plate me i ‘ Bt é : | 7 ~ .099 5 | | 7 = 985 2020 aos wy “SS f= 2 23 ,NUSIZZ2 = 227_ sr o = T50 - 158 7 = 98 7S 208 : Isbaxgir i3s isd 93 152 e 1 =.099, Nos{ 40-249 Vv oy ns i : — © f & 4 2 DAT & a 2a 654 ay 4 22>. (248 ae a 4 243 J 654 i944 7 \ e207 Se, aaa Be 245 250 i se 2 =497,Nbs.216-221 e as i eo ee Late Se Se 252 é - mt aig e aa Nae ea 254 Q _—_—— isa py 1 fal 160 ‘ a) [6S 255 163 162 | "203 Y=737, Nos.203 ~ 209° Gee aay St Se | | J = 329, Nos {71-183 T ———— 7) =.654,; Nos.184 - 194 d ee oe | fmnaentissh Zakeastiie oe en Se eee ey | 177 | oe | : | | | | , | | | | L ‘ | J 2 3 L 5 6 7 8 9 710 700 680 670 660 650 640 630 -620 -610 600 -580 Julius Bier & Co Lith, 140 110 100 80 70 60 50 Plate X 1, 2% he A. 5 6. 7. 8. a te os ite nae yy igacii-nae Gee bts tes pe pecekn erm Saa LAG VELOCITIES INj FEET 66 | | cern 5g i i RECTPANGUIAANR OPEN) CONDULES See: Se a ine x pee : ape! Thales > 38 DARCY-BAZIN PXPEREMENTS \\ ee poe 423 130 : ALLUSVR VEING BEET OT OF CLUANGES IN A get The Value of Wty written opposite cach eaperimentad pont, ie Ordinates Co-ctiterent win \on(rs 2 7 7 fbscisse Feloctty én teet., ; — f Ondtnates fein ti he geen Sch = is 120 Beales y Hieheber: O22 24 ee 2. w a APOSCEISG hyip) Piel Me ict pockechinmke 4 q Nea anja ie eer L we ees estate : See al lo : 100 a ight 19 an ok os = vadels Ens aan alee each i cea 2 a ee SnOSe fox So i 9 | [ee ; pod \ ot : [bese Le 80 P 70 255 : 60 : Z| ap pore =e hoop = =~ 42 \ S oe @ = 2 eeiteeey a0 29 =, ——-* ae . im tos rae Al 5. 6 7. 8. Sulius Rien & Co Lith 100 90 80 75 Plate NI 3 4 5 G VELOCITIES IN FEET ihe 100 KXPERIMENTS WITH PIPES 323 NEW ALMADIEN 1877 HAMILTON SMUPFH J* albsctsscc Velocities in leet. ee Ordinates €o-eslictent nv, in : ve nars)# ; & beng corrected for the : : ~* =o pe Mae pe ip pee nee, all i contractton and unpartatéon of veloctty 325 4 | 0 2 5 1 § -fsctsvee ue ee peed Ordenates Pusiut 2. 331 a 90 [pene 7 85 oa z ul oO iw Lau, Ww 1 ° 309 oO so 80 321 fl al ln ad a nit 75 314 315 70 335 338 Saini X mes a erles pea is a bored | 337 woe 339. pel Rhea Se f ijeaessa he. Gilets a a 65 i ! 3 : 1 2 » { o 6 Julius Kren & Co Lith Plate XV. AIEABURING APPARATUS AT COLUMBIA INE, EXPEKRIMERTS WITH WEEE AND ORIFICES. EXPERIMENTS WIVH WEIR AND ORIFICES, ILAMILTON SMITH Jr. Wig. 7 1 eee || ee eo me Splash Hoard HOOK| GAUGE Y . ij, at ea uw i mo $ BOTTOM oF leat ets i AS eee. eo wit i S OF WEIR aii L i Si © en Si uak x es RESERVOIR Ss i OVER-FALL . a c ey Neca cal i 2 | Mf s iis Ries PapRT 4 iReneeay ' MEASURING TANK | LONGUPTUDINAL SECTION ig << | Oo \ pcs a, als ae Scale for Figs, 7-6 oO 10 a8. 30 L r { his alt aiden (Perea eoreee iota : | we | ox 3a | at . as | Vege ee a " | Splash Board : ee it x i = = Se ap —— wi ° | T Over-fall qi} S c |B A eip ° = : $ : 5 ay G | ieee Over ie ra Wy | 5 frat | \ GAUGE MARKS PLAN. FORMS OF NOZZLES FOR EXPERIMENTS WITH ORIFICES WITH GREAT HEADS. HAMILTON SMITH Jr. Seale for Figs. 1-6 eG [ : 0 1 Pa 5 4 3 8 9 Ly icabbabies’ + 1 aie: = J. st Fie. A Fig 3 | ae — | | | : } | | ‘ | \ HI \ HI \ D for ring E=.0597 D » os FS0847 Juhus Bien &Co Lith 1 | i | STOP VALVE pe a | { fron hediig pipe | “ SCREEN HOLYOKE, 1884-1885. ILAMILTON SMITH Jr. LONGITUDINAL SECTION ‘ 1 Scale 36 Fig. 1 tees Fié.2 TANKS A AND B 3' WIDE IN CLEAR Plates set in center fe seeHLOr ey SRM ALES CE SE gt MRED SEIT SE Ca Ree Sk ODE ta 0 fh i fiidactnso—T ee Ee LA aziv HALF SIZE SECTION THROUGH PLATE APPATLATUS FOR EXPERIMENTS WITH ORIFICES “our Plate XVII. + 1837, DOAN ~ Fults Bien & Co Lid, o or ae etltem te Prey ot ye! baslctarar’ * 4 ahah? whose oo Reece ee nak te. nicks ices looser ee Tes HPD telat os 4a geal! ‘ bled aohd Vater tes arin Dai ve On i Ce crn oon Sm emcees i bdeieantl oho Conre ries Rests 5 re a tee iar Masala tae jet Hesse “egies “ he Dicer: Mrrererbtesesastt abr nee ta Hein Sentra = oeehrirestle ers Poh Core ‘ rh tiptet beak, Ley trons pate sake ey Dore baste rem Wright tier trtrtttcbas bd eek et ree, 6 iy att ‘3 seks rire Perera ie sas anA eT ta! ished a! by tkerestoenbs neg tittle “7 Meet aseas tosses Wet nis brsadcecer otters gnats Lares Sear geht nie Sete leivber tla. acs porn Det west ttre r Rees S Ditty Ge eiceaimat tue hereon dele Srna rer tanta beh ate Nee ook Sra oh tie adbreleahe if trpt both a et Lemna OT arden Cn fetal rat ut ei DG eran Phckteah a is baba olsk Forte Der ines tht ether neta, uals crate utereeeomerc terse ee Satyr a Born Hata a ue mre Sahat recat eet te Batak chat . cons g ane Corrs heer co Sra a peices tale Bit te fre ( r . ve ponueniri i ft a Sarthe aU iy iejede Mente cee oe bionteb atin Na rhe Weitere ba aT te beth 4 Gon pei ear tn tiatobar * peaat he Cairo veinh-imiaveresar gs eatdsie reise p te be Mie evi te tite ee Care ret St Spraritnienawetesaetoc Penton es tena ae : “ rere Eres Helene 7 anne Sone Mbtiaenee etree bith aesitie gst ity eee tee ool S ve Seatac yee Conn tye shale: of o babies: Pies * fue ere Pact pers okt nthe ia eens iat ‘abet et ete ink ritt dh tates de eae ain Sener pres Stories aan rn Sar Pine Sebo dah Srey robb tt page Lorient