61FT OP 
 ROBERT 
 BE1PHBR. 
 
PRACTICAL 
 
 HYDRAULICS 
 
 BY 
 
 P. M. KANDALL, 
 
 AUTHOR OF 
 
 QUARTZ OPERATORS' HAND-BOOK." 
 
 PUBLISHED AND SOLD BY 
 
 Y & Oo. 
 
 PROPRIETORS MINING AND SCIENTIFIC PRESS, 
 SAN FRANCISCO, CAL., 1886. 
 
Entered according to the Act of Congress, in the Librarian's Office, 
 Washington, D. C., 1886, by DBWEY & Co. 
 
PREFACE. 
 
 The present work is designed as a true exposition 01 the prin- 
 ciples and application of those branches of hydraulics, of which it 
 treats. 
 
 The necessity of such a work at this time will be obvious to 
 those who shall have compared the results deduced from the for- 
 mulas of nearly all our most noted authors on hydraulics, with the 
 results of observation. Thus, the formulas of DuBuat, Eytelwein, 
 Girard, Prony, D'Aubuisson, Neville, Leslie, Pole, Beardmore and 
 Hagen, enjoying the reputation of standard authors, give as by the 
 data at hand, with respect to the flow of water in open streams of 
 medium size, results varying from fifteen to one hundred and 
 twenty-five per cent in excess ot the observed results, and in 
 large streams, results varying from thirty to sixty-seven per cent 
 below those observed. These errors are radical. The defect! venesa 
 of these and other works heretofore regarded standard on this 
 highly important branch of hydraulics, is well portrayed by the 
 following extracts from an article in an Engliah periodical, "Engi- 
 neering" of Dec. 31, 1875, entitled "Hydraulic Experiments," viz : 
 " The tabulated velocities (in Neville's work, based upon DuBuat) 
 " though expressed in hundredths of an inch, are in reality but the 
 " wildest guesses at the actual velocities in irrigation canals of 
 " ordinary dimensions. Colonel Cautley relied upon DuBuat when 
 " he laid out the Ganges Canal, and found him but a rotten reed, 
 " for the water in every instance tore along at an unexpected velo- 
 " city, and the erosion of the bed and destruction of the works 
 " followed in its wake. Da Buat then must be put upon the top 
 " shelf of the book-case, and it will be just as well when the steps 
 
 164453 
 
IV PKEFACE. 
 
 " are there to carry up every English work, in which the names of 
 " Branning, Girard, Bossut, Prony, Eytelwein, or D'Aubuisson are 
 " continually recurring as authorities against whom no action can 
 " be taken. In this general clearance Baardmore, Downing, Box, 
 " and almost every other hydraulic text book compiled by English- 
 " men, will with more or less hesitation have been shelved." 
 
 Again, " in 1880," says L. D. A. Jackson, in his Hydraulic Man- 
 ual " the extensive experiments of Captain Allan Cunningham on 
 " the Ganges Canal, have substantiated the truth of Kutter's laws 
 " when applied to very large canals, and dealt the final blow to the 
 " velocity formulas of all the older hydraulicians." 
 
 In the main text of the present work it is stated that D'Arcy 
 in 1835, and Bazin, in 1865, published formulas better adapted than 
 any preceding for finding the flow of water in open streams and 
 pipes of medium size; that Humphreys and Abbot published in 
 1861, formulas suited to the determination of flow of large streams, 
 but not to the flow of small streams, and that the wide gap 
 between the formulas of D'Arcy and Bazin, and those <5f 
 Humphreys and Abbot were effectually closed up in 1870 by 
 the introduction of Kutter's formula. We will now add, that 
 this achievement with respect to hydraulic science seems to us 
 the masterpiece of the nineteenth century. The Kutter formula 
 applies equally well to small, medium sized and large s v reams. 
 Farther experiments may perchance require it to be somewhat mod- 
 ified; but so far as known at present, of all the formulas deduced 
 for like purposes, it seems the nearest approximate to perfection. 
 The principal tables computed by Herr Kutter, from his for- 
 mula under consideration, give the coefficients of velocity in terms 
 of metrical measures, thereby rendering their application a labor- 
 ious task in the determination of the velocities themselves in terms 
 of feet measures. 
 
 To obviate this task, Table 27 in the present work has been 
 computed from the same formula (Kutter's) giving in terms of feet 
 measures, the velocities of flow in open streams, differing in regime 
 and in slope, and varying from the size of a small ditch to that of 
 the Mississippi River. The table is nominally for open streams, 
 but is equally well adapted for determining the flow of water in 
 pipes. Table 17 computed for the flow in pipes only, will, for this 
 
PREFACE. V 
 
 purpose, be found, however, still more convenient. For this table 
 we are indebted in part to J. T. Fanning's very admirable treatise on 
 "Water Supply Engineering," which indebtedness we hereby re- 
 spectfully acknowledge. It will be noted however that we have 
 not only considerably enlarged the original table of Mr. Fanning, 
 but, among other things, conferred upon it a new and valuable 
 feature that of giving the quantity of flow in addition to the veloc- 
 ity. Each result set down in Tables 17 and 27, represents essen- 
 tially a mean of numerous observed results: hence must necessarily 
 coincide in practice with other results obtained under like con- 
 ditions. With respect to accuracy, scope of application and ease of 
 reference, these tables seem to meet more fully the requirements of 
 all concerned in this branch of hydraulic engineering, than any 
 others designed for similar purposes. 
 
 Tables 28 and 29 will be found very important auxiliaries to 
 Table 27, in the ready determination of the flow of water in beds 
 of different forms. 
 
 Tables, two relating to the flow of water in rectangular weirs, 
 four to quadrant weirs, seven to the flow through rectangular ori- 
 fices, and eight to the different values of the so-called " miner's inch," 
 will also be found of no little value in practice. The simplicity of the 
 quadrant weir, its cheapness and the assurances by Prof. Thompson 
 of its superiority over those of different forms, induced the author 
 of the work in hand to compute Table 4. 
 
 This form seems peculiarly well adapted to the measurement of 
 the flow of small quantities of water; for example, from two to 20 
 miner's inches. This table, however, greatly exceeds these limits. 
 The discussion of the subjects of " maximum work effected by 
 water on issuing under pressure from pipes," and of "minimum 
 weight and consequent minimum cost of an inverted siphon," is, so 
 far as the author is informed, new. By the application of the prin- 
 ciples here demonstrated, the greatest economy, the only proper 
 limit or standard of the truly practical, is attained. 
 
 The simple plan, pursued in the preparation of the present 
 work consists : 
 
 1st. In demonstrating concisely the principle, or principles, 
 involved in the subject matter, yet in a manner sufficiently ample 
 and clear to be readily followed by the student, or by the practi- 
 
VI PREFACE. 
 
 tioner desiring to refresh his mind, or to assure himself of the cor- 
 rectness of the results. 
 
 2nd. In expressing in words the simplest rule or rules cor- 
 responding to the formula or result of such demonstration. 
 
 3d. In applying the rule or rules so derived, to practical ex- 
 amples with full and clear explanations; or in applying the formula 
 direct to the examples, when it is too complex to be well expressed 
 in words. 
 
 4th. In providing tables, so far as feasible, to meet the re- 
 quirements of practice. 
 
 By means of these tables and the simple rules given therewith, 
 most of the problems likely to occur with respect to the measure- 
 ment of water in motion, as through vertical openings, over weirs, 
 in pipes, in open streams and through nozzles; with respect to the 
 quantity of water required for various mining purposes, and for the 
 purposes of irrigation; and with respect to the power of water as a 
 motor, are answered direct, or readily solved by anyone familiar 
 with common arithmetic only, as well as by the skilled engineer. 
 
 P. M. RANDALL. 
 San Francisco, March 17th, 1886. 
 
INDEX. 
 
 Formulas, and Formulas and Rules corresponding ; '* F" repre- 
 senting formula, and "R" rule ; page referring to rule : 
 
 F. R. PAOB. 
 
 (1) Acceleration of gravity at sea level in lat. 45" 2 
 
 (3) Acceleration of gravity, the latitude, elevation and acceleration of 
 
 gravity at lat. 45*. given 2 
 
 (4) Acceleration of gravity, the latitude, force of gravity at sea level and 
 
 elevation, given 3 
 
 (175)(176).40-Additional head for angular bend 162 
 
 (177) (178). 41-Additional head for curved bend 163 
 
 (94) (95). .28 -Coefficients of discharge ; of partial contraction 87 
 
 (98) 29 -Coefficient of contraction P9 
 
 (105) 30 -Coefficients of discharge under a head of water in motion 100 
 
 31-Coefficients for a short tube 103 
 
 (107) 32 Discharge through cylindrical tube - length to diameter 3:1 104 
 
 (157) 35 Discharge through clean pipes 155 
 
 (160) 38 - Diameter due head, discharge and the length of a pipe 155 
 
 39 - Discharge, head, length and diameter of pipe ; cases 1, 2, 3 and 4, by 
 
 Table 17 155 
 
 54 Discharge of a flume whose hydraulic mean depth equals that of a 
 
 given pipe ...200 
 
 55 Discharge of a flume by another method 201 
 
 (74) 23 Flow through a parabolic weir whose apex ia level with still water. . . 52 
 
 (284) Formula for finding slope of bank 218 
 
 (285) Formula for finding wet perimeter 218 
 
 (286) Formula for finding mean velocity from central surface velocity 239 
 
 ,287) Formula for finding flood-flow of streams 248 
 
 (8) 1 Head, due velocity and gravity 6 
 
 (11) 2 Head, due time and gravity 7 
 
 (158) 36-Head due dimensions and discharge of a pipe 155 
 
 (198) 46-Head, such that the weight of an inverted siphon shall be a minimum .180 
 
 (101) Imperfect contraction (circular) 93 
 
 (102) Imperfect contraction (rectangular) 93 
 
 (148) 34-Inlet head due velocity 150 
 
 (280) Kutter's Formula ...210 
 
Vlll , INDEX. 
 
 P. R. PAGE, 
 
 (282) (283). Resolved from (2SO) 211 
 
 Application of Kutter's Formula for finding velocity 211 
 
 g For finding diameter of pipe 213 
 
 c For finding coefficient of roughness 215 
 
 a For finding sine of slope 216 
 
 (159) 37-Length due head, discharge and diameter 155 
 
 42 Modifications for degrees of foulness of pipes 169 
 
 (189) 45 Mean pressure in pipe 179 
 
 (190) 47 Mean ordiuate due head and hydrostatic ordinate 181 
 
 (203) 48 Mean pressure per square inch for entire pipe 181 
 
 (204) 49 Minimum diameter of an inverted siphon 181 
 
 (205) 50 Minimum thickness of an inverted siphon 182 
 
 (2 11) 51 -Minimum weight of an inverted siphon 182 
 
 52 Most suitable form of a canal 192 
 
 (88) Orifice discharge (circular) 61 
 
 (90) Orifice discharge (semi-circularupper) 61 
 
 (91) Orifice discharge (semi-circularlower) 62 
 
 26 Orifice discharge (rectangular) 72 
 
 (92) 27-Orifice discharge (rectangular) 73 
 
 (187) 43 Pressure per square inch due he?d 170 
 
 (2) Radius of the earth at sea level, due lat 2 
 
 6 Rule for velocity and discharge of an open stream of water 230 
 
 (14) 5 -Time due velocity and gravity 8 
 
 (15) 6 -Time due head and gravity 8 
 
 (188) 44 Thickness due radL.s, pressure and modulus of strength 171 
 
 (9) 3 Velocity due time and gravity 7 
 
 (13) 4 Velocity due head and gravity 7 
 
 (24) 7 Velocity through a rectangular orifice 16 
 
 (25) 8 Velocity through a rectangular orifice 16 
 
 24) 9 Velocity over a weir 18 
 
 (25) 10 Velocity over a weir 18 
 
 (128) 33-Velocity through short pipe*; 12 J 
 
 (184) Velocity through nozzles 
 
 53 Velocity in a triangular or rectangular flume 199 
 
 (28) 11-Weir discharge (rectangular) 19 
 
 (30) 12-Weir discharge (corrected) 25 
 
 (31) 13-Weir discharge (crest three feet wide) 28 
 
 (37) 14 Weir discharge (quadrantal) 33 
 
 15 -Weir discharge (equilateral) 35 
 
 (34) (48) . . 16- -Weir discharge (triangular) 41 
 
 (54) (55) . . 17 Weir discharge (rectangular) 44 
 
 (61) 18 -Weir discharge (semi-circular) 47 
 
 (62) 19 -Weir discharge (semi-circular) 48 
 
 (63) 20 Weir discharge (semi-circular) 48 
 
 (72) 21 Weir discharge (parabolic open) 51 
 
 (73) 22 Weir discharge (parabolic open) 52 
 
 (78) 24-Weir discharge (triangul ar submerged) 54 
 
 81) 25- Weir discharge (triangular submerged) 58 
 
CONTENTS. 
 
 PRINCIPLES OF HYDRA ULICS. 
 
 Hydraulics Defined. Gravity the source of motion. Acceleration of gravity. Varia- 
 tion in intensity of gravity in different latitudes and at different altitudes. An- 
 alytical determination of the laws of hydraulics. Formulas and rules corre 
 spending. Examples and calculations. .... pp. 1-11 
 
 OPENINGS. 
 
 Submerged and Weir. Diagrams. Analysis of Flow. Comparisons of formulas, flow 
 through orifices: rectangular (pp. 11-16), triangular, trapezoidal, circular and 
 semi-circular, parabolic. Examples and calculations. . . pp. 46-73 
 
 WEIRS. 
 
 Discharge over rectangular, triangular, quadrantal, equilateral, trapezoidal, circular, 
 semi-circular, parabolic (pp. 18-52). Correction (p. 19). Correction due variable 
 coefficient (p. 25). Discharge over crest three feet wide (p. 28). Examples and 
 calcul ations. 
 
 MINER'S INCH. 
 
 Statutory. North Bloomfield. Smartsville. South Yuba Canal. - Examples and cal- 
 culations. ......... pp. 77-79 
 
 CONTRACTION. 
 
 Partial, for circular and rec' angular orifices. For given orifice and given head of 
 water. Imperfect. ....... pp. 84- 2 
 
 SHORT TUBES. 
 
 Cylindrical, convergent, divergent and compound. Examples and calculations. 
 
 pp. 102-108 
 
 PIPES. 
 
 Analysis of flow through pipes (pp. 112- 116). Empirical formulae coefficients of re- 
 sistance as found by Weisbach, Darcy and Fanning (pp. 116-124). Interpolation 
 in Table 16. Flow through pipes -short, long. Analytical determination of 
 maximum work, etc. Inlet head. Equations and rules for velocity, head, 
 length, diameter and volume of flow for clean iron pipes. Coefficient of flow. 
 Effect of bends angular, curved. Relative carrying capacities of clean, foul and 
 very foul pipes. Pressure ordinatea. Requisite thickness of pipe to withstand a 
 given pressure. Examples and calculations. ... pp. 124 - 179 
 
X CONTENTS. 
 
 NOZZLES. 
 
 Flow of water through. Coefficient of flow due convergence. Examples and calcula- 
 tions. . . . i'.'i . . . . pp. 166-169 
 
 INVERTED SIPHON. 
 
 D. termination of minimum values with respect to pressure, diameter, thickness and 
 weight. Hydrostatic pressure below the level of outlet. Special verification of 
 formulas. . ...... . , . . PP. 172 - 183 
 
 OP^V STREAMS. 
 
 Flow of water. Form of rectangle of maximum carrying capacity. Form of trape- 
 zoid of maximum carrying capacity. Most appropriate form of canal. -Formulas 
 for the flow of water in open streams. Kutter's velocity of water in a flume whose 
 hydraulic mean depth is equal to that of a pipe of given diameter. Coefficient of 
 roughness involved in Table 17. Discussion of Kutter's foimula. Application of 
 Kutter's formula as rendered by Equations (2S2 and 283). Application of Tables 
 27, 28 and 29. Interpolation in Table 27. Mean velocity corresponding to central 
 surface velocity. Examples and calculations. . . . pp. 187-239 
 
 MINING. 
 
 Hydraulic. Drift. Duty of miner's inch. Examples and calculations, pp. 241 - 243 
 
 IRRIGATION. 
 
 Quantity of water required. Unit of measure. .... p. 244 
 
 Measurement of the power of water as a motor. .... p. 246 
 
 Flood-flow of streams. . . . . . . p. 247 
 
TABLES. 
 
 No. PAGE. 
 
 9 ... Coefficients for Flow of Water Through Circular Orifices 85 
 
 10 .... Corrections of the Coefficients of Flow for Circular Orifices 94 
 
 11 ... .Corrections of the Coefficients of Flow for Rectangular Orifices 95 
 
 12. ... Corrections of the Coefficients of Flow Under Head of Water in Motion . . 100 
 13. . . .Coefficients of Discharge and Velocity for Flow Through Conically Con- 
 vergent Tubes 105 
 
 14 .... Coefficients of Di scharge Through Divergent Tubes 107 
 
 15 .... C oefficients of Discharge Through Compound Tubes 109 
 
 16. . . .Coefficients of Resistance to the Flow (Cf) of Water in Clean Pipes 113 
 
 19. ... Coefficients for Bend Resistances in Pipes 161 
 
 20. . . .Coefficients for Bend Resistances in Pipes with Circular Transverse Sec- 
 tions 163 
 
 21 Coefficients for Bend Resistances in Pipes with Rectangular Transverse 
 
 Sections . . 164 
 
 26....Coefficients(n) for Roughness of Stream Beds 196 
 
 25. . . .Dimensions of the Most Suitable Forms of Canals 191 
 
 28. . . .Dimensions of Water- Ways, etc 227-228 
 
 2. . . .Flow for Given Depths Over Each Linear Foot of a Rectangular Weir. ... 24 
 
 3. . . .Flow for Given Depths with Crest Three Feet Wide 28 
 
 4. . . .Flow for Given Depths Over a Quadrant Weir 32 
 
 7.... Flow of Water Per Second Through Rectangular Orifices * * * and 
 
 Coefficients 69-70 
 
 8. ...Flow of Water Per Second Due " Miners' Inches" 80 
 
 22. . . .Flow of Water Through Nozzles 167-168 
 
 31 .... Limiting Velocities in Open Streams 247 
 
 23. .. .Moduli of Strength, Working Load and Safety 172 
 
 30 .... Mean Velocity from Central Surface 240 
 
 32 .... Miscellanies 249-252 
 
 24. .. .Number, Thickness and Weight of One Square Foot of Sheet Iron 184 
 
 27.... Open Streams Flow of for Coefficients of Roughness n=.012, n=.017, 
 
 n=.025 and n=.035 219 -226 
 
 29.... Relations of Depth, Base and Slope of Bank 229 
 
 5. . . .Square Roots, Cubes of Square Roots and Fifth Powers of Square Roots 
 
 of Numbers 65 
 
 6. . . .Square Roots of Numbers 66-67 
 
 17.... Velocities and Quantities of Flow in Clean Iron Pipes Due Given Slopes 
 
 and Diameters 134-143 
 
 18 .... Velocities in Pipes and Corresponding Inlet Heads . . . .' 151 
 
 1. . . .Weir Coefficients ." 22 
 
WNWERS1TY 
 
 OF 
 
 PRINCIPLES OF HYDRAULICS. 
 
 The term hydraulics was originally applied to water 
 pipes, or to the conveyance of water through pipes. 
 It is now used in a wider sense to designate that 
 branch of engineering which treats of water in mo- 
 tion, the means of measuring, conveying, and raising 
 it, and its application to machinery as a prime motor. 
 
 The source of this motion is the force of gravity 
 a force which acts indiscriminately upon every parti- 
 cle of matter, and impresses upon each particle at 
 every instant the same degree of velocity in vacuo. 
 The fundamental principles or laws of hydraulics then, 
 are those of uniformly varied motion. 
 
 'These, with respect to a body falling from rest 
 through a certain hight in vacuo, are as follows : 
 
 1st The velocities acquired are proportional to the 
 times elapsed since the beginning of the motion. 
 
 2d The spaces fallen are proportional to the 
 squares of the times elapsed. 
 
 3d These spaces or hights are proportional to the. 
 squares of the velocities acquired at the end of each. 
 
 4th The velocity acquired at the end of the first 
 
2 PRACTICAL HYDRAULICS. 
 
 unit of time is equal to twice the distance fallen dur- 
 ing this time. 
 
 The intensity of the force of gravity varies in dif- 
 ferent latitudes, but for most purposes in hydraulic 
 calculations, it may be regarded constant. 
 
 The velocity which the force of gravity can gener- 
 ate in a second of time at the surface of the earth, is 
 usually designated by g, and is termed acceleration of 
 gravity. Its value, as given in Rankine's Applied Me- 
 chanics, p. 485, for Lat. 45 at sea-level, is g. 
 
 Thus #=32.1695 feet. (1) 
 
 For the most part in practice # = 32.2. 
 
 In case a high degree of accuracy is required, the ob- 
 lateness of the earth, the latitude and elevation of the 
 place at which the value of the force of gravity is 
 sought, have to be considered. 
 
 The formulas involving these elements (the two for- 
 mer deduced from numerous pendulum experiments 
 made in various parts of the world), are given in 
 Rankine's Applied Mechanics, pp. 485-486, as follows: 
 
 =20,887,540 feet (l + 0.00164 cos 2A (2) 
 
 g=g(l 0.00284 cos 2&) (l 2 ) (3) 
 
 In which R denotes the earth's radius at the lo- 
 cality of observation, I the latitude, g' the force of grav- 
 ity at sea-level, in Lat. 45, and g the force of gravity for 
 the elevation above sea-level in the given Lat. 1. The 
 
PRACTICAL HYDRAULICS. o 
 
 factor M } in the second member of equation (3) is 
 
 readily obtained from the well-established proposition 
 that the gravity of a body varies inversely as the 
 square of the distance from the center of the earth. 
 If in a given latitude I, g denote the force of gravity 
 at sea-level, and g" the force of gravity at an eleva- 
 tion, h, there will result: 
 
 g=g (l_*) (4) 
 
 y V \ R) 
 
 When h=Q, it is obvious that in equations (3) and 
 (4), the factor (l ^)=1. 
 
 When Z=45, cos 21=0. 
 
 Whence, (2), #^20,887,540 feet3956 miles. (2 ) 
 
 Example 1. What is the value of the force of 
 gravity at Presidio, San Francisco, in latitude 37, 
 47', 48", at sea-level? 
 
 Calculation. Employing formula (3). 
 
 Find cos 2 (37, 47', 48",)=0.2488. 
 
 Substitute this value, namely, 0.2488. and the value 
 of #'=32.1695, of equation (1) in equation (3), ob- 
 serving that &=0, #=32.1695 (10.00284X0.2488). 
 Whence, #=32.1468 ft. Answer. 
 
 Ex. 2. In latitude 37, 47', 48", as in Ex. 1, 
 what is the force of gravity at an elevation of 2 miles? 
 Cal.WQ find from (2) and (2,). 
 #=3956 (1+0.00164X.2488). 
 Whence, #=3957.6 miles. 
 
4 PRACTICAL HYDRAULICS. 
 
 Substituting this value of R and the value of g as 
 
 x 2 x 2 \ 
 
 found from Ex. 1, inequation 4, #" = 32.1468 (l 3 ^r ) 
 Reducing #"^32.1144. Ans. 
 
 Ex. 3. Required the force of gravity at an eleva- 
 tion of two miles above sea-level at the equator. 
 
 Col. When 1=0, find cos 2J=1. Substituting 
 this value of 21 in eqs. (2) and (2J, and reducing, .R--= 
 3962.5. Substituting the value of R=-- 3962.5, the 
 value of cos 2Z=1, the value of #'=-32. 1695, and 
 the given value of h 2 miles in formula (3), 
 
 #=32.1695 ( 10.00284") (l } 
 
 \ ) \ S962.5/ 
 
 Whence, #--32.0457. Ans. 
 
 Remark. Had we made R= 4000 in the preceding 
 examples, it would in no case have varied the result to 
 exceed .0003. We may, therefore, without sensible 
 error, regard the radius constant and equal to R=- 4000. 
 
 These refinements, with respect to variations in the 
 force of gravity under different conditions, though 
 highly important in establishing a standard of 
 measurement, and in various scientific investigations, 
 yet for the most part are little applied in practice. 
 
 Reverting to the subject of the laws of varied mo- 
 tion, it will be noted that the velocity acquired by 
 a falling body at the end of the first second of time, is 
 double the hight which the body has fallen during 
 that time. A body then, in vacuo, falls during the 
 first second of time 16.1 feet, or more accurately, 16.085 
 in Lat. 45. 
 
PRACTICAL HYDRAULICS. O 
 
 Denote in seconds, the times of a falling body in 
 vucuo by the consecutive numbers: 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, etc. 
 
 Then, according to the 2d law, the hights of the 
 fall are proportional to the squares of these times; 
 thus 1, 4, 9, 16, 25, 36, 49, 64, 81, etc.; and, accord- 
 ing to the 3d law, the hights are proportional to the 
 squares of the velocities acquired during these times. 
 
 If the hight of fall, as found by law 2d, due any 
 given time, be taken from the hight of fall due this 
 time increased by one second, the remainder will be 
 equal' to the space fallen during this increment of, 
 one second. 
 
 Thus the hights, so fallen in the natural order of 
 times, are 1, 3, 5, 7, 9, 11, 13, 15, 17, etc. 
 
 EXAMPLES AND CALCULATIONS. 
 
 Ex. 4. How many feet will a body, as water in 
 vacua, fall during the 5th second of its descent? 
 
 Gal. By the foregoing it will be seen that the fall 
 during the 1st second is 16.1 feet nearly, and during the 
 5th second is 9 times as much ; 
 
 Hence, 16.1 X 9-144.9 feet. Ans. 
 
 Ex. 5. What distance will water fall in vacuo in 
 5 seconds? 
 
 Gal. Note in accordance with law 2d, that a body 
 
6 PRACTICAL HYDRAULICS . 
 
 falls 25 times as far in 5 seconds as it does in 1 second ; 
 hence, 16.1X25=402.5 feet. Ans. 
 
 In further illustration: 
 
 Gal. Note that the hights fallen respectively dur- 
 ing each second of the given time of 5 seconds, are 
 1, 3, 5, 7, 9. The sum of which is 25, as found in the 
 preceding calculation; hence, 16.1X25=402.5 feet. 
 
 Ex. 6. "What will be the velocity of water falling, 
 in vacuo, at the end of the 5th second? 
 
 Gal. Observe that, according to the 1st law, the 
 velocity is 5 times as great at the end of the 5th sec- 
 ond as it was at the end of the first that is, 2X 5=10; 
 hence, 16.1X10=161. feet. 
 
 RULES WITH RESPECT TO THE RELATIONS OF SPACE, 
 TIME, VELOCITY, AND THE FORCE OF GRAVITY, INVOLVED 
 IN THE FALL OF A BODY, AS WATER, IN VACUO. 
 
 The velocity given to find the head or distance the 
 ivater has fallen. 
 
 Rule 1 . Divide the square of the given velocity by 
 twice the acceleration of gravity that is, by 64.4. 
 
 Ex. 7. The velocity is 150 feet per second. What 
 is the head or distance fallen? 
 
 Cal 150X 150-^-64.4=349.4 feet nearly. Ans. 
 
 To find the head or distance water will fall in a 
 given time 
 
PEACTICAL HYDRAULICS, t 
 
 Rule 2. Multiply the square of the time in sec- 
 onds by 16.1 feet. 
 
 Ex. 8. What distance will water fall in 4 seconds? 
 tfaZ. 4X4X 16.1= 257.6 feet. Ans. 
 
 The time given to find the velocity of water fatting 
 freely. 
 
 Rule 3. Multiply the time in seconds by the accel- 
 eration of gravity, namely, 32.2 feet. 
 
 Ex. 9. What velocity does water acquire in falling 
 7 seconds? 
 
 (7aZ. 32.2X7=225.4 feet. Ans. 
 
 The head, or distance of water fallen freely, given 
 to find the acquired velocity. 
 
 Rule 4. Extract the square root of twice the pro- 
 duct of the head and the acceleration of gravity, or 
 multiply the square root of the given head by the 
 square root of twice the acceleration of gravity 
 that is, by 8.025. 
 
 Ex. 10. What velocity will water acquire by fall- 
 ing freely 196 feet? In other words, with what ve- 
 locity will it flow under a pressure or head of 196 
 feet? 
 
 Cal. Square root of 196 feet is 14 feet: 
 
 Then 14x8.025-112.35 feet. Ans. 
 
PRACTICAL HYDRAULICS. 
 
 The velocity being given to find the time which a 
 body, as water, has fallen. 
 
 Rule 5. Divide the given velocity by the acceler- 
 ation of gravity, viz., 32.2 feet. 
 
 Ex. 11. The velocity is 322 feet per second. What 
 is the time fallen? 
 
 Cal 322-^32.2=10 seconds. Ans. 
 
 To find the time required for water to fall freelij 
 through a given distance. 
 
 Rule 6. Extract the square root of twice the given 
 distance, divided by 32.2, or, in other words, divide 
 the square root of the given distance by the square 
 root of one-half the acceleration of gravity that is, 
 by 4.012. 
 
 Ex. 12. What time will water require to fall freely 
 a distance of 144 feet? 
 
 Cal. The square root of 144 feet is 12 feet: 
 Hence, 12-^-4.012=3 seconds nearly. Ans. 
 
 Determination of these laws of hydraulics by 
 analysis. 
 
 To determine these laws let h represent the head or 
 distance in feet, through which the water acts, or has 
 fallen in a given time denoted by ; let v represent the 
 velocity acquired at the bottom of this head or dis- 
 tance fallen, and let g denote the acceleration of 
 
PRACTICAL HYDRAULICS. 
 
 gravity that is, the velocity which the force of 
 gravity can generate in a second of time at the sur- 
 face of the earth. 
 
 Then the expressions for velocity and acceleration 
 of gravity will be: 
 
 (5) 
 
 dt ^ } 
 
 and 0=! (6) 
 
 Eliminate dt from (5) and (6); 
 
 - (7) 
 
 9 
 
 Integrating (7), fc= (8) 
 
 Integrating (6), v^cjt (9) 
 
 Combining (5), and (9), dh^gtdt (10) 
 
 Integrating (10), h = ~ (11) 
 
 Combining (9), and (11), fc=~ (12) 
 
 Reduce (7), as to v, 
 
 v=i/w-=i/lXi/h (13) 
 
 Divide (9) by g, t=- (14) 
 
 ^ 
 
 Reduce (11), as to t t t=\/ ' h~^V\ (^) 
 
 By inspection it will be seen that in the text, the 
 given rules and the enunciations termed laws are de- 
 rived from these formulas, or are but expressions for 
 
10 PRACTICAL HYDRAULICS. 
 
 them ; and that the relations of space, time, velocity 
 and force of gravity are more clearly expressed by 
 formula than it was possible to do by words. 
 
 To facilitate this inspection, the following references 
 to the different forms of expression, but equivalent in 
 meaning, are given. Thus: 
 
 Laws. Rules. Formulas. 
 
 1st. 3d. (9), v=gt. 
 
 2d. 2d. (11), . . ' ft= 
 
 3d. 1st. (8), h= 9 - 
 
 4th. Modification of (12), v=2fc-5-l Sec. 
 
 4th. (13), reduced from (8), v=i/zjh 
 
 5th. (14), reduced from (9), t-= 
 
 6th. (15), reduced from (11), *= J- 
 
 \ g 
 
 The value of g being substituted in these formulas, 
 every possible question with respect to the free fall of 
 water or other body can be answered. 
 
 The formula of most frequent occurrence in hy- 
 draulics corresponds to the 3d law, 1st rule, and is 
 denoted in the column of formulas above by (8) that 
 
 is, h-. This formula, reduced with respect to v, is 
 denoted by (13); v=i/ii=i/Xi/A=8.025i/ji, in 
 which forms it frequently occurs in works in en- 
 
PRACTICAL HYDRAULICS. 11 
 
 gineering. In this, or these formulas, v is termed the 
 velocity due to a given hight, h, and h the hight due 
 to a given velocity, v that is, v denotes the distance 
 which water flowing freely under a pressure of h feet 
 in hight, will pass over in one second of time at the 
 bottom of this hight, h, which velocity is the same, 
 as water falling freely through the hight, h, would 
 acquire at its bottom. 
 
 FLOW OF WATER THROUGH OPENINGS. 
 
 Openings are of two classes the submerged and 
 the weir. 
 
 An opening having its top, as shown at B, Fig. 1, 
 beneath the water's surface, AD, is termed a sub- 
 merged orifice ; an opening having an open top, as 
 shown at C, the crest, in Fig. 2, is termed a weir. 
 In both, the form of outlet, for the most part in prac- 
 tice, is rectangular. This is more especially true with 
 respect to weirs.. 
 
 In Fig. 1, representing a vertical section through a 
 
 FIG. I. 
 
 FIG. 2. 
 
 rectangular opening, conceive the opening, BC, com- 
 posed of horizontal fluid layers indefinitely thin. 
 
12 PRACTICAL HYDRAULICS. 
 
 Let - horizontal length of opening. 
 & =AB head with respect to top of opening. 
 h = AC head with respect to bottom of opening. 
 #=head additional to h, and due any horizontal 
 fluid layer indefinitely thin in the opening BC. 
 dx= thickness of such fluid layer. 
 v= velocity due (h-\-x) per second. 
 Q= discharge of water in cubic feet per second. 
 
 Then by (13), v=(2g) $ (h+x) * (16) 
 
 dQ=l (20) i (h+x) i dx. (17) 
 
 Integrating (17), between the limits of x==Q, and 
 x=h h--=EG. 
 
 (18) 
 
 Equation (18), expresses the quantity of water, in 
 cubic feet, which will flow through a submerged ori- 
 fice under the general conditions imposed. It is, 
 however, alike applicable to weirs. 
 
 For, by making &=0, there results: 
 
 (19) 
 
 which expresses the quantity of water in cubic feet 
 that will flow over a weir under the general condi- 
 tions imposed. 
 
 The true mean velocity of a stream through the 
 submerged orifice under the given conditions, is, 
 
 (20) 
 
PRACTICAL HYDRAULICS. 1.3 
 
 Some hydra ulicians assume that the mean head is 
 equal to the distance between the surface and the mid- 
 dle of the submerged orifice; hence, deduce that the 
 mean velocity is by (13), 
 
 Comparing (20), and (21), omitting the common 
 factor (2#)i 
 
 (22) 
 
 In refutation of the assumption that the mean ve- 
 locity of a stream of water flowing from a sub- 
 merged orifice, takes place at the middle of the open- 
 ing, the following illustrations are presented. By 
 substituting the respective values of the given heads 
 and vertical widths in equation (22), there result: 
 
 When the vertical width is equal to one-half the 
 head above it, the velocity is found one- sixth of one 
 per cent too great ; when it is equal to the head above 
 it, the velocity is found one-half of one per cent too 
 great; 
 
 When it is equal to twice the head above it, the 
 velocity is found one and one-ninth per cent (.0111) 
 too great; 
 
 When it is equal to three times the head above it, 
 the velocity is found one and two-thirds of one per 
 cnt (.01645) too great; 
 
 And when we pass to the limit of the weir, by 
 
14 PEACTICAL HYDRAULICS. 
 
 making h=0, the velocity becomes six per cent too 
 great. 
 
 The assumption that the mean velocity takes place 
 at the middle of a rectangular opening, is thus shown 
 erroneous. It may, perhaps, approximate the truth 
 with sufficient accuracy for most practical purposes, 
 when the vertical width of the opening is less than the 
 head above it; but is inadmissible when the width in 
 comparison is greater. 
 
 The formula obtained on the assumption that the 
 mean velocity takes place at the middle of the open- 
 ing, is commended by its simplicity of expression. 
 But its application, in case the ratio of the head to the 
 vertical opening is large, involves the use of a co- 
 efficient of flow, varying with that ratio. The for- 
 mula so encumbered would evidently be more complex 
 and tedious of application than that, namely (20), 
 which it seeks to evade or displace. 
 
 Water in motion is subject to resistances, which 
 retard its velocity and diminish its volume of flow. 
 Modifications, therefore, are requisite to be made in 
 the theoretical formulas of hydraulics, so that they 
 shall embrace 'the measure of these resistances, and 
 thereby more fully meet the requirements of practice. 
 These modifications are effected for the most part by 
 coefficients, whose values have been determined by 
 experiment. A coefficient, as here used, corresponds 
 to that part of the theoretical quantity, as velocity 
 and volume, remaining after it has been diminished in 
 amount equal to the loss due resistances overcome. 
 
PRACTICAL HYDRAULICS. 15 
 
 With respect to the velocity of water flowing through 
 orifices, a multiplicity of experiments, during a period 
 of many years, have been made with extreme care by 
 the ablest hydraulicians. The tabulated results of these 
 experiments, differing with respect to head and size of 
 opening, vary from .572 to .795. 
 
 If the theoretical velocity of water flowing through 
 a rectangular opening with thin sides or lips be taken 
 as a unit, then will an average of the experiments 
 referred to, closely approximate .62. 
 
 This fraction is very generally adopted that is, for 
 the entire opening, the theoretical velocity is, to the 
 average experimental velocity, as 100 to 62. 
 
 Let it not be inferred that the actual velocity of the 
 flowing water is 62 per cent of the theoretical ve- 
 locity. Water, in flowing from a reservoir, ap- 
 proaches the opening or outlet in convergent lines. 
 This convergence continues a short distance beyond 
 and outside the outlet, as shown at M M /? Fig. 1, 
 where occurs the minimum cross section of the stream, 
 and where the velocity is nearly equal to the theoreti- 
 cal velocity. The area of the outlet being taken as 
 the unit, the area of this cross section is equal to 
 .637, while the velocity of the stream at this point is 
 equal to .974 of the theoretical velocity. The co- 
 efficient of discharge there is equal to the product of 
 these coefficients of cross section and velocity. Let 
 c=this coefficient of discharge; 
 
 Then c= .637 X .974^.62 nearly. (23) 
 
16 PRACTICAL HYDRAULICS. 
 
 In weir openings, the experiments of J. B. Francis, 
 C. E., make c=.622. The top contraction seems to 
 have no separate coefficient in the formula volume. 
 C, the coefficient of discharge, is also the coefficient 
 of average velocity, at and for the entire opening, but 
 not as hitherto remarked for obtaining the actual ve- 
 locity, as it occurs at M M y , Fig. 1. Introducing this 
 
 coefficient (.622), or modifier, (2^)^ into equations (20) 
 
 and (21), and making (%i)^=8.026 as found, there 
 results, 
 
 (25) 
 
 TO DETERMINE THE VELOCITY OF WATER FLOWING 
 THROUGH A RECTANGULAR OPENING. 
 
 Rule 7. From the square root of the cube of the 
 head of water on the bottom of the opening, subtract 
 the square root of the cube of the head on the top of 
 the opening. Divide the remainder by the difference 
 of these heads, and multiply this quotient by 3.33, 
 the product will be the required velocity.^ 
 
 Rule 7. Corresponds to equation (24). 
 
 Rule 8. Multiply the square root of one-half the 
 
PRACTICAL HYDRAULICS. 
 
 sum of the respective heads on the top and bottom of 
 the opening, by 4.99. 
 
 Rule 8. Corresponds to equation (25). 
 
 Ex. 13. In a rectangular opening, the head on the 
 top of the orifice is 2.25 feet, and on the bottom of the 
 opening is 4 feet. What is the average velocity of the 
 flow of water at the outlet? 
 
 Gal. by Rule 7, or formula (24). 
 
 Square root of the cube of the head on the bottom of 
 the opening (4)i~=8; 
 
 Square root of the cube of the head on the top of 
 the opening (2.25)1=3.375. 
 
 Difference between heads 4 2.25=1.75. 
 
 Thus: 3.33 (8. 3.375)-^-1.75=8.8 feet. Ans. 
 
 Gal. by Rule 8, or formula (25). 
 
 Half sum of given heads (4+2.25)^-2=3.125; 
 4.99 times the square root of this quotient. 
 
 4.99 (3.125)^=8.814 feet. Ans. 
 
 Dividing the result obtained under Rule 8 by result 
 obtained under Rule 7. 
 
 8.814-5-8.8=1.0016, 
 
 It is seen that in this case, Rule 8, corresponding to 
 formula (25) gives a velocity nearly one-sixth of one 
 per cent too great. 
 
18 PRACTICAL HYDRAULICS. 
 
 TO FIND THE AVERAGE VELOCITY OF WATER FLOWING 
 OVER A WEIR. 
 
 Rule 9. Multiply the square root of the head over 
 the crest by 3.33. 
 
 Rule 9 corresponds to formula (24) by making the 
 head on the top nothing. 
 
 Rule 10. Multiply the square root of one-half the 
 head over the crest by 4.99. 
 
 Rule 10 corresponds to formula (25) by making 
 the head on the top nothing. 
 
 Ex. 14. In a rectangular outlet, open top, the 
 head on the bottom of the opening is one foot. What 
 is the average velocity of the flow? 
 
 Gal. By Rule 9. 
 
 Square root of the given head (1)^=1. 
 
 Then 3.33X 1=3.33 feet. ^s. 
 
 Gal. By Rule 10. 
 
 Square root of one-half the given head: 
 
 (.5)i=.7072. 
 
 Then 4.99X .7072=3.53 feet. Ans. 
 Dividing result obtained under Rule 10 by that ob- 
 tained under Rule 9, 
 
 3.53-^3.33=1.06. 
 
 It is hereby seen that Rule 10 gives a velocity six 
 per cent too great. 
 
PEACTICAL HYDRAULICS. 19 
 
 Substituting the values, of (2^)2=8.025, 
 and c=. 622 of (23) in (18). 
 
 Q=3.33Z (hph* (26) 
 
 When hQ, (26) becomes 
 
 Q=3.33Z h* (2T) 
 
 Formula (26) is adapted to finding the discharge of 
 a rectangular submerged orifice, and formula (27), the 
 discharge of a weir. In the latter case, when the 
 depth of water on the crest exceeds three inches, and 
 does not exceed two feet, and the length of weir is not 
 less than three times this depth, J. B. Francis, C. E., 
 a most eminent experimentalist, determined by care- 
 ful experiments made on a large scale, and under the 
 most favorable circumstances, that the loss by means 
 of end contraction, is equal to one-tenth the depth of 
 water over the weir for each such contraction. 
 
 Introducing this modification in (27) and there re- 
 sults: 
 
 Q=3.33 (0.1 nh)hl (28) 
 
 In which n denotes the number of end contractions- 
 For a weir one foot in length with water one foot 
 
 deep, Nystrom's Mechanics makes the coefficient 
 
 3. 135 instead of 3.33. 
 
 TO FIND THE DISCHARGE OF WATER OVER A WEIR, 
 WITH CORRECTION MADE FOR DEPTH ON CREST. 
 
 Rule 11. Deduct from the length of the weir, one- 
 
20 PRACTICAL HYDRAULICS. 
 
 tenth the depth of water over the crest, for each and 
 every end contraction -(usually two) ; multiply the cor- 
 rected length so found by 3. 33 times the square root 
 of the cube of the depth or head of water on the 
 crest. 
 
 Ex. 15. The length of weir being 3.01 feet, cut 
 in two-inch planks, and the full depth h, over the bot- 
 tom of the notch 1.023, what is the discharge in cubic 
 feet per second? 
 
 Gal. By Rule 11. 
 
 Loss in this case by two end contractions: 
 
 1.023X.2=.2046. 
 
 Corrected length: 3. 01. 2046=2. 8054 feet. 
 Square root of the cube of the given head: 
 
 (1.023)1=1.085; hence, 
 
 <2=3.33x2.8054Xl.035=9.67 cubic feet. Ans. 
 
 Working this example by the weir formula given 
 in Weisbach's Mechanics, whence the example is taken, 
 there results: 
 
 Q=10.22 cubic feet flow per second, which is five 
 and two-thirds per cent more than obtained by Rule 
 11, derived from the formula in accordance with ex- 
 periments of Francis. 
 
 This discrepancy would seem to arise mostly from 
 the correction introduced in our rule to compensate 
 for end contraction. 
 
PRACTICAL HYDRAULICS. 21 
 
 The length of weir being 3.01 feet, and the depth 
 of water on crest .545 feet, the discharge by Rule 11 
 or formula (28) amounts to 3.887 cubic feet per sec- 
 ond, and by Weisbach's formula 4.253, which latter is 
 nine and one-half per cent greater than obtained by 
 Rule 11, or formula 28. 
 
 Again, the length of weir being 3.01 feet, and the 
 depth of water on the crest .189 feet, the discharge 
 per second, by Rule 11, amounts to .8133 cubic feet, 
 and by Weisbach's formula . 753 cubic feet, which latter 
 is nearly seven and one-half per cent less than the 
 discharge obtained by Rule 11 or formula (28). 
 
 In the foregoing examples, the length of weir, the 
 respective depth of water on the crest, and the cor- 
 responding coefficient of discharge, are given as the 
 data and results of actual experiments made by W. R. 
 Johnson, editor of the first American edition of Weis- 
 bach's Mechanics. 
 
 The experiments of Professor Johnson are entitled 
 to great consideration. Those of Mr. Francis, how- 
 ever, from which formula (28), or Rule (11) is derived, 
 were conducted on a large scale with extreme care and 
 with the aid of the most improved mechanical appli- 
 ances, so as to commend their results as the best 
 authority known at present in 'weir measurement. 
 
 In addition to the variation expressed by the factor 
 (/ O.Inh) in formula (28), the coefficient of dis- 
 charge is found to vary with the , head or depth of 
 water 'on the crest of the weir. 
 
22 
 
 PRACTICAL HYDRAULICS. 
 
 To compensate for this variation, Table 1 is given, 
 to be used with formula (28). 
 
 TABLE I. 
 
 WEIR COEFFICIENTS. 
 Water Supply- Engineering J. T. Fanning. 
 
 Depth in}}; t 
 
 Coefficient 
 
 12 
 1.000 
 
 3.339 
 
 1 
 .083 
 
 3.263 
 
 14 
 1.167 
 
 3.339 
 
 .124 
 3.274 
 
 16 
 1.333 
 
 3.340 
 
 2 
 .167 
 
 3.285 
 
 18 
 1.500 
 
 3.339 
 
 3 
 .25 
 
 3.301 
 
 20 
 1.667 
 
 3.339 
 
 4 
 .333 
 
 3314 
 
 24 
 2.000 
 
 3.338 
 
 6 
 .500 
 
 3.329 
 
 30 
 2.500 
 
 3.334 
 
 8 
 .667 
 
 3.336 
 
 40 
 3.333 
 
 3.331 
 
 10 
 .833 
 
 3.338 
 
 48 
 4.000 
 
 3.317 
 
 1 in 
 
 Depth m }- f '. 
 j ieezj. 
 
 Coefficient 
 
 
 The mean coefficient, as given in formula (28), is 
 3.33. The maximum, as seen by Table 1, is 3.34. 
 
 The mean coefficient 3. 33 corresponds to the depths 
 523 feet and 8.42 feet. 
 
 A comparison shows that the mean coefficient is 
 three-tenths of one per cent (.003) less than the maxi- 
 mum, two per cent greater than that corresponding to 
 a depth of one inch or 0.083 feet; four- tenths of one 
 per cent (.004) greater than that due a depth of four 
 feet; and nine-tenths of one per cent greater than 
 due a depth of one-fourth of a foot. The greatest 
 variation occurring in the coefficient of discharge for 
 different depths, between four feet and one-fourth of a 
 
PKACTICAL HYDRAULICS. 
 
 foot, is seen to be below one per cent. An equal vari- 
 ation, for the most part, in practice, is likely to occur 
 from various other causes, and too often elude the ob- 
 servation of the engineer. 
 
 Ex. 16. The length of a weir being six feet, and 
 the full depth of water over the crest two inches, 
 what is the discharge per second? 
 
 Col. By formula (28) modified by Table 1. 
 
 Loss in this case by two end contractions. 
 
 2inches=.167feet; . 167 X.2=. 0334. 
 
 6 .0334 5.9666 corrected length. 
 
 3 
 
 (.167)^=. 06824, square root of cube of given head. 
 3.285 coefficient as per Table 1, due head of two 
 inches. 
 
 <2=3.285X5.9666X. 06824=1.307 cu. feet. Ans. 
 
 Ex. 17. The length of weir being 10.8 feet, and 
 full depth of water over crest four feet, what is the 
 discharge per second? 
 
 Cdl.By formula (28) modified by Table 1. 
 10.8 4 X- 2=10 corrected length of weir. 
 (4)^=8, square root of cube of given head. 
 
 3.317=coefficient as per Table 1, due head of four 
 feet. 
 
 Q=3. 317X10X8=265. 36 cubic feet. Ans. 
 
24 PRACTICAL HYDRAULICS. , 
 
 TABLE II. 
 
 Flow for given depths over each linear foot of a 
 rectangular weir. 
 
 Head. 
 Feet. 
 
 Flow 
 Cubic Feet. 
 
 Head. 
 Feet. 
 
 Flow 
 Cubic Feet. 
 
 Head. 
 Feet. 
 
 Flow 
 Cubic Feet. 
 
 .04 
 
 .0261 
 
 .46 
 
 1.0386 
 
 .2 
 
 4.3904 
 
 .05 
 
 .0365 
 
 .48 
 
 1.1072 
 
 .3 
 
 4.9506 
 
 .06 
 
 .0480 
 
 .50 
 
 1.1771 
 
 .4 
 
 5.5311 
 
 .07 
 
 .0604 
 
 .52 
 
 1.2483 
 
 .5 
 
 6.1341 
 
 .08 
 
 .0737 
 
 .54 
 
 1.3209 
 
 .6 
 
 6.7558 
 
 .09 
 
 .0881 
 
 .56 
 
 .3951 
 
 .7 
 
 7.3987 
 
 .10 
 
 .1035 
 
 .58 
 
 .4724 
 
 .8 
 
 8.0516 
 
 .11 
 
 .1195 
 
 .60 
 
 .5506 
 
 .9 
 
 8.7317 
 
 .12 
 
 .1369 
 
 .62 
 
 .6286 
 
 2.0 
 
 9.4399 
 
 .13 
 
 .1536 
 
 .64 
 
 .7080 
 
 2.1 
 
 10.1460 
 
 .14 
 
 .1718 
 
 .66 
 
 .7888 
 
 2.2 
 
 10.8694 
 
 .15 
 
 .1906 
 
 .68 
 
 .8705 
 
 2.3 
 
 11.6189 
 
 .16 
 
 .2102 
 
 .70 
 
 .9599 
 
 2.4 
 
 12.3850 
 
 .17 
 
 .2303 
 
 .72 
 
 2.0380 
 
 2.5 
 
 13.1668 
 
 .18 
 
 .2512 
 
 .74 
 
 2.1237 
 
 2.6 
 
 13.9649 
 
 .19 
 
 .2726 
 
 .76 
 
 2.2118 
 
 2.7 
 
 14.7783 
 
 .20 
 
 .2951 
 
 .78 
 
 2.2996 
 
 2.8 
 
 15.6067 
 
 .22 
 
 .3407 
 
 .80 
 
 2.3883 
 
 2.9 
 
 16.4501 
 
 .24 
 
 .3882 
 
 .82 
 
 2.4788 
 
 3.0 
 
 17.3086 
 
 .26 
 
 .4377 
 
 .84 
 
 2.5699 
 
 3.1 
 
 18.1809 
 
 .28 
 
 .4892 
 
 .86 
 
 2.6620 
 
 3.2 
 
 19.0676 
 
 .30 
 
 .5445 
 
 .88 
 
 2.7557 
 
 3.3 
 
 19.9687 
 
 .32 
 
 .593:' 
 
 .90 
 
 5.8500 
 
 3.4 
 
 20.7953 
 
 .84 
 
 .6572 
 
 .92 
 
 2.9463 
 
 3.5 
 
 21.7194 
 
 .36 
 
 .7158 
 
 .94 
 
 3.0432 
 
 3.6 
 
 22.6568 
 
 .38 
 
 .7761 
 
 .96 
 
 3.1409 
 
 3.7 
 
 23.6074 
 
 .40 
 
 .8384 
 
 .98 
 
 3.2395 
 
 3.8 
 
 24.5710 
 
 .42 
 
 .9020 
 
 LOO 
 
 3.3390 
 
 3.9 
 
 25.5472 
 
 .44 
 
 .9717 
 
 1.1 
 
 3.8522 
 
 4.0 
 
 26.5360 
 
 To obviate the use of a formula so encumbered, 
 Table 2 has been computed. In which the 1st, 3d 
 and 5th columns represent the heads or depths in 
 feet, from the level of still water to the crest; the 2d, 
 
PRACTICAL HYDRAULICS. 25 
 
 4th and 6th columns represent the quantities dis- 
 charged per second, for the given depths over each 
 lineal foot of weir. These quantities are the products 
 of the unit length, cubes of the square roots of the 
 given depths, and their respective variable coefficients 
 found in Table 1. If Q t denote the tabulated quantity, 
 so discharged per lineal foot, h t , the given head, and 
 c y , the variable coefficient due that head, then we shall 
 have the formula by which Table Q, has been com- 
 puted, viz.: 
 
 e t =e,fc*. (29) 
 
 Multiplying (29) by (I 0.1 nh), the factor for cor- 
 recting length of weir as in (28), and putting Q equal 
 discharge over a weir whose length is three or more 
 times the depth of water on crest, and there results: 
 
 Q=Q t (l0.1nh,)=:c,'(l0.l nh)h? (30) 
 
 WHENCE TO FIND THE DISCHARGE OF WATER OVER A 
 WEIR WITH CORRECTIONS MADE DUE VARIABLE COEFFI- 
 CIENT, AND DEPTH ON CREST. 
 
 Rule 12. Deduct from the given length of the 
 weir one-tenth the depth of water on the crest, for 
 each and every end contraction, and multiply the 
 length so corrected by the quantity in "flow" column 
 opposite the given head in Table 2. 
 
 Rule 12 is derived from Eq. 30 middle right hand 
 member employed. The extreme right hand member 
 
26 PRACTICAL HYDRAULICS. 
 
 of the same equation expresses the value .of Q in terms 
 of the corrected length, and the head and the variable 
 coefficient as found in Table 1. 
 
 This, in fact, is the modified formula of (28), by 
 which examples 16 and 17 were solved. 
 
 Ex. 18. The depth of water being two feet over a 
 crest seven feet in length, what is the discharge? 
 
 Col. By Rule 12. 
 
 7 2 X- 2=6. 6 feet, corrected length, 
 
 BY TABLE 2. 
 
 9.4299 cubic feet= discharge due head of two feet. 
 Whence, 9.4399x6.6=62.30 cubic feet. Ans. 
 Cat. By formula (30), extreme right hand member. 
 7_2x.2=6.6 feet, corrected length. 
 
 (2)t=2.8284=square root of cube of depth. 
 
 3.338=coefficient by Table 1, due head of two feet. 
 
 Q=3.338X6. 6X2.8284=62.32 cubic feet. 4ws. 
 
 The calculation in which Table 2 is employed, is 
 seen to be much shorter and far more simple than 
 that in which Table 1 is employed. 
 
 Weirs are usually constructed with horizontal crests 
 and vertical ends, forming a notch, through which the 
 flow of water is measured in its passage from a reser- 
 voir, or other storing place; or in its passage over a 
 submerged dam across a flume, canal or natural 
 stream. A weir should be free from vibration. Its 
 crest and ends should be chamfered on the down- 
 stream side to an edge say one-tenth of an inch 
 
PEACTICAL HYDEAULICS. 27 
 
 thick. Its upstream face should be vertical, and its 
 downstream face, so inclined or fashioned as not to 
 resist the flow of water. On the upstream side, the 
 depth of water below the level of the crest should be 
 fully twice as great as the head of water on the crest. 
 The head or depth is the vertical distance, as AC, Fig. 
 2, between the crest and the level of still water at a 
 point some distance above the weir, as at a. In ordi- 
 nary practice, the head is measured with a common 
 rule or linear measuring scale, from the top of a post, 
 P, in Fig. 2, set level with the crest. If greater accu- 
 racy is required, the "Boy den Hook Gauge" should be 
 employed. Care must be taken that the flow over 
 the weir shall not be affected by the approaching 
 stream. The area of the weir opening should not 
 exceed a fifth part of that of the supply stream. With 
 proper care in taking the data, the weir affords very 
 accurate means of measuring the flow of water. 
 
 This, taken in connection with the weir's simplicity 
 and facility of construction, cheapness and wide appli- 
 cation, renders it of great practical importance, es- 
 pecially to those concerned in the measurement of flow- 
 ing water, in places of difficult access. A temporary 
 dam, in illustration, built across a natural stream, with 
 a crest board firmly fitted, level and vertically width- 
 wise to the dam's top, makes a good measuring weir. 
 In flumes and canals, measuring weirs can obviously 
 be constructed with equal facility. 
 
 The weir crest being about three feet wide and 
 level, with a rising incline to its receiving edge, Mr, 
 
28 
 
 PRACTICAL HYDRAULICS: 
 
 Francis offers the following formula, for approximate 
 measurements, for depths between six and eighteen 
 inches: 
 
 (2=3.01208 &/ (31) 
 
 As this formula is somewhat complicated, the writer 
 would present Table 3, computed from it, which will 
 be found far simpler and less tedious of application. 
 
 TABLE 3. 
 
 Plow for given depths over each lineal foot of weir, with 
 crest three feet wide. 
 
 Head. 
 
 Flow 
 Cubic Feet 
 
 Head. 
 
 Flow 
 Cubic Feet 
 
 Head. 
 
 Flow 
 Cubic Feet 
 
 6 inches. 
 7.5 " 
 9 
 
 1.043 
 1.467 
 1.939 
 
 10.5 inches 
 12 " 
 13.4 " 
 
 2.445 
 3.012 
 3.607 
 
 15. inches. 
 16.5 " 
 18 
 
 4.238 
 4.903 
 5.601 
 
 The depth or head being given. 
 
 TO FIND THE FLOW OF WATER OVER A WEIR WHOSE 
 CREST IS THREE FEET WIDE. 
 
 Rule 13. From "flow" column, opposite the give** 
 head, in Table 3, take the number representing the 
 discharge in cubic feet over one lineal foot, which 
 multiply by the given length of the weir. 
 
PEACTICAL HYDBAULICS. 29 
 
 Ex. 19. The head being 15 inches, and the length 
 of weir, whose crest is three feet, being ten feet, what 
 is the approximate discharge in cubic feet per second? 
 
 Gal. In Table 3, in "flow" column, opposite 15 
 inches, given head, find 4.238 cubic feet. 
 
 Whence, 4.238x10=42.38 cubic feet. Ans. 
 
 TBIANGULAB WEIBS. 
 
 A triangular form of measuring weir has been 
 employed with favorable results. 
 
 To determine the flow of water through a triangu- 
 lar weir: 
 
 In Fig. 3 let ft~BC represent the head of water in 
 feet, from the apex C to the level, AD, of still water. 
 
 p~ given ratio between the head, h, and the width 
 of the weir that is, let phAD, EF, or any width 
 taken at pleasure. 
 
 c= coefficient of discharge. 
 
 Q= quantity of flow in cubic feet. 
 
 #=any portion of the head h. 
 
 2<7=force of gravity=32.2. 
 
 v= velocity due (h x). 
 
 Then in general by (13) will v=c (2g)?(hx) (32) 
 
30 PRACTICAL HYDRAULICS. 
 
 dQ=pc (2g(hxx dx. (33) 
 
 Integrating (33) between the limits of a=0 and 
 x=h: 
 
 C= 2 V cubic feet. (34) 
 
 QUADRANTAL WEIR. 
 
 In a quadrantal weir, that is, a weir in which the 
 angle ACD 90 a right angle AD 2EC=2h= 
 ph; hence, p=2. Making c=.616; (20)J==8.025. 
 
 Substituting these values of pc and (2f/) 3 in (34). 
 
 Q2.6365 /J per second. (35) 
 
 If, in (35), k" representing inches be substituted for 
 h, which represents feet, and the unit of time be made 
 one minute, we shall have the following formula, 
 which is attributed to Professor James Thompson, of 
 Glasgow University: 
 
 Q=0. 317 h"* per minute. (36) 
 
 If a second be the unit of time, and the head be 
 in inches, we have: 
 
 Q^. 005385 h"* per second. (37) 
 
 Professor Thompson having experimented satisfac- 
 torily with the quadrantal weir, pronounces it more 
 simple and reliable than the rectangular weir, in that 
 the ratio between the head of water and the horizon- 
 tal width of notch is constant; in that the flow of 
 
PBACTICAL HYDRAULICS. 31 
 
 water through it is less effected by the "depth from 
 the crest to the bottom of the channel of approach," 
 and finally, in that the coefficient of discharge is con- 
 stant for different depths. In the experiments of 
 Prof. Thompson, "the volumes of water," says J. T. 
 Fanning, C. E., "varied from .033 to .6 cubic feet per 
 second." From this statement it is deducible that the 
 depths of water varied from two inches to 6.6 inches 
 (see Table 4), though the depths given by Mr. Fan- 
 ning vary from two to four inches. 
 
 The simplicity of this weir, its cheapness, and the 
 assurances of its superiority, as stated, have induced 
 the computation of Table 4 f or ^practical use. 
 
PRACTICAL HYDRAULICS. 
 
 TABLE 4. 
 
 Plow of Water per Second over a Quadrant Weir. 
 
 Head 
 
 Flow 
 
 Head 
 
 Flow 
 
 Head 
 
 Flow 
 
 Head 
 
 Flow 
 
 Inch. 
 
 Cub. Feet 
 
 [nch. 
 
 Cub. Feet 
 
 Inch. 
 
 Cub. Feet 
 
 Inch. 
 
 Cub. Feet 
 
 1. 
 
 .0053 
 
 4. 
 
 .1691 
 
 7. 
 
 .6852 
 
 10. 
 
 1.6713 
 
 .1 
 
 .0067 
 
 4.1 
 
 .1799 
 
 7.1 
 
 .7100 
 
 10.1 
 
 1.7134 
 
 .2 
 
 .0083 
 
 4.2 
 
 .1911 
 
 7.2 
 
 .7352 
 
 10.2 
 
 1.7562 
 
 .3 
 
 .0102 
 
 4.3 
 
 .2026 
 
 7.3 
 
 .7610 
 
 10.3 
 
 1.7995 
 
 .4 
 
 .0123 
 
 4.4 
 
 .2146 
 
 7.4 
 
 .7873 
 
 10.4 
 
 1.8435 
 
 .5 
 
 .0146 
 
 4.5 
 
 .2270 
 
 7.5 
 
 .8142 
 
 10.5 
 
 1.8882 
 
 .6 
 
 .0171 
 
 4.6 
 
 .2398 
 
 7.6 
 
 .8416 
 
 10.6 
 
 1.9334 
 
 1.7 
 
 .0199 
 
 4.7 
 
 .2531 
 
 7.7 
 
 .8696 
 
 10.7 
 
 1.9794 
 
 1.8 
 
 .0230 
 
 4.8 
 
 9668 
 
 7.8 
 
 .8981 
 
 10.8 
 
 20260 
 
 1.9 
 
 .0263 
 
 4.9 
 
 7^809 
 
 7.9 
 
 .9271 
 
 10.9 
 
 2.0732 
 
 2. 
 
 .0298 
 
 5. 
 
 .2954 
 
 8. 
 
 .9567 
 
 11. 
 
 2.1211 
 
 2.1 
 
 .0338 
 
 5.1 
 
 .3105 
 
 1 8.1 
 
 .9869 
 
 11.1 
 
 2.1696 
 
 2.2 
 
 .0379 
 
 5.2 
 
 .3259 
 
 8.2 
 
 1.0177 
 
 11.2 
 
 2.2187 
 
 2.3 
 
 .0424 
 
 5.3 
 
 .3418 
 
 8.3 
 
 1.0489 
 
 11.3 
 
 2.2686 
 
 2.4 
 
 .0472 
 
 5.4 
 
 .3593 
 
 8.4 
 
 1.0808 
 
 11.4 
 
 23192 
 
 2.5 
 
 .0522 
 
 5.5 
 
 .3750 
 
 8.5 
 
 1.1133 
 
 11.5 
 
 23703 
 
 2.6 
 
 .0576 
 
 5.6 
 
 .3922 
 
 8.6 
 
 1.1463 
 
 11.6 
 
 2.4222 
 
 2.7 
 
 .0633 
 
 5.7 
 
 .4100 
 
 8.7 
 
 1.1799 
 
 11.7 
 
 2.4748 
 
 2.8 
 
 .0693 
 
 5.8 
 
 .4282 
 
 8.8 
 
 1.2142 
 
 11.8 
 
 2.5280 
 
 2.9 
 
 .0757 
 
 5.9 
 
 .4468 
 
 8.9 
 
 1.2489 
 
 11.9 
 
 2.5818 
 
 3. 
 
 .0824 
 
 6. 
 
 .4661 
 
 9. 
 
 1.2843 
 
 12. 
 
 2.6365 
 
 3.1 
 
 .0894 
 
 6.1 
 
 .4857 
 
 9.1 
 
 1.3203 
 
 13. 
 
 3.2205 
 
 3.2 
 
 .0968 
 
 6.2 
 
 .5059 
 
 9.2 
 
 1.3568 
 
 17. 
 
 6.2979 
 
 3.3 
 
 .1046 
 
 6.3 
 
 .5266 
 
 9.3 
 
 1.3941 
 
 19. 
 
 8.3167 
 
 3.4 
 
 .1126 
 
 6.4 
 
 .5477 
 
 9.4 
 
 .4318 
 
 23. 
 
 13.4089 
 
 3.5 
 
 .1211 
 
 6.5 
 
 .5693 
 
 9.5 
 
 .4702 
 
 29. 
 
 23.9369 
 
 3.6 
 
 .1300 
 
 6.6 
 
 .5915 
 
 9.6 
 
 .5092 
 
 31. 
 
 28.2800 
 
 3.7 
 
 .1391 
 
 6.7 
 
 .6141 
 
 9.7 
 
 .5490 
 
 37. 
 
 44.0128 
 
 3.8 
 
 .1488 
 
 6.8 
 
 .6373 
 
 9.8 
 
 .5890 
 
 41. 
 
 56.8896 
 
 3.9 
 
 .1588 
 
 6.9 
 
 .6610 
 
 9.9 
 
 .6299 
 
 43. 
 
 64.0830 
 
 
 
 
 
 
 
 47. 
 
 80.0420 
 
 Table 4 has been computed from formula (37), ID 
 which the head is in inches, and the quantity dis- 
 charged per second is in cubic feet, The tabulated 
 
PRACTICAL HYDRAULICS. 33 
 
 discharges for depths of water over six inches requires 
 the confirmation of experiment; until that shall be 
 had, however, there seems good reason from the data 
 to regard them approximately correct. 
 
 APPLICATION OF TABLE 4. 
 
 Ex. 20. The depth of water in a quadrantal weir 
 being 2.1 inches, what is the discharge over it in cubic 
 feet per second? 
 
 Q a l t j n "head" column, Table 4, find the given 
 head 2.1 inches, opposite which, in "flow" column, 
 will be found .0338 cubic feet: the quantity sought. 
 
 To determine the flow of water over a quadrantal 
 weir, the head given being equal to the product of 
 two factors, each designating a head of ivater in 
 Table 4. 
 
 Rule 14. Multiply the product of the discharges 
 due the factor heads by 189.2. 
 
 Ex. 21. The head in a quadrantal weir being 15 
 inches, what is the discharge per second? 
 
 Cat. Observe that the factors of the given head 
 are three and five: 3X515. 
 
 By Table 4, flow due 5" head=.2954. 
 
 By Table 4, flow due 3" head=.0824. 
 
 Hence, .2954 X. 0824X189. 2 =4. 6054 cubic feet. 
 Ans, 
 
34 PRACTICAL HYDRAULICS. 
 
 Ex. 22. The head in a quadrantal weir being 42 
 inches, what is the discharge per second? 
 
 Gal. Observe that the factors of the given head 
 are six and seven: 6x7=42 inches. 
 
 By Table 4, flow due 6" head-. 4661. 
 
 By Table 4, flow due 7" head=.6852. 
 
 Hence, .4661X.6852xl89.2=60.4258 cubicfeet.- 
 A ns. 
 
 The application of Rule 14 to depths below 13 inches 
 will be unnecessary; the rule is given to avoid an 
 extended table. 
 
 EQUILATERAL WEIR. 
 
 To determine the flow of water over an equilateral 
 weir, EOF, in Figure 3, .make ^9 y. That is when 
 h is the hight of an equilateral triangle, its side or 
 width as in EOF, is EF=ph= -. Substituting this 
 
 1/3 
 
 value of p in Eq. (33J, there results: 
 
 Q= 1.52217 /J" per second. (38) 
 
 Were the head given in inches, and the quantity of 
 flow required in cubic feet per minute, then would 
 
 Q = 1.77 h* per minute. (39) 
 
 The heads being equal in the two forms of trian- 
 gular weirs, thus far considered, then will the dis- 
 charge in the equilateral form be to the discharge in 
 the quadrantal form as .57735 is to 1. Hence, to find 
 
PKACTICAL HYDEAULICS. 35 
 
 by Table 4, the discharge over an equilateral weir, 
 the head being given: 
 
 Rule 15. Find in Table 4 the discharge due the 
 given head over a quadrantal weir. Multiply the 
 quantity so found by .57735. 
 
 Ex. 23. The head of water in an equilateral weir 
 is eight inches. What is the discharge in cubic feet 
 per second? 
 
 Gal. By Table 4, it is seen that the "flow" due the 
 given head, eight inches, is .9567 cubic feet per second. 
 Hence, .9567 X. 57735 =.5524 cubic feet. Ans. 
 
 In Eq. (34) any value may be given p, as 1, 2, 3, 4, 
 5, etc., so as to indicate the relations existing between 
 the hight and width of triangular weirs. But since 
 c, the coefficient of discharge, varies with every differ- 
 ent condition imposed, the labor of determining the 
 theoretical flow due any considerable number of such 
 forms would necessarily be barren of practical results. 
 
 In general let (34) be reduced to its simplest form: 
 
 Q^Z.Upch?. (40) 
 
 In which h denotes feet, nd Q cubic feet. 
 
 If in (40) we make p=2 ^ind c=.616, the formula 
 becomes that of the quadrantal weir, as shown by 
 Eq. (35). 
 
 If we make p = -r=1.1547, and c = .6!6, the for- 
 mula becomes that of the equilateral weir, as shown 
 byEq. (38.) 
 
 If we make j>=2]/ 3 ^3.4641, and c = .616, the for- 
 
36 PRACTICAL HYDRAULICS. 
 
 mula becomes that of a weir, whose apex or angle 
 C=120; Fig. 3. 
 
 Q-4.5665 h*. (41) 
 
 And in this manner may special formulas be de- 
 duced from (40), to meet the various requirements of 
 triangular weirs. In theory the results obtained are 
 as they should be ; but in fact, experiment best de- 
 termines in what cases c is constant, and in what 
 variable. 
 
 TRAPEZOIDAL WEIRS. 
 
 To determine the flow of water over a trapezoidal 
 weir, let, in Fig. 3, ACD represent a triangular 
 weir, in which GI is a line indicating a division with 
 respect to flow of water over the weir. 
 
 The flow of the triangular portion, GCI, taken 
 from the entire flow due ACD, there remains that 
 portion of the flow due the trapezoid DAGI. The 
 mean velocity of the water in ACD is' evidently 
 greater than the mean velocity is in DAGI, else the 
 flow in the trapezoid could readily be determined from 
 that of ACD by the simple ratio of these areas. 
 
 Let, in the present solution, h, p, c, Q, g and v have 
 the same values or functions which they had in the 
 discussion hitherto of triangular weirs. In addition 
 put CL=?i/i; 
 
 Then BL^ (I n) h, (42) 
 
 the depth of the trapezoid DAGI. 
 
 Integrating equation (33) between the limits of a?=0 
 
PRACTICAL HYDRAULICS. 37 
 
 and xnh, and there results the flow per second due 
 the triangular portion of the weir GCI. 
 
 Q=pc (2/)-| (l-)+| (1 n)+ ,,&. (43) 
 Deducting (43) from (34), there remains: 
 
 -Ti)lAi. (44) 
 
 Formula (44) represents the discharge due the 
 brapezoid DAGI. 
 
 y -v . * 
 
 (44) 
 
 Ex. 24. The width of a trapezoidal weir being 
 bwo feet, the depth one-fourth (J) of a foot, the sides 
 inclining 45 to the horizon, and the coefficient of dis- 
 charge being .62, required the cubic feet flow over it 
 per second? 
 
 Gal. Since the width is two feet, and the inclina- 
 tion of the sides 45, were the sides produced down- 
 ward till they meet, the depth of the triangle so formed 
 would be equal to one-half the given width that is, 
 k\ foot, and j9=2. 
 
 The given depth of weir being J, hence, n(~L J) 
 
 __ 3 
 
 Making substitution of these values in (44), 
 2 + 3x1=4.25; 
 
38 PRACTICAL HYDRAULICS. 
 
 (1) *=1. We have 
 Q=1.07x2X.62xl25Xi=7.05 cubic feet. Ans. 
 
 Ex. 25. The width of a trapezoidal weir being 4.5 
 feet, the depth one foot, the sides inclining 45 to the 
 horizon, and the coefficient of discharge being .62, re- 
 quired the cubic feet flow over it per second. 
 
 Gal. 1st. The width being 4.5 feet, and the incli- 
 nation of the sides 45, if the sides be produced till 
 they meet, the depth of the triangle so formed will be 
 2.25=, feet. 
 
 2.251-1.25; 7i^f|-=f; l w = l f = {; p =_2. 
 
 Substituting these values in formula (44), 
 
 Q= 1.07X2X6.2 (2+V) (J)*()*=10.946 cubic 
 feet. Ans. 
 
 Gal. 2d. The bottom width 1.25X2=2.5 feet. 
 
 Regard the trapezoidal weir made up of a rectangular 
 weir, whose length is 2.5 feet and depth one foot, and 
 of a quadrantal weir, whose width is two feet and 
 depth one foot. Then: 
 
 The quantities of "flow" are found to be by Table 2, 
 for each linear foot of crest,. 3.339 cubic feet; hence, 
 for 2.5 feet, 3.339x2.58.3475 cubic feet, and by 
 Table 4, for "flow" over a quadrantal weir one foot 
 deep=2.6365 cubic feet. 
 
 Hence, 8.3475 + 2.6365=10.984 cubic feet. Ans. 
 
 The discrepancy between the results obtained by 
 the 1st and 2d calculations arises from the different 
 values assigned the coefficients of discharge. 
 
PRACTICAL HYDRAULICS. 39 
 
 By Gal. 1st the coefficient was taken, as proposed in 
 the given problem, at .62. 
 
 By Cal. 2d, the coefficient employed in computing 
 Table 2, was taken from Table 1, which will be seen 
 to be 3.339. So that the coefficient of discharge em- 
 ployed in Table 2, was, in fact, 3.39-s-J (8.025)= 
 .6241, instead of .62, as provided in the given propo- 
 sition. The coefficient of discharge employed in com- 
 puting Table 4, was .616, which was deduced from 
 the formula given by Prof. Thompson. Weisbach's 
 Mechanics and Engineering state that the coefficient 
 employed by Prof. Thompson, was .619, while J. W. 
 Stone, C. E., author of Hydraulic Formula, states 
 that it was .617. 
 
 Cal. 3d. The top width is given 4.5 feet. The 
 bottom width=(2.25 l)X2=-2.5 feet. Mean width 
 
 By Francis' formula (3.5 .1X2X1)^3.3, cor- 
 rected length. 
 
 By Table 2, flow over each linear foot, 3.339 cubic 
 feet. Hence, 3.339X3.3=11.008 cubic feet. Ans. 
 
 This result differs but little from those obtained 
 from more rigorous solutions. 
 
 FLOW OF WATER OVER TRAPEZOIDAL WEIRS IN WHICH 
 THE LENGTH OF THE CREST IS GREATER THAN THE TOP 
 WIDTH OF THE NOTCH. 
 
 Let ABCD, Fig. 4, represent a trapezoidal weir 
 
40 PRACTICAL HYDKAULICS. 
 
 in which the length of the crest AB=& is greater 
 than the width of the water surface CD=t. 
 
 FIG. 4. 
 
 Draw CE parallel to DB, and let fall on BA, the 
 perpendicular DG, CF. Then will the rhomboid, 
 ECDB, equal in area the parallelogram FCDG, and 
 the entire weir, ACDB, equal in area FCDG + ACE. 
 
 Let hthG head or depth of water between the 
 levels of the crest and still water; p=the ratio of the 
 base, EA, to hight, FC, of the triangle, ACE; c= co- 
 efficient of flow; #=any head not greater than h. 
 
 Then dQ=c (Zg)~*(t + px) x^dx. (45) 
 
 Integrating (45), observing that QQ, 
 When 05=0, 
 
 Q=c (2g)*(^t A* + ^> h%). (46) 
 
 Reducing (46), observing that 6=3 (t+ph), 
 
 (47) 
 
 Making bph, =0, that is, closing the top of the 
 weir, at the level of still water, and equation (46) 
 becomes 
 
 cubic feet. (48) 
 
 &0 
 
 Comparing equations (48) and (34), 
 
PRACTICAL HYDRAULICS, 41 
 
 0:0 ::!:!:: 8: 2. (49) 
 
 48 34 15 15 
 
 Equation (49J shows that, with respect to two tri- 
 angular weirs of equal size, the discharging ca- 
 pacity of the weir whose top is closed at the level of 
 
 still water, as E in Fig. 6, is, to the discharging ca- 
 pacity of the weir whose top is open, as AB in Fig. 5, 
 as 3 is to 2, or 1.5: 1. Table 4 was computed from an 
 open weir, as ABC, Fig. 5. To render it applicable 
 to a closed weir, as DEF : 
 
 Rule 16. Multiply the tabulated flow due any 
 head given in Table 4 by 1.5. 
 
 Ex. 26. In a trapezoidal weir, the head between 
 the crest and the level of still water being three (3) 
 inches, the length of the crest two (2) feet, and the 
 width of the opening at water level one and three- 
 fourths (1.75) feet, what is the flow in cubic feet per 
 second when the coefficient of discharge is .62 ? 
 
 Gal. Head 3 inches=J feet. 
 
 By formula (47), (J)*=J. 
 
 Three times bottom width: 2x3=6. 
 
 Twice top width: 1.75x2=3.5. 
 
 Whence, 1.07X-62 (3.5+6) =.7878 cubic feet, - 
 
 Ans. 
 
42 PRACTICAL HYDRAULICS. 
 
 Ex. 27. In a trapezoidal weir, the head being 
 twelve inches, the bottom width three feet, the top 
 width one foot, and the coefficient of discharge .624, 
 what is the flow of water over it in cubic feet per sec- 
 ond ? By formula (47). 
 
 Cal.lst. LOT X. 62 (1X2+3X3) Xl=T.'3445 cu- 
 bic feet. Ans. 
 
 Cat. 2d. Observe that the weir opening, ACDB, 
 Fig. 4, is resolvable into two parts, to wit: the part 
 CDBE, which is equal to the rectangle CDGF, 
 and the triangular part ACE, whose crest is AE, 
 and whose top is closed at C, at the level of still water. 
 Applying, in Example (27), Table 2, to the rectangular 
 part CDGF, substituted for the part CDBE, and 
 Table 4, to the triangular part ACE: 
 
 By Table 2, due one foot head, one foot crest, 3.3390 
 cubic feet. 
 
 By Table 4, due in quadrantal weir with open top, 
 12-inch head, 2.6365. 
 
 By Rule 16, 2.6365x1.5=3.9547. 
 
 Hence 3.3390+3.9547=7.2937 cubic feet. Ans. 
 
 The discrepancy between calculating 1st and 2d 
 arises from Table 4, in the computation of which 
 .616 on the authority of Prof. Thompson, was em- 
 ployed as the coefficient of discharge, instead of .624, 
 as proposed in the given example. 
 
PEACTICAL HYDRAULICS. 43 
 
 FLOW OF WATEK OVER A RECTANGULAR WEIR, HAV- 
 ING ITS ANGLES, HORIZONTAL AND VERTICAL, AND ITS 
 UPPERMOST ANGLE ON THE LEVEL OF STILL WATER. 
 
 Let ABCD, of Fig. 7, represent a rectangular 
 weir, having its vertical angle C, at the level of still 
 water. Through C, draw a horizontal line indefinitely. 
 Produce AB, AD, intersecting this horizontal line 
 in fi and F. Draw A.G=h perpendicularly to EF, 
 and bisecting A. 
 
 Let AB=m, and AD=7i. It is obvious from the im- 
 posed conditions, that AE=AF=m+7i; that EF= 
 2GE=2GF=2AG:=2X- that FD=DC=AB=m, 
 and BE=BC=AD=. 
 
 Observe that in Fig. 7 are represented three open 
 quadrantal weirs, FAE, FDC and CBE, and that the 
 given rectangle ABCD-=FAE FDG -CBE. (50) 
 
 Denote by Q u the flow of the given rectangular 
 weir. 
 
 By similar triangles, find the heads or depths as 
 follows; 
 
 ( mh ) 
 
 PD= \r\ 
 
 \m-\-n ) 
 
44 PEACTICAL HYDRAULICS. 
 
 ( nh ) 
 LB= (52) 
 
 Substituting these values, and the value of A.G~h 
 in (34), noting that for quadrantal weirs, p=2, 
 
 (53) 
 
 If in (53), the general formula for the flow of water 
 through a rectangular weir having its uppermost an- 
 gle vertical at the level of still water, we make m 
 equal n, and substitute the value of (2^)2 =8.025, and 
 c = .616, 
 
 ^=1.70435^*. (54) 
 
 In which case the rectangle becomes a square, as 
 represented in Fig. 7, by AIGN. 
 
 Comparing formula (54) with formula (35), which 
 is for the flow of water over a quad ran tal weir, we 
 have: 
 
 1.70435 
 
 To determine the flow of water through a square 
 weir, having its uppermost vertical angle at the level 
 of still water. 
 
 Rule 17. According to formula (54), multiply the 
 square root of the fifth power of the head or depth by 
 1.70435; or by formula (55), multiply the flow in 
 Table 4 for the given head by .6464. 
 
PRACTICAL HYDRAULICS. 45 
 
 Ex. 27. The head being 3 inches= J foot in a rec- 
 tangular weir, having its upper most vertical angle at 
 level of still water, what is the flow in cubic feet per 
 second? 
 
 Cal. 1st. By formula (54). 
 
 Fifth power of the square root of (J)" 2 ' = jV 
 
 1.70435XA=-- 05826 cubic feet.=4m 
 
 Cal. 2d By Rule 17, second part. 
 
 By Table 4, flow due 3-inch head =.0824. Then 
 .0824 X. 6464=. 05326 cubic feet. Ana. 
 
 Ex. 28. The sides of a rectangular weir, with its 
 angles vertical and horizontal being 2 feet and 1 foot, 
 the coefficient of discharge being .62, what is the flow 
 per second? 
 
 Cal. Employ formula (53). 
 
 Taking the given data m=2, n=l > 
 
 Then m + fi 3, and (see Fig. 7), 
 
 6.55376. 
 
 By Table 5, ml=(2)^ 
 
 By Table 5, (m+w)* = (3)^=15.590. 
 
 Substituting these values of m" 2 , ri* t 
 /J, (2#)i=8.025, and c=.62. Q, = T 8 y X- 62x8. 025 
 
 1559 
 
 Whence, Q /y -9.9644 cubic feet. Ans t 
 
46 PRACTICAL HYDRAULICS. 
 
 FLOW OF WATER THROUGH CIRCULAR AND SEMI- 
 CIRCULAR WEIRS. 
 
 Let Figs. 8, 9 and 10, represent respectively the 
 circular and semi-circular weirs, Fig. 8 touching the 
 water surface at A, Fig. 9 in a similar manner at B, 
 and Fig. 10 at its diameter CD. 
 
 Let r feet denote the radius with which the several 
 weirs are described, then in both Figs. 9 and 10 will r 
 denote the head, while in Fig. 8 the head (maximum) 
 will be 2r. 
 
 Let x in Fig. 8 denote any portion of the head, and 
 
 Q t the flow in cubic feet, (2#)" 2 ' acceleration of grav- 
 ity, and c coefficient of discharge. 
 
 Then dQ=2c(2g * 2 (r)i (lffixdx. (56) 
 
 Integrating (56) between limits of oj-=0, and x=2r, 
 and substituting the value of (20)i=8.025 t 
 
 Q,=24.2129 crl (57) 
 
PRACTICAL HYDRAULICS. 47 
 
 Let Q 2 ttie flow in Fig. 9 per second. Integrating 
 (56) between limits of x=0, and x=r, there results the 
 flow in that portion of Fig. 8 represented by FAE, 
 which is equal to HBG, Fig. 9, the discharge sought, 
 viz.: 
 
 (58) 
 
 Again in Fig. 10 : Let x denote any portion of the 
 head from A, and Q 3 the flow in cubic feet per second ; 
 Then 
 
 d Q s = (20)* (r x*fix%da>. (5 9) 
 
 Integrating (59) between limits #=0, and x=r, and 
 substituting value of (2#)i=8.025, 
 
 #3-7.6932 crl (60) 
 
 Comparing equations (60) and (35) and making 
 c=.616, and r=/i, 
 
 (61) 
 
 To find by Table 4 the flow through a semi-circular 
 weir: Open at the top as represented by CD Fig. 10. 
 
 Rule 18. Multiply the flow in Table 4 for the 
 given head or radius by 1.79. See formula (61). 
 
 The triangle CA 2 D, inscribed in the semi-circular 
 weir, Fig. 10, represents a quadrantal weir whose 
 flow is Q, while the flow of the semi-circular weir 
 CA 2 D is Q 3 . 
 
 Comparing equations (58) and (35) and making 
 c=.616, and r=h, 
 
 (62) 
 
48 PRACTICAL HYDRAULICS. 
 
 To find by Table 4, the flow through a semi-circular 
 weir closed at the top, as represented at B Fig. 9. 
 
 Rule 19. Multiply the flow in Table 4 for the 
 given head or radius by 2.1568 (62). 
 
 To find by Table 4 the flow through a circular 
 weir touching the water surface at A as represented 
 in Fig. 8. 
 
 Comparing equations (57) and (35) and making 
 c=.616, andr=fc, 
 
 $,=5.6566 Q. (63) 
 
 Rule 20. Multiply the flow in Table 4 for the head 
 or depth equal to the given radius of the circular weir 
 or opening by 5.6566. See formula (63). 
 
 Ex 29. In a semi-circular weir, with open top, as 
 represented by Fig. 10, the head or radius is ten inches. 
 What is the discharge in cubic feet per second? 
 
 Cal. By Table 4 the flow due a head of 10 inches 
 is 1.6713 cubic feet. 
 
 By Rule 18 we have 
 
 1.6713X1.79 = 2.9916 cubic feet. Ans. 
 
 Ex. 30. In a semi-circular weir or opening with 
 closed top, as represented by Fig. 9, the head or radius 
 being ten inches, what is the flow in cubic feet per 
 second ? 
 
 Gal. By Table 4 the flow due a head of ten inches 
 is 1.6713 cubic feet. 
 
 By Rule 19 there results 
 
 1.6713X2.1568=3.6047 cubic feet. Ans. 
 
 Ex. 31. In a circular weir or opening touching the 
 
PBACTICAL HYDRAULICS. 
 
 49 
 
 water surface, as at A, Fig. 8, the radius is eight inches, 
 required the cubic feet flow per second. 
 
 Gal. 1st. By Table 4 the flow due a head of eight 
 inches is .9567 cubic feet. 
 
 By Rule 20 we have 
 
 .9567X56.566=-5:4117 cubic feet. Ans. 
 
 Gal. 2d. By formula (57) : 8 inches^ f feet. 
 
 By Table 5, (f)*=^=. 3629 nearly: 
 24.2129X.616X.3629=5.4117 cubic feet. Ans. 
 
 FLOW OF WATER THROUGH PARABOLIC WEIRS. 
 
 Let Figs. 11 and 12 represent parabolic weirs, touch- 
 ing the water surface, Fig. 11, at its apex, A, and Fig. 
 12 at its inverted base, EF. 
 
 Let h, in each weir, as AD or HA, denote the head, 
 and let b denote the base, as BD, DO or EH, FH. 
 
 Let c=the coefficient of discharge. 
 
 Q 4 = the quantity discharged in cubic feet per second, 
 and (2#)i -=8.025 due gravity. 
 
 Let a^any part of the head, estimated from A, in 
 Fig. 11. 
 
50 PRACTICAL HYDBAULICS. 
 
 The equation of the parabola, in which x and y are 
 co-ordinates, is: 
 
 y*-=2px; whence y = (2px. (64) 
 
 The equation of flow is: 
 
 dQ 4 =c (20)i(2p)iadte. ' (65) 
 
 Integrating (65) between limits x=Q, and x=h; 
 and substituting the values of (2jo)i=p, (2#)i -8.025; 
 
 (66) 
 Let Q 5 the flow in weir represented by Fig. 12. 
 
 (67) 
 
 Integrating (67) between limits o?=0, and # &, and 
 substituting the values of (2p)^= and (2#)i=8.025 ; 
 
 Q 5 -3.1546 c b h*. (68) 
 
 Assume any ratio, n, to exist between the base b, 
 and the hight or head, h: 
 
 As b=nh. (69) 
 
 Then Q 5 =-3.1546 w h*. (70) 
 
 This formula is adapted to the finding of the flow 
 of water over both shallow and deep weirs. Thus by 
 
PBACTICAL HYDRAULICS. 51 
 
 making n successively equal to 1, 2, 3, 4, 5, 6, 7, etc., 
 the represented flow in (76) becomes correspondingly 
 affected. To accomplish a similar result by the semi- 
 circular weir, would be no easy task, requiring the 
 employment of a very intricate and unwieldy formula 
 or extensive table. 
 
 Making in Eq. 70, n=2, there results: 
 
 g 5 =r6.3092cfci (71) 
 
 In this case, bZk, and hence: 
 
 Comparing equations (71) and (35), and making c 
 = .616, 
 
 Q 5 =1.4734e. (72) 
 
 To FIND, BY TABLE 4, THE FLOW OF WATER OVER A 
 
 PARABOLIC WEIR, WITH AN OPEN TOP THE WIDTH BE- 
 ING EQUAL TO TWICE THE DEPTH OR HEAD. 
 
 Rule 21. Multiply the flow in Table 4 for the head 
 or depth, equal the given head, by 1.4734. See for- 
 mula (72). 
 
 Ex. 32. In a parabolic weir, with open top, the 
 head is 11 inches, and the width 22 inches. What is 
 the discharge of water through it in cubic feet per 
 second ? 
 
 Gal. by Table 4. Flow corresponding to head of 11 
 inches, 2.1696 cubic feet. 
 Then by Rule 21: 
 
52 PRACTICAL HYDRAULICS. 
 
 2.1696X1.4734=3.1967 cubic te&t.Ans. 
 Comparing equations (70J and (35), and making 
 c-:.6l6, 
 
 (73) 
 
 To FIND, BY TABLE 4, THE FLOW OF WATER OVER A 
 
 PARABOLIC WEIR, WITH OPEN TOP THE WIDTH BEING A 
 GIVEN NUMBER, n TIMES, THE HEAD OR DEPTH. 
 
 Rule 22. Multiply the flow in Table 4, for the 
 head or depth, equal to the given head, by .73705 
 times the ratio between the given head and width. 
 See formula (73). 
 
 Ex. 33. In an open parabolic weir, the head being 
 10 inches, and the width 50 inches that is, 5 times 10 
 inches (71=0), required the cubic feet flow per second. 
 
 CalBy Table 4, flow due 10 inches, 1.6713. 
 
 By Rule 22. 1.6713X .73705X 5=6.1592 cubic 
 feet. Ans. 
 
 Comparing equations (66) and (35), making b=nh, 
 as in (69), and c=. 616, 
 
 (74) 
 
 TO FIND THE FLOW OF WATER THROUGH A PARABOLIC 
 WEIR WHOSE APEX, A, FlG. 11, IS AT THE LEVEL OF 
 STILL WATER. 
 
 Rule 23. Multiply the flow in Table 4, due the 
 head or depth equal to the given head, by .9375 times 
 the ratio, n, between the given head and width. 
 
PRACTICAL HYDRAULICS. 53 
 
 Ex. 34. In a parabolic opening or weir, whose 
 apex reaches the surface of still water, the head or 
 depth being 23 inches, and the width 230 inches 
 that is, 10 times 23 inches (n~LQ), required the flow 
 in cubic feet per second. 
 
 Col. By Table 4, flow due 23 inches, 13.4089. 
 
 By Rule 23, 13.4089X .9375X 10 =125.71 cubic 
 feet. Ans. 
 
 FLOW OF WATER THROUGH A SUBMERGED TRIANGULAR 
 OPENING, HAVING ITS VERTEX BELOW TE BASE. 
 
 Let, in Fig. 13, BCD represent the opening, in 
 which 6=BD, the width at top; h= AE, head on the 
 top of orifice; h t AC, head on its bottom; c=coefti- 
 
 cient of discharge; (2^f)^=acceleration of gravity; 
 Q=discharge in cubic feet per second; &=any part of 
 EC, estimated from E; a=EC, depth of opening. 
 
 Then dQ=c (2g)*(h+xdx. (75) 
 
54 PRACTICAL HYDRAULICS. 
 
 Integrating (75) between the limits x=Q, and x 
 h,h, 
 
 If in (76), we make h=Q, and b=2a, there results: 
 
 *. (77) 
 
 Equation (77), derived from the general equation 
 (76), is seen to be identical with equation (35), for 
 the flow of water over a quadrantal weir. 
 
 In equation (76), denote the ratio between the head 
 on the bottom and the head on the top of the triangu- 
 
 lar opening by m; thus -=m, and substitute the 
 
 h, 
 
 value of (2#)i=8.025. 
 
 l (78J 
 
 To FIND THE FLOW OF WATER THROUGH A SUBMERGED 
 TRIANGULAR ORIFICE, HAVING ITS VERTEX BELOW THE 
 BASE. 
 
 Rule 24. From J, subtract \ of the ratio of the 
 given heads on the bottom and top of the orifice, and 
 multiply this difference by the cube of the square root 
 of the same ratio; subtract the product from T 2 T ; 
 
PBACTIOAL HYDKAULICS. 55 
 
 multiply the remainder by 16.05 times the product of 
 the ratio between the depth and width of the orifice, 
 the fifth power of the square root of the head on the 
 bottom, and the coefficient of discharge. 
 
 Ex. 35. In a submerged triangular orifice, repre- 
 sented by Fig. 13, the head, AC, on the bottom =h = 
 2.25 feet; the head, AE, on the iop=h=l foot; the 
 width, BD=6=5 feet; the depth EC=a=1.25 feet; 
 and the coefficient of discharge c .616. What is the 
 flow in cubic feet per second? 
 
 Gal 1st. By formula (78) and Rule 24, derived 
 therefrom : 
 
 Ratio of heads, ^=m=- s ^ v =%', 
 
 Difference (H?)=U; 
 
 Cube of square root of ratio, m~2 (f)" 2 / T . 
 
 Product of this difference, and the cube of the 
 square root of the ratio of the given heads, 
 
 ITTT- 
 Difference, -^1^%- 
 
 Q=16.05X .616X T .VT X ^ X W = 18 - 29 cubic 
 feet. Ans. 
 
 Gal. 2d. Assuming that the effective head is the 
 mean of the given heads on the top and bottom of the 
 orifice, then will the velocity be as per equation (25) 
 or Rule (8). 
 
 ^.616X8.025 (*i+*)i=(ii|i)i=6.3 feet per sec- 
 ond. 
 
 Area orifice=5X 1.25-^2=3.125 square feet. 
 
56 PRACTICAL HYDBAULICS. 
 
 Discharge equal to the product of the velocity and 
 the area of the orifice, Q=6.3x 3.125=19.69 cubic 
 feet per second. 
 
 Gal. 3. Assume that the true head is on the center 
 of gravity of the opening geometrically considered, 
 la a triangle, the center of gravity is at the inter- 
 section of right lines drawn from any two angles and 
 Usecting the opposite sides. Its distance, estimated, 
 from an angle, is equal to two-thirds the length of the 
 bisecting line; or estimated from the middle of a side, 
 is equal to one-third the length of the bisecting line. 
 
 In Fig. 13, AG=//=+f=l + i-|l==1.4167. 
 
 By formula (13), modified by coefficient ^=.616= 
 8.025 (1.4167)i=5.8836 feet per second, area of ori- 
 fice, 5X1.25-^2=3.125 square feet. 
 
 Discharge equal to the product of the velocity and 
 the area of orifice, Q=5.8836X3.125=18.39 cubic 
 feet per second. 
 
 Comparing these results, it is seen 'that the second is 
 seven and six-tenths per cent (.076) too great, and the 
 third fifty-four one hundredth of one per cent (.0054) 
 too great. 
 
 The rule generally adopted is, that "in all cases when 
 the center of gravity of an orifice lies at least as deep 
 below the fluid surface as the figure is high," the 
 depth h t (that is, the depth at the center of gravity), 
 of this point may be regarded the head of water. 
 This rule may approximate the truth sufficiently close 
 for ordinary practice, but is not to be employed when 
 a high degree of accuracy is required. 
 
PRACTICAL HYDRAULICS, 
 
 57 
 
 FLOW OF WATER THROUGH A SUBMERGED TRIANGULAR 
 ORIFICE HAVING ITS VERTEX ABOVE THE BASE. 
 
 Let, in Fig. 14, BCD represent the opening, in 
 which fr=BD, the width at bottom; h=A.C, head or 
 vertex; h J =A'E, head or bottom; a=EC, depth of 
 
 opening; (2(/^)=acceleration of gravity; c=coefficient 
 of discharge; Q flow in cubic feet per second; x= 
 any part of a EC, estimated from C : 
 
 (79) 
 
 Then dQ=c- -(h+x)*x dx. 
 
 Integrating (79J between the limits x=0, and x=a 
 
 (80) 
 
 Denote the ratio between the head on the bottom 
 and that on the vertex by m: 
 
 h 
 
 m=-; h=mh r 
 
58 PRACTICAL HYDRAULICS. 
 
 Substitute the value of h in (80), and the value of 
 
 (81) 
 
 a . 15 
 
 If in the general equation (80), we make h=0, and 
 b =2 a, there results: 
 
 Q=*c(2g)*h?, (82) 
 
 which is identical with formula (48) for the flow of 
 water through a quadrantal weir having its apex at 
 the water's surface. 
 
 To FIND THE FLOW OF WATER THROUGH A SUBMERGED 
 TRIANGULAR ORIFICE HAVING ITS VERTEX ABOVE THE 
 BASE. 
 
 Rule 25. From subtract J of the ratio of the 
 head on the bottom to that on the apex of the orifice ; 
 add this difference to the T 2 7 part of the fifth power of 
 the square root of the same ratio; multiply this sum 
 by 16.. 05 times the product of the ratio between the 
 depth and width of the orifice, and the fifth power of 
 the square root of the head on the bottom. Derived 
 from formula (81). 
 
 Ex. 36. In a submerged triangular orifice, repre- 
 sented by Fig. 14, the head, AC, on the apex=/i=l 
 
PBACTICAL HYDRAULICS. 59 
 
 foot ; the head, AE, on the bottom=& / 2.25 ; the 
 width, BD=6=5 feet; the depth,. CE=a=1.25; and 
 the coefficient of discharge, c=.616. What is the flow, 
 Q t , in cubic feet per second? 
 
 Gal 1st. By Rule 25, or formula (81). 
 
 Ratio of head on bottom to that on apex, ra= 
 
 h __ l _4 
 TT- 
 
 Difference (f- iX|)= 
 
 Sum 
 
 Q = 16.05 X T.VB- X .616 X ATT X W - 20. 84 cubic 
 feet per second. 
 
 Cal. 2d. Assuming that the effective head is the 
 mean of the given heads on the apex and bottom of 
 the orifice, then will the flow be the same as found by 
 Cal. 2d, for Ex. 35, viz., Q=19.69 cubic feet per 
 second. 
 
 Cal. 3d. Assume that the true head is on the cen- 
 ter of gravity of the opening, geometrically con- 
 sidered. 
 
 The center of gravity, as shown in Fig. 14, is at 
 the intersection of CE and DF; DF bisecting BC in F. 
 
 CG:CE::2:3; CG=^; 
 
 AG=AC + CG='=1 + fxl.25=1.8334. 
 
 Area of opening=5X 1.25-^-2=3. 125. 
 
 Q=.616 X 8.025 X (1 .8334)2 x 3. 125 = 20.92 cubic 
 feet flow per second. 
 
 By inspection it is seen that the result by Cal. 2d 
 
60 
 
 PKACTICAL HYDRAULICS. 
 
 is five and one-half (.055) of one per cent too small, 
 and the result by Gal. 3d is nearly four-tenths (.0039) 
 of one per cent too large. Neither of these empirical 
 methods then would satisfy the requirements of any 
 considerable accuracy. 
 
 FLOW OF WATER THROUGH A SUBMERGED CIRCULAR 
 ORIFICE. 
 
 A A 
 
 FIG. 15. 
 
 Let, in Fig. 15, EDH represent a submerged circu- 
 lar orifice ; hAC, the head on center ; n angle 
 DOE; r=CD=CE,the radius; A=area; c=coefficient 
 of discharge; ( 2g) 2 = acceleration of gravity; Q=flow 
 in cubic feet; #=any part of r, estimated from C. 
 Then, in general, 
 
 A=r 2 x\ (83) 
 
 Differential (83), dA=2xdx. (84) 
 
 Head at any point, B A y B=A x cos n. (85) 
 
PRACTICAL HYDEAULICS. 61 
 
 dQ= 2cfi(2^)^i x (l ~^dx. (86) 
 
 Integrating (86) between the limits x~0, and x=v: 
 
 r3 cos 3ra 5rt cos 4?i 
 40/i3 384/1* 
 
 etc.) (87) 
 
 Observing that the sum of cosines of a complete 
 circle 2n is=0, and substituting the value of (2g) 2 = 
 8.025, and n=3.l416 in (87): 
 
 (88) 
 
 Making in (88), h=r, Q=-24.2129c r*, (89) 
 
 which is identical with (57). 
 
 FLOW OF WATER THROUGH SEMI-CIRCULAR ORIFICES. 
 
 Let ii 2 the mean of all the cosines of the first quad- 
 rant, and n 2 =the mean of all cosines of the second 
 quadrant; then will the mean of the first and second 
 quadrants vanish. 
 
 To determine the flow of water through the upper 
 semi-circle of a submerged circular orifice, substitute 
 in one-half of (87), the mean value of cos 7i=ii 2 . 
 
 Q=12.6056 C rl-----etc. (90) 
 
62 PKACTICAL HYDRAULICS. 
 
 To determine the flow of water through the lower 
 semi-circle of a submerged circular orifice, substitute 
 in one-half of (87), the mean value of cos n= n 2 . 
 
 Q=12.6056c AV (l + J^ + ^-^+etc.) (91 
 Ex. 37. Th? radius of a circular opening is one 
 
 foot, the head on the center of the opening four feet, 
 
 and the coefficient of discharge .616. What is the 
 
 flow in cubic feet per second? 
 
 Gal 1st. Substituting the values of r, h and c in 
 
 formula (88), 
 
 C=25.2113X .616X 2 (1- 
 
 105 
 
 (TnrxT* = -0019531 \ 
 
 =.0000191 V 
 = _. 0000004 1 =.0019726. 
 
 t 
 
 1_.0019726=.9980274. 
 Q=i25.2113X.616X2X. 9980274=31. 00 cubic feet. 
 
 . 2d. Assume that the true head is on the cen- 
 ter of gravity of the opening, geometrically con- 
 sidered; then will the discharge be: 
 
 Q=.616X8.025X2X3.14l6=31.06 cubic feet. 
 Ans. 
 
 Ex. 38. The radius of a circular opening is one 
 foot, the head on the center of the opening four feet, 
 and the coefficient of discharge .616. What is the 
 flow in cubic feet per second in the upper semi-circle? 
 
PRACTICAL HYDRAULICS. 63 
 
 Gal. Substituting the values of r, h and c and ii= 
 3.1416 in formula (90). 
 
 =-0530516. 
 =.0019531. 
 .0001245. 
 =.0000020j =.0551312. 
 
 Q'=12. 6056 X. 616 X 2 X. 94487=14. 6739 cubic feet. 
 Ans. 
 
 Ex. 39. The radius of a circular opening is one 
 foot, the head on the center of the opening four feet, 
 and the coefficient of discharge .616. What is the 
 flow in cubic feet per second in the lower semi-circle? 
 
 Gal. Substituting the values of r, h, c and n= 
 3.1416 in formula (91 J, 
 
 + 1. 1.0000000 
 
 + 12x3 2 1416 =+ .0530516 
 
 .0019531 
 
 + ^6o7ki4r6 == + .0001245 
 
 -1024^256 ~.0000020y 
 
 .1.0531761 -.0019551=1.051221. 
 
 Q"=12.6056X. 616X2X1.051221=16.3254 cubic 
 feet. Ans. 
 
64 PRACTICAL HYDRAULICS. 
 
 The value of Q" might have been readily found as 
 follows: 
 
 Q"=Q Q'=31.0 14.6739=16.326. 
 
 It will be noted that formula (91) expressed the dis- 
 charge of water through the lower semi-circle of a 
 submerged lateral orifice, in which the head is on the 
 center ; whereas formula (60) expresses the discharge 
 of water through a semi-circular weir, represented by 
 Fig. 10, in which the head OA is on the bottom, care 
 will need be taken that these formulas be not con- 
 founded. This fact is rendered more apparent by 
 making in (91), h=r, when there results approxi- 
 mately : 
 
 While with respect to formula (60), 
 Q s =7.693cr*. 
 
PKACTICAL HYDRAULICS. 
 
 65 
 
 TABLE 5. 
 
 Square Boots, Cubes of Square Roots, and Fifth Powers of 
 Square Roots of Numbers. 
 
 No. 
 
 Square 
 Roots. 
 
 Cubes of 
 Square 
 Roots. 
 
 5th Power 
 of Square 
 Roots. 
 
 No. 
 
 Square 
 Roots. 
 
 Cubes 
 of Square 
 Roots. 
 
 5th Power 
 of Square 
 Roots. 
 
 .1 
 
 .316 
 
 .032 
 
 .003 
 
 1. 
 
 1.000 
 
 1.000 
 
 1.000 
 
 .125 
 
 .353 
 
 .044 
 
 .006 
 
 1.25 
 
 1.118 
 
 1.397 
 
 1.747 
 
 .15 
 
 .387 
 
 .058 
 
 .009 
 
 1.50 
 
 1.225 
 
 1.837 
 
 2.756 
 
 .175 
 
 .418 
 
 .073 
 
 .013 
 
 1.75 
 
 1.304 
 
 2.315 
 
 4.051 
 
 .2 
 
 .447 
 
 .089 
 
 .018 
 
 2. 
 
 1.414 
 
 2.822 
 
 5.657 
 
 .225 
 
 .474 
 
 .107 
 
 .024 
 
 2.25 
 
 .500 
 
 3.375 
 
 7.594 
 
 .25 
 
 .500 
 
 .125 
 
 .031 
 
 2.50 
 
 .581 
 
 3.953 
 
 9.882 
 
 .275 
 
 .524 
 
 .144 
 
 .040 
 
 2.75 
 
 .658 
 
 4.560 
 
 12.541 
 
 .3 
 
 .548 
 
 .164 
 
 .049 
 
 3. 
 
 .732 
 
 5.196 
 
 15.590 
 
 .325' 
 
 .570 
 
 .185 
 
 .060 
 
 3.25 
 
 .803 
 
 5.869 
 
 19.041 
 
 .35 
 
 .592 
 
 .207 
 
 .072 
 
 3.50 
 
 .871 
 
 6.547 
 
 22.918 
 
 .375 
 
 .612 
 
 .230 
 
 .086 
 
 3.75 
 
 .936 
 
 7.262 
 
 27.232 
 
 .4 
 
 .633 
 
 .253 
 
 .101 
 
 4. 
 
 2.000 
 
 8.000 
 
 32.000 
 
 .425 
 
 .652 
 
 .277 
 
 .118 
 
 4.25 
 
 2.061 
 
 8.761 
 
 37.236 
 
 .45 
 
 .671 
 
 .302 
 
 .136 
 
 4.5 
 
 2.121 
 
 9.546 
 
 42.957 
 
 .475 
 
 .689 
 
 .327 
 
 .156 
 
 4.75 
 
 2.179 
 
 10.352 
 
 49.174 
 
 .5 
 
 .707 
 
 .353 
 
 .177 
 
 5. 
 
 2.236 
 
 11.180 
 
 55.901 
 
 .525 
 
 .724 
 
 .380 
 
 .200 
 
 5.25 
 
 2.291 
 
 12.029 
 
 63.153 
 
 .55 
 
 .742 
 
 .408 
 
 .224 
 
 5.5 
 
 2.345 
 
 12.898 
 
 70.942 
 
 .575 
 
 .758 
 
 .436 
 
 .251 
 
 5.75 
 
 2.398 
 
 13.783 
 
 79.281 
 
 .6 
 
 .775 
 
 .465 
 
 .279 
 
 6. 
 
 2.449 
 
 14.697 
 
 88.181 
 
 .625 
 
 .791 
 
 .494 
 
 .309 
 
 6.25 
 
 2.500 
 
 15.625 
 
 97.656 
 
 .650 
 
 .806 
 
 .524 
 
 .341 
 
 6.5 
 
 2.550 
 
 16.572 
 
 107.71 
 
 .675 
 
 .821 
 
 .555 
 
 .374 
 
 6.75 
 
 2.598 
 
 17.537 
 
 118.37 
 
 .7 
 
 .837 
 
 .586 
 
 .410 
 
 7. 
 
 2.646 
 
 18.520 
 
 129.64 
 
 .725 
 
 .852 
 
 .617 
 
 .448 
 
 7.5 
 
 2.739 
 
 20.539 
 
 154.04 
 
 .75 
 
 .866 
 
 .650 
 
 .486 
 
 8. 
 
 2.828 
 
 22.627 
 
 181.02 
 
 .775 
 
 .880 
 
 .682 
 
 .529 
 
 8.50 
 
 2.915 
 
 24.781 
 
 210.64 
 
 .8 
 
 .894 
 
 .715 
 
 .572 
 
 9. 
 
 3.000 
 
 27.000 
 
 243.00 
 
 .825 
 
 .908 
 
 .749 
 
 .618 
 
 10. 
 
 3.162 
 
 31.623 
 
 316.23 
 
 .85 
 
 .922 
 
 .784 
 
 .666 
 
 11. 
 
 3.317 
 
 36.483 
 
 401.31 
 
 .875 
 
 .936 
 
 .818 
 
 .717 
 
 12. 
 
 3.464 
 
 41.569 
 
 498.83 
 
 .9 
 
 .949 
 
 .854 
 
 .768 
 
 13. 
 
 3.606 
 
 46.872 
 
 609.33 
 
 .925 
 
 .956 
 
 .890 
 
 .823 
 
 14. 
 
 3.742 
 
 52.383 
 
 733.36 
 
 950 
 
 .975 
 
 .926 
 
 .858 
 
 15. 
 
 3.873 
 
 58.094 
 
 871.41 
 
 .975 
 
 .987 
 
 .963 
 
 .939 
 
 16. 
 
 4. 
 
 64.000 
 
 1024.0 
 
66 
 
 PBACTICAL HYDRAULICS. 
 
 TABLE 6. 
 
 Square Boots of Numbers. 
 
 No. 
 
 Square 
 Roots. 
 
 No. 
 
 Square 
 Roots. 
 
 No. 
 
 Square 
 Roots. 
 
 No. 
 
 1 Square 
 j Roots. 
 
 .00 
 
 1.000 
 
 8. 
 
 2.828 
 
 19. 
 
 4.359 
 
 95. 
 
 9.747 
 
 .05 
 
 1.025 
 
 8.1 
 
 2.846 
 
 19.2 
 
 4.382 
 
 96. 
 
 9.798 
 
 .1 
 
 1.049 
 
 8.2 
 
 2.864 
 
 19.4 
 
 4.405 
 
 97. 
 
 9.849 
 
 .15 
 
 1.072 
 
 8.3 
 
 2.881 
 
 19.6 
 
 4.427 
 
 98. 
 
 9.899 
 
 .2 
 
 1.095 
 
 8.4 
 
 2.898 
 
 19.8 
 
 4.45.0 
 
 99. 
 
 9.950 
 
 .25 
 
 1.118 
 
 8.5 
 
 2.915 
 
 20. 
 
 4.472 
 
 100. 
 
 10.000 
 
 .3 
 
 1.140 
 
 8.6 
 
 2.933 
 
 21. 
 
 4.583 
 
 102. 
 
 10.100 
 
 .35 
 
 1.162 
 
 8.7 
 
 2.950 
 
 22. 
 
 4.690 
 
 104. 
 
 10 198 
 
 .4 
 
 1.183 
 
 8.8 
 
 2.966 
 
 23. 
 
 4.796 
 
 106. 
 
 10.295 
 
 .45 
 
 1.204 
 
 8.9 
 
 2.983 
 
 24. 
 
 4.899 
 
 108. 
 
 10.392 
 
 .5 
 
 1.225 
 
 9. 
 
 3. 
 
 25. 
 
 5.000 
 
 110. 
 
 10.488 
 
 .55 
 
 1.245 
 
 9.1 
 
 3.017 
 
 26. 
 
 5.099 
 
 112. 
 
 10.583 
 
 .6 
 
 1.265 
 
 9.2 
 
 3.033 
 
 27. 
 
 5.196 
 
 114. 
 
 10.677 
 
 .65 
 
 1.285 
 
 9.3 
 
 3.050 
 
 28. 
 
 5.292 
 
 116. 
 
 10.770 
 
 .7 
 
 1.304 
 
 - 9.4 
 
 3.066 
 
 29. 
 
 5.385 
 
 118. 
 
 10.863 
 
 .75 
 
 1.323 
 
 9.5 
 
 3.082 
 
 30. 
 
 5.477 
 
 120. 
 
 10.954 
 
 .8 
 
 1.342 
 
 9.6 
 
 3.098 
 
 31. 
 
 5.568 
 
 122. 
 
 11.045 
 
 1.85 
 
 1.360 
 
 9.7 
 
 3.114 
 
 32. 
 
 5.657 
 
 124. 
 
 11.136 
 
 1.9 
 
 1.378 
 
 9.8 
 
 3.130 
 
 33. 
 
 5.745 
 
 126. 
 
 11.225 
 
 1.95 
 
 1.396 
 
 9.9 
 
 3.146 
 
 34. 
 
 5.831 
 
 128. 
 
 11.314 
 
 2. 
 
 1.414 
 
 10. 
 
 3.162 
 
 35. 
 
 5.916 
 
 130. 
 
 11.402 
 
 2.1 
 
 1.449 
 
 10.1 
 
 3.178 
 
 36. 
 
 6.000 
 
 132. 
 
 11.489 
 
 2.2 
 
 1.483 
 
 10.2 
 
 3.194 
 
 37. 
 
 6.083 
 
 134. 
 
 11.576 
 
 2.3 
 
 1.517 
 
 10.3 
 
 3.209 
 
 38. 
 
 6.164 
 
 136. 
 
 11.662 
 
 2.4 
 
 1.549 
 
 10.4 
 
 3.225 
 
 39. 
 
 6.245 
 
 138. 
 
 11.747 
 
 2.5 
 
 1.581 
 
 10.5 
 
 3.240 
 
 40. 
 
 6.325 
 
 140. 
 
 11.832 
 
 2.6 
 
 1.612 
 
 10.6 
 
 3.256 
 
 41. 
 
 6.403 
 
 142. 
 
 11.916 
 
 2.7 
 
 1.643 
 
 10.7 
 
 3.271 
 
 42. 
 
 6481 
 
 144. 
 
 12000 
 
 2.8 
 
 1.673 
 
 10.8 
 
 3.286 
 
 43. 
 
 6.557 
 
 146. 
 
 12.083 
 
 2.9 
 
 1.703 
 
 10.9 
 
 3.302 
 
 44. 
 
 6.333 
 
 148. 
 
 12.166 
 
 3. 
 
 1.732 
 
 11. 
 
 3.317 
 
 45. 
 
 6.708 
 
 150. 
 
 12.247 
 
 3.1 
 
 1 761 
 
 11.1 
 
 3.332 
 
 46. 
 
 6.782 
 
 155. 
 
 12.450 
 
 3.2 
 
 1.789 
 
 11.2 
 
 3.347 
 
 47. 
 
 6856 
 
 160. 
 
 12.649 
 
 3.3 
 
 1.817 
 
 11.3 
 
 3362 
 
 48. 
 
 6.928 
 
 165. 
 
 12.845 
 
 3.4 
 
 1.844 
 
 11.4 
 
 3.376 
 
 49. 
 
 7.000 
 
 170. 
 
 13.038 
 
 3.5 
 
 1.871 
 
 11 5 
 
 3.391 
 
 50. 
 
 7.071 
 
 175. 
 
 13.229 
 
 3.6 
 
 1.897 
 
 11.6 
 
 3 406 
 
 51. 
 
 7.141 
 
 180. 
 
 13417 
 
 3.7 
 
 1.924 
 
 11.7 
 
 3.421 
 
 52. 
 
 7.211 
 
 185. 
 
 13.601 
 
 3.8 
 
 1.949 
 
 11.8 
 
 3.435 , 
 
 53. 
 
 7.280 
 
 190. 
 
 13.784 
 
 3.9 
 
 1.975 
 
 11.9 
 
 3.450 
 
 54. 
 
 7.348 
 
 195. 
 
 13.964 
 
PKACTICAL HYDRAULICS. 
 
 67 
 
 TABLE 6. CONTINUED. 
 
 Square Roots of Numbers. 
 
 No. 
 
 Square 
 Roots. 
 
 No. 
 
 Square 
 Roots. 
 
 No. 
 
 Square 
 Roots. 
 
 I Square 
 Roots. 
 
 No. 
 
 4. 
 
 2.000 
 
 12. 
 
 3.464 
 
 55. 
 
 7.416 
 
 200. 
 
 14.142 
 
 4.1 
 
 2.025 
 
 12.1 
 
 3.479 
 
 56. 
 
 7.483 
 
 205. 
 
 14.318 
 
 4.2 
 
 2.049 
 
 12.2 
 
 3.493 
 
 57. 
 
 7.550 
 
 210. 
 
 14.491 
 
 43 
 
 2.074 
 
 12.3 
 
 3.507 
 
 58. 
 
 7.616 
 
 215. 
 
 14.663 
 
 4.4 
 
 2098 
 
 12.4 
 
 3.521 
 
 59. 
 
 7.681 
 
 220. 
 
 14.832 
 
 4.5 
 
 2.121 
 
 12.5 
 
 3.536 
 
 60. 
 
 7.746 
 
 225. 
 
 15.000 
 
 4.6 
 
 2 145 
 
 12.6 
 
 3.550 
 
 61. 
 
 7.810 
 
 230. 
 
 15.166 
 
 4.7 
 
 2.168 
 
 12.7 
 
 3.564 
 
 62. 
 
 7.874 
 
 235. 
 
 15.330 
 
 4.8 
 
 2.191 
 
 12.8 
 
 3.578 
 
 63. 
 
 7.937 
 
 240. 
 
 15.492 
 
 4.9 
 
 2.214 
 
 12.9 
 
 3.592 
 
 64. 
 
 8.000 
 
 245. 
 
 15.652 
 
 5. 
 
 2.236 
 
 13. 
 
 3.606 
 
 65. 
 
 8.062 
 
 250. 
 
 15.811 
 
 5.1 
 
 2.258 
 
 13.2 
 
 3.633 
 
 66. 
 
 8.124 
 
 260. 
 
 16.125 
 
 5.2 
 
 2.280 
 
 13.4 
 
 3.661 
 
 67. 
 
 8.185 
 
 270. 
 
 16.432 
 
 5.3 
 
 2.302 
 
 13.6 
 
 3.688 
 
 68. 
 
 8.246 
 
 280. 
 
 16.733 
 
 5.4 
 
 2.324 
 
 13.8 
 
 3.715 
 
 69. 
 
 8.307 
 
 290. 
 
 17.029 
 
 5.5 
 
 2.345 
 
 1 14. 
 
 3.742 
 
 70. 
 
 8.367 
 
 300. 
 
 17.320 
 
 5.6 
 
 2.366 
 
 14.2 
 
 3.768 
 
 71. 
 
 8.426 
 
 310. 
 
 17.607 
 
 5.7 
 
 2.387 
 
 14,4 
 
 3.795 
 
 72. 
 
 8.485 
 
 320. 
 
 17.889 
 
 5.8 
 
 2.408 
 
 14.6 
 
 3.821 
 
 73. 
 
 8.544 
 
 330. 
 
 18.166 
 
 5.9 
 
 2.429 
 
 14.8 
 
 3.847 
 
 74. 
 
 8.602 
 
 340. 
 
 18.439 
 
 6. 
 
 2.449 
 
 15. 
 
 3.873 
 
 75. 
 
 8.660 
 
 350. 
 
 18.708 
 
 6.1 
 
 2.470 
 
 15.2 
 
 3.899 
 
 76. 
 
 8.718 
 
 360. 
 
 18.974 
 
 6.2 
 
 2.490 
 
 15.4 
 
 3.924 
 
 77. 
 
 8.775 
 
 370. 
 
 19.235 
 
 6.3 
 
 2.510 
 
 1 15.6 
 
 3.950 
 
 78. 
 
 8.832 
 
 380. 
 
 19.494 
 
 6.4 
 
 2.530 
 
 15.8 
 
 3.975 
 
 79. 
 
 8.888 
 
 390. 
 
 19.748 
 
 6.5 
 
 2.550 
 
 16. 
 
 4.000 
 
 80. 
 
 8.944 
 
 400. 
 
 20.000 
 
 6.6 
 
 2.569 
 
 16.2 
 
 4.025 
 
 81. 
 
 9.000 
 
 410. 
 
 20.248 
 
 6.7 
 
 2.588 
 
 16.4 
 
 4,050 
 
 82. 
 
 9.055 
 
 420. 
 
 20.494 
 
 6.8 
 
 2.608 
 
 16.6 
 
 4.074 
 
 83. 
 
 9.110 
 
 430. 
 
 20.736 
 
 6.9 
 
 2.627 
 
 16.8 
 
 4.099 
 
 84. 
 
 9.165 
 
 440. 
 
 20.976 
 
 7. 
 
 2.646 
 
 17. 
 
 4.123 
 
 85. 
 
 9.220 
 
 450. 
 
 21.213 
 
 7.1 
 
 2.665 
 
 17.2 
 
 4.147 
 
 86. 
 
 9.274 
 
 460. 
 
 21.448 
 
 7.2 
 
 2.683 
 
 17.4 
 
 4.171 
 
 87. 
 
 9.327 
 
 470. 
 
 21.679 
 
 7.3 
 
 2.702 
 
 17.6 
 
 4.195 
 
 88. 
 
 9.380 
 
 480. 
 
 21.909 
 
 7.4 
 
 2.720 
 
 17.8 
 
 4.219 
 
 89. 
 
 9.434 
 
 490. 
 
 22.136 
 
 7.5 
 
 2.739 
 
 18. 
 
 4.243 
 
 90. 
 
 9.487 
 
 500. 
 
 22.361 
 
 7.6 
 
 2.757 
 
 18.2 
 
 4266 
 
 91. 
 
 9.539 
 
 525. 
 
 22.913 
 
 7.7 
 
 2.775 
 
 18.4 
 
 4.290 
 
 92. 
 
 9.592 
 
 550. 
 
 23.452 
 
 7.8 
 
 2.793 
 
 18.6 
 
 4.313 
 
 93. 
 
 9.644 
 
 575. 
 
 23.979 
 
 7.9 
 
 2.811 18.8 
 
 4.336 
 
 94. 
 
 9.695 
 
 600. 
 
 24.495 
 
68 PRACTICAL HYDRAULICS. 
 
 FLOW OF WATER THROUGH VERTICAL, RECTANGULAR 
 ORIFICES IN THIN PARTITIONS. 
 
 The assumption that the mean velocity of a stream 
 of water flowing through a vertical rectangular ori- 
 fice is at the middle of the opening, has been shown 
 by equation (22) to be not strictly true. But, owing 
 to its simplicity of application, and its close approxi- 
 mation to the truth, hydraulicians, for the most part, 
 are wont to adopt it, and to correct the error involved 
 by coefficients obtained by experiment. 
 
 Table 7, derived from Fanning's Hydraulic En- 
 gineering, embraces a w T ide range of coefficients so 
 determined. Thus, it is suited to heads of water 
 from two-tenths (.2) of a foot to fifty (50) feet, and 
 to orifices one foot wide, whose hights vary from four 
 (4) feet to one and one-half (1J) inches. 
 
PRACTICAL HYDRAULICS. 
 
 69 
 
 TABLE 7. 
 
 Flow of Water per second through rectangular orifices in thin 
 
 vertical partitions, and the coefficients employed 
 
 in the computation. 
 
 Head 
 on 
 Center. 
 
 Coeffi- 
 cient. 
 
 4 feet 
 high; 1 
 ft. wide 
 
 Coeffi- 
 cient. 
 
 2 feet 
 high; 1 
 ft. wide. 
 
 Coeffi- 
 cient. 
 
 1 feet 
 high ; 1 
 ft. wide 
 
 Coeffi- 
 cient. 
 
 Ifoot 
 high; 1 
 ft. wide 
 
 0.6 
 
 
 
 
 
 
 
 .598 
 
 3.72 
 
 0.7 
 
 
 
 
 
 
 
 .599 
 
 4.02 
 
 0.8 
 
 
 
 
 
 .613 
 
 6.60 
 
 .600 
 
 4.31 
 
 0.9 
 
 
 
 
 
 .613 
 
 7.01 
 
 .601 
 
 4.57 
 
 1.0 
 
 
 
 
 
 .614 
 
 7 39 
 
 .601 
 
 4.87 
 
 1.25 
 
 
 
 .619 
 
 11.11 
 
 .614 
 
 8.26 
 
 .602 
 
 5.29 
 
 1.50 
 
 
 
 .619 
 
 12.16 
 
 .614 
 
 9.06 
 
 .603 
 
 5.92 
 
 1.75 
 
 
 
 .619 
 
 13.13 
 
 .615 
 
 9.79 
 
 .603 
 
 6.40 
 
 2.00 
 
 
 
 .618 
 
 14,04 
 
 .614 
 
 10.45 
 
 .604 
 
 6.85 
 
 2.25 
 
 
 
 .618 
 
 14.89 
 
 .614 
 
 10.96 
 
 .604 
 
 7.27 
 
 2.50 
 
 .629 
 
 31.92 
 
 .618 
 
 15.67 
 
 .614 
 
 11.66 
 
 .604 
 
 7.67 
 
 2.75 
 
 .628 
 
 33.43 
 
 .617 
 
 16.43 
 
 .614 
 
 12.24 
 
 .605 
 
 8.05 
 
 3.00 
 
 .627 
 
 34.75 
 
 .617 
 
 17.15 
 
 .613 
 
 12.78 
 
 .605 
 
 8.41 
 
 3.50 
 
 .625 
 
 37.54 
 
 .616 
 
 18.49 
 
 .612 
 
 13.79 
 
 .605 
 
 9.08 
 
 4.00 
 
 .625 
 
 40.09 
 
 .615 
 
 19.74 
 
 .611 
 
 14.71 
 
 .605 
 
 9.97 
 
 4.50 
 
 .623 
 
 42.39 
 
 .614 
 
 20.90 
 
 .610 
 
 15.58 
 
 .604 
 
 10.29 
 
 5.00 
 
 .621 
 
 44.55 
 
 .612 
 
 21.98 
 
 .609 
 
 16.48 
 
 .604 
 
 10.84 
 
 6.00 
 
 .616 
 
 48.42 
 
 .609 
 
 23.96 
 
 .606 
 
 17.88 
 
 .602 
 
 11.84 
 
 7.00 
 
 .612 
 
 52.23 
 
 .606 
 
 25.75 
 
 .604 
 
 19.23 
 
 .601 
 
 12.76 
 
 8.00 
 
 .609 
 
 55.29 
 
 .604 
 
 27.39 
 
 .602 
 
 20.50 
 
 .601 
 
 13.64 
 
 9.00 
 
 .606 
 
 58.35 
 
 .602 
 
 28.98 
 
 .601 
 
 21.72 
 
 .601 
 
 14.47 
 
 10.00 
 
 .604 
 
 61.26 
 
 .602 
 
 30.53 
 
 .601 
 
 22.88 
 
 .601 
 
 15.25 
 
 15.00 
 
 .604 
 
 75.09 
 
 .602 
 
 37.42 
 
 .601 
 
 28.02 
 
 .601 
 
 18.68 
 
 20.00 
 
 .605 
 
 86.78 
 
 .602 
 
 43.24 
 
 .601 
 
 32.37 
 
 .601 
 
 21.50 
 
 25.00 
 
 .605 
 
 99.06 
 
 .603 
 
 48.39 
 
 .601 
 
 36.19 
 
 .601 
 
 24.12 
 
 30.00 
 
 .605 
 
 106.46 
 
 .603 
 
 53.34 
 
 .602 
 
 39.68 
 
 .601 
 
 26.43 
 
 35.00 
 
 .606 
 
 115.08 
 
 .694 
 
 57.35 
 
 .602 
 
 4288 
 
 .601 
 
 28.55 
 
 40.00 
 
 .607 
 
 123.13 
 
 .605 
 
 61.36 
 
 .603 
 
 45.86 
 
 .602 
 
 30.53 
 
 45.00 
 
 .606 
 
 130.39 
 
 .605 
 
 65.14 
 
 .603 
 
 48.68 
 
 .602 
 
 32.39 
 
 50.00 
 
 .609 
 
 138.12 
 
 .606 
 
 67.21 
 
 .603 
 
 51.36 
 
 .602 
 
 34.15 
 
 
 Mean. 
 
 
 Mean. 
 
 
 Mean. 
 
 
 Mean. 
 
 
 
 .614 
 
 
 
 .610 
 
 
 
 .608 
 
 
 
 .602 
 
 
 
70 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 7. 
 
 Plow of Water per second through rectangular orifices In thin 
 
 vertical partitions, and the coefficients employed 
 
 in the computation. 
 
 Head 
 on 
 Center. 
 
 Coeffi- 
 cient. 
 
 9 fett 
 high; 1 
 t. wide, 
 cu. ft. 
 
 Coeffi- 
 cient. 
 
 6 feet 
 high; 1 
 ft. wide, 
 cu. ft. 
 
 Coeffi- 
 cient. 
 
 3 feet 
 high; i 
 ft. v ide 
 cu. ft. 
 
 foeffi- 
 c ent. 
 
 li feet 
 high; 1 
 ft. wide, 
 cu. ft. 
 
 0.2 
 
 
 
 
 
 
 
 .633 
 
 .28 
 
 0.3 
 
 
 
 
 
 .629 
 
 .69 
 
 .633 
 
 .35 
 
 0.4 
 
 
 
 .614 
 
 1.56 
 
 .631 
 
 .80 
 
 .633 
 
 .40 
 
 0.5 
 
 .605 
 
 2.57 
 
 .615 
 
 1.74 
 
 .631 
 
 .89 
 
 .633 
 
 .45 
 
 0.6 
 
 .606 
 
 2.83 
 
 .616 
 
 1.91 
 
 .632 
 
 .98 
 
 .633 
 
 .49 
 
 0.7 
 
 .607 
 
 3.06 
 
 .616 
 
 207 
 
 .632 
 
 .06 
 
 .633 
 
 .53 
 
 0.8 
 
 .608 
 
 3.27 
 
 .617 
 
 2.21 
 
 .632 
 
 .14 
 
 .633 
 
 .59 
 
 0.9 
 
 .609 
 
 3.48 
 
 .617 
 
 2.35 
 
 .632 
 
 .20 
 
 .632 
 
 .60 
 
 1.0 
 
 .609 
 
 3.67 
 
 .617 
 
 2.48 
 
 .632 
 
 .26 
 
 .632 
 
 .63 
 
 1.25 
 
 .610 
 
 4.02 
 
 .617 
 
 2.71 
 
 .632 
 
 .39 
 
 .631 
 
 .69 
 
 1.50 
 
 .610 
 
 4.50 
 
 .617 
 
 3.03 
 
 .631 
 
 1.55 
 
 .630 
 
 .77 
 
 1.75 
 
 .610 
 
 4.86 
 
 .617 
 
 3.27 
 
 .631 
 
 1.67 
 
 .630 
 
 .83 
 
 2.00 
 
 .610 
 
 5.20 
 
 .617 
 
 3.50 
 
 .630 
 
 1.79 
 
 .629 
 
 .89 
 
 2.25 
 
 .610 
 
 5.51 
 
 .616 
 
 3.71 
 
 .629 
 
 1.89 
 
 .629 
 
 .95 
 
 2.50 
 
 .610 
 
 5.81 
 
 .616 
 
 391 
 
 .628 
 
 1.99 
 
 .628 
 
 1.00 
 
 2.75 
 
 .610 
 
 6.09 
 
 .616 
 
 4.10 
 
 .627 
 
 2.09 
 
 .627 
 
 1.04 
 
 3.00 
 
 .610 
 
 6.36 
 
 .615 
 
 4.27 
 
 .627 
 
 2.18 
 
 .627 
 
 1.09 
 
 3.5 
 
 .609 
 
 6.36 
 
 .615 
 
 461 
 
 .625 
 
 2.35 
 
 .625 
 
 1.17 
 
 4.00 
 
 .609 
 
 7.32 
 
 .614 
 
 4.92 
 
 .624 
 
 2.50 
 
 .624 
 
 1.25 
 
 4.5 
 
 .607 
 
 7.75 
 
 .613 
 
 5.21 
 
 .622 
 
 2.65 
 
 .622 
 
 1.32 
 
 5.00 
 
 .606 
 
 8.16 
 
 .611 
 
 549 
 
 .620 
 
 2.78 
 
 .620 
 
 1.39 
 
 6.00 
 
 .604 
 
 8.91 
 
 .609 
 
 5.98 
 
 .615 
 
 3.03 
 
 .615 
 
 1.51 
 
 7.00 
 
 .603 
 
 9.61 
 
 .606 
 
 6.43 
 
 .611 
 
 3.24 
 
 .611 
 
 1.62 
 
 8.00 
 
 .602 
 
 10.25 
 
 .603 
 
 6.84 
 
 .607 
 
 3.45 
 
 .609 
 
 1.71 
 
 9.00 
 
 .602 
 
 10.86 
 
 .602 
 
 7.25 
 
 .605 
 
 3.64 
 
 .607 
 
 1.83 
 
 10.00 
 
 .601 
 
 11.44 
 
 .601 
 
 7.62 
 
 .603 
 
 3.83 
 
 .606 
 
 1.92 
 
 15.00 
 
 .601 
 
 14.01 
 
 .601 
 
 9.34 
 
 .603 
 
 4.69 
 
 .607 
 
 2.36 
 
 20.00 
 
 .601 
 
 16.18 
 
 .602 
 
 10.80 
 
 .604 
 
 5.42 
 
 .607 
 
 2.72 
 
 25.00 
 
 .602 
 
 18.10 
 
 .602 
 
 12.08 
 
 .604 
 
 6.06 
 
 .608 
 
 3.05 
 
 30.00 
 
 .602 
 
 19.84 
 
 .603 
 
 13.47 
 
 .604 
 
 6.64 
 
 .609 
 
 3.35 
 
 35.00 
 
 .602 
 
 21.44 
 
 .603 
 
 14.31 
 
 .605 
 
 7.18 
 
 .610 
 
 3.62 
 
 40.00 
 
 .603 
 
 22.94 
 
 .604 
 
 15.32 
 
 .606 
 
 7.68 
 
 .611 
 
 3.79 
 
 45.00 
 
 .603 
 
 24.35 
 
 .604 
 
 16.26 
 
 .606 
 
 8.16 
 
 .613 
 
 4.12 
 
 50.00 
 
 .604 
 
 25.68 
 
 .605 
 
 17.16 
 
 .607 
 
 8.61 
 
 .614 
 
 4.35 
 
 
 Mean. 
 
 
 Mean. 
 
 
 Mean. 
 
 
 Mean. 
 
 
 
 .606 
 
 
 .611 
 
 
 
 .620 
 
 
 
 .622 
 
 
PEACTICAL HYDRAULICS. 
 
 An inspection of Table 7 discloses that the coeffi- 
 cient of flow is variable, both with respect to the 
 head of water and form of orifice. 
 
 Thus, the orifice being ' 'four feet high," the maxi- 
 mum coefficient .629, is due a head of 2.50 feet ; 
 thence the coefficient gradually diminishes to . 604, as 
 the head increases to 10 feet; thence is constant to 15 
 feet: thence gradually increases to .609, with the in- 
 crease of the head to 50 feet. 
 
 In the other given orifices, variations obtain, but to 
 a less extent. 
 
 With respect to the variation of coefficients arising 
 from the form of orifice, it will be seen, by running 
 the eye horizontally to the right, from and for any 
 given head, that the values of the coefficients diminish 
 as the hights of the orifices decrease from four feet to 
 one foot, and increase as the hights of the orifice de- 
 crease from one foot to one and one-half (1J) inches. 
 In illustration take several heads, as 3, 10, 25, 50 feet, 
 and the coefficients due the several forms of orifice. 
 
 Head. 
 Feet. 
 
 4'xl' 
 Coef. 
 
 2'xl' 
 Coef. 
 
 li' x 1' 
 Coef. 
 
 1'xl' 
 Coef. 
 
 9" x 1' 
 Coef. 
 
 6x1' 
 Coef. 
 
 3" x 1' 
 Coef. 
 
 H'xl" 
 Coef. 
 
 3 
 10 
 25 
 50 
 
 .627 
 .604 
 .605 
 .69 
 
 .617 
 .602 
 .603 
 .fi6 
 
 .613 
 .601 
 .601 
 .603 
 
 .605 
 .601 
 .601 
 .602 
 
 .610 
 .601 
 .602 
 .604 
 
 .615 
 .601 
 .602 
 .605 
 
 .627 
 .603 
 .604 
 .607 
 
 .627 
 .606 
 .608 
 .614 
 
72 PRACTICAL HYDRAULICS. 
 
 TO FIND THE FLOW OF WATER IN CUBIC FEET PER 
 SECOND THROUGH VERTICAL RECTANGULAR ORIFICES IN 
 THIN VERTICAL PARTITIONS BY TABLE 7, THE HEAE ON 
 CENTER AND SIZE OF OPENING BEING MADE. 
 
 Rule 26. In "head on center" column, Table 7, 
 find the given head, opposite which, in column headed 
 by the given hight of orifice, will be found the flow 
 for an orifice one foot wide, which multiply b}^ the 
 given width in feet. 
 
 Ex. 40. The head being ten feet, and orifice four 
 feet wide and nine inches high, what is the flow per 
 second? 
 
 Cal. In column "9" high 1 foot wide," opposite 10 
 feet in "head on center" column, will be found 11.44 
 cubic feet, which, multiplied by four feet, the given 
 width, gives: 
 
 11.44X4=45.76 cubic feet. Ans. 
 
 The heads, sizes of orifices, and the computed flow 
 of water given in Table 7, will be found highly con- 
 venient for ready reference in a great number of cases, 
 but are seen to be too limited to fully meet the re- 
 quirements of practice. Indeed, a table sufficiently 
 ample for that purpose would be too un wieldly for use. 
 
 The general formula for the low of water per sec- 
 ond through vertical rectangular orifices in thin par- 
 titions, is: 
 
 (92) 
 
 In which Q denotes the flow in cubic feet ; c, coefficient 
 
PKACTICAL HYDRAULICS. 73 
 
 of discharge; a, the area of the orifice in square feet; 
 and ti, the head on the center of the orifice: h! is equal 
 to the half sum of the respective heads on the bottom 
 and top of the orifice, as seen in equation (21). 
 
 In case the hight of the orifice and the head on its 
 top are given, then h r is equal to the sum of the given 
 head and half the hight of the opening; or if the hight 
 of the opening and the head on its bottom are given, 
 then h r is equal to the difference between the given head 
 and half the hight of the orifice. 
 
 TO FIND THE FLOW OF WATER IN CUBIC FEET PER 
 SECOND, THROUGH VERTICAL RECTANGULAR ORIFICES IN 
 
 THIN PARTITIONS. 
 
 * 
 
 Rule 27. Multiply 8.025 times the square root of 
 the head on the center of the orifice, by the product of 
 the area of the orifice and the coefficient of discharge. 
 
 Rule 27 corresponds to formula (92). 
 
 With respect to the "square root of the head," and 
 "the coefficient of discharge," contemplated in Rule 
 27, it will be remembered that Table 6 gives the square 
 roots of numbers likely to be required, and Table 7, 
 the coefficients of discharge. In finding a proper co- 
 efficient of discharge, in case the given hight of orifice 
 is found in Table 7, the coefficient corresponding to 
 that hight and to the given head is to be employed; 
 but in case the given hight of orifice is an inter- 
 mediate, or lies between the hights contained in the 
 
74 PRACTICAL HYDRAULICS. 
 
 table, its coefficient will need be computed. The 
 tabulated coefficients are, in fact, ordinates of curves, 
 determined by experiment. 
 
 In determining these intermediate ordinates or co- 
 efficients between any two adjacent hights in Table 7, 
 as 4 feet and 2 feet, 1.5 feet and 1 foot, no appreciable 
 error will occur by substituting a right line for a 
 curve. The determination of the intermediate coeffi- 
 cients will then be effected by arithmetical differences. 
 In illustration: let it be required to find the coefficient 
 due a head of 2.5 feet, and orifice 3.5 feet high. 
 
 Now 3.5 is between the adjacent hights, 4 and 2 
 feet, in Table 7. The respective coefficients due a 
 head of 2.5 feet are .629 in "4 feet high" column, and 
 .618 in "2 feet high" column. 
 
 Difference of hights, 42=2 feet. 
 
 Difference of greater and given hights, 4 3.5 .5 feet. 
 Quotient of these differences, 2-v-.5 4 divisor. 
 
 Difference of coefficients, .629 .618=. Oil 
 
 Arithmetical difference sought, .Oll-j-4=.003 nearly. 
 Coefficient due 3.5 feet, .629 .003=.626 
 
 The intermediate coefficients corresponding to 3 
 feet and 2.5 feet are now readily found. Thus: 
 Coefficient due 3 feet, .626 .003=.623 
 
 Coefficient due 2.5 feet, .623 ;003=.620 
 
 EXAMPLES AND CALCULATIONS. 
 Ex. 41. An orifice is 3,5 feet wide, 1.25 feet high 
 
PRACTICAL HYDRAULICS. 75 
 
 and the head on its center is 7 feet. What is the flow 
 in cubic feet per second? 
 
 Cal By Table 6, square root of 7 feet=2.646. 
 
 By Table 7, coefficient due 7 feet; for orifice, 1.5 
 feet high=.604; for orifice, 1 foot high^ .601. 
 Difference of tabulated hights, 1 . 5 1 = . 5 
 
 Difference of greater and given hights, 1.5 1.25=. 25 
 Quotient of these differences, . 5 -=- .25 2 div . 
 
 Difference of coefficients, . 604. 60l=.003 
 
 Arithmetical difference, .003-^-2=.0015 
 
 Coefficient due 1.25 feet, .604 .0015=. 6025 
 
 Area of orifice', 3.5X1.25=4.375 square feet. 
 
 Flow=8. 025 X . 6025 X4. 375 X 2. 646=55. 97 cubic 
 feet. Ans. 
 
 Ex. 42. Given the head on the bottom of a rec- 
 tangular orifice 12 feet, the head on its top 11 feet, 
 and the width of orifice 4 feet, what is the flow in 
 cubic feet per second? 
 
 Cal. Head on center =^=11. 5 feet. 
 
 By Table 6, square root of head on center =3. 391 
 feet. 
 
 Hight of orifice=12 11=1 foot. 
 
 By Table 7, coefficient due head of 11.5 feet, and 
 orifice 1 foot high =.601. 
 
 It will be observed that the coefficient is constant 
 for head from 10 to 1 5 feet, inclusive. 
 
 Area of orifice=4Xl=4 square feet. 
 
 Flow=8.025X.60lx4x3.391=65.42 cubic feet. 
 Ans. 
 
76 PRACTICAL HYDRAULICS. 
 
 Ex. 43. The head on the top of a rectangular ori- 
 fice 6 inches high and 6 feet wide being 7.25 feet, 
 what is the flow in cubic feet per second? 
 
 CaL Half the hight of orifice 6"-*-2= 3" .25 feet, 
 
 Head on center =7. 25 + . 25=7.5 feet. 
 
 By Table 7, coeflicient due head of 7 feet=.606. 
 
 Coefficient due head of 8 feet=.603. 
 
 Mean coefficient on that due 7.5 feet=.6045. 
 
 By Table 6, square root, 7.5 feet 2.739. 
 
 Area of orifice, 6X .5=3 square feet. 
 
 Flow=8. 025 X .6045 X 3 X 2.739=29.89 cubic feet. 
 Ans. 
 
 Ex. 44. The head on the bottom of a rectangular 
 orifice 9 inches high and 3 feet wide, being 15.875 
 feet, what is the flow in cubic feet per second ? 
 
 CM. Half the hight of orifice, 9"-^2=4.5"=.375 
 feet. 
 
 Head on center =15. 97 5 .375=15.6 feet. 
 
 By Table 7, coefiicient due head of 15.875 feet, and 
 orifice 9" high=.601. 
 
 Observe that the coefficient is constant from 10 feet 
 to 20 feet, inclusive. 
 
 By Table 6, square root of 15.6=3.95 feet. 
 
 Area of orifice=3X.75=2.25 square feet. 
 
 Flow=8.025X. 601X2.25X3. 95=42.86 cubic feet. 
 Ans. 
 
 The preferable unit for measuring the flow of water 
 is 1 cubic foot, but so widely is the "miner's inch," 
 employed in California as a unit of measure, that we 
 cannot well pass it in silence. 
 
PRACTICAL HYDRAULICS. 77 
 
 "MINER'S INCH." 
 
 The term "miner's inch" is employed to express that 
 quantity of water which, under a given head or pres- 
 sure, as 4, 7, 9, etc., inches, will flow through each 
 square inch of a discharge opening ; or, in other words, 
 which will flow through each square inch of cross sec- 
 tion of a stream of water. 
 
 The quantity of water so flowing in a minute, an 
 hour, 24 hours, etc., is designated minute inch, hour 
 inch, 24:-hour inch, etc., according to the length of 
 time specified. 
 
 STATUTORY MINER'S INCH. 
 
 Under the head, "Water Rights," the Civil Code of 
 the State of California, Sec. 1415, provides in these 
 words, "That he (the locator) claims the water there 
 flowing to the extent of (giving the number) inches, 
 measured under a four-inch pressure." 
 
 On this data, the value of the statutory miner's 
 inch, the mean coefficient of discharge being in prac- 
 tice .6216, is as follows: 
 
 For one second (second inch), 0.02 cubic feet. 
 
 For one minute (minute inch), 1.20 cubic feet. 
 
78 PRACTICAL HYDRAULICS. 
 
 For one hour (hour inch), 72.00 cubic feet. 
 
 For 24 hours (24-hour inch), 1728 cubic feet. 
 
 If a cubic foot be divided by the flow in one sec- 
 ond, there will result the number of miner's inches 
 whose discharge is equal to a cubic foot per second. 
 Thus, 1^-. 02=50 statutory miner's inches; that is, 
 fifty statutory miner's inches are equal to one cubic 
 foot flow per second. 
 
 NORTH BLOOMFIELD, ETC., MINER'S INCHES. 
 
 At the North Bloomfield, Milton and Columbia Hill 
 hydraulic mines, the water is measured in its flow 
 through a rectangular orifice 50 inches long, 2 inches 
 wide, and under a pressure of 7 inches on the center of 
 the opening. The flow per square inch of orifice, for 
 24 hours, due this head, as given me by Hamilton 
 Smith, Jr., C. E., formerly chief engineer of the 
 North Bloomfield Works, and president of the Miners' 
 Association, is 2230 cubic feet; of which the coefficient 
 of discharge is found to be .6064. Mr. Smith's ex- 
 periments made with a module of equal dimensions 
 under a 7-inch head, at Columbia Hill in 1874, found, 
 as stated by Aug. J. Bowie, Jr., M. E., in an article 
 entitled "Bowie on the Measurement and Flow of 
 Water," found the value of the 24-hour inch to be 
 2260.8 cubic feet, and the coefficient of discharge to be 
 .616. Mr. Bowie, in the article referred to, gives, in 
 
PBACTICAL HYDRAULICS. 79 
 
 addition, substantially the following data, with respect 
 to the 
 
 SMARTSVILLE MINER'S INCH. 
 
 Hight of orifice, 4 inches; head on center, 9 inches; 
 value of 24-hour inch, 2534.4 cubic feet; coefficient of 
 discharge, .6078. 
 
 SOUTH YUBA CANAL INCH. 
 
 Hight of orifice, 2 inches; head on center, 6 inches. 
 
 And with respect to a series "of experiments made 
 by himself at La Grange with an orifice 12 inches 
 high, 12.75 inches wide, under a pressure of 6 inches 
 on the top of the orifice, or head of one foot on the 
 center. The mean of which experiments gave as the 
 value of otfe miner's inch for 24 hours, 2159.146 cubic 
 feet; effective coefficient of efflux, .5905. The flow 
 through this module was assumed equal to 200 miner's 
 inches. 
 
 A comparison shows that these coefficients of dis- 
 charge approximate closely those given in Table 7, 
 obtained on equal data. The results of these experi- 
 ments also clearly show that the value of a coefficient 
 of discharge depe.nds, among other things, upon the 
 form of the module. In illustration, the module be- 
 
80 
 
 PRACTICAL HYDRAULICS. 
 
 ing 50 inches long, 2 inches wide, the coefficient of 
 discharge is found to be .5905. Should we estimate 
 the effect of the difference between the given heads 
 (7 inches and 1 foot) on the coefficients of discharge, 
 there would result .5885 instead of .5905. 
 
 The variety of values comprised in the term, 
 miner's inch, as employed in California, is often a 
 source of no little annoyance and confusion. To aid 
 in overcoming this difficulty, Table 8 prepared from 
 Table 7, is given. Each result so obtained is a mean 
 of the experiments of the world's ablest hydraulicians. 
 
 TABLE 8. 
 
 Flow of water through rectangular orifices due Miner's 
 Inches of different values. 
 
 Head 
 
 Orifice. 
 
 | ec. inch 
 
 Min. inch 
 
 Hour inch 
 
 24-hour 
 
 1 Cub. Ft. 
 
 on 
 Cent. 
 
 High 
 
 Wide. 
 
 Coef. j Flow. 
 
 F.ow. 
 
 Flow. 
 
 inch Flow. 
 
 Flow 
 j;er Sec. 
 
 In. 
 
 In. 
 
 In. 
 
 
 Cub. Feet. 
 
 Cub. Feet. 
 
 CuH. Feet. 
 
 Cub. Pfeet. 
 
 Miner'sln. 
 
 3 
 
 2 
 
 12. 
 
 .631 
 
 .01758 
 
 .055 
 
 63.29 
 
 1519. 
 
 58.87 
 
 4 
 
 2 
 
 12. 
 
 .631 
 
 .02030 
 
 .218 
 
 73.10 
 
 1754. 
 
 49.24 
 
 5 
 
 2 
 
 12. 
 
 .632 
 
 .02274 
 
 .364 
 
 81.85 
 
 1964. 
 
 43.98 
 
 6 
 
 2 
 
 12. 
 
 .632 
 
 .02490 
 
 .494 
 
 89.65 
 
 2152. 
 
 40.15 
 
 7 
 
 2 
 
 12. 
 
 .632 
 
 .02690 
 
 .614 
 
 96.84 
 
 2324. 
 
 37.17 
 
 8 
 
 2 
 
 12. 
 
 .632 
 
 .02876 
 
 .725 
 
 103.53 
 
 2484. 
 
 34.77 
 
 9 
 
 4 
 
 12. 
 
 .624 
 
 .03011 
 
 1.807 
 
 108.42 
 
 2602. 
 
 33.20 
 
 10 
 
 6 
 
 12. 
 
 .617 
 
 .03139 
 
 1.883 
 
 113.00 
 
 2711. 
 
 31.85 
 
 11 
 
 9 
 
 12. 
 
 .609 
 
 .03249 
 
 1.949 
 
 116.98 
 
 2807. 
 
 30.77 
 
 *12 
 
 12 
 
 12.75 
 
 .601 
 
 .02562 
 
 1.537 
 
 92.24 
 
 2214. 
 
 39.03 
 
 *The flow due the given opening, 12" x 12 75" = 153 square inches, 
 divided by 200, has been proposed, as hereinbefore stated, as the 
 standard miner's inch. Its adoption seems to be but local. 
 
PRACTICAL HYDRAULICS. 81 
 
 EXAMPLES AND CALCULATIONS. 
 
 Ex. 45. A water right location is made for 6000 
 miner's inches. What is the equivalent flow in cubic 
 feet per second? 
 
 Gal. 1st. The statutory miner's inch is estimated, 
 as stated, under a 4-inch pressure. 
 
 By Table 8, opposite 4 inches in "head on center" 
 column, find 49.24 miner's inches in "1 cubic foot flow 
 per second" column, 6000^-49.24=121.6 cubic feet. 
 Ans. 
 
 CaL 2d. For the most part in practice, 50 miner's 
 inches measured under a 4-inch pressure, are adopted 
 as 'equal to a flow of one cubic foot of water per sec- 
 ond. This results, as shown in discussing the statu- 
 tory miner's inch, from taking the mean coefficient 
 .6216 for different heads, instead of the tabulated co- 
 efficient .632 for the given 4-inch head. 
 
 Whence, 6000-^50=120 cubic feet. Ana. 
 
 Ex. 46. In a water right claim of 5000 miner's 
 inches, measured under a 4-inch pressure, are how 
 many North Bloomfield miner's inches miner's inches 
 measured under a 7- inch head? 
 
 Col. 1st. By Table 8, the value of a second inch, 
 under a 4-inch head, is .0203 cubic feet flow; and the 
 value of a second inch, under a 7-inch head, is .0269 
 cubic feet flow. 
 
82 PRACTICAL HYDRAULICS. 
 
 Whence, .0203X5000-^.0269=3773 miner's inches 
 Ans. 
 
 Cal. 2d. In the discussion of the miner's inch, it 
 has been shown that in common practice the value of 
 the 24-hour inch, under a 4-inch head, is 1728 cubic 
 feet; and under a 7-inch head at North Bloomneld is 
 2230 cubic feet. 
 
 Whence, 1728x5000-+- 2230=- 3874 miner's inches. 
 Ans. 
 
 Ex. 47. In 2000 miner's inches, through a rectan- 
 gular opening 2 inches high, and under a 6-inch pres- 
 sure, as employed at the South Yuba canal, are how 
 many miner's inches flowing through a rectangular 
 opening 4 inches high and under a 9-inch pressure, as 
 adopted at Smartsville? 
 
 Cal. 1st. As the result will be the same, whether 
 the calculation be made in second, minute, hour, or 
 24-hour miner's inches, let the 24-hour inch be em- 
 ployed ; then by Table 8, under a 6 -inch pressure 
 through an opening 2 inches high, the value is 2152 
 cubic feet ; and under a 9-inch pressure through an open- 
 ing 4 inches high, the value is 2602 cubic feet; whence, 
 2000x2152-^-2602=1654 Smartsville miner's inches. 
 
 Cal. 2d. Under the heading Smartsville, Bowie 
 "On Measurement and Flow of Water," makes that 
 miner's inch, 2534.4, due coefficient of discharge .6078. 
 instead of .624, as adopted in Table 8. 
 
 Whence, 2000X2152 + 2534.4=1698 Smartsville 
 miner's inches. Ans. 
 
 Cal. 3d. Table 8 shows that the coefficient due a 
 
PRACTICAL HYDRAULICS. 83 
 
 6 -inch head, and opening 2 inches high, is .632, and 
 that the coefficient due a 9-inch head, and opening 4 
 inches high, is .624. Now, as commonly practiced, 
 the "mean" coefficient .62 would be employed; so that, 
 the result sought would depend upon the square root 
 of the ratio of the given heads, 9 inches=.75 feet, and 
 6 inches-=.5 feet; thus, by Table 5, 
 
 .8165X2000=1633 Smartsville miner's inches.- 
 Ans. 
 
 EXAMPLES AND CALCULATIONS. 
 
 Ex. 48. The head being 2.25 feet on the center of 
 a circular orifice .0328 feet diameter, what is the dis- 
 charge in cubic feet per second? 
 
 Gal. 1st. Rule 27 is equally applicable to rectangu- 
 lar and circular orifices. 
 
 By Table 6 the square root of given head 2.25 feet 
 1.5 feet. 
 
 Area of given orifice .0323 feet diameter is equal to 
 the square of the diameter, multiplied by .7854 ; 
 (.0328) 2 X. 7854= .000845 square feet. 
 
 By Table 9, coefficient of discharge due a head of 
 2.25 feet, according to Castel, is approximately equal 
 to .673 ; then by Rule 27, 
 
 .873X8.025Xl.5X. 000845=. 00685 cubic feet. 
 Ans. 
 
 Cal. 2d. According to Weisbach,the coefficient due 
 a head of 2.25 feet, and orifice .0328 feet diameter, is 
 
84 PRACTICAL HYDRAULICS . 
 
 approximately .628. Employing this coefficient in- 
 stead of .673, 
 
 .628X 8.025X1. 5X- 000845=. 00639 cubic feet. 
 Ans. 
 
 Weisbach observes, that "for square orifices from 1 
 to 9 square inches area, wrth from 7 to 21 feet head of 
 water, according to the experiments of Bossut and 
 Michelotti, the mean coefficient of efflux is ?7i=.610; 
 for circular ones of from J to 6 inches diameter, with 
 from 4 to 21 feet head of water, m=.6l5." 
 
 A mean of the coefficients of Table 9 is equal to .62 
 nearly. 
 
 In ordinary practice this is employed. When 
 greater accuracy is required, reference will need be had 
 to Table 9. 
 
 PARTIAL CONTRACTION. 
 
 Experiments show that if contraction be suppressed, 
 the flow of water through an orifice will be increased 
 accordingly. 
 
 Let 7i- the ratio between the entire perimeter of an 
 orifice and the part suppressed that is, if p denote 
 the entire perimeter, and p t the part suppressed, then 
 
 *=? 
 
 c=coefficient of discharge due perfect contraction. 
 
 c n coefficient of discharge due partial contraction. 
 
 #= a number deduced from experiment, which, be- 
 ing multiplied by the product of the ratio, n, and the 
 coefficient, c, gives cxn, the increase due partial con- 
 traction; thus, Cn==c (1+ajw) . 
 
PRACTICAL HYDRAULICS. 
 
 85 
 
 TABLE 9. 
 
 Coefficients for the Flow of Water through circular orifices. 
 Extracts from D'Aubiussoc, Fannies and Weisbach. 
 
 OBSERVERS. 
 
 Diam 1 Heads. 
 Feet. ' Feet. 
 
 Mariotti 0.0223 5.8712 
 
 .0223 25.9120 
 
 Castel ! .0328 2.1320 
 
 J .0328 1.0168 
 
 0492 0.4526 
 
 .0492 0.9840 
 
 Eytelwine .* 0856 2.3714 
 
 Bossut 0889 42640 
 
 Michelo^ti | .0889 7 3144 
 
 Castel i .0984 05510 
 
 Veatari '....! .1345 2.8864 
 
 Bossut I .1771 12.4968 
 
 Michelotti 1771 7.2160 
 
 .2657 73472 
 
 .2657 12.4968 
 
 .2657 22.1728 
 
 .5314 6.9208 
 
 .5314 12.0048 
 Mean. 
 
 Gen. Ellis 2 1.7677 
 
 .2 5.8269 
 
 .2 96381 
 
 .1 1.1470 
 
 .1 10.8819 
 
 .1 17.7400 
 
 .5 2.1516 
 
 I .5 9.0600 
 
 .5 17.2650 
 
 Weisbach | .0328 2.0000 
 
 | .0656 20000 
 
 '. I .0984 2.0000 
 
 n .1312 2.0000 
 
 .0328 .8333 
 
 .0656 .8333 
 
 .0984 .8333 
 
 .1312 .8333 
 Mean. 
 
PRACTICAL HYDRAULICS. 
 
 An inspection of Table 9 shows that the coefficient 
 of flow for small orifices and for small velocities, is 
 greater than it is for large orifices and for great 
 velocities. 
 
 It will also be observed that the results of experi- 
 ments differ considerably, though the data employed 
 is approximately similar. 
 
 Thus Castel finds the coefficient of flow for an ori- 
 fice .0328 feet diameter, under a head of 2.132 feet, to 
 be .673: while Weisbach finds it, for an orifice .0328 
 feet diameter, under a head of 2 feet, to be .628. 
 
 Bidone's experiments give, for circular orifices, x= 
 0.128, and for rectangular orifices, x= 0.1 25. 
 
 Weisbach's experiments give, for rectangular ori- 
 fices, #=0.134. 
 
 Weisbach, however, employs for rectangular ori- 
 fices the mean between these results that is, 
 
 Substituting these values in Eq. (93) there results, 
 for circular orifices: 
 
 .12871). (94) 
 
 And for rectangular orifices: 
 
 (95) 
 
PEACTICAL HYDRAULICS. 87 
 
 TO FIND THE COEFFICIENT OF DISCHARGE OF PARTIAL 
 CONTRACTION FOR CIRCULAR AND FOR RECTANGULAR 
 ORIFICES. 
 
 Rule 28. Case 1st The orifice being circular, add 
 1. to .128 times the ratio of the entire perimeter to the 
 part suppressed, and multiply this sum by the coeffi- 
 cient of discharge of perfect contraction. 
 
 Case 2d The*" orifice being rectangular, add 1 to 
 0.143 times the ratio of the entire perimeter to the 
 part suppressed, and multiply this sum by the coeffi- 
 cient of discharge of perfect contraction. 
 
 Ex. 49. A rectangular orifice being 1 foot wide, 
 6 inches high, and the head 10 feet, what is the coeffi- 
 cient of discharge if the contraction at one end be 
 suppressed? 
 
 Gal. By Table 7, coefficient of perfect contraction 
 for the given head and given orifice is -=.601. 
 
 Part suppressed 6 inches .5 feet. 
 
 Entire perimeter 1 + 1-J-.5 + .5 3 feet. 
 
 Ratio of entire perimeter to paTt suppressed / 
 0. 143 times this ratio ; 0. 143 X V = . 024. 
 
 Sum of 1 and this product = 1.024. 
 
 This sum, multiplied by .601, the coefficient of per- 
 fect contraction, .60l)<1.024=- .615, the coefficient of 
 partial contraction.= ^.7is. 
 
 Ex. 50. A rectangular orifice being 1 foot wide, 
 
88 PRACTICAL HYDRAULICS. 
 
 6 inches high, and the head 10 feet, what is the co- 
 efficient of partial contraction if the contraction at 
 both ends be suppressed? 
 
 Gal. By Table 7, the coefficient of perfect con- 
 traction, for the given head and given orifice, is .601. 
 
 Part suppressed 6" + 6"=12"=l foot. 
 
 Entire perimeter, 1 + 1-f .5+.5=3 feet. 
 
 Ratio of entire perimeter to part suppressed=J, 
 0.143 times the ratio; 0.143X J=0.048. 
 
 Sum of 1 and this product= 1.048. 
 
 This sum, multiplied by .601, the coefficient of per- 
 fect contraction. 
 
 .601X1. 048=. 630, the coefficient of partial con- 
 traction. Ans. 
 
 TO DETERMINE THE COEFFICIENT OF CONTRACTION 
 FOR A GIVEN ORIFICE AND GIVEN HEAD OF WATER. 
 
 Let a=the hight of orifice; & the breadth of ori- 
 fice; ft head of water. 
 
 And let c, c n , n, p, p t and x have the same offices 
 as assigned them under the heading, "Partial Con- 
 traction;" c 6 =the coefficient of contraction due the 
 breadth. 
 
 In Table 7, the hights of the orifice vary from 4 
 feet to 0.125 feet, while the breadth of each orifice is 
 1 foot. It is evident, if the contraction be suppressed 
 at both ends of any orifice given in Table 7, the con- 
 
PRACTICAL HYDRAULICS. 89 
 
 traction due the horizontal lips only, each 1 foot in 
 length, will obtain. Now if the lips be increased any 
 given number of times 1 foot, the contraction will be 
 proportionately increased. This being done, if the 
 contraction due the ends be restored, and the result 
 divided by the length of the elongated orifice, or by 
 the given number of times that the lips were increased 
 in length, the quotient will express the mean contrac- 
 tion due 1 foot breadth of the given orifice. 
 
 For an orifice in Table 7, whose hight is a, and 
 whose head of water is h, if the contraction of both 
 ends be suppressed, the ratio n~~ 2-Ka> an( i the co ~ 
 efficient of partial contraction: 
 
 Multiplying both sides of Eq. (96) by b, the breadth 
 of the given orifice, restoring the end contraction, 
 2lil> dividing the result by the breadth 6, substituting 
 c b for left hand member, and reducing, 
 
 /-. . 2ax \ 2cax /Q7\ 
 
 Substituting in (97) the values of #=0.143, and 
 
 c b = c (1 + 14371) ^=s. (98) 
 
 Rule 29. Find, as by Rule 28, the value of the co- 
 efficient of partial contraction for an orifice of the 
 given hight, 1 foot wide, and having the contraction 
 at both ends suppressed. 
 
 From the value so found deduct the quotient arising 
 
90 PBACTICAL HYDRAULICS. 
 
 from dividing 0.143 times the product of the coeffi- 
 cient of perfect contraction, and the ratio of the entire 
 perimeter of the orifice 1 foot wide, to the part sup- 
 pressed, by the breadth, of the given orifice. The re- 
 mainder will be the coefficient of contraction due the 
 given orifice. Rule 29 is derived from formula (98). 
 
 Ex. 51. A rectangular orifice being 5 feet wide, 3 
 inches high, and the head of water 3 feet, what is the 
 coefficient of contraction? 
 
 Gal. By Table 9, the coefficient of perfect con- 
 traction, for an orifice 1 foot wide, 3 inches high, un- 
 der a head of 8 feet, is=.607. 
 
 Part suppressed=3"+3"=6"==.5 feet. 
 
 Entire perimeter=l-f-l-{-.5-=2.5 feet. 
 
 Ratio of entire perimeter to part suppre8sed^~^ fi T = 
 .2; 0.143 times this ratio; 0.143X .2-= .0286. 
 
 Sum of 1 and this product= 1.0286. 
 
 This sum, multiplied by .607, the coefficient of per- 
 fect contraction, gives the value of the coefficient of 
 partial contraction, when the contraction of both 
 ends is suppressed, 
 
 c n =1.0286X.607=-.6244. 
 
 By Rule 29, 0.143 times the product of the coeffi- 
 cient of perfect contraction, and the ratio of the en- 
 tire perimeter of the orifice 1 foot wide, to the part 
 suppressed, divided by the breadth of the given ori- 
 fice, 0.l43x.2X.607-f 5=.0035; 
 
 c 6 =.6244 .0035-. 621. Ans. 
 
PRACTICAL HYDRAULICS. 91 
 
 Ex. 52. A rectangular orifice being 2 feet wide, 
 4 feet high, and under a head on center of 2.5 feet, 
 what is the coefficient of discharge? 
 
 CaL By Table 7, the coefficient of perfect con- 
 traction, as determined by experiment, for an orifice 1 
 foot wide, and otherwise conforming to the given con- 
 ditions, is -=.629. 
 
 Part suppressed (both ends) 4+48 feet. 
 
 Entire perimeter, 1 + 1 + 810 feet. 
 
 Ratio of entire perimeter to part suppressed =^-5- 
 -=.8; 0.143 times this\ratio; 0.143X.8-0.1144. 
 
 Sum of 1 and this product= 1.1144. 
 
 This sum multiplied by .629: 1.1144X.629 = .70l. 
 
 By Rule 29, .0.143 times the product of the coeffi- 
 cient of perfect contraction, and the ratio of the entire 
 perimeter of the orifice 1 foot wide, to the part sup- 
 pressed, divided by the breadth of the given orifice; 
 0.l43x.8X.629-+-2=-0.036, 
 
 Cb= . t 701 .036 -=.665. Ans. 
 
 Formulas (95) and (98), and Rules 28 and 29, based 
 upon the mean results of the experiments of Bidone 
 and Weisbach, give but approximations to the true 
 coefficients sought. They are, however, sufficiently 
 accurate for most cases occurring in practice. Ex- 
 ample 52 is an extreme case.- Yet the coefficient .665, 
 determined from its solution, seems practically correct, 
 or not too large, in presence of the fact that the area 
 of the given orifice is twice as great as that of the 
 tabulated orifice whose coefficient of discharge is .629; 
 
92 PKACTICAL HYDBAULICS. 
 
 while the perimeter or contracting boundary of the 
 former is to that of the latter as 12 is to 10. Still it 
 is to be admitted that, in determining the coefficient 
 for a given orifice, the result is more satisfactory when 
 the hight of the tabulated orifice employed does not 
 much exceed' its breadth. 
 
 IMPERFECT CONTRACTION. 
 
 A G 
 
 FIG. 16. 
 
 In the flow of a stream from an orifice, the head of 
 water being nominally still, and the orifice small, in 
 relation to the side of the vessel in which it lies, the 
 contraction is called perfect; the water arriving with 
 considerable velocity at the orifice, as through a con- 
 duit, A G F E, Fig. 16, which cross section varies 
 from 1 to 20 times that of the orifice, the contraction 
 thence is termed imperfect. 
 
 Let c coefficient of perfect contraction; c n coeffi- 
 cient of imperfect contraction; n ratio of the cross 
 section of the conduit, A G F E, Fig. 16, through A E, 
 to the area of the orifice O; A area of orifice; A, 
 area cross section of conduit. 
 
PKAOTICAL HYDEAULICS 93 
 
 The values of imperfect contraction given by Weis- 
 bach, as determined by his experiments and calcula- 
 tions, are: 
 
 1st. For circular orifices: 
 
 c n =^c [1 + 0.04564 (14.821" 1) ]. (99) 
 
 2d. For rectangular orifices: 
 
 c n =c [1 + 0.76 (9l)]. (100) 
 
 Equation (99) for circular orifices is readily resolved 
 into this form : 
 
 1^=0.04564 (14.821" 1), (101 ) 
 
 c 
 
 And equation for rectangular orifices into this: 
 
 =^=0.076 (9 n 1). (102) 
 
 C 
 
 The length of the conduit or adjunct is assumed to 
 be three times its diameter, or not sufficiently great 
 for the flow of water to be sensibly affected by side 
 friction, as occurs in long pipes. 
 
 A^ 
 
 By giving fractional values to n=-r- , or values not 
 
 A / 
 greater than 1, numerical values corresponding, are 
 
 f> Q 
 
 found for the expression - - in equations (101) and 
 
 (102). 
 
 In illustration : assume the areas of the orifice equal 
 
94 
 
 PRACTICAL HYDRAULICS. 
 
 to 1 square foot, and the area of the cross section of 
 the conduit, A G F E, equal to 2 square feet; then n= 
 
 Substituting the value of n in Eq. 102, 
 ^=^-0.076 (941). 
 
 (103) 
 
 Now the | power of 9, in other words the square 
 
 root of 9=3; 31=2; hence ^==^ 0.076X2 -.152, 
 
 c 
 
 correction found for n J 0.5, in Table 2. 
 
 In the computation of Tables 10 and 11, different 
 values from 71-=. 05 the common difference being 
 .05 to Ti 1, are employed. 
 
 TABLE 10. 
 
 Corrections of the Coefficients of Plow for Circular Orifices. 
 Weisbach. 
 
 n 
 
 0.05 
 
 0.10 
 
 0.15 
 
 0.20 
 
 0.25 
 
 0.30 
 
 0.35 
 
 0.40 
 
 0.45 
 
 0.50 
 
 -* _ 
 
 0.007 
 
 0.014 
 
 0.023 
 
 0.034 
 
 0.045 
 
 0.059 
 
 0.075 
 
 0.092 
 
 0.112 
 
 0.134 
 
 c 
 
 
 
 
 
 
 
 
 
 
 
 H 
 
 0. 55 
 
 0.60 
 
 0.65 
 
 0.70 
 
 0.75 
 
 0.80 
 
 0.85 
 
 0.900 
 
 0.95 
 
 1.00 
 
 <!,,"- 
 
 01 fii 
 
 01 SQ 
 
 ft 99^ 
 
 ft 9fift 
 
 ft W* 
 
 
 Odftfi 
 
 ft 4-71 
 
 OKAfi 
 
 n i o 
 
 C ' 
 
 
 
 
 
 
 
 
 
 
 If n has any value not found in Table 10 or Table 
 11, substitute such value in equation 99 in case the 
 
PRACTICAL HYDRAULICS. 
 
 95 
 
 given orifice is circular, or in equation (100) in case 
 the orifice is rectangular, and solve by means of loga- 
 rithms. 
 
 TABLE 11. 
 
 Corrections of the Coefficients of Flow for Rectangular Ori- 
 fices. Weisbach. 
 
 n 
 
 0.05 
 
 0.10 
 
 0.15 
 
 0.20 
 
 0.25 
 
 0.30 
 
 0.35 
 
 0.40 
 
 0.45 
 
 0.50 
 
 c n -c 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 0.009 
 
 0.019 
 
 0.030 
 
 0.042 
 
 0.056 
 
 0.071 
 
 0.088 
 
 0.107 
 
 0.128 
 
 0.152 
 
 n 
 
 0.55 
 
 0.60 
 
 0.65 
 
 0.70 
 
 0.75 
 
 0.80 
 
 0.85 
 
 0.90 
 
 0.95 
 
 1.00 
 
 CM C 
 
 
 
 
 
 
 
 
 
 
 
 C 
 
 0.178 
 
 0.208 
 
 0.241 
 
 0.278 
 
 0.319 
 
 0.365 
 
 0.416 
 
 0.473 
 
 0.537 
 
 0.608 
 
 EXAMPLES AND CALCULATIONS ILLUSTRATING THE 
 USE OF TABLES 10 AND 11. 
 
 Ex. 53. The diameter of a circular orifice being 6 
 inches .5 feet, the head on center 9.06 feet, the area 
 of the orifice one-fourth (.25), that of the cross sec- 
 tion of the conduit A G F E, Fig. 16, what is the co- 
 efficient of discharge? 
 
 Gal. By Table 9, the coefficient of discharge 
 through a circular orifice 6 inches diameter . 5 feet in 
 a thin partition under a head of 9.06 feet of water, 
 nominally still, as observed by Gen. Ellis, is=. 60191; 
 say c-=.602. 
 
96 PRACTICAL HYDRAULICS. 
 
 By Table 10, the value corresponding to n=.25, the 
 given ratio, is: 
 
 c 
 Solving this equation for c n , 
 
 Substituting the value of c=.602 in the last equa- 
 tion, there results: 
 
 c M =1.045X.602=.629. Ans. 
 
 Ex. 54. A rectangular orifice being 9 inches high 
 and 1 foot wide in the end of a conduit, as A G F E, 
 Fig. 16, 1 foot high, 1.25 feet wide, and 3 feet long, 
 under a head of 3.5 feet on center, of water nominally 
 still in tank, B A D C, what is the coefficient of dis- 
 charge ? 
 
 By Table 7, the coefficient of discharge due the 
 given orifice and given head of water nominally still is 
 
 c=.609, 
 
 Area of orifice, . 95 X 1 = . 95, 
 Area of cross-section of conduit, 1.25X1=1.25. 
 Ratio of transverse sections -ft ^-^=.6. 
 By Table 11, the value corresponding to ti=.6, the 
 
 {* - - f* 
 
 ratio of transverse sections, is =0.208; whence 
 
 c 
 
 c n =1.208c. 
 
 Substituting in last equation the value of c=.609. 
 <V=1.208X.609=.736. -Ana. 
 
PBACTICAL HYDRAULICS. 97 
 
 Ex. 55. An orifice 2 feet square in the end of a 
 conduit, AG FE, Fig. 16, 2.5 feet square, 6 feet long, 
 under a head of 5 feet on center of water, nominally 
 still in tank, B A D C, what is the discharge in cubic 
 feet per second? 
 
 Gal. 1st. By Table 7, the coefficient of perfect con- 
 traction applicable to an orifice 1 foot wide, 2 feet 
 high, under a head of five feet of water nominally 
 still, is c=.612. 
 
 By Rules 28 and 29: 
 
 Part suppressed (both ends) 2 + 2=4 feet. 
 
 Entire perimeter (tabulated orifice) 1 -f 1 + 4=6 feet. 
 
 Ratio of entire perimeter to part suppressed =f, 
 0.143 times this ratio; 0.143X|=.0953. 
 
 Sum of 1 and this product= 1.0953. 
 
 This sum, multiplied by tabulated coefficient, 
 
 1.0953X.612=.670. 
 
 0.143 times the product of the coefficient of perfect 
 contraction, and the ratio of the entire perimeter of 
 the orifice 1 foot wide to the part suppressed, divided 
 by the breadth of the given orifice, 
 
 0.143X. 612X1-2=0.029. 
 
 Coefficient due given orifice (2 feet square) under 
 head of still water: 
 
 Cfc =.670 .029=. 641. 
 Area of given orifice, 2X2=4 square feet. 
 
98 PRACTICAL HYDRAULICS. 
 
 Area of cross section of conduit, 2.5X2.5=6.25 
 square feet. 
 
 Ratio of transverse sections, n=^ T =.64. 
 
 By Table 11, the value corresponding to 71=. 6 5 
 (nearest to .64), the ratio of transverse sections is 
 
 c 
 
 By interpolation between values corresponding to 
 n .60, and n=.65, there results approximately: 
 
 ^ _ /i 
 
 =.234; whence, 
 
 c n =1.234c. 
 
 Substituting the value of c&=.641, as before found, 
 for c in the last equation, and there results: 
 
 <V=1.234X.641=.791. 
 
 By Table 6, square root of head i/ 5 =2.236. 
 
 By Rule 27, Q=. 791X8.025X4X2.236=56.78 cu- 
 bic feet. Ans. 
 
 Gal. 2d. By Table 7, the discharge found for an 
 orifice 1 foot wide, 2 feet high, under a head of 5 feet, 
 is=21.98 cubic feet per second. 
 
 If it be assumed, that for practical purposes, the 
 discharge through the given orifice 2 feet square, in 
 Example 55, will be proportionate to the tabulated 
 discharge, there will result: 
 
 Q=21.98X2=43.96 cubic feet per second. 
 
 By comparison, it is seen the result by Cal. 1 ig 
 
PRACTICAL HYDRAULICS. 99 
 
 nearly 30 per cent greater than that obtained by 
 Gal 2d. 
 
 This discrepancy seems to illustrate the necessity of 
 careful investigation in essaying the determination of 
 problems of practical hydraulics, however tedious the 
 process may be. 
 
 COEFFICIENT OF THE FLOW OF WATER THROUGH A 
 VERTICAL RECTANGULAR ORIFICE, UNDER A HEAD IN 
 MOTION. 
 
 FIG. 17. 
 
 The case in which the head of water is in motion, 
 occurs for the most part in open channels. In Fig. 
 17, ABCD represents a vertical section lengthwise 
 of a stream of water in an open channel. B C a dam 
 across the stream, in which is an orifice, E F in hight. 
 The dam is assumed to act as a restraint, but not 
 sufficient to sensibly affect the mean velocity of the 
 stream of water above it. 
 
 Let c= coefficient of discharge under a head of 
 water, nominally still. 
 
 c n = coefficient of discharge under a head of water 
 in motion, and dependent for its value on the ratio n. 
 
100 
 
 PEACTIOAL HYDKAULICS. 
 
 A=area of orifice. 
 
 A y =area of cross section of canal or channel. 
 
 A 
 
 n == ratio of these areas, not exceeding J. 
 A, 
 
 Weisbach gives as the result of his experiments, the 
 head being measured 1 meter=3.28 feet above the 
 dam. 
 
 n C f\ s* A + "I -*--*- I f\ n A <+ *> f~lf\A\ 
 
 =0.641 ! =0.641 n\ 
 
 c A 
 
 Whence, 
 
 (105) 
 
 To FIND THE COEFFICIENT OF DISCHAEGE UNDER A 
 HEAD OF WATER IN MOTION. 
 
 Rule 30. Add 1 to .641 times the square of the 
 ratio of transverse sections that of the canal to that 
 of the orifice and multiply this sum by the coeffi- 
 cient of discharge due the given orifice and given 
 head as though it were in still water. 
 
 Rule derived from formula (105J. 
 
 TABLE 12. 
 
 Corrections of the Coefficients of Plow Through Rectangular 
 Orifices Under a Head of Water in Motion. Weisbach. 
 
 n 10.05 
 Cn ~ c 00 
 
 0.10 
 0.006 
 
 0.15 [0.20 [0.25 
 0.014|o.026|0.040 
 
 0.30 
 0.058 
 
 0.35 
 0.079 
 
 0.40 (0.45 
 0.1030.130 
 
 0.50 
 0.160 
 
 c 1 
 
PEACTICAL HYDKAULICS- 101 
 
 Ex. 56. A dam containing a rectangular orifice 5 
 feet wide, 1 foot high, put across a flume 6 feet wide, 
 raises the water 5 feet in hight above the bottom of 
 the flume, and 3.5 feet above the lower edge of the 
 orifice. What is the discharge in cubic feet per sec- 
 ond ? 
 
 Gal. 1st. Half hight of orifice=.5 feet. 
 
 Head on center 3.5 .5=3 feet. 
 
 Area of orifice 5X1=5 square feet. 
 
 Cross section of flume, 6 X 5=30 square feet. 
 
 Ratio of transverse sections ^V~i- 
 
 Square of ratio () 2 =0.0278. 
 
 By Table 7. Coefficient of perfect contraction for 
 an orifice 1 foot wide, 1 foot high, under a head of 3 
 feet is=.605. 
 
 By Rule 28. Part (both ends) suppressed 1 + 1=2 
 feet. Entire perimeter (tabulated) =4 feet. Ratio of en- 
 tire perimeter to part suppressed, =.5 ; 0.143 times this 
 ratio; .143X .5=.0715. This sum multiplied by .605, 
 the coefficient of perfect contraction, gives the value 
 of the coefficient of partial contraction when the con- 
 traction of both ends is suppressed. 
 
 c.=1.0715X .605^.648. 
 
 By Rule 29. 0.143 times the product of the coefii- 
 cient of perfect contraction and the ratio of the entire 
 perimeter of the orifice 1 foot wide, to the part sup- 
 pressed, divided by the breadth of the given orifice; 
 
 0.143x.5X.605-^-5=.009. 
 
102 PRACTICAL HYDRAULICS. 
 
 Whence, 
 
 c 6= .648 .009=.639. 
 
 Substitute the value of c & =.639 for c, and the value 
 of the square of the ratio; () 2 =.0278 informula (105) 
 or employ Eule 30. 
 
 c n =(l + .64lX.0278)X.639=.650. 
 
 By Table 6: 
 
 Square root of given head of 3 feet=1.732. 
 By Rule 27: 
 
 #=.650X8.025X5xr.732=45.17 cubic feet per 
 second. Ans. 
 
 Gal 2d. By Table 7: The discharge found for an 
 orifice 1 foot wide, 1 foot high, under a head of 3 feet, 
 is 8. 41 cubic feet per second. If it be assumed that 
 for practical purposes the discharge through the given 
 orifice, 5 feet wide, 1 foothi^h, in Example 56, will be 
 proportionate to the tabulated discharge, there will 
 result : 
 
 8.41X5=42.05. Ans. 
 
 A discrepancy of 3.12 cubic feet, or 7 T 4 per cent. 
 FLOW OF WATER THROUGH SHORT TUBES. 
 
 Short tubes or adjutages are cylindrical, conical or 
 compound in form. 
 
 Cylindrical Tubes. The length of a cylindrical 
 
UNIVERSITY 
 
 OF vV "^ 
 
 PBAOTICAL HYDBAULICS. 103 
 
 tube being from 2.5 to 3 times its diameter, the mean 
 coefficient of flow through it as determined by the ex- 
 periments of Bidone, Eytelwine, D'Aubuisson and 
 Weisbach, is .815, while under otherwise similar cir- 
 cumstances, the mean coefficient of discharge through 
 an orifice in a thin plate is .615. The ratio of .815 to 
 .615 is 1.325; that is, the discharge through a short 
 tube of the given proportions (2.6 to 1), is 1.325 times 
 as much as the discharge through an orifice of equal 
 diameter in a thin plate. For practical purposes this 
 ratio may be assumed general in its application with- 
 out material error. 
 
 Hence, to find the coefficient for a short tube, hav- 
 ing given the coefficient of an orifice in a thin plate, 
 of equal diameter, and under an equal head. 
 
 Rule 31. Multiply the given coefficient of the ori- 
 fice by 1.325. 
 
 Ex. 57. The diameter of a short tube length to 
 diameter as 2.6 to 1 being 6 inches, and the head 
 9.06 feet, what is the coefficient of discharge? 
 
 CaL Given diameter 6"=. 5 feet. 
 
 By Table 9, coefficient due orifice, .5 feet diameter, 
 under 9.06 feet head=.602. 
 
 By Rule 31, . 602 X 1.325=. 798. Ans. 
 
 In case there are no experiments on which to rely, 
 the mean coefficient .815 is to be employed. 
 
 Let d== diameter in feet of a cylindrical tube whose 
 length is from 2.5 to 3 times the diameter. 
 
 &=head of water on center. 
 
 c=.8l5, coefficient of discharge. 
 
104 PEACTICAL HYDBAULICS. 
 
 a=.7854c? 2 , area of cross section of tube. 
 (2=discharge in cubic feet per second. 
 Substituting the values of a, c and h=h t in equa- 
 tion (92). 
 
 Q=.815X8.025X.7854dV*. (106) 
 
 Whence, 
 
 Q=5.137dV*. (107) 
 
 To find the flow of water through a cylindrical tube 
 whose length is from 2.5 to 3 times the diameter. 
 
 Rule 32. Multiply the square root of the head of 
 water on center by 5.137 times the square of the di- 
 ameter of the tube. 
 
 Rule 32 derived from equation (107). 
 
 Ex. 58. A tube being 3 inches in diameter and 8 
 inches long, and the head of water in the center being 
 5 feet, what is the discharge in cubic feet per second? 
 
 Gal. Diameter 3"=. 25 feet. 
 
 Square of diameter .25X .25=. 0625 square feet. 
 
 By Table 6: Square root of head, i/ 5 =2.236. 
 
 Q=5.137X.0625X2.236=.718 cubic feet.=^7is. 
 
 In case the proportion of length to diameter is 
 much changed, as 1 to 1, the coefficient of flow is 
 nearly the same as that for a thin plate, or if the 
 length be much increased over three times the diame- 
 ter, the coefficient .815 becomes diminished according 
 to the occurrence of friction of the sides of the 
 lengthened tube, which is termed a pipe. 
 
 Conical Tubes. Conical tubes are convergent or 
 
 divergent. The outer orifice being smaller than the 
 " 
 
PRACTICAL HYDRAULICS. 
 
 105 
 
 inner, the tube is convergent: but if larger, the tube 
 is divergent. 
 
 Convergent Tubes. Extensive experiments have been 
 made by D' Aubuisson and Castel on the flow of water 
 through convergent tubes. These were made with 
 tubes of various sizes and proportions; but mostly 
 with those .61 inches diameter at the discharging end, 
 1 .59 inches at the inlet end, and under a head of water 
 9.84 feet. The results of their experiments, as stated 
 by Weisbach, are given in the following table: 
 
 TABLE 13. 
 
 Coefficients of discharge and velocity for How through 
 conlcally convergent tubes. 
 
 Smaller diameter=.61 inches. 
 
 Angle of 
 Convergence 
 
 Coefficient 
 of Flow. 
 
 Coefficient 
 of Velocity. 
 
 Angle of 
 Convergence 
 
 Coefficient 
 of Flow. 
 
 Coefficient 
 of Velocity. 
 
 0' 
 
 0829 
 
 0.829 
 
 13 24' 
 
 0.946 
 
 0.963 
 
 1 36' 
 
 0.866 
 
 0.867 
 
 14 28' 
 
 0941 
 
 0.966 
 
 3 10' 
 
 0895 
 
 0.894 
 
 16 36' 
 
 0.938 
 
 0.971 
 
 4 10' 
 
 0.912 
 
 0.910 
 
 19 28' 
 
 0.924 
 
 0.970 
 
 5 26' 
 
 0.924 
 
 0.919 
 
 21 0' 
 
 0.919 
 
 0.972 
 
 7 52' 
 
 0.930 
 
 0.932 
 
 23 0' 
 
 0914 
 
 0.974 
 
 8 58' 
 
 0.934 
 
 0.942 
 
 29 58' 
 
 0.895 
 
 0.975 
 
 10 20' 
 
 0.938 
 
 0.951 
 
 40 20' 
 
 0.870 
 
 0.980 
 
 12 4' 
 
 0942 
 
 0955 
 
 48 0' 
 
 0.847 
 
 0.984 
 
 Ex. 59. The smaller diameter of a conically con- 
 vergent tube being 6 inches, the angle of convergence 
 
106 PEAOTICAL HYDEAULICS: 
 
 5 26' and the head of water on center 9.06 feet, 
 what is the flow of water in cubic feet per second? 
 
 Gal. Diameter 6 inches=.5 feet. 
 
 By Table 9 the coefficient corresponding to the 
 given diameter and head= .602, and coefficient cor- 
 responding to .61 inches on which Table 13 is based 
 =.618. 
 
 By Table 13 the coefficient corresponding to the 
 given angle of convergence 5 26' is=.924. 
 
 Ratio of coefficients .602-.618=.974. 
 
 Then coefficient of flow due the given diameter 
 .924X .974=.900. 
 
 And cross section of tube . 5 X. 5 X. 7854 1963 
 square feet. 
 
 By Table 6, square root of head=y / 9jo6=3.01 
 nearly. 
 
 By Rule 27, Q=.900X8.025X. 1963X3.01=4.27 
 cubic feet. Ans. 
 
 Divergent Tubes. Experiments show that the flow 
 of water through a shor divergent tube is similar to 
 that in a 'thin plate, the coefficients of which are 
 given in Table 9. In ordinary practice .62 is em- 
 ployed. 
 
 In case a vacuum is formed in a divergent tube, the 
 flow is greatly increased, so that it may then even ex- 
 ceed the theoretical flow due the force of gravity, 
 through an orifice in a thin plate, whose diameter is 
 equal to that of the smaller diameter of the divergent 
 tube ; in other words, its coefficient of flow becomes 
 greater than unity. The conditions effecting this re- 
 
PKACTICAL HYDRAULICS. 
 
 107 
 
 suit are a high velocity of flow in a tube of small di- 
 vergence, and whose length is several times its smaller 
 diameter. Thus, the smaller diameter of a divergent 
 tube being 1.32 inches, the length 9 times this diame- 
 ter:^ 1.88 inches, the included angle of the tube 
 equal to 5 6', and the head of water 2.89 feet, Ven- 
 turi found the coefficient of flow equal to 1.46, or 2.4 
 times that of an equal orifice in a thin plate. If the 
 entry end of an otherwise similar tube be bell- 
 mouthed in form, the coefficient of flow estimated for 
 the smaller diameter will evidently exceed that ob- 
 tained by Venturi. The principle of the formation of 
 a vacuum by flowing water at a high rate of velocity, 
 through a divergent tube, and thereby greatly increas- 
 ing the volume of discharge, was known to the 
 ancients. D'Aubuisson states that the application of 
 the principle, at a distance less than 52.5 feet from the 
 public conduits of Rome, by Roman citizens having 
 grants of water, was prohibited by Roman law . 
 
 TABLE 14. 
 
 Coefficients of the flow of water through divergent tubes. 
 
 
 Length of 
 
 
 
 Lenffth of 
 
 
 Angle. 
 
 Tube. 
 
 Coefficient. 
 
 Angle. 
 
 Tube. 
 
 Coefficient. 
 
 
 Feet. 
 
 
 
 Feet. 
 
 
 3 30' 
 
 0.364 
 
 0.93 
 
 5 44' 
 
 .193 
 
 .82 
 
 4 38' 
 
 1.095 
 
 1.21 
 
 i 10 16' 
 
 .865 
 
 .91 
 
 4 38' 
 
 1.508 
 
 1.21 
 
 10 16' 
 
 .147 
 
 .91 
 
 4 38' 
 
 1.508 
 
 1.34 
 
 14 14' 
 
 .147 
 
 .61 
 
 5 44' 
 
 0.57 
 
 1.02 
 
 
 
 
108 PRACTICAL HYDEAULICS. 
 
 Ex. 60. In a divergent tube the smaller diameter 
 being .61 of an inch, the length 1.508 feet, the angle 
 included between its sides 4 38', and the head on cen- 
 ter 2.89 feet, what is the volume of flow in cubic feet 
 for 24 hours? 
 
 Cat. Diameter .61 inches=.0508 feet. 
 
 By Table 14, mean coefficient of discharge (1 .21 + 
 1.34)^2=1.275. 
 
 Area of cross section of tube, . 0508 X- 0508 X. 7854 
 = .002027 square feet. 
 
 By Table 6, square root of head 1/^9= 1.7. 
 By Rule 27, volume of discharge per second, 
 Q=1.275X8.025X.002027X1.7=. 03525. 
 
 In 24 hours are 86,400 seconds; hence, .03525 X 
 86400=3046.05 cubic feet. Ans. 
 
 Ex. 61. In a compound tube, Fig. 18, the cylin- 
 drical part, P, is .0853 feet in diameter, 2.0605 feet in 
 length; the convergent part, C, .2559 feet long; the 
 divergent part, D, .7667 feet in length; and the head 
 2.3642 feet. What will be the discharge in 10 hours? 
 
 CaL By Table 15, the coefficient of flow due 
 CPD=.905. 
 
 Compound Tubes. Compound tubes are of various 
 forms. Eytelwine, as stated by J. T. Fanning, after 
 experimenting with cylindrical tubes of uniform 
 diameter and different lengths, placed between them 
 and the reservoir convergent tubes of the form of the 
 contracted vein, and renewed the experiments; then 
 
PRACTICAL HYDRAULICS. 
 
 109 
 
 added to the discharge end a divergent tube with 
 5 6' angle. Fig. 18. 
 
 D 
 
 FIG. 18. 
 
 In Fig. 18, C represents the conically convergent 
 part of the tube of the form of the contracted vein; 
 P, the cylindrical part of uniform diameter, but of 
 different lengths, and the conically divergent part 
 with 5 6' angle. The results obtained are given in 
 the following table: 
 
 '% < 
 
 TABLE 15. 
 
 Coefficients of the flow of water through compound tubes. 
 
 Head. 
 Feet. 
 
 Diameter 
 of P 
 Feet. 
 
 Liength of 
 Pin 
 Diameter. 
 
 Length of 
 Pin 
 Feet. 
 
 Coefficient 
 for P. 
 
 Coefficient 
 forCP. 
 
 Coefficient 
 forCPD. 
 
 2.3642 
 
 0.0853 
 
 0.038 
 
 0.0033 
 
 0.62 
 
 
 
 2.3642 
 
 0.0853 
 
 1.000 
 
 0.0853 
 
 .62 
 
 .967 
 
 
 2.3642 
 
 0.0853 
 
 3.000 
 
 0.2559 
 
 .82 
 
 .943 
 
 1.107 
 
 2.3642 
 
 00853 
 
 12.077 
 
 1.0302 
 
 .77 
 
 .870 
 
 .978 
 
 2.3642 
 
 0.0853 
 
 24.156 
 
 2.0605 
 
 .73 
 
 .803 
 
 .905 
 
 2.3642 
 
 0.0853 
 
 36.233 
 
 3.0907 
 
 .68 
 
 .741 
 
 .836 
 
 2.3642 
 
 0.0853 
 
 48.272 
 
 4.1176 
 
 .63 
 
 .687 
 
 ;762 
 
 2.3642 
 
 0.0853 
 
 60.116 
 
 5.1479 
 
 .60 
 
 .648 
 
 .702 
 
110 PBACTICAL HYDEAULICS. 
 
 Area of cross-section of tube P, .0853X-0853X 
 .7854=. 005761 square feet. 
 
 By Table 6, the square root of head 1/21*642= 1.5 4 
 nearly. 
 
 By Rule 27, Q=.905X8.025x.0057X1.54=.06367 
 cubic feet per second. 
 
 In 10 hours are 3600X10=36,000 seconds; hence, 
 .06367X36000=2292.07 cubic feet. Ans. 
 
 Divergent and Compound Tubes. These tubes sel- 
 dom find a place in practice. The lessons, however, 
 which they teach, are of interest, and serve to stimu- 
 late the vigilance of the engineer, lest irregularities 
 occurring from design or otherwise, shall elude his ob- 
 servation in matters of importance. 
 
PRACTICAL HYDRAULICS. 
 
 Ill 
 
 FLOW OF WATER THROUGH PIPES. 
 
 'p^f^r 
 
 :Tx~ft 
 
 3) gggZk-J* ft 
 
 FIG. 19. 
 
 The flow of water through a pipe is estimated to 
 begin at a point where the stream, after contraction, 
 expands so as to fill the pipe, as at F G, Fig. 19. 
 
 The part, D F G ; performing the office of a short 
 tube, is, as hereinbefore shown, from 2 .6 to 3 times the 
 diameter of the pipe. 
 
 The total head, AD, consists of three parts: AB, 
 which generates the velocity ; B C, which overcomes 
 the resistance of entry; and C D, which overcomes all 
 resistances in the pipe, FGE. CE is termed the 
 hydraulic gradient, and is such, if the pipe is running 
 
112 PRACTICAL HYDRAULICS. 
 
 full, the water will rise to this grade through tubes, 
 as a b, cd and ef. 
 
 Short and Long Pipes. A pipe, exclusive of the 
 tube portion described, in case its length does not ex- 
 ceed a thousand times its diameter, is termed a short 
 pipe, and in case its length exceeds a thousand times 
 its diameter, is termed a long pipe. 
 
 Let, in Fig. 19: 
 
 h=A~D, the total head. 
 
 h v ==A.l$, the velocity head, or head necessary to 
 generate the velocity v. 
 
 h e ~B C, the entry head, or head necessary to over- 
 come the resistances of entry. 
 
 h^=h v +h e = A C, the inlet head, or head necessary 
 to generate the velocity, v, in the pipe, and to over- 
 come the resistances of entry. 
 
 h f =C D, the head necessary to overcome the resist- 
 ances within the pipe. 
 
 v=the measured velocity of discharge. 
 
 ^=the theoretical velocity due the head h t =A. C. 
 
 c=the coefficient of flow in a short tube (length to 
 diameter as 3 to 1), as determined by experiment. 
 
 c v =c, the coefficient of velocity, as the stream, 
 after contraction, fills the pipe. 
 
 c e the coefficient of entry. 
 
 c y a variable coefficient for the resistances within 
 pipes, as determined by experiment. 
 
 c=internal diameter of pipe. 
 j=perimeter or internal contour of pipe. 
 
 a=area of cross section of pipe. 
 
PEACTICAL HYDRAULICS. 113 
 
 =length of pipe. 
 s=sine of slope. 
 r=hydraulic mean radius. 
 /amount of resistances to flow in the pipe. 
 w= weight of water discharged during the time of 
 resistance to its flow. 
 Then there results: 
 
 Equation of total head, h=h v + h e + h f . (108) 
 
 Equation of entry head, h e =h i h v . (109) 
 
 By equation (8), velocity head, ^ p = . 
 
 &9 
 
 ?; 2 
 
 By equation (8), inlet head, h i = f ^. (Ill) 
 
 49 
 
 Equation of theoretical velocity due inlet head, 
 
 *,= (112) 
 
 Substituting the value of v ( of (112) in (111), 
 
 Substituting the values of h t of (113), and of h v of 
 (110) in (109), 
 
 (114J 
 
 Putting c .= 
 
 | 1^1 (U5) 
 
114 PRACTICAL HYDRAULICS. 
 
 Substituting c e for J 5 1 j- in (114), 
 K J 
 
 /> -1 2 
 
 (116) 
 
 The work performed by the weight, w, falling ver- 
 tically by the force of gravity through the distance, 
 hj, in one second, is 
 
 ~F=wh f , "foot pounds." (117) 
 
 Experiments show that the amount of resistances 
 occurring from friction of the internal surfaces of a 
 pipe, and from other causes, varies nearly as the s'quare 
 of the velocity, v. 
 
 Experiments also show that the amount of resist- 
 ances increases directly as the length, I, of the pipe, 
 and inversely as its diameter, d, or hydraulic mean 
 
 ,. a 
 
 radius, r= . 
 P 
 
 The work performed by the force of gravity, in 
 overcoming these resistances, so as to effect the dis- 
 charge of the weight, w, of water with the velocity, 
 v, per second, as first proposed by Chezy, and sub- 
 quently adopted by most authors on hydraulics, is: 
 
 a 
 
 in which c f is a variable coefficient whose values are 
 determined by experiment. 
 
PBACTICAL HYDEAULICS, 115 
 
 v 2 
 Substituting the value of h v =- of (HO) in (118), 
 
 and equating (117) and (118), 
 
 Dividing (119) by w, V=- ( 12 ) 
 
 Substituting the values of h v of (HO), h e of (116), 
 and h f of 120, in (108), 
 
 ' 
 
 Factoring (121), fc=| 1 + c +^}^- 
 
 Transposing (122) with respect to v, 
 I tgh U 
 
 a n ( 2 d 
 Hydraulic radius, r== ~ == ~ == ' 
 
 Substituting I for of (124) in (123), 
 
 U 
 
 ,rrw^ (125) 
 
 Factoring (125), 
 
 H 1 U 
 
 - 7l (126) 
 
 Under the heading, "flow of water through short 
 
116 PRACTICAL HYDRAULICS. 
 
 tubes," the value of the mean coefficient of discharge 
 has been shown to be c='.815. But c==c w =.815; that 
 is, the coefficient of flow at the inlet orifice of a short 
 tube, is equal to the coefficient of velocity in the pipe, 
 estimating the beginning at that point, where the 
 stream, after contraction, expands so as to fill the pipe . 
 
 Substituting the value of c t; =.815 in equation (115), 
 
 c e =.505. (127) 
 
 Substituting the value of c =.505 in (125), 
 
 r J 
 
 In determining the velocity of flow in a pipe, whose 
 length exceeds a thousand times the diameter, the 
 value of (l+c e ) (in 125), being small in comparison 
 
 with the value of is usually omitted as insignifi- 
 
 cant;* or, more direct, let equation (120) be transposed 
 
 1 
 
 with respect to v, and - be substituted for -: 
 
 x (129 ) 
 
 c f 
 
 Substituting s=~, the sine of slope, CED, in 
 (129), 
 
 Hydraulicians have given different empirical formu- 
 las for the determination of the values of c f . Thus: 
 Weisbach, assuming that the resistance of friction in- 
 
PEACTICAL HYDRAULICS. 
 
 117 
 
 creases simultaneously as the square, and as the square 
 root of the cube of the velocity finds as follows: 
 
 3=(4 c,)=0.01439+ <L.<Li;u.5.. (131) 
 
 V v 
 
 This formula, as claimed by its author, agrees more 
 accurately with observations than do those of the older 
 hydraulicians. In the experiments from which it was 
 derived, the velocity varied from 0.14 feet to 15.25 
 feet per second, and the pipes from 1.06 inches to 
 5.31 inches in diameter. 
 
 H. Darcy's formula for velocity, resolved with re- 
 spect to this coefficient, gives: 
 
 This formula was deduced from very extensive ex- 
 periments. In. these the variation, with respect to 
 velocity, was from 0.29 feet to 16.24 feet per second, 
 and with respect to diameters of pipes, from 3 inches 
 to 20 inches nearly. 
 
 Weisbach remarks of this formula, that it "is not 
 sufficiently accurate for small velocities." 
 
 J. T. Fanning's "Series of Coefficients of Flow (m)" 
 [ m c/ ] "of water in clean pipes, under pressure, at 
 different velocities, and in pipes of different diame- 
 ters" from which the following table is extracted 
 seem more simple and comprehensive on this subject 
 than can well be rendered in a single formula. These 
 coefficients are deduced from experiments, in which 
 the variation in velocities was from 0.18 feet to 46.7 
 feet per second, and in which the diameters of the 
 pipes were from \ inch to 3 feet. 
 
118 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 16. 
 
 Coefficients of resistance to the Plow (cy) of Water in Clean 
 Pipes. Extracts from Fanning. 
 
 Velocity. 
 Feet per 
 Second. 
 
 DIAMETERS. 
 
 i-iuch 
 Coef. 
 
 1-inch 
 Coef. 
 
 3-inch 
 Coef. 
 
 6-inch 
 Coef. 
 
 12-inch 
 Coef 
 
 24-inch 
 Coef. 
 
 48 -inch 
 Coef. 
 
 96-inch 
 Coef. 
 
 .1 
 
 .0150 
 
 .0119 
 
 .0080 
 
 .0073 
 
 .0067 
 
 
 
 
 .2 
 
 .0143 
 
 .0116 
 
 .0079 
 
 .0072 
 
 .0066 
 
 
 
 
 .3 
 
 .0137 
 
 .0113 
 
 .0078 
 
 .0072 
 
 .0066 
 
 .0055 
 
 
 
 .4 
 
 .0133 
 
 .0110 
 
 .0078 
 
 .0071 
 
 .0065 
 
 .0054 
 
 
 
 .5 
 
 .0128 
 
 .0107 
 
 .0077 
 
 .0071 
 
 .0065 
 
 .0054 
 
 .0040 
 
 
 .6 
 
 .0124 
 
 .0104 
 
 .0077 
 
 .0070 
 
 .0064 
 
 .0054 
 
 .0040 
 
 .0029 
 
 .7 
 
 .0120 
 
 .0102 
 
 .0076 
 
 .0070 
 
 .0064 
 
 .0053 
 
 .0040 
 
 .0029 
 
 .8 
 
 .0116 
 
 .0100 
 
 .0075 
 
 .0069 
 
 .0063 
 
 .0053 
 
 .0040 
 
 .0029 
 
 .9 
 
 .0113 
 
 .0097 
 
 .0075 
 
 .0069 
 
 .0063 
 
 .0053 
 
 .0040 
 
 .0029 
 
 1.0 
 
 .0110 
 
 .0095 
 
 .0074 
 
 .0068 
 
 .0062 
 
 .0053 
 
 .0040 
 
 .0029 
 
 1.1 
 
 .0107 
 
 .0093 
 
 .0074 
 
 .0068 
 
 .0062 
 
 .0052 
 
 .0039 
 
 .0029 
 
 .2 
 
 .0104 
 
 .0091 
 
 .0073 
 
 .0067 
 
 .0062 
 
 .0052 
 
 .0039 
 
 .0029 
 
 .3 
 
 .0101 
 
 .0090 
 
 .0073 
 
 .0067 
 
 .0061 
 
 .0052 
 
 .0039 
 
 .0029 
 
 .4 
 
 .0099 
 
 .0088 
 
 .0072 
 
 .0067 
 
 .0061 
 
 .0051 
 
 .0039 
 
 .0028 
 
 .5 
 
 .0096 
 
 .0087 
 
 .0072 
 
 .0066 
 
 .0061 
 
 .0051 
 
 .0039 
 
 .0028 
 
 .6 
 
 .0094 
 
 .0085 
 
 .0072 
 
 .0066 
 
 .0060 
 
 .0051 
 
 .0039 
 
 .0028 
 
 .7 
 
 .0092 
 
 .0084 
 
 .0071 
 
 .0066 
 
 .0060 
 
 .0051 
 
 .0039 
 
 .0028 
 
 .8 
 
 .0090 
 
 .0083 
 
 .0071 
 
 .0065 
 
 .0060 
 
 .0051 
 
 .0039 
 
 .0028 
 
 1.9 
 
 .0088 
 
 .0082 
 
 .0070 
 
 .0065 
 
 .0060 
 
 .0050 
 
 .0039 
 
 .0028 
 
 2. 
 
 .0086 
 
 .0081 
 
 .0070 
 
 .0065 
 
 .0059 
 
 .0050 
 
 .0038 
 
 .0028 
 
 2.25 
 
 .0084 
 
 .0079 
 
 .0069 
 
 .0064 
 
 .0059 
 
 .0050 
 
 .0038 
 
 .0028 
 
 2.5 
 
 .0080 
 
 .0077 
 
 .0068 
 
 .0063 
 
 .0058 
 
 .0049 
 
 .0038 
 
 .0028 
 
 2.75 
 
 .0078 
 
 .0075 
 
 .0068 
 
 .0063 
 
 .0058 
 
 .0049 
 
 .0038 
 
 .0028 
 
 3. 
 
 .0075 
 
 .0073 
 
 .0067 
 
 .0062 
 
 .0057 
 
 .0048 
 
 .0038 
 
 .0028 
 
 3.5 
 
 .0073 
 
 .0071 
 
 .0066 
 
 .0061 
 
 .0056 
 
 .0048 
 
 .0037 
 
 .0028 
 
 4. 
 
 .0072 
 
 .0070 
 
 .0065 
 
 .0061 
 
 .0055 
 
 .0047 
 
 .0037 
 
 .0027 
 
 5. 
 
 .0070 
 
 .0068 
 
 .0064 
 
 .0059 
 
 .0054 
 
 .0047 
 
 0037 
 
 .0027 
 
 6. 
 
 .0069 
 
 .0067 
 
 .0062 
 
 .0058 
 
 .0053 
 
 .0046 
 
 ,0036 
 
 .0027 
 
 7. 
 
 .0068 
 
 .0066 
 
 .0061 
 
 .0057 
 
 .0053 
 
 .0045 
 
 .0036 
 
 .0027 
 
 8. 
 
 .0066 
 
 .0065 
 
 .0060 
 
 .0056 
 
 .0052 
 
 .0045 
 
 .0036 
 
 .0026 
 
 9. 
 
 .0065 
 
 .0064 
 
 .0059 
 
 .0056 
 
 .0051 
 
 .0045 
 
 .0036 
 
 .0026 
 
 10. 
 
 .0064 
 
 .0063 
 
 .0058 
 
 .0055 
 
 .0051 
 
 .0044 
 
 .0035 
 
 .0026 
 
 12. 
 
 .0063 
 
 .0061 
 
 .0058 
 
 .0054 
 
 .0050 
 
 .0044 
 
 .0035 
 
 .0026 
 
 14. 
 
 .0062 
 
 .0061 
 
 .0057 
 
 .0053 
 
 .0049 
 
 .0043 
 
 .0035 
 
 .0026 
 
 16. 
 
 .0062 
 
 .0060 
 
 .0057 
 
 .0053 
 
 .0049 
 
 .0043 
 
 
 
 18. 
 
 .0062 
 
 .0060 
 
 .0057 
 
 .0053 
 
 
 
 
 
 20. 
 
 .0062 
 
 .0060 
 
 .0057 
 
 
 
 
 
 
 
 
 
 
 
PRACTICAL HYDRAULICS. 119 
 
 COMPARISON or THE VALUES OF THE COEFFICIENT 
 OF RESISTANCES, C ft AS FOUND BY WEISBACH, DARCY 
 AND FANNING. 
 
 Ex. 62. The velocity, in a clean pipe J foot di- 
 ameter, being 1 foot, what is the coefficient of resist- 
 ances ? 
 
 Gal. 1st. By Weisbach's formula (131), square root 
 of given velocity: 
 
 l/v=l. 
 
 Substituting the value of ]/v in (131), z4cc~ 
 0. 01439 +iL-OJ T T_i_5_5 ; whence, 
 
 c/ =.0079. Ans. 
 
 Gal. 2d. By Darcy's formula ( 132), hydraulic 
 mean radius r=5.= 1 ^ r . 
 
 Substituting value of r in (132), c =.0066. Ans. 
 
 Gal. 3. By Table 16, from Fanning's series of co- 
 efficients, cy .0074. Ans. 
 
 Ex. 63. The velocity in a clean pipe 2 feet di- 
 ameter, being 4 feet per second, what is the coefficient 
 of resistance? 
 
 Gal. 1st. By Weisbach, formula (131), square root 
 of given velocity: 
 
 1 /'U=: 1 /4=2. 
 
 Substituting value of i/v in (131), 2=4 c/^0.01439 
 + jL-_o_yjj>_5; whence, 
 
 Cf== .0057. Ans. 
 
120 PRACTICAL HYDRAULICS. 
 
 Cat. 2d. By Darcy's formula (132), hydraulic mean 
 radius r=J=-|=J.. 
 
 Substituting value of r in (132), 
 
 c/ =.0052. -Ans. 
 Gal 3d. By Table 16, Fanning's: 
 c,=.0047. Ans. 
 
 Ex. 64. The velocity in a clean pipe 4 feet diame- 
 ter, being 9 feet per second, what is the coefficient of 
 resistance? 
 
 Gal. 1st. By Weisbach's formula (131), square root 
 of velocity, 
 
 } /v= l /9=S. 
 
 Substituting value of ^v in (131), 2=4c/=0.01439 
 5 ; whence, 
 
 c/ --=.0050. Ans. 
 
 Gal. 2d. By Darcy's formula (132), hydraulic mean 
 radius r=.==l. 
 
 Substituting value of r in (132), 
 
 c / =.0051. Ans. 
 Gal. 3d. By Table 16, 
 
 c/ =.0037. Ans. 
 
 Ex. 65. The velocity in a clean pipe 8 feet diame- 
 ter, being 9 feet per second, what is the coefficient of 
 resistance? 
 
PRACTICAL HYDRAULICS, 12X 
 
 Cat, 1st, By Weisbach's formula (131), square root 
 of velocity, 
 
 /v=]/9:=3. 
 
 Substituting value of ]/v in (131), z=^4i ^=0. 01439 
 4-jL_o_yj_5_5; whence, 
 
 <V-=.0050. Ans. 
 
 Cal. 2d. By Darcy's formula (132), hydraulic mean 
 radius r=|-=f =2. 
 
 Substituting value of r in (132), 
 
 Gal. 3d. By Table 16, from Fanning's series of 
 coefficients, 
 
 c/^.0026. Ans. 
 
 Weisbach states that his formula for tta coefficient 
 of resistance (z) "is founded upon the assumption that 
 the resistance of friction increases at the same time 
 with the square and with the square root of the cube 
 of the velocity." He further says "that the values 
 from newer experiments show that the coefficient of 
 resistance (z) for the friction of water in tubes de- 
 creases not only as the velocity (v) increases, but also, 
 although more slowly, as the diameter of the pipe be- 
 comes greater." He omits, however, to amend his 
 formula so as to embrace this element. 
 
 The coefficient of resistance (c/ in our notation) as 
 deduced from Darcy's formula for velocity, decreases as 
 the diameter of the pipe increases. 
 
 An inspection of the results obtained by Darcy's 
 
122 PRACTICAL HYDEAULICS. 
 
 formula for Examples 64 and 65 would show that this 
 diminuition practically ceases, when the diameter ex- 
 ceeds 4 feet. Thus, for a pipe 4 feet diameter, the co- 
 efficient found is .0051, and for a pipe 8 feet diameter, 
 it is .0050. The results for large pipes, by Darcy's 
 formula, are not in conformity with those obtained by 
 later experiments. 
 
 By Table 16, from Fanning's series, the coefficients 
 under the imposed conditions, as seen, are .0037 for a 
 4-foot pipe, and .0026 for an 8-foot pipe. 
 
 It is to be noticed that, in general, the results of 
 
 later experiments with respect to the velocity of water 
 
 in large pipes closely approximate those tabulated by 
 
 Mr. Fanning in terms of the coefficient of resistance. 
 
 Thus F. P. Stearns, M. Am. Soc. C. E., in a paper 
 
 read before the American Society of Civil Engineers, 
 
 October 1, 1884, states that three experiments made 
 
 with the " Sudbury Conduit," a cast-iron pipe 4 feet 
 
 diameter and 1747 feet in length, coated with a coal tar 
 
 preparation, and in good condition, resulted as follows: 
 
 Mean velocity, 4.966 feet. 
 
 Mean coefficient of velocity, 142.11 feet. 
 
 Mean value of ri t 0.001221 feet. 
 
 Substituting the values of the mean velocity here 
 given, i>=4.966 feet, and the value of rirs~ 
 0.001221, in equation (130) and resolving with respect 
 to the coefficient of resistance, 
 
 C f=Q. 003188. 
 Referring to Table 16, we find the coefficient of re- 
 
PRACTICAL HYDRAULICS. 123 
 
 sistance corresponding to a velocity of 5 feet (nearest 
 approximate to 4.966 feet) in a pipe 4 feet diameter, 
 
 (y=0.0037. 
 
 Substituting the value of ^=0.0037 in Eq. (130), 
 and resolving with respect to the coefficient of velocity, 
 c, as employed by Mr. Stearns, there results: c=131.9, 
 as compared with 142.11. 
 
 Mr. Stearns further states, in the paper referred to, 
 as follows : 
 
 "The experiments of Hamilton Smith, Jr., M. Am. 
 Soc. C. E. (transactions for April, 1883), give curves 
 of coefficients for pipes up to 30 inches diameter. Ex- 
 tending these curves would give for a 48 -inch pipe, 
 with velocity of 5 feet per second, a coefficient of 128, 
 as compared with 142.1 above." 
 
 " The experiments of S. N. Tubbs, M. Am. Soc. C. 
 E., on pipes of 2 and 3 feet diameter, would give by 
 extending the curves a coefficient about the same as 
 that in the average above." 
 
 "The experiments of Dr. Lampe, at Danzig, give 
 results somewhat higher than those of Mr. Hamilton 
 Smith, Jr." 
 
 The formulas applied to Example 3, to find the 
 values of the coefficient of resistance for a 3-inch pipe, 
 give by 
 
 Weisbach, c/=0.0079. 
 
 And by Darcy, c,=0.0066. 
 
 The value by Fanning is <?/=(). 0074. 
 
 We thus, perceive concerning the values of this co- 
 
124 PRACTICAL HYDRAULICS. 
 
 efficient for small pipes, that Fanning differs less than 
 7 per cent from Weisbach, whose experiments were 
 confined, as hitherto shown, to this class of pipes, 
 while Darcy differs nearly 20 per cent from him. It 
 must not be understood, however, that these differences 
 obtain to the same extent in estimating the velocity. 
 Reference to Eq. (130) shows that the coefficient of 
 velocity depends not upon the direct value of the co- 
 efficient of resistance, but upon the square root of it. 
 So that in estimating the velocity or quantity of flow, 
 the real difference, in the case cited, between the re- 
 sults of Weisbach and Fanning would amount to only 
 about 3J per cent. 
 
 INTERPOLATION IN TABLE 16. 
 
 The general formula of Darcy furnishes a simple 
 means of interpolation with respect to the coefficients 
 of resistance, when the difference between the ex- 
 tremes is not large. Thus the general formula is: 
 
 (133) 
 
 In which r represents the mean hydraulic radius m 
 and n auxilliary numbers whose values are determined 
 by substituting the tabulated values of r and c f in 
 (133) between which latter intermediate values are 
 sought corresponding to different values of r'. The 
 values thus found for m and n are employed as con- 
 
PRACTICAL HYDRAULICS. 125 
 
 stants in the determination of these intermediate 
 values of q/>. 
 
 For example, the velocity of flow being 3 feet per 
 second, let it be required to determine the values of c/ 
 due the diameters 30, 36, 42 and 44 inches, between 
 24 and 48 inches. 
 
 The hydraulic mean radii of 24 inches and 48 inches, 
 are respectively |=| and =1. By Table 16, the 
 value of Cf due a pipe 24 inches diameter is .0048, and 
 that due a pipe 48 inches diameter is .0038. Substi- 
 tuting these values of r and c/ in Eq, (133), there re- 
 sults ; 
 
 (134) 
 
 and m + 7i=. 0038. (135) 
 
 Subtracting (135) from (134), 
 
 7i=.0010 ; (133) 
 
 Substituting value of n of (136) in (134), 
 
 m=.0028. (137) 
 
 The hydraulic mean radii corresponding to the 
 given intermediate diameters, are respectively f , f , | 
 and If 
 
 Substituting the values of m=.0028, n.QQlQ, and 
 r=f , |, f and \\ in (133), there results for the given 
 diameters : 
 
 30", C/ =.0028+ I (.0010)=. 0044; (138) 
 
 36", c/=.0028+ 4 (.0010)=. 0041; (139) 
 
126 PRACTICAL HYDRAULICS: 
 
 42", c f =. 0028+ 4 (.0010)=.0039; (160) 
 
 44", c/=.0028 + H- (.0010)^.0039; (141) 
 
 to be interpolated as required. 
 
 Thus, by interpolation, may Table 16 be completed 
 with the approximate values of the coefficient (c/) of 
 resistance. 
 
 To FIND THE VELOCITY OF WATER FLOWING THROUGH 
 SHORT PIPES. 
 
 Rule 33. Extract the square root of 64.4 times 
 the given head (total head) in feet, divided by 1.505, 
 increased by the product of the length of the pipe in 
 feet, and the coefficient of resistance coefficient found 
 in and computed from Table 16 due the given di- 
 ameter, divided by the hydraulic mean radius. 
 
 Rule 33 corresponds Eq. (128). 
 
 EXAMPLES AND CALCULATIONS WITH RESPECT TO THE 
 
 FLOW OF WATER THROUGH SHORT PIPES. 
 
 Ex. 66. A pipe being 1 foot in diameter, 10 feet 
 long, and the head of water being one hundred feet, 
 what will be the velocity of flow per second? 
 
 Gal. As the pipe is very short, it is evident that 
 a large portion of the head will be expended in gener- 
 ating the velocity. Let it be assumed that not less 
 
PEACTICAL HYDRAULICS, 127 
 
 than one-half the total head will be so expended. In 
 which case the velocity will exceed 50 feet per sec- 
 ond. Turning to Table 16, we perceive that the co- 
 efficient of resistance due a velocity of 16 feet in a 
 pipe 1 foot diameter is .0049, and that the decrease of 
 the preceding coefficients in the 1 foot diameter col- 
 umn, corresponding to the increase of velocity, is very 
 small. Let, then, the least coefficient in column be 
 taken, c/=.0049. 
 
 Hydraulic mean radius r=J. 
 
 Substituting these values of c/=.0049, r=J; also, 
 the values of 2^=64.4, h=lOO feet, and 1= 10 feet in 
 Eq. (128) ; or, in other words, apply Rule 33 : 
 
 64.4X100 
 
 Whence, v=6l.53 feet. Ans. 
 
 It will be remembered that equation (128), or Rule 
 33, is to be employed in finding the velocity when the 
 given length of the pipe is less than one thousand 
 times its diameter. 
 
 Ex. 67. The head being 10 feet, what will be the 
 velocity of water flowing through a pipe 2 feet in di- 
 ameter and 1000 feet long? 
 
 Cat. Assume by way of trial that the velocity will 
 be 9 feet per second. 
 
 By Table 16, the coefficient of resistance due 9 feet 
 velocity in a 2-foot pipe is c/=.0045. 
 
 Hydraulic mean radius r=|=J. 
 
 Substituting the values of c/=,0045, r=J, h=10, 
 
128 PRACTICAL HYDRAULICS. 
 
 =100, and 2</=64.4, in equation (128); or applying 
 Bule 33: 
 
 64.4X10 U 
 
 L.505+.0045X1000-5-JJ 
 
 Whence, v=7.83 feet, trial result. 
 
 Turning again to Table 16, we find that, corres- 
 ponding to 8 feet velocity nearest approximate to our 
 trial result 7.83 feet, the coefficient of resistance is 
 c/=,0045, the same as employed in our calculation. 
 
 Therefore we have: 
 
 v=7.83 feet. Ans. 
 
 Had the tabulated coefficient of resistance corres- 
 ponding to the velocity found by trial, not have been 
 equal to, or closely approximate to, that employed in 
 our calculation, it would have been necessary to repeat 
 the operation, using this new coefficient. 
 
 Ex. 68. The head being 20 feet, the pipe 4 feet 
 diameter, and 4,000 feet long, what is the velocity of 
 flow per second? 
 
 Gal. 1st. Assume the velocity for the purpose of 
 trial 9 feet. Then: 
 
 By Table 16, c^O. 035. 
 
 Hydraulic mean radius r= 1. 
 
 Substituting the values of c/- .0036, r=l, A=20 
 feet, =4,000 feet, and 2(/=64.4 in Eq. (128); or ap- 
 plying Rule 33: 
 
 f 64.4X20 U 
 
 ~"\1.505 + .0036X4000-5-l j 
 
 Whence, v=8.999 feet. Ans. 
 
PBACTICAL HYDRAULICS. 129 
 
 Gal. 2. Employing formula (130), which is for a 
 long pipe, there results: 
 
 _( 64.4X20 } 
 ~ "\.0036X4000-f-lj 
 
 Whence; v=9A2S feei.Ans. 
 
 Comparing these results, the difference is seen to be 
 .429 feet velocity per second. It appears quite evi- 
 dent, then, that 1.505, the first term in the denomina- 
 tor of the right hand member of Eq. (128) an 
 equation for velocity of water in short pipes cannot 
 be omitted, where accuracy is required, even if the 
 length of the pipe, as in the given example, is 1,000 
 times the diameter. 
 
 Ex. 69. The head being 12 feet, the pipe 15 inches 
 diameter, 1,200 feet long, let it be required to find: 
 
 The velocity of flow per second; 
 
 The discharge in cubic feet per second; 
 
 The loss of head due velocity; 
 
 The loss of head due resistance of entry; 
 
 The head expended in overcoming the resistances 
 within the pipe; 
 
 The sine of slope; 
 
 The fall per mile due resistances in pipe ; 
 
 And the total fall per mile. 
 
 Cal. Assume by way of trial the velocity of flow= 
 6 feet per second. 
 
 By Table 16, the coefficients of flow due a velocity 
 of 6 feet in a 12-inch pipe, are c/=.0053; and in a 
 24-inch pipe, c/=.0046. 
 
130 PRACTICAL HYDRAULICS. 
 
 The hydraulic mean radii are as follows: 
 
 Forl2", r=J; 
 And for 24", r=|=J. 
 
 Substituting the values of <y=.0053, and of r=J, 
 in Eq. (133); also, the values of c/= .0046, and of r= 
 J in the same equation, there results: 
 
 m+4n=.0053. (a) 
 
 m + 27i=. 0046. (6) 
 
 Whence, 7i=.00035. (c) 
 
 Substituting value of (ri) in Eq. (a), 
 
 m=.0039. (c?) 
 
 Substituting the values of m of (d), and of -n. of (c) 
 inEq. (133), 
 
 The hydraulic mean radius due 15" r= r 5 K . 
 Substituting this value of r= T 6 7 in Eq. (e), 
 
 c/-=. 0039 + .00035-s- T 8 T =. 0050. (f) 
 
 Substituting the values of <y=:0050, r = -^j-, ^^=12 
 feet, i = l,200 feet, and 2^-64.4, in Eq. (128), or 
 applying Rule 33, 
 
 .505+. 0050 X 1 200-5- ^J 
 Whence, ^=6.11 feet. Ans. (h) 
 
PRACTICAL HYDRAULICS. 131 
 
 The value of v of Eq. (h) is so near the assumed 
 value, 6 feet, it will be unnecessary to repeat the oper- 
 ation for finding a nearer approximate to the true 
 value. 
 
 Area of cross-section of given pipe, 
 
 a=(<-f ) 2 X. 7854=1.227 square feet. (i) 
 
 Quantity of flow is equal to the product of the area 
 of cross-section and the velocity; hence, 
 
 Q=at;=1.227x6.11=7.5 cubic feet Ans. (j) 
 
 To find the loss of head due velocity, substitute the 
 value of v of Eq. (h), and the value of 2$ =64. 4, in 
 Eq. (110), 
 
 h= ( l'l^= .58 feet. Ans. (k) 
 
 To find the loss of head due the resistance of entry, 
 substitute the value of c e -=.505, of Eq. (127), and the 
 value of V=-=.58, of Eq. (k), in (116), 
 
 h e =.5SX .505-=. 29 feet. Ans. (I) 
 
 To find the head expended in overcoming the re- 
 sistances within the pipe, substitute the values of h v = 
 .58 of Eq. (k), of h e =.2S of Eq. (I), and of h= 
 12 feet, the total or given head in Eq. (108), and 
 transposing: 
 
 ^=12 .58 .29 = 11.13 feet. Ans. (m) 
 
 To find the sine of slope, divide the head, hf, ex- 
 pended in overcoming the resistances in the pipe by 
 the length of the pipe. Thus, the sine of the angle of 
 slope, C E D, Fig. 19, is in the given example: 
 
132 PRACTICAL HYDRAULICS. 
 
 8=-^=ftjj=-. 009275. Ans. (n) 
 
 To find the fall per mile: 
 
 Let F fall in feet per mile: 
 
 1 mile=5280 feet. (o) 
 
 Then 1 : .009275 : : 5280 : F. (p) 
 
 Whence, F =-48.972 feet. Ans. (q) 
 
 When 1.505, the leading term in the denominator 
 of Eq. (128), or, in other words, when Eq. (130) is em- 
 ployed instead of Eq. (128), it is assuming that in 
 Fig. 19, A E is the slope or hydraulic gradient instead 
 of C E, which is erroneous. 
 
 But when the pipe is very long in comparison with 
 its diameter, the error is insignificant. 
 
 Let F m the entire fall per mile. 
 
 Substitute the values of /^ .58 of Eq. (&), and h e 
 .29 of Eq. (I) in Eq. (109), 
 
 Then F w =F+^48.972+.8T=49.842 feet. Ans. 
 
 FLOW OF WATER IN LONG PIPES. 
 
 For a given or determined velocity, the inlet head 
 hi, remains constant, regardless of the length of the 
 pipe. 
 
 In general, if F denote the fall per mile fall re- 
 
PEACTICAL HYDKAULICS. 133 
 
 quisite to overcome the resistances within the pipe 
 (n) the number of miles, length of the pipe, and F t the 
 total fall, then will 
 
 In the computation of the following table for the 
 velocity and quantity of flow in long pipes, Eq. (130), 
 
 - 
 
 in which the value of s- is given, has been em- 
 
 ployed. The inlet head, h it if required, will be de- 
 termined from the velocity. 
 
134 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 17. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes due 
 given Slopes and Diameters. 
 
 
 Sine 
 
 DIAMETERS. 
 
 Fall per 
 
 of 
 
 
 
 Slope 
 
 "=.03125 feet. 
 
 ^"=.04167 ft. 
 
 f"=.0625 ft. 
 
 1"=.0833 ft. 
 
 Mile. 
 
 h 
 
 Velo'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Feet. 
 
 = f 
 
 Ft per 
 
 Feet per 
 
 ?t. per 
 
 Ft. per 
 
 Ft. per 
 
 Ft. per 
 
 Ft. per 
 
 Ft. per 
 
 
 t 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 21.12 
 
 .004 
 
 
 
 
 
 
 
 
 
 26.40 
 
 .005 
 
 
 
 
 
 
 
 
 
 31.68 
 
 .006 
 
 
 
 
 
 
 
 
 
 36.96 
 
 .007 
 
 
 
 
 
 
 
 
 
 42.24 
 
 .008 
 
 
 
 
 
 
 
 .04 
 
 .0057 
 
 47.52 
 
 .009 
 
 
 
 
 
 
 
 .13 
 
 .0062 
 
 52.80 
 
 .010 
 
 
 
 
 
 1.03 
 
 .0032 
 
 24 
 
 .0068 
 
 63.36 
 
 .012 
 
 0.698 
 
 0.00054 
 
 0.89 
 
 .0012 
 
 1.14 
 
 .0035 
 
 .43 
 
 .0078 
 
 73.92 
 
 .014 
 
 0.763 
 
 0.00059 
 
 0.91 
 
 .0013 
 
 1.23 
 
 .0038 
 
 .54 
 
 .0084 
 
 84.48 
 
 .016 
 
 0.826 
 
 0.00063 
 
 0.99 
 
 .0014 
 
 1.34 
 
 .0041 
 
 .64 
 
 .0089 
 
 95.04 
 
 .018 
 
 0.888 
 
 0.00069 
 
 1.05 
 
 .0014 
 
 1.45 
 
 .0045 
 
 .76 
 
 .0096 
 
 105.6 
 
 .02 
 
 0.947 
 
 0.00073 
 
 1.10 
 
 .0015 
 
 1.52 
 
 .0047 
 
 .81 
 
 .0099 
 
 158.4 
 
 .03 
 
 1.18 
 
 00090 
 
 1.44 
 
 .0020 
 
 1.92 
 
 .0059 
 
 2.28 
 
 .0125 
 
 211.2 
 
 .04 
 
 1.38 
 
 0.0011 
 
 1.77 
 
 .0024 
 
 2.30 
 
 .0071 
 
 2.73 
 
 .0149 
 
 264.0 
 
 .05 
 
 1.58 
 
 0.0013 
 
 2.04 
 
 .0028 
 
 2.60 
 
 .0080 
 
 3.05 
 
 .0167 
 
 316.8 
 
 .06 
 
 1.79 
 
 0.0014 
 
 2.31 
 
 .0032 
 
 2.85 
 
 .0087 
 
 3.40 
 
 .0186 
 
 369.6 
 
 .07 
 
 1.98 
 
 0.0015 
 
 2.49 
 
 .0034 
 
 3.10 
 
 .0095 
 
 3.64 
 
 .0199 
 
 422.4 
 
 .08 
 
 2.13 
 
 0.0016 
 
 2.68 
 
 .0037 
 
 3.30 
 
 .0101 
 
 3.92 
 
 .0214 
 
 475.2 
 
 .09 
 
 2.36 
 
 0.0018 
 
 2.85 
 
 .0039 
 
 3.54 
 
 .0109 
 
 4.18 
 
 .0228 
 
 528.0 
 
 .10 
 
 2.49 
 
 0.0019 
 
 3.04 
 
 .0041 
 
 3.73 
 
 .0114 
 
 4.44 
 
 .0242 
 
 633.0 
 
 .12 
 
 2.76 
 
 0.0021 
 
 3.32 
 
 .0045 
 
 4.18 
 
 .0128 
 
 4.90 
 
 .0268 
 
 739.2 
 
 .14 
 
 3.04 
 
 0.0023 
 
 3.46 
 
 .0047 
 
 4.50 
 
 .0138 
 
 5.29 
 
 .0289 
 
 844.0 
 
 .16 
 
 3.30 
 
 0.0025 
 
 3.84 
 
 .0052 
 
 4.83 
 
 .0148 
 
 5.64 
 
 .0308 
 
 950.4 
 
 .18 
 
 3.50 
 
 0.0027 
 
 4.09 
 
 .0056 
 
 5.11 
 
 .0157 
 
 6.00 
 
 .0328 
 
 1050. 
 
 .20 
 
 3.71 
 
 0.0028 
 
 4.31 
 
 .0059 
 
 5.40 
 
 .0166 
 
 6.33 
 
 .0346 
 
 1320. 
 
 .25 
 
 4.15 
 
 0.0032 
 
 4.83 
 
 .0066 
 
 6.10 
 
 .0187 
 
 7.14 
 
 .0390 
 
 1584. 
 
 .30 
 
 4.58 
 
 0.0035 
 
 5.36 
 
 .0073 
 
 6.73 
 
 .0206 
 
 7.90 
 
 .0432 
 
 2112. 
 
 .40 
 
 5.32 
 
 0.0041 
 
 6.26 
 
 .0086 
 
 7.79 
 
 .0239 
 
 9.13 
 
 .0499 
 
 2640. 
 
 .50 
 
 5.99 
 
 0.0048 
 
 7.07 
 
 .0097 
 
 8.82 
 
 .0271 
 
 10.34 
 
 .0565 
 
 3168. 
 
 .60 
 
 6.62 
 
 0.0051 
 
 7.80 
 
 .0107 
 
 9.79 
 
 .0300 
 
 11.57 
 
 .0632 
 
 3696. 
 
 .70 
 
 7.20 
 
 0.0055 
 
 8.47 
 
 .0116 
 
 10.76 
 
 .0330 
 
 12.71 
 
 .0694 
 
 4224. 
 
 .80 
 
 7 75 
 
 0.0059 
 
 9.14 
 
 .0125 
 
 11.72 
 
 .0357 
 
 
 
 4752. 
 
 .90 
 
 8.22 
 
 0.0063 
 
 9.80 
 
 .0134 
 
 12.60 
 
 .0379 
 
 
 
 5280. 
 
 1.00 
 
 8.74 
 
 0.0067 
 
 10.39 
 
 .0142 
 
 
 
 
 
 
 
 
 
PRACTICAL HYDRAULICS. 
 
 135 
 
 TABLE 17. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes due 
 given Slopes and Diameters. 
 
 Fall per 
 Mile. 
 Feet. 
 
 Sine 
 of 
 Slope 
 
 _i 
 
 DIAMETERS. 
 
 li"=.125 feet. 
 
 If "=.1458 ft. 
 
 2"=. 1667 ft. 
 
 2i"=.4167 ft. 
 
 Velo' 
 Ft per 
 Sec. 
 
 Cubic 
 Feet per 
 Sec. 
 
 Velo'y 
 Ft. per 
 Sec. 
 
 Cubic 
 Ft. per 
 Sec. 
 
 Velo'y 
 Ft. per 
 Sec. 
 
 Cubic 
 Ft. per 
 Sec. 
 
 Velo'y 
 Ft. per 
 Sec. 
 
 Cubic 
 Ft. per 
 Sec. 
 
 21 12 
 
 .004 
 
 
 
 
 
 1.18 
 
 .0258 
 
 .35 
 
 ,0460 
 
 26.40 
 
 005 
 
 
 
 .21 
 
 .0201 
 
 1 34 
 
 .0292 
 
 .52 
 
 0518 
 
 3L68 
 
 !ooe 
 
 1.19 
 
 .0146 
 
 ]36 
 
 !0227 
 
 li-60 
 
 ]0327 
 
 .68 
 
 !0573 
 
 36.96 
 
 .007 
 
 1.29 
 
 .0158 
 
 .45 
 
 .0243 
 
 1.60 
 
 .0349 
 
 .81 
 
 .0617 
 
 42.24 
 
 .008 
 
 1.39 
 
 .0171 
 
 .58 
 
 .0264 
 
 1.73 
 
 .0378 
 
 .04 
 
 .0662 
 
 47.52 
 
 .009 
 
 1.48 
 
 .0182 
 
 .70 
 
 .0284 
 
 1.87 
 
 .0408 
 
 2.07 
 
 .0706 
 
 52.80 
 
 .010 
 
 1.60 
 
 .0196 
 
 .79 
 
 .0299 
 
 1.98 
 
 .0432 
 
 2.20 
 
 .0750 
 
 63.36 
 
 .012 
 
 1.73 
 
 .0212 
 
 .95 
 
 .0326 
 
 2.22 
 
 .0484 
 
 2.44 
 
 .0832 
 
 73.92 
 
 .014 
 
 1.86 
 
 .0228 
 
 2.13 
 
 .0356 
 
 2.36 
 
 .0515 
 
 2.62 
 
 .0003 
 
 84.48 
 
 .016 
 
 2.01 
 
 .0247 
 
 2.22 
 
 .0371 
 
 2.50 
 
 .0546 
 
 2.78 
 
 .0948 
 
 95.04 
 
 .018 
 
 2.10 
 
 .0258 
 
 2.35 
 
 0392 
 
 2.63 
 
 .0574 
 
 2.93 
 
 .0999 
 
 105.6 
 
 .02 
 
 2.28 
 
 .0279 
 
 2.53 
 
 .0422 
 
 2.80 
 
 .0611 
 
 3.15 
 
 .1074 
 
 158.4 
 
 .03 
 
 2.81 
 
 .0346 
 
 3.10 
 
 .0518 
 
 3.39 
 
 .0740 
 
 3.81 
 
 .1299 
 
 211.2 
 
 .04 
 
 3.37 
 
 .0413 
 
 3.69 
 
 .0617 
 
 4.00 
 
 .0873 
 
 4.51 
 
 .1538 
 
 264.0 
 
 .05 
 
 3.73 
 
 .0458 
 
 4.28 
 
 .0715 
 
 5.02 
 
 .1095 
 
 5.34 
 
 .1821 
 
 316.8 
 
 .06 
 
 4.11 
 
 .0504 
 
 4.69 
 
 .0783 
 
 5.50 
 
 . 1200 
 
 5.85 
 
 .1995 
 
 369.6 
 
 .07 
 
 4.42 
 
 .0542 
 
 5.02 
 
 .0838 
 
 5.90 
 
 .1288 
 
 6.34 
 
 .2162 
 
 422.4 
 
 .08 
 
 4.73 
 
 .0580 
 
 5.36 
 
 .0895 
 
 6.30 
 
 .1375 
 
 6.77 
 
 .2309 
 
 475.2 
 
 .09 
 
 5.05 
 
 .0619 
 
 5.63 
 
 .0940 
 
 6.61 
 
 .1442 
 
 7.19 
 
 .2442 
 
 528.0 
 
 .10 
 
 5.48 
 
 .0672 
 
 6.01 
 
 .1003 
 
 6.98 
 
 . 1523 
 
 7.61 
 
 .2595 
 
 633.0 
 
 .12 
 
 6.03 
 
 .0740 
 
 6.65 
 
 .1110 
 
 7.49 
 
 .1634 
 
 8.27 
 
 .2820 
 
 739 2 
 
 .14 
 
 6.54 
 
 .0802 
 
 7.19 
 
 ,1200 
 
 8.01 
 
 .1748 
 
 8.89 
 
 .3031 
 
 840.0 
 
 .16 
 
 7.01 
 
 .0862 
 
 7.70 
 
 .1285 
 
 8.50 
 
 .1855 
 
 9.48 
 
 .3233 
 
 950.0 
 
 .18 
 
 7.50 
 
 .0923 
 
 8.22 
 
 .1372 
 
 8.96 
 
 .1955 
 
 10.04 
 
 .3424 
 
 1056. 
 
 .20 
 
 7.88- 
 
 .0969 
 
 8.69 
 
 .1450 
 
 9.38 
 
 .2047 
 
 10.63 
 
 .3624 
 
 1320. 
 
 .25 
 
 8.77 
 
 .1079 
 
 9.69 
 
 .1617 
 
 10.43 
 
 .2276 
 
 11.86 
 
 .4054 
 
 1584. 
 
 .30 
 
 9.65 
 
 .1187 
 
 10.62 
 
 .1773 
 
 11.38 
 
 .2483 
 
 13.15 
 
 .4084 
 
 2112. 
 
 .40 
 
 11.23 
 
 .1380 
 
 12.28 
 
 .2050 
 
 12.98 
 
 .2833 
 
 
 
 2640. 
 
 .50 
 
 12.60 
 
 .1550 
 
 
 
 
 
 
 
 
 
 
 
 
 
136 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 17. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes due 
 given Slopes and Diameters. 
 
 
 Sine 
 
 DIAMETERS. 
 
 Fall per 
 
 of 
 
 
 Mile 
 
 ? 
 
 3"=. 25 feet. 
 
 4" =.333 ft. 
 
 6" =.5 feet. 
 
 8" =.667 Feet. 
 
 Feet. 
 
 4 
 
 Veloc'y 
 Ft. per 
 Sec. 
 
 Cubic 
 Ft. per 
 Sec. 
 
 Velo'y 
 Ft per 
 Sec. 
 
 Cubic 
 Ft per 
 Sec. 
 
 Velo'y 
 Ft per 
 Sec. 
 
 Cubic 
 Ft per 
 Sec. 
 
 Velo'y 
 Ft per 
 Sec. 
 
 Cubic 
 Ft. per 
 Sec. 
 
 8 448 
 
 .0016 
 
 
 
 
 
 
 
 64 
 
 573 
 
 8.976 
 
 .0017 
 
 
 
 
 
 
 
 .68 
 
 .586 
 
 9 504 
 
 .0018 
 
 
 
 
 
 1 44 
 
 .282 
 
 75 
 
 611 
 
 10 032 
 
 .0019 
 
 
 
 
 
 1 48 
 
 290 
 
 80 
 
 628 
 
 10 560 
 
 0020 
 
 
 
 1 25 
 
 1091 
 
 1 52 
 
 298 
 
 83 
 
 639 
 
 11 616 
 
 .0022 
 
 
 
 1 31 
 
 1144 
 
 1 60 
 
 314 
 
 93 
 
 659 
 
 12.672 
 
 .0024 
 
 1.10 
 
 .0540 
 
 1.36 
 
 .1187 
 
 1.68 
 
 .330 
 
 2.02 
 
 .703 
 
 13.728 
 
 .0026 
 
 1.16 
 
 .0569 
 
 1.42 
 
 .1235 
 
 1.76 
 
 .346 
 
 2.11 
 
 .737 
 
 14784 
 
 .0028 
 
 1.22 
 
 .0599 
 
 1.48 
 
 .1298 
 
 1 83 
 
 .359 
 
 2.22] .768 
 
 15.840 
 
 .0030 
 
 .28 
 
 .0630 
 
 1.53 
 
 .1335 
 
 1.92 
 
 .377 
 
 2.32| .808 
 
 18.480 
 
 .0035 
 
 .41 
 
 .0692 
 
 1.68 
 
 .1465 
 
 2.10 
 
 .395 
 
 2.511 .876 
 
 21.120 
 
 .004 
 
 .53 
 
 .0749 
 
 1.79 
 
 .1562 
 
 2.26 
 
 .444 
 
 2.69 .931 
 
 26.40 
 
 .005 
 
 .71 
 
 .0839 
 
 203 
 
 .1771 
 
 2.53 
 
 .496 
 
 1 2.99 1.015 
 
 31.68 
 
 .006 
 
 .86 
 
 .0915 
 
 2.20 
 
 .1923 
 
 2.79 
 
 .548 
 
 3.32^ 1.157 
 
 36.96 
 
 .007 
 
 2.02 
 
 .0992 
 
 2.46 .2146 
 
 2.79 
 
 .589 
 
 [ 3.32 1.262 
 
 42.24 
 
 .008 
 
 2.16 
 
 .1060 
 
 2.67 .2339 
 
 3.00 
 
 .631 
 
 3.62! 1.344 
 
 47.52 
 
 .009 
 
 228 
 
 .1119 
 
 2.82 
 
 .2460 
 
 322 
 
 .672 
 
 1 3.85 1.424 
 
 52.80 
 
 .010 
 
 2.43 
 
 .1190 
 
 2.96 
 
 .2582 
 
 3.431 .721 
 
 1 4.08 1.496 
 
 63.36 
 
 .012 
 
 2.68 
 
 .1313 
 
 3 23!. 2893 
 
 3.67 
 
 .784 
 
 4.29 
 
 1.614 
 
 73.92 
 
 .014 
 
 2.88 
 
 .1413 
 
 3.48 
 
 .3036 
 
 3.99 
 
 .858 
 
 4.71 
 
 1.782 
 
 84.48 
 
 .016 
 
 3.07 
 
 .1507 
 
 3.71 
 
 .3237 
 
 4.37 
 
 .922 
 
 5.11 
 
 1.916 
 
 94.04 
 
 .018 
 
 3.24 
 
 .1593 
 
 3.91 
 
 .3412 
 
 4.70 
 
 .975 
 
 5.49 
 
 2.033 
 
 105.6 
 
 .02 
 
 3.50 
 
 .1717 
 
 413.3607 
 
 497 
 
 1.022 
 
 5.83 2.155 
 
 158.4 
 
 .03 
 
 4.24 
 
 .2081 
 
 5.16 
 
 .4502 
 
 521 
 
 1.263 
 
 6.181 2.667 
 
 211.2 
 
 .04 
 
 5.03 
 
 .2469 
 
 6.11 
 
 5331 
 
 6.43 
 
 1.484 
 
 7.64| 3.145 
 
 264.0 
 
 .05 
 
 5.67 
 
 .2785 
 
 6.83 
 
 .5954 
 
 7.56 
 
 1.665 
 
 9.01 
 
 3.513 
 
 316.8 
 
 .06 
 
 621 
 
 ,3049 
 
 7.32 
 
 .6390 
 
 8.48 
 
 1.929 
 
 10.06 
 
 3.847 
 
 369.0 
 
 .07 
 
 6.79 
 
 .3331 
 
 7.99 
 
 .6967 
 
 9.83 
 
 1.976 
 
 [11.02 4.196 
 
 4220 
 
 .08 
 
 7.25 
 
 .3559 
 
 8.60 
 
 .7506 
 
 10.062.114 
 
 12 02'.. 
 
 4752 
 
 .09 
 
 7.78 
 
 .3816 
 
 9.13 
 
 .7960 
 
 10.922.274 
 
 
 5280 
 
 .10 
 
 8.24 
 
 .4043 
 
 9.70 
 
 .8467 
 
 11.582 399 
 
 
 
 633.6 
 
 12 
 
 905 
 
 .4440 
 
 10.63 
 
 .9270 
 
 12.21 
 
 
 
 
 739 2 
 
 .14 
 
 9 77 
 
 .4977 
 
 11.53 
 
 1.006 
 
 
 
 
 
 844.8 
 
 .16 
 
 10.46 
 
 .5131 
 
 1238 
 
 1.081 
 
 
 
 
 
 950.4 
 
 .18 
 
 11.08 
 
 .5436 
 
 
 
 
 
 
 
 1056. 
 
 .20 
 
 11.88 
 
 .5832 
 
 
 
 
 
 
 
 1320. 
 
 .25 
 
 13.29 
 
 .6523 
 
 
 
 
 
 
 
PKACTICAL HYDRAULICS. 
 
 137 
 
 TABLE 17. 
 
 Velocities and Quantities of Plow in Clean Iron Pipes due 
 given Slopes and Diameters. 
 
 
 Sine 
 
 DlAMETBRS. 
 
 Fall per 
 
 
 
 
 of 
 Slope. 
 
 9"=.75 feet. 
 
 10"=. 833 ft. 
 
 11"=. 917 ft. 
 
 "=1 foot. 
 
 Mile. 
 
 h 
 
 Vdoci'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Veloci'y 
 
 Cubic 
 
 Feet. 
 
 -i 
 
 Feet per 
 Sec. 
 
 Feet per 
 Sec. 
 
 Ft per 
 
 Sec. 
 
 Ft per 
 
 Sec. 
 
 Ft per 
 Sec. 
 
 Ft per 
 Sec. 
 
 Feet 
 Per Sec. 
 
 Ft per 
 Sec. 
 
 4.752 
 
 .0009 
 
 
 
 
 
 
 
 1.54 
 
 1.210 
 
 5.280 
 
 .0010 
 
 
 
 
 
 
 
 1.61 
 
 1.265 
 
 5.808 
 
 [0011 
 
 
 
 
 
 1.62 
 
 1.069 
 
 1.68 
 
 .319 
 
 6.336 
 
 .'0012 
 
 
 
 1.61 
 
 .878 
 
 1.70 
 
 1.122 
 
 1.79 
 
 .402 
 
 6 864 
 
 .0013 
 
 
 
 1.69 
 
 .922 
 
 1.79 
 
 1.181 
 
 1.88 
 
 .476 
 
 7.392 
 
 .0014 
 
 1.66 
 
 .735 
 
 1.76 
 
 .960 
 
 1.85 
 
 1.221 
 
 1.94 
 
 .489 
 
 7.920 
 
 .0015 
 
 1.72 
 
 .761 
 
 1.84 
 
 .984 
 
 1.92 
 
 1.267 
 
 2.01 
 
 .579 
 
 8.448 
 
 .0016 
 
 1.78 
 
 .788 
 
 1.92 
 
 1.047 
 
 2.00 
 
 1.320 
 
 2.08 
 
 .634 
 
 8.976 
 
 0017 
 
 1.83 
 
 .810 
 
 1.99 
 
 1.085 
 
 2.07 
 
 1.366 
 
 2.15 
 
 .689 
 
 9.504 
 
 .0018 
 
 1.89 
 
 .836 
 
 2.04 
 
 1.110 
 
 2.12 
 
 1.399 
 
 2.20 
 
 .728 
 
 10.032 
 
 .0019 
 
 1.96 
 
 .867 
 
 2.13 
 
 1.162 
 
 2.20 
 
 1.452 
 
 2.27 
 
 .783 
 
 10.560 
 
 .0020 
 
 2.01 
 
 .890 
 
 2.19 
 
 1.194 
 
 2.26 
 
 1.492 
 
 2.33 
 
 1.826 
 
 11.616 
 
 .0022 
 
 2.12 
 
 .938 
 
 2.32 
 
 1.265 
 
 2.39 
 
 1.577 
 
 2.47 
 
 1.940 
 
 12.672 
 
 .0024 
 
 2.22 
 
 .982 
 
 2.43 
 
 1.325 
 
 2.50 
 
 1.650 
 
 2.58 
 
 2.026 
 
 13.728 
 
 .0026 
 
 2.32 
 
 1.025 
 
 2.53 
 
 1.377 
 
 2.61 
 
 1.723 
 
 2.70 
 
 2.117 
 
 14.784 
 
 .0028 
 
 2.41 
 
 1.065 
 
 2.61 
 
 1.423 
 
 2.71 
 
 1.789 
 
 2.81 
 
 2.207 
 
 15.840 
 
 .0030 
 
 250 
 
 1.105 
 
 2.70 
 
 1.470 
 
 2.81 
 
 1.855 
 
 2.93 
 
 2.297 
 
 18.480 
 
 .0035 
 
 2.71 
 
 1.198 
 
 2.91 
 
 1.587 
 
 3.02 
 
 1.993 
 
 3.14 
 
 2.466 
 
 21.120 
 
 .004 
 
 2.88 
 
 1.273 
 
 3.09 
 
 1.683 
 
 3.24 
 
 2.138 
 
 3.39 
 
 2.662 
 
 26.400 
 
 .005 
 
 320 
 
 1.414 
 
 3.42 
 
 1.865 
 
 3.63 
 
 2.176 
 
 3.85 
 
 3.020 
 
 31.680 
 
 .006 
 
 3.54 
 
 1.565 
 
 3.78 
 
 2.059 
 
 4.00 
 
 2.640 
 
 4.22 
 
 3.310 
 
 36.960 
 
 .007 
 
 3.84 
 
 1.697 
 
 4.08 
 
 2.222 
 
 4.33 
 
 2.858 
 
 4.58 
 
 3.601 
 
 42.240 
 
 008 
 
 4.11 
 
 1.816 
 
 4.37 
 
 2.383 
 
 4.64 
 
 3.062 
 
 4.91 
 
 3.856 
 
 47.520 
 
 .009 
 
 4.34 
 
 1.918 
 
 4.61 
 
 2.514 
 
 4.90 
 
 3.234 
 
 5.19 
 
 4.072 
 
 52.80 
 
 010 
 
 4.58 
 
 2.024 
 
 4.88 
 
 2.662 
 
 5.18 
 
 3.419 
 
 5.48 
 
 4.305 
 
 63.36 
 
 012 
 
 5.04 
 
 2.228 
 
 5.38 
 
 2.932 
 
 5.70 
 
 3.762 
 
 6.02 
 
 4.728 
 
 73.92 
 
 014 
 
 5.50 
 
 2.431 
 
 5.89 
 
 3.210 
 
 6.18 
 
 4.079 
 
 6.49 
 
 5.094 
 
 84.48 
 
 016 
 
 5.91 
 
 2.612 
 
 6.33 
 
 3.450 
 
 6.65 
 
 4.389 
 
 6.98 
 
 5.482 
 
 95.04 
 
 018 
 
 6.29 
 
 2.780 
 
 6.75 
 
 3.679 
 
 7.09 
 
 4.676 
 
 7.44 
 
 5.838 
 
 105.6 
 
 02 
 
 6.62 
 
 2.926 
 
 7.07 
 
 3.856 
 
 7.45 
 
 4.917 
 
 7.84 
 
 6.100 
 
 158.4 
 
 03 
 
 8.18 
 
 3.616 
 
 8.73 
 
 4.762 
 
 9.22 
 
 6.085 
 
 9.72 
 
 7.630 
 
 211.2 
 
 04 
 
 9.60 
 
 4.243 
 
 10.20 
 
 5.563 
 
 10.74 
 
 7.088 
 
 12.28 
 
 8.860 
 
 264.0 
 
 05 
 
 10.81 
 
 4.778 
 
 12.29 
 
 6.704 
 
 12.49 
 
 8.243 
 
 12.69 
 
 9.967 
 
 316.8 
 
 06 
 
 11.85 
 
 5.238 
 
 
 
 
 
 
 
138 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 17. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes, due 
 Given Slopes and Diameters. 
 
 
 Sine 
 
 DIAMETERS. 
 
 Fall per 
 
 of 
 
 
 Milp 
 
 Slope. 
 
 T 
 
 14"=1.167 Feet- 
 
 15"=1.25 Ft. 
 
 16"=1.333Ft. 
 
 ! 18"= 1.5 Feet. 
 
 Julie* 
 
 Feet. 
 
 '=7 
 
 Veloci'y 
 Feet per 
 Sec. 
 
 Cubic 
 Ft. per 
 Sec. 
 
 Velo'y 
 Ft per 
 Sec. 
 
 Cubic 
 Ft per 
 Sec. 
 
 Velo'y 
 Ft per 
 Sec. 
 
 Cubic 
 Ft per 
 Sec. 
 
 (Veloc'y 
 Ft. per 
 Sec. 
 
 Cubic 
 Ft per 
 Sec. 
 
 2.112 
 
 .0004 
 
 
 
 
 
 
 
 1.46 
 
 2.61 
 
 2.640 
 
 .0005 
 
 
 
 
 
 
 
 1.56 
 
 2.79 
 
 3.168 
 
 .0006 
 
 
 
 
 
 1.50 
 
 2.04 
 
 1.66 
 
 2.97 
 
 3.696 
 
 .0007 
 
 
 
 1.56 
 
 1.91 
 
 1.61 
 
 2.25 
 
 1.76 
 
 3 10 
 
 4.224 
 
 .0008 
 
 .61 
 
 1.71 
 
 1.67 
 
 2.05 
 
 1.74 
 
 2.43 
 
 1.85 
 
 3.27 
 
 4.752 
 
 .0009 
 
 .71 
 
 1.83 
 
 1.78 
 
 2.19 
 
 1.86 
 
 2.59 
 
 1.98 
 
 3.49 
 
 5.280 
 
 .0010 
 
 .80 
 
 1.91 
 
 1.87 
 
 2.30 
 
 1.95 
 
 2.72 
 
 2.07 
 
 3.66 
 
 5.808 
 
 .0011 
 
 .90 
 
 2.02 
 
 1.98 
 
 2.43 
 
 2.07 
 
 2.88 
 
 2.20 
 
 3.88 
 
 6.336 
 
 .0012 
 
 .98 
 
 2.11 
 
 2.07 
 
 2.54 
 
 2.16 
 
 3.02 
 
 2.30 
 
 4.06 
 
 6.864 
 
 .0013 
 
 2.04 
 
 2.18 
 
 2.16 
 
 2.65 
 
 2.28 
 
 3.18 
 
 2.40 
 
 4.23 
 
 7.392 
 
 .0014 
 
 2.13 
 
 2.27 
 
 2.24 
 
 2.75 
 
 2.35 
 
 3.28 
 
 2.49 
 
 440 
 
 7.920 
 
 .0015 
 
 2.20 
 
 2.35 
 
 2.31 
 
 284 
 
 2.43 
 
 3.39 
 
 2.61 
 
 4.61 
 
 8.448 
 
 .0016 
 
 2.29 
 
 2.44 
 
 2.39 
 
 2.94 
 
 2.50 
 
 3.49 
 
 2.69 
 
 4.75 
 
 8.976 
 
 .0017 
 
 2.38 
 
 2.54 
 
 2.48 
 
 3.08 
 
 2.59 
 
 3.62 
 
 2.78 
 
 4.90 
 
 9.504 
 
 .0018 
 
 2.43 
 
 2.59 
 
 2.53 
 
 3.11 
 
 2.64 
 
 3.69 
 
 2.85 
 
 5.03 
 
 10.032 
 
 .0019 
 
 2.50 
 
 2.67 
 
 2.61 
 
 3.21 
 
 2.73 
 
 3.81 
 
 2.93 
 
 5.17 
 
 10.560 
 
 .0020 
 
 2.55 
 
 2.72 
 
 2.68 
 
 3.29 
 
 2.81 
 
 3.92 
 
 3.00 
 
 5.30 
 
 11.616 
 
 .0022 
 
 2.70 
 
 2.88 
 
 2.82 
 
 3.47 
 
 2.95 
 
 4.12 
 
 3.19 
 
 5.63 
 
 12.672 
 
 .0024 
 
 2.83 
 
 3.02 
 
 2.96 
 
 3.63 
 
 3.10 
 
 4.32 
 
 3.32 
 
 5.87 
 
 13.728 
 
 .0026 
 
 2.95 
 
 3.15 
 
 3.09 
 
 3.79 
 
 3.23 
 
 4.51 
 
 3.50 
 
 6.18 
 
 14.784 
 
 .0028 
 
 3.08 
 
 3.29 
 
 3.22 
 
 3.95 
 
 3.36 
 
 4.68 
 
 361 
 
 6.38 
 
 15.840 
 
 .0030 
 
 3.20 
 
 3.42 
 
 3.34 
 
 4.11 
 
 3.49 
 
 4.87 
 
 3.76 
 
 6.64 
 
 18.48 
 
 .0035 
 
 3.47 
 
 3.62 
 
 3.63 
 
 4.46 
 
 3.80 
 
 5.31 
 
 4.06 
 
 7.17 
 
 21.12 
 
 .004 
 
 3.74 
 
 3.99 
 
 3.90 
 
 4.78 
 
 4.06 
 
 5.67 
 
 4.33 
 
 7.65 
 
 26.40 
 
 .005 
 
 4.18 
 
 4.46 
 
 4.38 
 
 5.37 
 
 4.58 
 
 6.39 
 
 4.90 
 
 8.66 
 
 31.68 
 
 .006 
 
 4.60 
 
 4.91 
 
 4.81 
 
 5.91 
 
 5.03 
 
 7.02 
 
 5.40 
 
 9.54 
 
 3696 
 
 .007 
 
 5.03 
 
 5.37 
 
 5.26 
 
 6.45 
 
 5.49 
 
 7.66 
 
 5.84 
 
 10.33 
 
 42.24 
 
 .008 
 
 5.40 
 
 5.77 
 
 5.62 
 
 6.90 
 
 5.85 
 
 8.16 
 
 6.28 
 
 11.09 
 
 47.52 
 
 .009 
 
 5.73 
 
 6.11 
 
 5.95 
 
 7.31 
 
 6.19 
 
 8.64 
 
 6.63 
 
 11.71 
 
 52.80 
 
 .010 
 
 6.03 
 
 644 
 
 6.27 
 
 7.70 
 
 6.52 
 
 9.10 
 
 7.00 
 
 12.37 
 
 63.36 
 
 .012 
 
 6.56 
 
 7.00 
 
 6.84 
 
 8.39 
 
 7.12 
 
 9.95 
 
 7.73 
 
 13.65 
 
 73.92 
 
 .014 
 
 7.12 
 
 7.60 
 
 7.45J 9.15 
 
 7.79 
 
 10.87 
 
 8.35 
 
 14.75 
 
 84.48 
 
 .016 
 
 7-66 
 
 8.17 
 
 7.99! 9.81 
 
 8.33 
 
 11.63 
 
 8.97 
 
 1584 
 
 95.04 
 
 .018 
 
 847 
 
 8.93 
 
 8.03 10.47 
 
 8.90 
 
 12.43 
 
 9.57 
 
 1690 
 
 105.6 
 
 .02 
 
 8.67 
 
 9.26 
 
 9.04 11.09 
 
 9.41 
 
 13.14 
 
 10.10 
 
 17.85 
 
 158.4 
 
 .03 
 
 10.69 
 
 11.39 
 
 11.1313.66 
 
 11.58 
 
 16.17 
 
 12.37 
 
 21.86 
 
 211.2 
 
 .04 
 
 12.38 
 
 13.22 
 
 12.91 15.84 
 
 13.45 
 
 18.77 
 
 
 
PRACTICAL HYDRAULICS. 
 
 139 
 
 TABLE 17. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes due 
 given Slopes and Diameters. 
 
 Fall per 
 Mile. 
 Feet. 
 
 Sine 
 of 
 Slope. 
 
 h 
 
 4 
 
 DIAMETERS. 
 
 20"= 1.667 feet. 
 
 22"=1.833 feet. 
 
 24"=2feet. 
 
 27"= 2. 25 feet 
 
 Veloci'y 
 Feet per 
 Sec 
 
 Cubic 
 Feet per 
 Sec. 
 
 Veloci'y 
 Feet per 
 Sec. 
 
 Cubic 
 Feet pei 
 Sec. 
 
 Velo'y 
 Ft per 
 Sec. 
 
 Cubic 
 Ft per 
 Sec. 
 
 Velo'y 
 Ft per 
 Sec. 
 
 2.00 
 
 Cubic 
 Ft per 
 Sec. 
 
 2.112 
 
 .0004 
 
 
 
 
 
 
 
 7.95 
 
 2 640 
 
 .0005 
 
 
 
 
 
 1.78 
 
 5 59 
 
 2.08 
 
 8.27 
 
 3.168 
 
 .0006 
 
 1.66 
 
 3.61 
 
 1.80 
 
 4.61 
 
 1.94 
 
 6.10 
 
 2.10 
 
 8.51 
 
 3.696 
 
 .0007 
 
 1.86 
 
 4.07 
 
 1.99 
 
 5.25 
 
 2.12 
 
 6.64 
 
 2.29 
 
 8.91 
 
 4.224 
 
 .0008 
 
 2.00 
 
 4.35 
 
 2.13 
 
 5.62 
 
 2.27 
 
 7.13 
 
 2.39 
 
 9.30 
 
 4.752 
 
 .0009 
 
 2.15 
 
 4.68 
 
 2.28 
 
 6.01 
 
 2.41 
 
 7.56 
 
 2.58 
 
 10.26 
 
 5.280 
 
 .0010 
 
 226 
 
 4.92 
 
 2.39 
 
 6.32 
 
 2.53 
 
 7.95 
 
 2.70 
 
 10.74 
 
 5.808 
 
 .0011 
 
 2.36 
 
 5.15 
 
 2.51 
 
 6.62 
 
 2.66 
 
 8.34 
 
 2.88 
 
 11.45 
 
 6.336 
 
 .0012 
 
 2.48 
 
 5.40 
 
 2.63 
 
 6.94 
 
 2.79 
 
 8.75 
 
 3.00 
 
 11.93 
 
 6.864 
 
 .0013 
 
 2.58 
 
 5.62 
 
 2.74 
 
 7.24 
 
 2.91 
 
 9.14 
 
 3.16 
 
 12.54 
 
 7.392 
 
 .0014 
 
 2.68 
 
 5.82 
 
 2.85 
 
 7.51 
 
 3.02 
 
 9.47 
 
 3.26 
 
 12.96 
 
 7.920 
 
 .0015 
 
 2.78 
 
 6.05 
 
 2.95 
 
 7.78 
 
 3.12 
 
 9.80 
 
 3.40 
 
 13.49 
 
 8.448 
 
 .0016 
 
 2.88 
 
 6.27 
 
 3.05 
 
 8.05 
 
 3.23 
 
 10.13 
 
 3.52 
 
 13.98 
 
 8.976 
 
 .0017 
 
 2.97 
 
 6.48 
 
 3.17 
 
 8,36 
 
 3.37 
 
 10.57 
 
 3.63 
 
 14.41 
 
 9.504 
 
 .0018 
 
 3.05 
 
 6.65 
 
 324 
 
 8.55 
 
 3.43 
 
 10.77 
 
 3.73 
 
 14.81 
 
 10.032 
 
 .0019 
 
 3.17 
 
 6.92 
 
 3.35 
 
 8.85 
 
 3.54 
 
 11.10 
 
 3.83 
 
 15.21 
 
 10.560 
 
 .0020 
 
 3.23 
 
 7.05 
 
 3.43 
 
 9.07 
 
 3.64 
 
 11.43 
 
 3.93 
 
 15.63 
 
 11.616 
 
 .0022 
 
 3.40 
 
 7.42 
 
 3.62 
 
 9.55 
 
 3.84 
 
 12.05 
 
 4.14 
 
 16.44 
 
 12.672 
 
 .0024 
 
 3.57 
 
 7.79 
 
 3.79 
 
 10.01 
 
 4.02 
 
 1261 
 
 4.34 
 
 17.23 
 
 13.728 
 
 .0026 
 
 3.73 
 
 8.14 
 
 3.97 
 
 10.48 
 
 4.21 
 
 13.23 
 
 4.53 
 
 18.01 
 
 14.784 
 
 .0028 
 
 3.89 
 
 8.48 
 
 4.14 
 
 10.91 
 
 4.39 
 
 13.79 
 
 4.72 
 
 18.75 
 
 15.84 
 
 .0030 
 
 4.02 
 
 8.77 
 
 4.28 
 
 11.29 
 
 4.54 
 
 14.25 
 
 4.91 
 
 19.50 
 
 18.48 
 
 .0035 
 
 4.35 
 
 9.49 
 
 4.64 
 
 12.25 
 
 4.94 
 
 15.50 
 
 5.32 
 
 21.13 
 
 21.12 
 
 .004 
 
 4.66 
 
 10.16 
 
 4.97 
 
 13.12 
 
 5.29 
 
 16.62 
 
 5.69 
 
 22.62 
 
 26.40 
 
 .005 
 
 5.24 
 
 11.43 
 
 5.60 
 
 14.78 
 
 5.96 
 
 18.71 
 
 6.37 
 
 25.34 
 
 31.68 
 
 .006 
 
 5.77 
 
 12.59 
 
 6.10 
 
 16.20 
 
 6.50 
 
 20.42 
 
 6.98 
 
 27.74 
 
 36.96 
 
 .007 
 
 6.26 
 
 13.66 
 
 6.64 
 
 17.53 
 
 7.02 
 
 22.05 
 
 7.52 
 
 29.96 
 
 42.24 
 
 .008 
 
 6.72 
 
 14.66 
 
 7.12 
 
 18.78 
 
 7.52 
 
 23.61 
 
 8.05 
 
 31.99 
 
 47.52 
 
 ,009 
 
 7.13 
 
 15.54 
 
 7.55 
 
 19.93 
 
 7.98 
 
 25.07 
 
 8.55 
 
 33.97 
 
 52.80 
 
 .010 
 
 7.55 
 
 16.47 
 
 7.98 
 
 21.06 
 
 8.41 
 
 26.42 
 
 9.03 
 
 35.89 
 
 63.36 
 
 .012 
 
 8.25 
 
 17.99 
 
 8.74 
 
 23.07 
 
 9.24 
 
 29.03 
 
 10.00 
 
 39.76 
 
 73.92 
 
 .014 
 
 8.94 
 
 19.49 
 
 9.48 
 
 2468 
 
 10.03 
 
 31.49 
 
 10.87 
 
 43.22 
 
 84.48 
 
 .016 
 
 9.64 
 
 21.03 
 
 10.21 
 
 26.97 
 
 10.79 
 
 33.90 
 
 11.72 
 
 46.57 
 
 95.04 
 
 .018 
 
 10.30 
 
 22.45 
 
 10.91 
 
 29.70 
 
 11.52 
 
 36.18 
 
 12.09 
 
 48.06 
 
 105.6 
 
 .02 
 
 10.80 
 
 23.56 
 
 11.52 
 
 31.15 
 
 12.24 
 
 38 45 
 
 
 
 158.4 
 
 .03 
 
 12.23 
 
 28.86 
 
 12.59 
 
 33.21 
 
 
 
 
 
 211.2 
 
 .04 
 
 13.50 
 
 2958 
 
 
 
 
 
 
 
140 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE IT. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes, due 
 Slopes and Diameters. 
 
 Fall per 
 
 Sine 
 of 
 Slope. 
 
 DIAMETERS. 
 
 30"=2.5 Feet. 
 
 1=2.75 Ft. 
 
 36"=3. Feet. 
 
 40"=3.333 Ft. 
 
 i e. 
 
 h 
 
 Veloci'yi Cubic 
 
 o'yl Cubic 
 
 Velo'y 
 
 Cubic 
 
 Velo'y 1 Cubic 
 
 Feet. 
 
 8 
 
 Feet per Feet pe 
 
 pei 
 
 Ft per 
 
 Ft per 
 
 Ft per 
 
 Ft Per 
 
 Ft. per 
 
 
 I 
 
 Sec. 
 
 i Sec. 
 
 . 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 1.056 
 
 .0002 
 
 1.34 
 
 6.58 
 
 45 
 
 8.61 
 
 1.46 
 
 10.29 
 
 1.59 
 
 13.88 
 
 1.584 
 
 .0003 
 
 1.59 
 
 7.78 
 
 68 
 
 10.21 
 
 1.80 
 
 12.70 
 
 1.95 
 
 17.00 
 
 2.112 
 
 .0004 
 
 1.83 
 
 8.99 
 
 96 
 
 11.65 
 
 206 
 
 14.56 
 
 2.26 
 
 19.68 
 
 2.640 
 
 .0005 
 
 2.09 
 
 10.24 
 
 18 
 
 12.92 
 
 2.31 
 
 16.35 
 
 2.53 
 
 22.08 
 
 3.168 
 
 .0006 
 
 224 
 
 10.97 
 
 36 
 
 13.99 
 
 255 
 
 18.02 
 
 280 
 
 24.43 
 
 3.696 
 
 .0007 
 
 2.43 
 
 11.90 
 
 55 
 
 15.14 
 
 2.80 
 
 19.76 
 
 3.01 
 
 26.27 
 
 4.224 
 
 .0008 
 
 2.62 
 
 12.84 
 
 76 
 
 16.36 
 
 2.95 
 
 20.85 
 
 3.23 
 
 28.14 
 
 4.752 
 
 .0009 
 
 2.75 
 
 13.48 
 
 96 
 
 17.58 
 
 3.16 
 
 22.30 
 
 3.42 
 
 29.80 
 
 5.280 
 
 .0010 
 
 2.90 
 
 14.21 
 
 16 18.74 
 
 3.32 
 
 23.47 
 
 3.61 
 
 31.46 
 
 5.808 
 
 .0011 
 
 3.07 
 
 15.05 
 
 29 19.54 
 
 3.53 
 
 24.91 
 
 3.81 
 
 33.25 
 
 6.336 
 
 .0012 
 
 3.22 
 
 1581 
 
 4220.28 
 
 3.70 
 
 26.12 
 
 3.98 
 
 34.68 
 
 6.864 
 
 .0013 
 
 3.36 
 
 16.47 
 
 59121 29 
 
 3.85 
 
 27.20 
 
 4.15 
 
 36.21 
 
 7.392 
 
 .0014 
 
 3.50 
 
 17.18 
 
 7022.20 
 
 4.00 
 
 28.24 
 
 4.31 
 
 37.57 
 
 7.920 
 
 .0015 
 
 3.66 
 
 17.94 
 
 . ..8823.01 
 
 4.13 
 
 29.19 
 
 4.49 
 
 39.18 
 
 8.448 
 
 .0016 
 
 3.79 
 
 18.58 
 
 4.0023.76 
 
 4.29 
 
 30.29 
 
 4.65 
 
 40.54 
 
 8.976 
 
 0017 
 
 3.92 
 
 19.21 
 
 4.12J24.47 
 
 4.45 
 
 31.42 
 
 4.80 
 
 41.88 
 
 9.504 
 
 0018 
 
 4.01 
 
 19.66 
 
 4.2525.22 
 
 4.60 
 
 32.48 
 
 4.94 
 
 43.07 
 
 10.032 
 
 0019 
 
 4.14 
 
 20.32 
 
 4.40 
 
 26.14 
 
 4.73 
 
 33.40 
 
 5.08 
 
 44.28 
 
 10.560 
 
 .0020 
 
 4.24 
 
 20.79 
 
 4.54 
 
 26.94 
 
 4.88 
 
 34.49 
 
 5.18 
 
 45.20 
 
 11.616 
 
 .0022 
 
 4.45 
 
 21.80 
 
 4.76 
 
 28.27 
 
 5.12 
 
 36.15 
 
 5.52 
 
 48.12 
 
 12672 
 
 .0024 
 
 4.65 
 
 22.83 
 
 5.0029.02 
 
 5.34 
 
 37.74 
 
 5.79 
 
 50.48 
 
 13.728 
 
 0026 
 
 4.88 
 
 23.93 
 
 5.23 
 
 31.06 
 
 5.58 
 
 39.40 
 
 6.04 
 
 52.67 
 
 14.784 
 
 0028 
 
 5.07 
 
 24.86 
 
 5.44 
 
 32.28 
 
 5.78 
 
 40.86 
 
 6.31 
 
 55,04 
 
 15.84 
 
 0030 
 
 5.27 
 
 25.87 
 
 5.66 
 
 33.62 
 
 5.98 
 
 42.28 
 
 6.46 
 
 56.33 
 
 18.48 
 
 0035 
 
 5.70 
 
 27.96 
 
 6.09 
 
 36 17 
 
 6.50 
 
 45.95 
 
 7.00 
 
 61.09 
 
 21.12 
 
 004 
 
 6.08 
 
 29.84 
 
 6.49 
 
 38.57 
 
 6.91 
 
 48.83 
 
 7.50 
 
 65.41 
 
 26.40 
 
 005 
 
 6.84 
 
 33.55 
 
 726 
 
 43.12 
 
 7.77 
 
 54.89 
 
 8.38 
 
 73.09 
 
 31.68 
 
 006 
 
 7.50 
 
 36.79 
 
 7.98147.40 
 
 8.48 
 
 59.95 
 
 9.21 
 
 80.32 
 
 36.96 
 
 007 
 
 8.08 
 
 39.66 
 
 8.65J51.35 
 
 9.22 
 
 65.17 
 
 9.94 
 
 86:70 
 
 42.24 
 
 008 
 
 8.64 
 
 42.39 
 
 9.2554.91 
 
 9.88 
 
 69.80 
 
 10.61 
 
 92.58 
 
 47.52 
 
 009 
 
 9.22 
 
 45.23 
 
 9.8058.20 
 
 10.52 
 
 74.33 
 
 11.23 
 
 9800 
 
 52.80 
 
 010 
 
 9.72 
 
 47.71 
 
 10.3861.62 
 
 11.10 
 
 78.46 
 
 11.92 
 
 104.00 
 
 63.36 
 
 012 
 
 10.78 
 
 52.91 
 
 11.4568.00 
 
 11.72 
 
 82.84 
 
 
 
 73.92 
 
 014 
 
 11.75 
 
 57.65 
 
 12.45173.95 
 
 
 
 
 
PRACTICAL HYDRAULICS' 
 
 141 
 
 TABLE 17. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes due 
 Slopes and Diameters. * " 
 
 Fall per 
 Mile. 
 
 Sine 
 of 
 Slope. 
 k 
 f 
 
 DIAMETERS. 
 
 44"=3.607 feet. 
 
 4S"=4 feet 
 
 54"=4.5 feet. 
 
 eo"=5 feet. 
 
 Veloci'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Velo'y 
 
 Cubic 
 
 Feet. 
 
 8= '- 
 I 
 
 Feet per 
 Sec. 
 
 Feet per 
 Sec. 
 
 Ft per 
 Sec. 
 
 Ft per 
 Sec. 
 
 Ft per 
 Sec. 
 
 Ft per 
 tec. 
 
 Ft per 
 Sec. 
 
 Feet per 
 Sec. 
 
 0.528 
 
 .0001 
 
 1.22 
 
 12.89 
 
 1.28 
 
 16.08 
 
 1.38 
 
 21.96 
 
 1.52 
 
 29.77 
 
 1.056 
 
 .0002 
 
 1.72 
 
 18.15 
 
 1.83 
 
 2298 
 
 1.95 
 
 30.97 
 
 208 
 
 40.84 
 
 1.584 
 
 .0003 
 
 2.10 
 
 2222 
 
 222 
 
 27.89 
 
 2.42 38.53 
 
 265 
 
 52.09 
 
 2 112 
 
 .0004 
 
 2.42 
 
 25.55 
 
 2.62 
 
 32.93 
 
 2.8445.12 
 
 3.01 
 
 59.04 
 
 2.640 
 
 .0005 
 
 2.74 
 
 28.87 
 
 2.95 
 
 37.00 
 
 3.16J50.23 
 
 3.44 
 
 67.56 
 
 3.168 
 
 .0006 
 
 2.98 
 
 31.46 
 
 320 
 
 40.21 
 
 3.49i55.51 
 
 3.79 
 
 74.32 
 
 3.696 
 
 .0007 
 
 3.27 
 
 3447 
 
 3.48 
 
 4367 
 
 3.79160.21 
 
 4.10 
 
 80.51 
 
 4.224 
 
 .0008 
 
 3.51 
 
 37.05 
 
 3.73 
 
 46.81 
 
 4.00163.61 
 
 4.40 
 
 86.30 
 
 4.752 
 
 .0009 
 
 3.70 
 
 39.01 
 
 3.90 
 
 49.06 
 
 4. 24! 67. 20 
 
 4.69 
 
 91.99 
 
 5.280 
 
 .0010 
 
 3.89 
 
 41.06 
 
 4.15 
 
 52.15 
 
 4.5517237 
 
 4.94 
 
 96.98 
 
 5.808 
 
 .0011 
 
 408 
 
 42.09 
 
 4.38 
 
 5495 
 
 4 76J75.71 
 
 5.22 
 
 102.4 
 
 6.336 
 
 .0012 
 
 426 
 
 4497 
 
 4.57 
 
 57.36 
 
 4.98179.13 
 
 5.47 
 
 107.3 
 
 6.864 
 
 .0013 
 
 4.43 
 
 46.77 
 
 4.78 
 
 60.07 
 
 5.1982.54 
 
 5.68 
 
 115.5 
 
 7.392 
 
 ,0014 
 
 4.63 
 
 48.83 
 
 4.94 
 
 62.02 
 
 5.40 85.90 
 
 594 
 
 116.5 
 
 7.920 
 
 .0015 
 
 4.80 
 
 50.62 
 
 5.13 
 
 6447 
 
 5.63J89.52 
 
 6.10 
 
 119.7 
 
 8.448 
 
 .0016 
 
 4.97 
 
 52.46 
 
 5.30 
 
 66.53 
 
 5.82i92.47 
 
 6.30 
 
 123.7 
 
 8.976 
 
 .0017 
 
 5.12 
 
 54.04 
 
 5.45 
 
 68.50 
 
 6.0095.35 
 
 6.50 
 
 127.6 
 
 9.504 
 
 .0018 
 
 5.26 
 
 55.48 
 
 5.62 
 
 70.62 
 
 6.1497.65 
 
 6.69 
 
 131.3 
 
 10.032 
 
 .0019 
 
 5.40 
 
 57.01 
 
 5.79 
 
 72.75 
 
 6.30J100.2 
 
 6.87 
 
 134.8 
 
 10.560 
 
 .0020 
 
 5.58 
 
 5885 
 
 5.92 
 
 74.44 
 
 6.53 
 
 103.8 
 
 7.07 
 
 138.8 
 
 11.616 
 
 .0022 
 
 585 
 
 61.71 
 
 6.23 
 
 78.29 
 
 6.84 
 
 108.8 
 
 7.44 
 
 146.0 
 
 12.672 
 
 .0024 
 
 6.10 
 
 6435 
 
 6.50 
 
 81.68 
 
 7.14 
 
 113.5 
 
 7.77 
 
 1526 
 
 13.728 
 
 .0026 
 
 634 
 
 66.87 
 
 6.78 
 
 85.20 
 
 7.45 
 
 118.5 
 
 8.08 
 
 158.7 
 
 14.784 
 
 .0028 
 
 6.59 
 
 69 57 
 
 7.04 
 
 8846 
 
 7.74 
 
 123.1 
 
 838 
 
 164.5 
 
 15.84 
 
 .0030 
 
 6.85 
 
 72.32 
 
 7.30 
 
 91.73 
 
 8.06 128.2 
 
 8.68 
 
 1704 
 
 18.48 
 
 .0035 
 
 7.98 
 
 77.95 
 
 7.99 
 
 100.4 
 
 8.741138.9 
 
 9.37 
 
 184.0 
 
 21.12 
 
 .004 
 
 7.92 
 
 83.60 
 
 8.43 
 
 105.9 
 
 9.30 
 
 147.9 
 
 10.06 
 
 197.5 
 
 26.40 
 
 .005 
 
 8.85 
 
 93.37 
 
 950 
 
 119.3 
 
 10.43 
 
 165.8 
 
 11.30 
 
 220.0 
 
 31.68 
 
 .006 
 
 9.78 
 
 103.3 
 
 10.42 
 
 130.9 
 
 11.47 
 
 182.4 
 
 12.44 
 
 244.3 
 
 36.96 
 
 .007 
 
 10.59 
 
 111.7 
 
 11.31 
 
 142.1 
 
 12.45 
 
 190.0 
 
 
 
 4224 
 
 .008 
 
 11.36 
 
 119.9 
 
 12.25 
 
 153.9 
 
 
 
 
 
 
 47.52 
 
 .009 
 
 12.15 
 
 128.3 
 
 
 
 
 
 
 
 
142 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 17. 
 
 Velocities and Quantities of Flow in Clean Iron Pipes, due 
 Slopes and Diameters. 
 
 Fall per 
 
 *?! 
 
 Sine 
 of 
 Slope. 
 
 DIAMETERS. 
 
 72"=6 feet. 
 
 84"=7 feet. 
 
 96" =8 feet. 
 
 120 =10 feet. 
 
 M Ie. 
 
 fl 
 
 f 
 
 Velcci'y 
 
 Cubic 
 
 Veloc'y 
 
 Cubic 
 
 Vel'cy 
 
 Cubic 
 
 Vel'cy 
 
 Cubic 
 
 Feet. 
 
 s=L 
 
 ?eetper 
 
 Feet per 
 
 Ft per 
 
 Ft per 
 
 Ft per 
 
 Ft per 
 
 Ft per 
 
 Ft per 
 
 
 i 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 Sec. 
 
 0.528 
 
 .0001 
 
 1.66 
 
 4699 
 
 1.91 
 
 75.43 
 
 214 
 
 107.8 
 
 2.52 
 
 198.1 
 
 1.056 
 
 .0002 
 
 2.04 
 
 57.65 
 
 2.72 
 
 104.6 
 
 3.03 
 
 152.5 
 
 3.65 
 
 286.5 
 
 1.584 
 
 0003 
 
 2.92 
 
 82.53 
 
 3.28 
 
 126.2 
 
 3.75 
 
 188.5 
 
 4.49 
 
 352.3 
 
 2.112 
 
 .0004! 
 
 3.40 
 
 95.99 
 
 3.78 
 
 145.4 
 
 4.35 
 
 218.8 
 
 5.20 
 
 408.5 
 
 2.640 
 
 .0005 
 
 387 
 
 109.4 
 
 4.23 
 
 1628 
 
 4.88 
 
 245.3 
 
 5.80 
 
 458.7 
 
 3.168 
 
 .0006 
 
 4.30 
 
 121 6 
 
 4.60 
 
 177.0 
 
 5.32 
 
 267.4 
 
 6.41 
 
 503.6 
 
 3.696 
 
 .0007 
 
 4.67 
 
 132.0 
 
 4.99 
 
 1920 
 
 5.78 
 
 290.5 
 
 6.94 
 
 545.1 
 
 4.224 
 
 .0008 
 
 4.95 
 
 140.0 
 
 5.40 
 
 207.8 
 
 6.19 
 
 310.9 
 
 7.42 
 
 582.7 
 
 4.752 
 
 .0009 
 
 526 
 
 1487 
 
 5.78 
 
 222.4 
 
 6.60 
 
 324.2 
 
 7.87 
 
 618.1 
 
 5.280 
 
 .0010 
 
 5.58 
 
 157.8 
 
 6.11 
 
 235.1 
 
 6.97 
 
 350.5 
 
 8.30 
 
 651.4 
 
 5.808 
 
 .0011 
 
 5.87 
 
 166.0 
 
 658 
 
 253.3 
 
 7.29 
 
 366.2 
 
 8.70 
 
 6832 
 
 6.336 
 
 .0012 
 
 6.12 
 
 173.0 
 
 6.88 
 
 264.8 
 
 7.60 
 
 382.0 
 
 9.09 
 
 713.5 
 
 6.864 
 
 .0013 
 
 634 
 
 179.3 
 
 7.15 
 
 275.2 
 
 7.92 
 
 397.9 
 
 9.48 
 
 744.3 
 
 7.392 
 
 .0014 
 
 66.3 
 
 1875 
 
 7.48 
 
 287.7 
 
 8.25 
 
 414.7 
 
 9.84 
 
 772.5 
 
 7.920 
 
 .0015 
 
 6.86 
 
 1939 
 
 7.70 
 
 296.4 
 
 8.51 
 
 427.8 
 
 10,18 
 
 799.6 
 
 8.448 
 
 .0016 
 
 7.08 
 
 2002 
 
 8.00 
 
 307.9 
 
 882 
 
 443.1 
 
 10.51 
 
 825.7 
 
 8.976 
 
 0017 
 
 7.30 
 
 206.4 
 
 8.22 
 
 316.2 
 
 9.10 
 
 457.4 
 
 10.84 
 
 850.4 
 
 9.504 
 
 .0018 
 
 7.50 
 
 212 1 
 
 8.49 
 
 3267 
 
 9.36 
 
 470.5 
 
 11.15 
 
 875.9 
 
 10.302 
 
 .0019 
 
 7.70 
 
 217.7 
 
 8.73 
 
 335.8 
 
 9.58 
 
 481.5 
 
 11.45 
 
 899.9 
 
 10.560 
 
 .0020 
 
 7.97 
 
 225.2 
 
 905 
 
 348.3 
 
 9.88 
 
 496.4 
 
 11.78 
 
 925.3 
 
 11.616 
 
 .0022 
 
 8.33 
 
 235.5 
 
 9.48 
 
 364.9 
 
 10.40 
 
 522.8 
 
 12.35 
 
 971.4 
 
 12.672 
 
 .0024 
 
 8.72 
 
 246.4 
 
 9.88 
 
 389.1 
 
 10.89 
 
 547.4 
 
 12.90 
 
 1013.6 
 
 13.728 
 
 .0026 
 
 9.06 
 
 256.2 
 
 10.28 
 
 394.4 
 
 11.34 
 
 570.0 
 
 13.43 
 
 1057.4 
 
 14.784 
 
 .0028 
 
 9.45 
 
 267.2 
 
 10.61 
 
 408.4 
 
 11.78 
 
 592.1 
 
 13.94 
 
 1094.8 
 
 15.84 
 
 .0030 
 
 9.83 
 
 277.9 
 
 11.00 
 
 423.4 
 
 12.18 
 
 612.0 
 
 
 
 18.48 
 
 .0035 
 
 10.62 
 
 299.7 
 
 12.55 
 
 4830 
 
 
 
 
 
 21.12 
 
 .004 
 
 11.34 
 
 320.7 
 
 
 
 
 
 
 
 26.40 
 
 .005 
 
 1268 
 
 358.5 
 
 
 
 
 
 
 
PRACTICAL HYDRAULICS. 143 
 
 APPLICATION OF TABLE 17. 
 
 Ex. 70. The diameter of a pipe being 4 feet, and 
 the fall per mile 5.28 feet, what is the discharge in 
 cubic feet per second? 
 
 Gal In "fall per mile" column, Table 17, find 5.28 
 feet, opposite which, in right hand column, "4t feet 
 diameter," will be found 52.15 cubic feet, the dis- 
 charge sought. 
 
 Ex. 71. The fall per mile being 10.56 feet, a pipe 
 of what diameter will be requisite to discharge 100 
 cubic feet per second? 
 
 Gal. ID. "fall. per mile" column, Table 17, find 
 10.56 feet, opposite which find 100 cubic feet, or near- 
 est approximate thereto: 103.8 cubic feet is found in 
 right hand column, headed 4.5 feet diameter, which, 
 in practice, is sufficiently near the diameter sought in 
 most cases. 
 
 Ex. 72. The flow of 30 cubic feet of water per 
 second through a pipe 2.5 feet diameter, 5 miles long, 
 is employed at a hydraulic mine, whose elevation is 400 
 feet below the elevation of the inlet end of the pipe. 
 What is the effective head of the water at the mine? 
 
 Gal. In "2.5 feet" diameter column, Table 17, find 
 29.84 cubic feet nearest approximate to the given flow 
 of 30 feet, opposite which, in the "fall per mile" col- 
 
144 PRACTICAL HYDKAULICS. 
 
 umn, is found 21.12 feet, the loss of head per mile. 
 The loss in 5 miles, the given length of pipe, will be: 
 
 21. 12X5=105.60 feet. 
 
 400105.60=294.4 feet, effective head Ans. 
 
 Ex. 73. The data being the same as in Ex. 72, ex- 
 cept that the diameter of the pipe is 3 feet instead of 
 2.5, what is the effective head of the water *at the 
 mine? 
 
 CaL Find in "3 feet," right hand diameter column, 
 Table 17, 30.29 cubic feet, nearest approximate to 30 
 cubic feet, the given quantity, opposite which, in "fall 
 per mile" column, is found 4.448 feet, the loss per mile. 
 
 Then 8.448x5=42.24 feet loss of head in 5 miles; 
 40042.24=357.76 feet effective head.=4ws. 
 
 Comparing results with respect to the 2.5-foot pipe 
 of Ex. 72, and the 3-foot of Ex. 73: 
 
 357.76294.40=62.36 feet, it is seen that the loss 
 of head in the 3-foot pipe is 63.36 feet less than in the 
 2.5-foot pipe, which, in matters of economy, is of no 
 little importance. 
 
 Ex. 74. The elevation of the Guenoc Reservoir Site 
 of the Feather River Water Co. is 1015 feet above the 
 city base of San Francisco. The measured distance of 
 pipe-line between the reservoir and the city is 104.83 
 miles. How many gallons of water a day (24 hours) 
 will a pipe 40 inches diameter, whose inlet is at the 
 reservoir, and outlet at* San Francisco, deliver at an 
 elevation of 350 feet above city base? 
 
 Gal. 1015 350=665 feet, -total fall; 665-^104.83 
 
PRACTICAL HYDRAULICS. 145 
 
 =6.34 feet fall per mile. In "fall per mile" column, 
 Table 17, the nearest approximate fall to 6.34 feet is 
 6.336 feet, opposite which, in "40 inches," right hand 
 column "diameters," is found 34.68 cubic feet, the dis- 
 charge per second. 
 
 24X60X60 =-86,400 seconds in 24 hours; 34.68X 
 86,400=2,996,352 cubic feet discharge in 24 hours. 
 
 In 1 cubic foot are 7.5 gallons nearly; 2,996,352X 
 7.5=22,472,640 gallons. Ans. 
 
 Ex. 75. At a quartz mill, requiring 225 effective 
 horse-power, the efficiency of the water wheel, in 
 excess of the loss by nozzle resistance, is 60 per cent; 
 the length of the pipe line is 3 miles, and the eleva- 
 tion of the reservoir, at point of water supply, is 500 
 feet above the point of application of the water at the 
 mill. Required the diameter of the pipe to carry the 
 requisite quantity of water? 
 
 Col. 1st. 225-^.60 = 375, total horse-power. 
 
 375X550=206,250 "footpounds;" 206,250-^-62.5 = 
 3300 cubic feetXl foot, which we will term foot vol- 
 ume. ^ 
 
 Assume the loss of head by the internal surface 
 resistances of pipe 10.56 feet per mile; then 10.56X3 
 = 31.68 feet, total loss of head. 
 
 50031.68=468.32 feet, effective fall; 3300+- 
 468.32=7.05 cubic feet flow required per second. 
 
 In "fall per mile" column, Table 17, find 10.56 feet, 
 opposite which, in right hand column, "diameters," find 
 7.05 cubic feet. This is found under heading "20 
 
146 PEACTICAL HYDRAULICS. 
 
 inches." Hence the diameter of the required pipe is 
 20 inches. Ans. 
 
 Gal. 2d. By Cal. 1st, the "foot volume," corres- 
 ponding to the "foot pounds," required at the mill, is = 
 3300 cubic feetX 1 foot. 
 
 Assume 4.224 feet, the loss of head per mile clue 
 pipe resistances; then 4.224X3 -=12.672, total loss of 
 head; 500 12.672=487.328 feet, effective fall; 3300 
 -^487.3286.77 cubic feet required per second. 
 
 In "fall per mile" column find 4.224 feet, opposite 
 which, in right hand column, "diameters," find 7.32 
 cubic feet. This is found under heading "24 inches," 
 Hence the diameter sought is 24 inches. Ans. 
 
 Cal. 3d. By Cal. 1st, the "foot volume," corres- 
 ponding to the "foot pounds," required at the mill, is 
 = 3300 cubic feet XI foot. 
 
 Assume 21.12 feet, the loss of head per mile due 
 pipe resistances; then 21.12X3 = 63.36 feet, total loss 
 of head; 500 63.36=436.64 feet, effective head; 3300 
 -^-436.64^=7.55 cubic feet required per second. 
 
 In fall per mile column find 21.12 feet, opposite 
 which, in right hand column, "diameters," is found 
 7.65 cubic feet a very near approximate to 7.55 
 cubic feet, the required quantity. This is found under 
 heading "18 inches." Hence the diameter sought is 
 18 inches. Ans. 
 
 Maximum Work. To determine the maximum 
 work, which water subjected to flow through a long 
 pipe under pressure will perform, on issuing from the 
 pipe: 
 
PEACTICAL HYDRAULICS. 147 
 
 Let ^=the total head, exclusive of the inlet head, 
 which, rarely in practice exceeds 1.5 feet. 
 
 xhf, the friction head, expended in overcoming 
 the resistances within the pipe. 
 
 d diameter of pipe. 
 
 I length of pipe. 
 
 <r= ., hydraulic mean radius regarded constant. 
 
 cy coefficient of friction of resistances within the 
 pipe regarded constant. 
 
 g acceleration of gravity. 
 
 v velocity of water in the pipe per second. 
 
 n ratio of circumference to diameter of pipe. 
 
 h t -h x, head for effective work. 
 
 W= total weight of water discharged per second. 
 
 w weight of a cubic foot of water. 
 
 n<i 2 . _ . 
 
 a -j-, area 01 cross section of the pipe. 
 
 u maximum work performed by the water per 
 second. Then, 
 
 u^Wh^avwh r (142) 
 
 Substituting in (142) , the values of 
 
 = and h,h x\ 
 4 
 
 f(Eq. 
 
 x (h-x). (143) 
 4 
 
 Differentiating (142), omitting the constant factors 
 n, d, $) Cf I, 2 and 4, 
 
148 PEACTICAL HYDBAULICS. 
 
 Transposing and reducing (144), 
 
 = 
 Differentiating (144), 
 
 As the second differential coefficient is negative, the 
 function u representing the work performed is a max- 
 imum where #=, as found in Eq. (145). 
 
 Thus it is shown that, in theory, the work performed 
 by water after flowing through a pipe, is a maximum 
 when the head expended in overcoming the resistances 
 of the pipe is equal to one-third of the total head, ex- 
 clusive of the inlet head. A modification occurs, how- 
 ever, with respect to this result, owing to the experi- 
 mental coefficient of resistance being variable by 
 undetermined law, instead of constant, as assumed in 
 our solution. Another source of variation is evidently 
 due to difference in diameters of pipes. An inspection 
 of Table 17 shows, that in practice the ratio of the 
 head expended in overcoming the resistances in a clean 
 iron pipe is approximately equal to f , as a mean, in- 
 stead of J, as found, of the total head. The remain- 
 ing part of the total head amounting to nearly f as a 
 mean, applies to effective work. 
 
 Col. 4th. ByCal. 1st, the "foot volume" (substitu- 
 ted for ' ' foot pounds " for convenience of calculation) 
 required at the mill, is 3300 cubic feet X 1 foot. 
 
PRACTICAL HYDRAULICS. 149 
 
 Applying the ratio, f , proposed in the preceding 
 article, there results: 500 X f 187.5 feet, loss of head 
 by friction; 1875-^. 3=62.5 feet, loss of head per mile. 
 
 In " fall per mile" column, the nearest approximate 
 to 62.5 feet is 63.36 feet, loss of head per mile; then 
 63.36X3=190.08 feet, total loss of head; 500190.08 
 =309.92 feet effective head; 3300-^309.92=10.65 
 cubic feet flow required per second. 
 
 Opposite 63.36 feet in "fall per mile column," 
 Table 17, is found 13.65 cubic feet, nearest approxi- 
 mate to 10.65 cubic feet, the required quantity. This 
 is found in " 18 inches " column of contents for " diam- 
 eters." Hence the pipe meeting the requirements 
 is=l8 inches diameter. Ans. 
 
 Remark. In "16-inch column," opposite 6 3. 36 feet 
 ' 'fall per mile," is found 9.95 cubic feet, which is less 
 than the required quantity. Hence a 16-inch pipe is 
 too small. The velocity in a 17-inch pipe is v= 
 (.^H^X .012X11)^=7.46 feet per second for 63.36 
 feet fall per mile. The discharge per second in a 17- 
 inch pipe will be (1J) 2 X.7854X7.46=11.76 cubic feet 
 per second. 
 
 This amount, 11.76, exceeds the required amount of 
 10.65 cubic feet. Hence a 17-inch pipe will carry 
 sufficient water to do the work. The margin of safety, 
 11.76 10.65=1.11 cubic feet, however, is small. 
 
 An 18-inch pipe affording a margin of safety of 
 13.65 10.65=3 cubic feet, seems by no means large. 
 Even a 20-inch pipe, with a fall of 63.36 feet per 
 mile, and affording a margin of 7.34 cubic feet per 
 
150 PRACTICAL HYDRAULICS. 
 
 second, would not exceed the limits imposed by 
 D'Aubuisson, in his advice to engineers, if, indeed, it 
 would the limits of true economy. 
 
 Reverting to the results obtained by 1st, 2nd and 
 3rd calculations for Ex. 75 , and to the tabulated re- 
 sults corresponding respectively to these, or approxi- 
 mately so, it will be seen that in the first case there is 
 no margin of safety, in the second .55 cubic feet, and 
 in the third .10 cubic feet per second. This in 
 practice would be inadmissible. A margin of 33 per 
 cent is none too large. So that if on portions of the 
 lines steeper grades could not be had, it would be bet- 
 ter to employ a " 22-inch" pipe in the first case, a 
 " 27-inch" in the second, and a "20-inch" in the 
 third case. 
 
 Inlet Head. The inlet head is equal to the sum of 
 the head expended in generating the velocity in a pipe, 
 and the head expended in overcoming the resistance 
 of entry. 
 
 Thus transposing Eq. (109), we have the inlet head, 
 
 (147) 
 
 V* 
 
 Substituting the values of h v = of Eq. (110), and 
 
 c7 
 
 of h e =c e v* of Eq. (116), noting that c e =.5Q5 of Eq. 
 (127), 
 
 i ~~ ' %g~ 
 
 Rule 34. The inlet head is equal to .0234 times the 
 square of the velocity in the pipe. 
 
PBACTIG1& HYDRAULICS, 
 
 151 
 
 TABLE 18. 
 
 Velocities in Pipes and Corresponding Inlet Heads. 
 
 Veloci'y 
 Feet. 
 
 Inlet 
 Head. 
 
 Feet 
 
 Velocity. 
 Feet 
 
 Inlet 
 Head. 
 Feet. 
 
 Velocity 
 Feet. 
 
 Inlet 
 Head. 
 Feet. 
 
 Velocity 
 Feet. 
 
 Inlet 
 Head. 
 Feet. 
 
 .80 
 
 .015 
 
 3.68 
 
 .316 
 
 6.32 
 
 .933 
 
 10.00 
 
 2.34 
 
 .90 
 
 .019 
 
 3.76 
 
 .331 
 
 6.37 
 
 .948 
 
 10.50 
 
 2.58 
 
 .00 
 
 .023 
 
 3.85 
 
 .346 
 
 6.42 
 
 .963 
 
 11.00 
 
 2.83 
 
 .13 
 
 .030 
 
 3.93 
 
 .361 
 
 6.47 
 
 .978 
 
 11.50 
 
 3.10 
 
 .27 
 
 .038 
 
 4.00 
 
 .374 
 
 6.52 
 
 .993 
 
 12.00 
 
 3.37 
 
 .39 
 
 .045 
 
 4.09 
 
 .391 
 
 6.57 
 
 1.01 
 
 12.50 
 
 3.66 
 
 .50 
 
 .053 
 
 4.17 
 
 .406 
 
 6.61 
 
 1.02 
 
 13.00 
 
 3.96 
 
 .60 
 
 .060 
 
 4.25 
 
 .421 
 
 6.66 
 
 1.04 
 
 13.50 
 
 4.27 
 
 .70 
 
 .068 
 
 4.32 
 
 .436 
 
 6.71 
 
 1.05 
 
 14.00 
 
 4.59 
 
 .79 
 
 .075 
 
 4.39 
 
 .452 
 
 6.76 
 
 1.07 
 
 14.50 
 
 4.92 
 
 .88 
 
 .083 
 
 4.47 
 
 .467 
 
 6.81 
 
 1.08 
 
 15.00 
 
 5.27 
 
 1.97 
 
 .090 
 
 4.54 
 
 .482 
 
 6.86 
 
 1.10 
 
 15.50 
 
 5.62 
 
 2.00 
 
 .094 
 
 4.61 
 
 .497 
 
 6.91 
 
 1.11 
 
 16.00 
 
 5.99 
 
 2.04 
 
 .098 
 
 4.68 
 
 .512 
 
 6.95 
 
 1.13 
 
 16.50 
 
 6.38 
 
 2.12 
 
 .105 
 
 4.75 
 
 .527 
 
 7.00 
 
 1.15 
 
 17.00 
 
 6.76 
 
 2.20 
 
 .113 
 
 4.81 
 
 .542 
 
 7.04 
 
 1.16 
 
 17.50 
 
 7.17 
 
 2.27 
 
 .120 
 
 4.87 
 
 .557 
 
 7.09 
 
 1.17 
 
 18.00 
 
 7.58 
 
 2.34 
 
 .128 
 
 4.94 
 
 .572 
 
 7.13 
 
 1.19 
 
 18.50 
 
 7.79 
 
 2.41 
 
 .135 
 
 5.00 
 
 .585 
 
 7.18 
 
 1.20 
 
 19.00 
 
 8.45 
 
 2.47 
 
 .143 
 
 5.07 
 
 .606 
 
 7.22 
 
 1.22 
 
 19.50 
 
 8.90 
 
 2.54 
 
 .150 
 
 5.14 
 
 .617 
 
 7.26 
 
 1.23 
 
 20.00 
 
 9.36 
 
 2.60 
 
 .158 
 
 5.20 
 
 .632 
 
 7.31 
 
 1.25 
 
 20.50 
 
 9.83 
 
 2.66 
 
 .166 
 
 5.26 
 
 .647 
 
 7.35 
 
 1.26 
 
 21.00 
 
 10.32 
 
 2.72 
 
 .173 
 
 5.32 
 
 .662 
 
 7.40 
 
 1.28 
 
 21.50 
 
 10.82 
 
 2.78 
 
 .181 
 
 5.38 
 
 .677 
 
 7.44 
 
 1.29 
 
 22.00 
 
 11.33 
 
 2.84 
 
 .188 
 
 5.44 
 
 .692 
 
 7.48 
 
 1.31 
 
 22.50 
 
 11.85 
 
 2.89 
 
 .196 
 
 5.50 
 
 .707 
 
 7.53 
 
 1.32 
 
 23.00 
 
 12.38 
 
 2.95 
 
 .202 
 
 5.56 
 
 .722 
 
 7.57 
 
 1.34 
 
 23.50 
 
 12.92 
 
 3.00 
 
 .211 
 
 5.62 
 
 .737 
 
 7.61 
 
 1.35 
 
 24.00 
 
 13.48 
 
 3.05 
 
 .218 
 
 5.67 
 
 .753 
 
 7.65 
 
 1.37 
 
 24.50 
 
 14.05 
 
 3.11 
 
 .226 
 
 5.73 
 
 .768 
 
 7.70 
 
 1.38 
 
 25.00 
 
 14.63 
 
 3.16 
 
 .232 
 
 5.79 
 
 .783 
 
 7.74 
 
 1.40 
 
 25.50 
 
 15.22 
 
 3.21 
 
 .241 
 
 5.85 
 
 .798 
 
 7.78 
 
 1.41 
 
 26.00 
 
 15.82 
 
 3.26 
 
 .248 
 
 5.90 
 
 .813 
 
 7.82 
 
 1.43 
 
 27.00 
 
 17.05 
 
 3.31 
 
 .256 
 
 5.95 
 
 .828 
 
 7.86 
 
 1.44 
 
 28.00 
 
 18.35 
 
 3.36 
 
 .263 
 
 6.00 
 
 .843 
 
 7.90 
 
 1.46 
 
 29.00 
 
 19.78 
 
 3.40 
 
 .271 
 
 6.06 
 
 .858 
 
 7.94 
 
 1,47 
 
 30.00 
 
 21.06 
 
 3.45 
 
 -278 
 
 6.11 
 
 .873 
 
 S.OO 
 
 1.50 
 
 35.00 
 
 28.67 
 
 3.50 
 
 .286 
 
 6.17 
 
 .888 
 
 8.50 
 
 1.69 
 
 40.00 
 
 37.34 
 
 3.55 
 
 .293 
 
 6.22 
 
 .903 
 
 9.00 
 
 1.90 
 
 45.00 
 
 47.39 
 
 3.59 
 
 .301 
 
 6.28 
 
 .918 
 
 9.50 
 
 2.11 
 
 50.00 
 
 58,50 
 
152 PBACTICAL HYDKAULICS: 
 
 Ex. 76. The velocity in a 14-inch pipe is 4. 18 feet. 
 What is the total head, the pipe being one mile long? 
 
 Gal. 1st. Find in velocity column, Table 17, for 
 "14 inches diameters" of pipes, the given velocity, 
 4.18 feet, opposite which, in "fall per mile" column, is 
 found 26.40 feet, the head required to overcome the 
 resistances of the pipe. 
 
 In velocity column, Table 18, find 4. 17 feet, nearest 
 approximate to the given velocity, opposite which, in 
 "inlet head" column, is found 4.06 feet. Then 
 26.40+ .406=26.806 feet, total head. Ans. 
 
 Gal 2d. Find, as by Cal. 1st, the fall 26.40 feet. 
 
 By Rule 37: 
 
 (4.18) 2 X. 02337=. 4083 feet, .inlet head; 26.40 + 
 .4083=26.8083 feet, total head. Ans. 
 
 Ex. 77. A pipe, 33 inches diameter, being 5 miles 
 long, and the velocity of flow in it 9.80 feet per sec- 
 ond, what is the total head? 
 
 Cal. 1st. In Table 17, opposite 9.80 feet, velocity 
 for a "33-inch" pipe, find in "fall per mile column" 
 47.52x5=237.6 feet, head due resistances in pipe. 
 
 In Table 18, in velocity column, the nearest ap- 
 proximate to the given velocity is 9.83 feet, opposite 
 which, in "inlet head" column, is found 2.257 feet; 
 then 237.6+2.257=239.857 feet, total head. Ans. 
 
 Gal. 2d. Find, as in Cal. 1st, the friction head= 
 237. 6 feet. 
 
 By Rule 34: 
 
 (9.8) 2 X.02337=2.244 feet, inlet head; 237.6 + 
 2.244=239.844 feet, total head. Ans. 
 
PKACTICAL HYDRAULICS. 153 
 
 Remark. The inlet head, except in case of great 
 velocity, is, in practice, usually omitted as insignifi- 
 cant. Thus applying it in Ex. 75, the velocity due 
 the friction head, 63.36 feet, employed in Cal. 4th, is 
 by Table 17, 7.73 feet for an "18-inch" pipe. 
 
 By Table 18, the inlet head, due 7.74 feet velocity 
 nearest approximate to 7.73 feet, is 1.40 feet, which is 
 seen to be small in comparison with the given head of 
 500 feet. 
 
 EQUATIONS AND RULES FOR VELOCITY, HEAD, LENGTH, 
 DIAMETER, AND VOLUME, OF FLOW FOR CLEAN PIPES. 
 
 The general equation for the volume of flow is: 
 
 q=av. (149) 
 
 Substituting in (149), the value of a= - (area of 
 cross section of pipe), and the value of v of (129), 
 noting that ^=j, 
 
 For convenience of notation put, 
 
 Then, 2=0 (152) 
 
154 PRACTICAL HYDRAULICS. 
 
 Transposing (152) successively with respect to h/, d, 
 Z, and reducing, 
 
 (154, 
 
 d=JJ^_K (155) 
 
 Substituting in Eq. (151), the values of n=3.1416, 
 2c/=64.4, and of c/=. 00644, as a mean coefficient of 
 resistance within the pipe: 
 
 c,= 39.27. (156) 
 
 By reference to Table 16, the coefficient of resist- 
 ance .00644, employed in finding the value of c t of 
 Eq. (151), is due a velocity of 2.25 feet per second in 
 a 6-inch pipe, 5 feet velocity in a 3- inch pipe, and .7 
 feet velocity in a 12-inch pipe. Its range thus appears 
 too limited for general application. 
 
 Substituting the value of c y of (156) in Eqs. (152), 
 ( 153), (154) and (155), there results: 
 
 2=39. 27 f^ 5 ]*; (157) 
 
 I *' J 
 
 \ d'* \ } 
 7=1542.13 j^- 5 j; (159) 
 
 }*,- (160) 
 
PRACTICAL HYDRAULICS. 155 
 
 Equations (157), (158), (159) and (160), expressed 
 as written rules are as follows: 
 
 Rule 35. The quantity of flow in cubic feet per 
 second, iu a clean pipe, is equal to 39.27 times the 
 square root of the quotient arising from dividing the 
 product of ohe head, and the 5th power of the diame- 
 ter, both in feet measure, by the length of the pipe in 
 feet. 
 
 Rule 35 corresponds to Eq. (157). 
 
 Rule 36. The head is equal to .000648 times the 
 quotient arising from dividing the product of the 
 length of pipe in feet, and the square of the discharge 
 in cubic feet per second, by the 5th power of the 
 diameter of the pipe. 
 
 Rule 36 corresponds to Eq. (158). 
 
 Rule 37. The length of a pipe is equal 1542.13 
 times the quotient arising from dividing the product 
 of the head and the 5th power of the diameter by the 
 square of the discharge in cubic feet per second. 
 
 Rule 37 corresponds to Eq. (157). 
 
 Rule 38. The diameter is equal to .'23034 times the 
 5th root of the quotient arising from dividing the pro- 
 duct of the length of the pipe in feet, and the square 
 of the discharge in cubic feet by the head in feet. 
 
 Rule 38 corresponds to Eq. (160). 
 
 Rule 39. The values of quantity of flow, the head, 
 length of pipe, and diameter of pipe are found by 
 Table 17. 
 
156 PKACTICAL HYDRAULICS." 
 
 Case 1st. Quantity of flow being required, di- 
 vide the given head by the given length, both in feet. 
 Find in "sine of slope" column, the sine equal to this 
 quotient, opposite which, in discharge column for 
 the given diameter, will be found the quantity of 
 flow sought. 
 
 Case 2d. The head being sought, find the given 
 discharge, or nearest approximate thereto, in the dis- 
 charge column for the given diameter, opposite which, 
 in "sine of slope" column, will be found the proper 
 sine, which, multiply by the given length of pipe. The 
 product will be the head sought. 
 
 Case 3d. The length of pipe being required, find in 
 the discharge column for the given diameter, the given 
 quantity of flow, opposite which, in "sine of slope" 
 column, will be found the proper sine. Divide the 
 given head by this sine, the quotient will be the length 
 of pipe required. 
 
 Case 4th. The diameter being required, divide 
 the given head by the given length of pipe. Find in 
 "sine of slope" column the sine equal to this quotient, 
 opposite which, in discharge column, find the given 
 quantity of flow, or nearest approximate thereto. 
 The diameter for this discharge will be the diameter 
 sought. 
 
 Ex. 78. The head being 40 feet, the pipe 6 inches 
 diameter, 10,000 feet long, what is the discharge in 
 cubic feet per second? 
 
 Cal. 1st. Substituting the given values d=6"=.5 
 feet, 7?/=40 feet, and =10,000 feet in Eq. (157); 
 
PBACTICAL HYDEAULICS. 157 
 
 ^ 5 =(.5) 5 X40^-10,000=.000125 1 
 
 39.27X(.000125)i=.440 cubic feet. Ana. 
 
 Employ Rule 39, case 1st. 
 
 Gal. 2d. Dividing the given head by the given 
 length of pipe, s=4(H 10,000=. 004, sine of slope. 
 
 In Table 17, find the sine of slope, opposite which, 
 in discharge column, for 6 -inch pipe, is found the 
 quantity sought=.444 cubic feet. Ans. 
 
 Ex. 79. A pipe 3 inches diameter, 10,000 feet long, 
 discharges .247 cubic feet per second, what is the 
 head ? 
 
 Gal. 1st. Substituting the given values, d== 
 3 inches, =10,000 feet, and g=247 cubic feet in Eq. 
 (US)-. 
 
 .000648X61.74=400.05 feet head. Ans. 
 
 Employ Rule 39, case 2. 
 
 Gal. 2d. In Table 17, find in 3-in. discharge column, 
 .2469 cubic feet, nearest approximate to the given 
 quantity .247, opposite which, in sine of slope column, 
 is found .04. Then, h=ls. 
 
 .04X10,000=400 feet head. Ana. 
 
 Ex. 80. If, under a head of 42 feet, a pipe 4 inches 
 diameter discharges .3 of a cubic foot of water per 
 second, what is the pipe's length? 
 
 Gal. 1st. Substituting the given values of <i=4"= 
 J foot, &=42, and 9=. 3 cubic feet in Eq. (159), 
 
158 PRACTICAL HYDRAULICS. 
 
 2=1542. 13 X42 X (i) 5 - (.3) 2 =2961.6 feet. Ans. 
 
 Employ Rule 39, case 3. 
 
 Cat. 2d. In Table 17, find in discharge column for 
 4 -inch pipe, .3036 cubic feet, nearest approximate to 
 the given quantity .3 cubic feet, opposite which, in 
 
 sine of slope column, is found .014; then l=-\ 1= ./ T 2 T 
 
 s 
 
 =3000 feet. Ans. 
 
 Ex. 81. The head of water being 280 feet, re- 
 quired the diameter of a pipe 4000 feet long, that will 
 discharge 80,000 gallons in 24 hours? 
 
 Col. 1st. q -80,000- (21X60X60 X 7.5) --=.124 
 cubic feet, discharge per second. 
 
 Substituting the given values, g=.l24 cubic feet, 
 7i/=280 feet, and 2=4000 feet in Eq. (160); 
 
 d=. 23034 (4000X(.124) 2 -280)^=.17 feet=2.04 
 inches diameter. Ans. 
 Employ E-ule 39, case 4. 
 
 Cat. 2d. s=j=J/$\= .01 , sine of slope. 
 
 In Table 17, find sine of slope .07, opposite which, 
 in discharge column, find .1286 cubic feet per second, 
 nearest approximate to the given volume .124. This 
 is found under heading "2 inches" diameter. The 
 diameter required then is=2 inches. Ans. 
 
 COEFFICIENT OF FLOW. 
 When the coefficient of flow in a long pipe is c = 
 
PRACTICAL HYDRAULICS. 159 
 
 39.27, the coefficient of velocity deduced from Eq. 
 (130) or (129), is 
 
 f Ct ,.\ 1 
 
 :100. (161) 
 
 In which case Eq. (130) becomes 
 
 i;=:100 (r s)i (162) 
 
 and Eq. (129), by observing that f=-r, 
 
 becomes V =5<) (163) 
 
 Several standard authors on hydraulics find as fol- 
 lows: 
 
 Chezy finds c=100. (164) 
 
 Eytelwein finds c=100. (165) 
 
 Leslie finds c=100. (166) 
 
 D'Aubuisson finds c=95.6. (167) 
 
 Blackwell finds c=95.83. (168) 
 
 Hawksley finds c=96.09 . (169) 
 
 Bartlett finds c=95.88. (1TO) 
 
 Jackson finds (small pipes) c=100. (171) 
 
 Fanning finds (mean for small pipes), 
 
 c=100. (172) 
 
 D'Arcy finds (for larger pipes) c=113.8. (173) 
 
 Substituting the value of c=113.8, as found by 
 
160 PRACTICAL HYDRAULICS. 
 
 D'Arcy, the value of n=3.14l6, in Eq. (150), and 
 reducing, 
 
 d *H (174) 
 
 COMPARISON OF RESULTS OBTAINED BY EQUATION 
 (174) AND BY TABLE 17. 
 
 If the head of water be 100 feet, the pipe 1 foot 
 diameter, and 10,000 feet long, the quantity of flow 
 by Eq. (174), will be g=4.469 cubic feet per second, 
 and by Table 17, g=4.305 cubic feet per second. 
 
 The head being 16 feet, the pipe 4 feet diameter, 
 and 10,000 feet long, the quantity of flow by Eq. 
 (174) will be <?=57.20 cubic feet per second, and by 
 Table 17, g=68.50 cubic feet per second. 
 
 When the head is 100 feet, the pipe 6 inches diame- 
 ter, and 10,000 feet long, the discharge by Eq. (174) 
 will be g=.79 cubic feet per second, and by Table 17 
 <?= .72 cubic feet per second. 
 
 The head being 20 feet, the pipe 8 feet diameter, 
 and 10,000 feet long, the quantity of flow by Eq. 
 (174) will be 5=361.8 cubic feet per second, and by 
 Table 17 q =49 6. 4 cubic feet per second. 
 
 These comparisons show that the coefficient, 113.8, 
 proposed as a mean, is too large for small pipes and 
 too small for large pipes; that it is adapted to a pipe 
 1 foot diameter, with a limited range for either 
 smaller or larger diameters, 
 
PEACTICAL HYDRAULICS' 
 
 161 
 
 They illustrate, also, that the engineer, in the prac- 
 tice of his profession, cannot safely venture far from 
 an established fact in hydraulics, without experiment 
 as a guide. 
 
 BENT PIPES. 
 
 Bends occurring in pipes resist the flow of water 
 through them. 
 
 To determine the additional head requisite to over- 
 come this resistance, Weisbach (partly on the experi- 
 ments of Du Buat, but chiefly on his own experi- 
 ments), gives substantially the following formulas and 
 tables compiled from them : 
 
 ANGULAR BENDS. 
 
 Let & y =the additional head required to overcome 
 one angular bend. 
 
 m=one-half the angle of deflection of the bend. 
 z~i}\Q coefficient of bend or knee resistance. 
 v= velocity of flow. Then, 
 
 ' ; (175) 
 
 0=0. 9457 sin. 2 m + 2.047 sin.* m. (176) 
 
 Table 19 is calculated from Eq. (176). 
 
 TABLE 19. 
 
 Coefficients for Bend Resistances in Pipe. 
 
 m 
 
 z 
 
 10 
 .046 
 
 20 
 .139 
 
 30 
 .364 
 
 40 
 .740 
 
 45 
 
 .984 
 
 50 
 1.260 
 
 55 
 1.556 
 
 60 
 1.861 
 
 65 
 2.158 
 
 70 
 2.43 
 
162 PRACTICAL HYDRAULICS. 
 
 Rule 40 . The additional head required to overcome 
 one angular bend is, in case the head generating the 
 velocity be given, equal to the product of the given 
 head, and the coefficient of bend or knee resistance, 
 due the given angle of deflection found in Table 19 ; 
 and is, in case the velocity be given, equal to the pro- 
 duct of said coefficient and the square of the given 
 velocity, divided by 64.4. 
 
 Ex. 82. The velocity of water in a pipe, in which 
 occurs one rectangular bend, is 10 feet per second. 
 What additional head will be requisite to overcome the 
 resistance of the head? 
 
 Gal. By Rule 40, square of velocity divided by 
 
 QAO 
 
 64.4=10X10-64.4=1.553 feet; m=^-=45, one- 
 half the angle of deflection. 
 
 By Table 19, the value of z, corresponding to 45, 
 is .984; then 1.553X. 984=1. 528 feet, additional 
 head. Ans. 
 
 CURVED BENDS. 
 
 Let h== additional head required to overcome the 
 resistance of curvature, 
 
 6=angle of curvature of the pipe, 
 R=radius of curvature of the bend, 
 r=radius of the pipe, 
 ^=coefficient of resistance;] 
 
PRACTICAL HYDRAULICS. 
 
 163 
 
 (178) 
 
 Eq. (178), from which Table 20 is computed, is for 
 pipes with circular cross sections. 
 
 TABLE 20. 
 
 Coefficients of Resistance of Curvature with Circular Trans- 
 verse Sections. 
 
 i 
 
 z, 
 
 0.1 
 .131 
 
 0.2 
 .138 
 
 0.3 
 
 .158 
 
 0.4 
 .206 
 
 0.5 
 .294 
 
 0.6 
 .440 
 
 0.7 
 .661 
 
 0.8 
 .977 
 
 0.9 
 1.408 
 
 1.0 
 1.978 
 
 Rule 41. The additional head required to overcome 
 the resistance of curvature of a bent pipe, is equal to 
 
 the product of the head, & j o~ r > generating the ve- 
 locity, the ratio j TQQO ~ f f 1 80 or n > ^ the an ~ 
 
 gle of deflection or length of bend, and the coefficient 
 of resistance, corresponding to the ratio of the radius 
 of curvature to the radius of the pipe. 
 
 Ex. 83. The velocity of water in a pipe 2 inches 
 diameter, is 16 feet per second: what additional head 
 will be requisite to overcome the resistances of a bend 
 in the pipe, whose radius of curvature is 2 inches, and 
 angle of deflection 90? 
 
 Oal. Head generating velocity, equal to the square 
 
164 
 
 PRACTICAL HYDBAULICS. 
 
 of the velocity, divided by 64.4, as ^=16X16-^-64.4 
 =3.97 5 
 
 v 90 
 
 p=i=.5, ratio of given radii; 6=r-Q-Q=J, ratio 
 * loU 
 
 of 180, to given angle of deflection. 
 
 By Table 20, the coefficient corresponding to .5, the 
 ratio of radii is .294. Then by Rule 41, 
 
 3.975 X J X .294=.584 feet. Ana. 
 
 V=0. 124+3.104 j^U. (179) 
 
 In Eq. (179), from which Table 21 is computed, r 
 represents half the width of a rectangular pipe; and 
 R, the radius of curvature of the axis. 
 
 TABLE 21. 
 
 Coefficients of Resistance of Curvature in Pipes with Rectan- 
 gular Transverse Sections. 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 E,"" 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0..7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 z /r . . . 
 
 t.124 
 
 .135 
 
 .180 
 
 .250 
 
 .398 
 
 .643 
 
 1.015 
 
 1.546 
 
 2.271 
 
 3.228 
 
 Remark. Rule 41 applies to finding the additional 
 head required to overcome the resistance of curvature 
 of a bent pipe, whose cross section is rectangular, by 
 employing Table 21 instead of Table 20. 
 
 Total head. Let in general: 
 
PRACTICAL HYDRAULICS. 165 
 
 represent the additional head requisite to overcome 
 the resistances of n, given bends angular or 
 curved in a pipe, the form of whose cross section is 
 given circular or rectangular. 
 
 Combining (180) and (122), observing that c=.505 
 
 of (127), ^= and 2#=64.4, 
 
 Ex. 84.- A pipe 1 foot diameter, 500 feet long, has 
 five quadrant or right angled bends: the radius 
 of curvature of each bend is 1 foot; the velocity of 
 water through the pipe is 8. 025 feet per second. What 
 is the total head ? 
 
 Gal. By Table 16, the coefficient corresponding to 
 a velocity of 8 feet per second in a pipe whose diame- 
 ter is d=l ft., is c/=.0052. 
 
 Square of velocity divided by 64.4 ; 8.025X8.025 
 
 -^-64.4=1. In the present case n 
 
 r .5 
 =2.5. Ratio of the given radii, -=.==.5. 
 
 By Table 20, when-^-=.5, s=-!=.294. 
 
 Aw Ou 
 
 Substituting in (181) the values of c,=.0052, d= 
 1 foot, Z=500 feet, %=2.5, and a; =0 1 =. 
 
 feet. Ans. 
 
166 PRACTICAL HYDRAULICS. 
 
 FLOW OF WATER THROUGH NOZZLES. 
 
 The resistance to the flow of water in conically con- 
 vergent tubes, estimated for the smaller orifice, is less 
 than the resistance in a cylindrical tube with equal 
 orifice. Thus, Table 13 shows that the coefficient of 
 velocity from a tube converging with an angle 3 10' 
 is .894: and that the coefficient for a cylindrical tube 
 of equal diameter is .829. Let t represent this ra- 
 tio: ; j' 
 
 . .894 
 
 -329- 
 
 Let i> the experimental velocity of water in a cyl- 
 indrical pipe, the ratio of whose diameter to length is 
 as I'-W') v 1 =ihe theoretical velocity in same pipe; 
 cy= coefficient of resistance, varying from .006 to .004; 
 c /l =mean coefficient of velocity with respect to the 
 smaller orifice of the nozzles: 
 
 c n =t^~ J. (183) 
 
 Substituting the values of <y=.004, d=l, andZ=10 
 
 in (128), 
 
 i ( \ i i 
 
 ; (184) 
 
 V!=(20&)*, asperEq. (8). (185) 
 
 Substituting the values of v, v lt and , in (183), 
 c n =:.836, nozzle coefficient (186). 
 
PRACTICAL HYDRAULICS. 
 
 167 
 
 TABLE 22. 
 
 Flow of Water Through Nozzles. 
 
 Smaller diameter : to length : : 1 : 10 ; angle of con- 
 vergence, 3 10. 
 
 Head. 
 Feet. 
 
 DIAMETERS OP NOZZLES. 
 
 1 inch. 
 Cu. ft. 
 
 1.5" 
 Cu. ft. 
 
 2." 
 Cu. ft. 
 
 2.5" 
 Cu. ft. 
 
 3." 
 Cu. ft. 
 
 3.5" 
 Cu. ft. 
 
 4." 
 Cu. ft. 
 
 4.5" 
 Cu. ft. 
 
 10. 
 
 .115 
 
 .258 
 
 .458 
 
 .715 
 
 1.03 
 
 1.39 
 
 1.92 
 
 2.32 
 
 12.5 
 
 .128 
 
 .288 
 
 .510 
 
 .797 
 
 1.16 
 
 1.56 
 
 2.05 
 
 2.60 
 
 15. 
 
 .131 
 
 .315 
 
 .562 
 
 .875 
 
 1.26 
 
 1.72 
 
 2.25 
 
 2.84 
 
 17.5 
 
 .151 
 
 .340 
 
 .605 
 
 .94 
 
 1.36 
 
 1.85 
 
 2.42 
 
 3.06 
 
 20. 
 
 .162 
 
 .364 
 
 .647 
 
 .02 
 
 1.46 
 
 1.99 
 
 2.59 
 
 3.28 
 
 22.5 
 
 .171 
 
 .386 
 
 .686 
 
 .08 
 
 1.54 
 
 2.10 
 
 2.75 
 
 3.48 
 
 25. 
 
 .182 
 
 .407 
 
 .725 
 
 .13 
 
 1.63 
 
 2.26 
 
 2.90 
 
 3.67 
 
 27.5 
 
 .190 
 
 .426 
 
 .768 
 
 .19 
 
 1.71 
 
 2.32 
 
 3.03 
 
 3.84 
 
 30. 
 
 .204 
 
 .446 
 
 .811 
 
 .26 
 
 1.81 
 
 2.48 
 
 3.24 
 
 4.10 
 
 32,5 
 
 .208 
 
 .464 
 
 .825 
 
 .30 
 
 1.87 
 
 2.53 
 
 3.30 
 
 4.18 
 
 35. 
 
 .214 
 
 .482 
 
 .857 
 
 .35 
 
 1.93 
 
 2.63 
 
 3.43 
 
 4.34 
 
 40. 
 
 .230 
 
 .520 
 
 .92 
 
 .43 
 
 207 
 
 2.81 
 
 3.67 
 
 4.64 
 
 45. 
 
 .242 
 
 .553 
 
 .97 
 
 .52 
 
 2.18 
 
 2.97 
 
 3.88 
 
 4.93 
 
 50. 
 
 .256 
 
 .583 
 
 1.04 
 
 1.60 
 
 2.30 
 
 3.13 
 
 4.09 
 
 5.25 
 
 60. 
 
 .273 
 
 .638 
 
 1.13 
 
 1.77 
 
 2.56 
 
 3.43 
 
 4.49 
 
 5.75 
 
 70. 
 
 .307 
 
 .690 
 
 1.22 
 
 1.92 
 
 2.75 
 
 3.75 
 
 4.88 
 
 6.16 
 
 80. 
 
 .328 
 
 .737 
 
 1.31 
 
 2.04 
 
 2.95 
 
 4.01 
 
 5.26 
 
 6.64 
 
 90. 
 
 .347 
 
 .778 
 
 1.39 
 
 2.20 
 
 3.06 
 
 5.54 
 
 5.54 
 
 7.00 
 
 100. 
 
 .366 
 
 .824 
 
 1.47 
 
 2.29 
 
 3.30 
 
 5.87 
 
 5.87 
 
 7.41 
 
 125. 
 
 .410 
 
 .92 
 
 1.64 
 
 2.56 
 
 3.67 
 
 6.75 
 
 6.55 
 
 8.26 
 
 150. 
 
 .449 
 
 1.01 
 
 1.80 
 
 2.80 
 
 4.03 
 
 7.20 
 
 7.20 
 
 9.07 
 
 175. 
 
 .485 
 
 1.09 
 
 1.94 
 
 3.02 
 
 4.36 
 
 7.78 
 
 7.78 
 
 9.80 
 
 200. 
 
 .518 
 
 1.16 
 
 2.07 
 
 3.23 
 
 4.59 
 
 8.28 
 
 8.28 
 
 10.45 
 
 250. 
 
 .584 
 
 .31 
 
 2.32 
 
 3.62 
 
 5.22 
 
 9.29 
 
 9.29 
 
 11.75 
 
 300. 
 
 .635 
 
 .43 
 
 2.54 
 
 3.96 
 
 5.67 
 
 10.15 
 
 10.15 
 
 12.88 
 
 350. 
 
 .686 
 
 .54 
 
 2.74 
 
 4.28 
 
 6.16 
 
 11.98 
 
 10.98 
 
 1385 
 
 400. 
 
 .733 
 
 .65 
 
 293 
 
 4.58 
 
 6.59 
 
 11.74 
 
 11.74 
 
 14.82 
 
 450. 
 
 .778 
 
 .75 
 
 3.11 
 
 4.86 
 
 6.98 
 
 12.46 
 
 12.46 
 
 15.71 
 
 500. 
 
 .820 
 
 .84 
 
 3.28 
 
 5.12 
 
 7.38 
 
 13.10 
 
 13.10 
 
 16.60 
 
 550. 
 
 .859 
 
 .89 
 
 3.44 
 
 5.36 
 
 7.56 
 
 13.75 
 
 13.75 
 
 17.01 
 
 600. 
 
 .899 
 
 201 
 
 3.59 
 
 561 
 
 8.13 
 
 14.36 
 
 14.36 
 
 18.06 
 
 700. 
 
 .95 
 
 2.21 
 
 3.92 
 
 6.11 
 
 8.86 
 
 15.70 
 
 15.70 
 
 19.93 
 
 800. 
 
 1.04 
 
 2.32 
 
 4.14 
 
 6.47 
 
 9.29 
 
 16.56 
 
 16.56 
 
 20.90 
 
 900. 
 
 1.10 
 
 2.48 
 
 4.39 
 
 6.87 
 
 9.90 
 
 17.57 
 
 17.57 
 
 22.27 
 
 1000. 
 
 1.16 
 
 260 
 
 4.64 
 
 7.24 
 
 10.40 
 
 18.58 
 
 18.58 
 
 23.40 
 
168 
 
 PEACTICAL HYDRAULICS. 
 
 TABLE 22. 
 
 Flow of Water Through Nozzles. 
 
 Smaller diameter : to length : : 1: 10 ; angle of con- 
 vergence 3 10. 
 
 Head 
 Feet. 
 
 DIAMETERS OF NOZZLE. 
 
 5 inch. 
 Cu. Ft. 
 
 5.5" 
 Cu. ft. 
 
 6. 
 Cu. ft. 
 
 7." 
 Cu.ft. 
 
 8." 
 Cu. ft. 
 
 9." 
 Cu. ft 
 
 10." 
 Cu. ft. 
 
 12." 
 Cu. ft. 
 
 10. 
 
 286 
 
 3.46 
 
 4.13 
 
 5.60 
 
 7.69 
 
 9.29 
 
 11.44 
 
 16.48 
 
 125 
 
 3.19 
 
 3.86 
 
 4.62 
 
 6.26 
 
 1.18 
 
 10.40 
 
 12.79 
 
 18.44 
 
 15. 
 
 3.51 
 
 4.23 
 
 5.05 
 
 6.86 
 
 9.00 
 
 11.36 
 
 14.10 
 
 20.20 
 
 17.5 
 
 3.78 
 
 4.57 
 
 5.44 
 
 7.42 
 
 9.67 
 
 1224 
 
 15 14 
 
 21.75 
 
 20. 
 
 4.04 
 
 4.89 
 
 583 
 
 7.93 
 
 1035 
 
 13.12 
 
 16.18 
 
 23.32 
 
 22.5 
 
 4.31 
 
 5.17 
 
 6.18 
 
 8.41 
 
 10.98 
 
 13.92 
 
 17.17 
 
 24.74 
 
 25. 
 
 4.53 
 
 5.47 
 
 6.51 
 
 9.13 
 
 11.59 
 
 1464 
 
 18.08 
 
 25.95 
 
 27.5 
 
 4.75 
 
 5.74 
 
 6.82 
 
 9.30 
 
 12.13 
 
 15.35 
 
 18.97 
 
 27.30 
 
 30. 
 
 5.07 
 
 6.14 
 
 7.29 
 
 9.94 
 
 12.98 
 
 16.39 
 
 20,27 
 
 29.14 
 
 32.5 
 
 5.15 
 
 6.33 
 
 7.38 
 
 10 10 
 
 13.20 
 
 16.72 
 
 20.62 
 
 29.72 
 
 35.0 
 
 5.35 
 
 6.47 
 
 7.72 
 
 10.49 
 
 13.73 
 
 17.36 
 
 21.40 
 
 30.86 
 
 40. 
 
 5.73 
 
 6.91 
 
 8.25 
 
 11.24 
 
 14.65 
 
 18.66 
 
 22.88 
 
 33.00 
 
 45. 
 
 6.06 
 
 7.34 
 
 8.75 
 
 11.89 
 
 15.59 
 
 19.79 
 
 24.26 
 
 34.98 
 
 50. 
 
 6.39 
 
 7.83 
 
 9.11 
 
 1263 
 
 16.35 
 
 20.82 
 
 25.58 
 
 36.83 
 
 60. 
 
 7.00 
 
 8.48 
 
 10.11 
 
 13.73 
 
 17.92 
 
 22.71 
 
 28.03 
 
 40.39 
 
 70. 
 
 766 
 
 9.27 
 
 11.02 
 
 14.48 
 
 19.51 
 
 24.47 
 
 30.27 
 
 42.56 
 
 80. 
 
 8.19 
 
 9.91 
 
 11.81 
 
 16.06 
 
 21.02 
 
 26.57 
 
 32.36 
 
 46.65 
 
 90. 
 
 8.68 
 
 10.42 
 
 12.46 
 
 17.03 
 
 22 18 
 
 28.03 
 
 34.75 
 
 49.82 
 
 100. 
 
 9.15 
 
 11.08 
 
 13.18 
 
 17.95 
 
 23.47 
 
 29.65 
 
 36.63 
 
 52.70 
 
 125. 
 
 10.24 
 
 12.38 
 
 14.69 
 
 20.07 
 
 26.21 
 
 33.05 
 
 40.96 
 
 58.75 
 
 150. 
 
 11.21 
 
 13.57 
 
 16.13 
 
 21.98 
 
 28.80 
 
 36.29 
 
 44.86 
 
 64.51 
 
 175. 
 
 12.11 
 
 13.76 
 
 17.42 
 
 23.75 
 
 31.10 
 
 39.20 
 
 48.47 
 
 69.70 
 
 200. 
 
 12.91 
 
 15.76 
 
 18.58 
 
 25.38 
 
 33.12 
 
 41.80 
 
 51.80 
 
 74.30 
 
 250. 
 
 15.48 
 
 17.52 
 
 20.88 
 
 28.39 
 
 37.15 
 
 46.98 
 
 57.92 
 
 83.52 
 
 300. 
 
 15.86 
 
 19.20 
 
 22.90 
 
 31.09 
 
 40.61 
 
 51.52 
 
 63.45 
 
 90.58 
 
 350. 
 
 17.14 
 
 19.84 
 
 24.57 
 
 33.59 
 
 4392 
 
 5540 
 
 68.54 
 
 97.5 
 
 400. 
 
 18.31 
 
 22.16 
 
 26.35 
 
 35.90 
 
 46.94 
 
 59.29 
 
 73.27 
 
 105.4 
 
 450. 
 
 19.43 
 
 23.51 
 
 27.94 
 
 38.08 
 
 49.82 
 
 6?. 86 
 
 77.72 
 
 111.8 
 
 500. 
 
 20.47 
 
 24.79 
 
 29.52 
 
 39.60 
 
 52.42 
 
 66.42 
 
 81.92 
 
 118.1 
 
 550. 
 
 21.47 
 
 25.99 
 
 3024 
 
 42.10 
 
 5501 
 
 68.04 
 
 85.91 
 
 121.0 
 
 600. 
 
 22.44 
 
 27.14 
 
 32.11 
 
 43.97 
 
 57.46 
 
 72.25 
 
 89.74 
 
 128.5 
 
 700. 
 
 24.46 
 
 29.59 
 
 35.42 47.93 
 
 62.78 
 
 7970 
 
 9.79 
 
 141.7 
 
 800. 
 
 25.89 
 
 31.43 
 
 37.15 ! 50.76 
 
 66.24 
 
 83.52 
 
 1036 
 
 148.6 
 
 900. 
 
 26.47 
 
 33.25 
 
 39.60 
 
 53.89 
 
 70.27 
 
 89.10 
 
 109.8 
 
 158.4 
 
 1000. 
 
 2895 
 
 35.04 
 
 41.62 
 
 56.75 
 
 74.30 
 
 93.6 
 
 1159 
 
 166.5 
 
X 
 PKACTICAL HYDRAULICS. 169 
 
 Ex. 85. A nozzle being 6 inches diameter at its 
 discharge end, 5 feet long, will discharge under a 
 head of 150 feet how many cubic feet of water per 
 second? 
 
 Col. In "6-inch" diameter column, opposite 150 
 feet in "head" column, Table 22, the quantity sought 
 is found=l6.l3 cubic feet. 
 
 RELATIVE CARRYING CAPACITIES OF CLEAN, FOUL 
 AND VERY FOUL WATER PIPES. 
 
 In the preceding discussion with respect to the 
 carrying capacities of water pipes, they have been 
 considered clean. 
 
 Assuming the coefficient of resistance for a clean 
 pipe, .00644; for a rough or foul pipe, .0082; and for 
 a very foul pipe, .012, then will the relative volumes 
 of flow be: 
 
 For ctean pipes, 1 . 0000 ; 
 
 For foul pipes, .8863; 
 
 For very foul pipes, .7325. : 
 
 Rule 42. In case a water pipe is rough and foul, 
 multiply the flow for a clean pipe in Table 17 (with 
 same diameter, length and head), by 9, 8, 7, etc., ac- 
 cording to the degree of foulness, as judgment shall 
 dictate. 
 
 Ex. 89. A long pipe, 6 inches diameter, has a fall 
 of 18.48 feet per mile. What will be the flow in cubic 
 
170 PRACTICAL HYDRAULICS. 
 
 feet per second, if its coefficient of discharge with re- 
 spect to a clean pipe be .8 ? 
 
 Gal In Table 17, find opposite the given fall 18.48 
 feet, in discharge column, for a 6-inch pipe, .395 cubic 
 feet. This is for a clean pipe . 
 
 Then by Rule 42, .395 X. 8-=. 316 cubic feet. Ans. 
 
 Pressure Ordinates. A mean pressure ordinate is 
 the vertical distance between the axis of the pipe and 
 the hydraulic gradient. Thus in Fig. 19, CE is the 
 hydraulic gradient, and (afc + r), (cd-fr), or (ef+r), 
 is a mean pressure ordinate. If the axis of the pipe 
 be depressed below the base, as represented in Fig. 19, 
 by the dotted line, D a, c, e, E; a b, cd or ef, will be 
 the mean pressure ordinate for the given point, a, c 
 or e. To find the value of the ordinate in pounds 
 pressure: 
 
 Let A =the pressure ordinate, or hight of water 
 column in feet, as a 6; jo pounds pressure per square 
 inch of the ordinate or column; whence 
 
 fc=-434 h pounds nearly. (187) 
 
 Rule 43. The pressure in pounds of a vertical col- 
 umn of water, whose cross section is one square inch, 
 is equal to .434 times the hight of the column or or- 
 dinate in feet. Rule 43 corresponds to Eq. (187). 
 
 THICKNESS or THIN PIPES REQUISITE TO WITHSTAND 
 
 A GIVEN PRESSURE IN POUNDS PER SQUARE INCH. 
 
 Let p=pounds pressure per square inch; =thiek- 
 
PKACTICAL HYDRAULICS. 171 
 
 ness of pipe in inches; r=radius of pipe; &=modulus 
 of strength, working load, or safety, as shall be given. 
 By Bartlett's Mechanics, page 507, 
 
 =-. (188) 
 
 Rule 44. The thickness in inches of a thin pipe, 
 requisite to withstand a given pressure per square 
 inch is equal to the quotient arising from dividing the 
 product of the pounds pressure per square inch, and 
 the radius of the pipe in inches by the given modulus 
 of the material in the pipe. Rule 44 corresponds 
 to Eq. (188). 
 
 Ex. 90. The water pipe, 34.19 inches in diameter, 
 at the hydraulic works of the Spring Valley Com- 
 pany, Butte County, California, laid in 1870, now in 
 perfect condition, as reported by the engineer in 
 charge, is subjected to a working strain of 17,549 
 pounds per square inch. The greatest pressure ordi- 
 nate is 887 feet in hight. What is the thickness of 
 the iron in the pipe? 
 
 Cal.By Rule 43, p =^887 X. 434= 385 pounds pres- 
 sure per square inch, nearly; r=34. 19-^-2^= 17. 095, 
 radius of pipe. 
 
 By Rule 44, =385X 17.095-^-17,549=. 3754 inches 
 thickness, or f of an inch. Ans. 
 
 Remark. The modulus &=17,549 pounds, work- 
 ing load, seems rather too high for ordinary plate iron. 
 
172 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 23. 
 
 Moduli of Strength, Working Load and Safety Weisbach. 
 
 Names cf Sutstances. 
 
 Modulus of 
 Strength. 
 Pounds. 
 
 Modulus of 
 Working load. 
 Pounds. 
 
 Modulus of 
 Safety. 
 Pounds. 
 
 Iron iii bars . . 
 
 59 805 
 
 20,622 
 
 10,311 
 
 Iron in plates 
 
 56 712 
 
 
 9 280 
 
 Cast iron 
 
 19 592 
 
 14,436 
 
 3094 
 
 Steel . 
 
 123 700 
 
 37,120 
 
 20 62 
 
 Copptr 
 
 38 151 
 
 
 6 187 
 
 Lead 
 
 
 
 329 
 
 Box, oak, fir, firm Scotch fir, 
 pine (American) Bartlett.. 
 
 12,373 
 
 3,094 
 
 1,237 
 
 INVERTED SIPHON. 
 
 FIG. 20. 
 
 A pipe, as represented by I /y C G O, Fig. 20, em- 
 ployed in conveying water across a valley or other 
 depression between two more elevated points, as I y and 
 O, is termed an inverted siphon. It is a matter of no 
 little importance in practice to determine the relation 
 of the hights of inlet and outlet of an inverted si- 
 
PRACTICAL HYDRAULICS. 173 
 
 phon. Thus the elevation, O, being given, it is demon- 
 strable that if I y/ be the point of inlet, the weight of 
 the pipe will be greater than if the inlet be taken at 
 a lower elevation. It is also demonstrable if 1 lu be the 
 point of inlet, the weight of the pipe will be greater 
 than if the inlet were taken at a higher elevation. 
 
 For example, having given the elevations of the 
 several stations, and distances between them, of an 
 inverted siphon, whose length is I feet, let it be re- 
 quired to find the hight of the inlet point, above the 
 outlet point, so that the weight, "W, pounds of the 
 material employed in the construction of the siphon, 
 carrying q, cubic feet of water per second, shall be a 
 minimum? 
 
 Let 1= length of siphon in feet; 
 
 W~ weight of siphon in pounds; 
 
 g= cubic feet flow of water per second; 
 
 2r 
 
 d=^ diameter of siphon in feet; 
 
 t= thickness of shell of siphon in inches; 
 
 w= weight of a cubic foot of material in siphon; 
 
 h working modulus of strength of material; 
 
 j9=mean pressure in pounds per square inch; 
 
 x hydraulic head, such that the weight, W, shall 
 be a minimum. 
 
 Let, in Fig. 20, O represent the point of outlet; 
 O C (approximately equal to O CT), a horizontal line, 
 intersecting the pipe line in C; O A=a, determined 
 from the given levelling data, the ordinate of mean 
 
174 PBACTIOAL HYDKAULICS. 
 
 hydrostatic pressure, in the part, O G C, of the siphon 
 filled to the level, O C; then 
 
 } 
 
 U 
 
 Let the perpendicular, C Ix, so that the area of 
 the triangle, O C I, shall equal the area of O G I y . 
 
 Then will the mean pressure ordinate in feet for 
 the entire pipe, be: 
 
 (190) 
 
 In most cases, without considerable error, C I may 
 be taken equal to C, I y , thereby assuming that the 
 area, C P I y , is equal to. the area, O P I. 
 
 If greater accuracy be required, make O A of such 
 a hight that the trapezoid, G A B y I t , shall equal the 
 area embraced by the pipe line, I y C G O, and the hy- 
 draulic gradient, OI y . In case the exact length of 
 the pipe is not given, but the approximate horizontal 
 distance, OC y =AB y , is known approximately, put 
 
 AB y -l ^-^r =^, length of pipe. 
 
 Then 
 
 Substituting the value of E F=& in (187), 
 
PEACTICAL HYDRAULICS. 175 
 
 Substituting the values of p of (191), and of r= 
 
 in (188), 
 
 t _1.808d 
 
 1C 
 
 The obvious equation for the weight of the siphon 
 
 W= fi< /-' ; . (193) 
 
 Substituting the value of t of (192) in (193), 
 
 Squaring both members of Eq. (155), observing 
 that x=h f> 
 
 Substituting the .value of df of (195) in (194), 
 . 1085 n w $ 
 
 c*k \ xt ) 
 
 Differentiating (196), omitting the constant factor 
 outside the parenthesis, 
 
 ^ = _^-i + | .-f = 0. (197) 
 
 doc o 
 
 A n 
 
 Reducing (197), x=-f. (198) 
 
 Decomposing (197) into factors, and differentiating 
 
176 PRACTICAL HYDRAULICS. 
 
 that factor which reduces to (zero), the second dif 
 ferential becomes: 
 
 d 2 W 4a- 
 
 which, being positive, determines that "W is a mini- 
 mum, when x=-^, as found in Eq. (198). 
 
 o 
 
 SPECIAL VERIFICATION. 
 
 To verify, by special example, the correctness of 
 the result, showing that the weight, W, of an in- 
 verted siphon, carrying a given quantity, q, of water, 
 
 is a minimum, when the head, hf=x=-^-: substitute 
 
 o 
 the value of #==- , also different values, one greater 
 
 o 
 
 and one less than -~-, as x=-n-, and # = - in Eq. 
 
 o O o 
 
 (196). 
 
 For convenience of notation, let / ; =the constant 
 factor outside of the parenthesis. The substitutions 
 being made: 
 
 ~~3' 
 
 W=2.971/,a*; (200) 
 
 \\r-i. Z> a 
 
 When #,=--, 
 
 W / =2.989/ / or; (201) 
 
PRACTICAL HYDRAULICS. 177 
 
 Sa 
 And when x 4t ~-n-a, 
 
 W y/ =3/>l * (202) 
 
 Since the literal factors in Eqs. (200), (201) and 
 (202) are identical, the numerical factors in these 
 equations express the relative weights of W, W y and 
 W /; in the given order. 
 
 These factors conclusively show that the greatest 
 economy is attained in the weight of the siphon, when 
 
 4a 
 
 the value of x-~- t as determined in Eq. (198). 
 o 
 
 MINIMUM WITH RESPECT TO PRESSURE ; DIAMETER, 
 THICKKESS, AND WEIGHT OF AN INVERTED SIPHON 
 MINIMUM MEAN PRESSURE PER SQUARE INCH. 
 
 Substituting the value of OS=-Q- in Eq. (191), 
 
 (203) 
 MINIMUM DIAMETER. 
 
 a 
 Substituting in (155), the values of x=- of (198), 
 
 =h f , 
 (204) 
 
 c,=3.l514-T Y of ( 151 )' and recollecting that ii=h f , 
 
178 
 
 MINIMUM MEAN THICKNESS. 
 
 Substituting the values of fi=-q-, and of d, of (204 1 ) 
 in (192), 
 
 (205) 
 
 MINIMUM WEIGHT. 
 
 Substituting the values of d t of (204), and of t of 
 (205) in (193), 
 
 (206) 
 
 In obtaining the value of (206), the inner diameter 
 of the pipe has been employed the same as used with 
 respect to discharge whereas the shell of the pipe 
 being exterior to this diameter, its weight is somewhat 
 greater than represented by the value of W (206), as 
 will readily appear by the following: 
 
 Let A=area of cross section of pipe with respect to 
 internal diameter. 
 
 A 7 =area of cross section with respect to external 
 diameter. 
 
 Then A=5rf,'; (207) 
 
PEACTICAL HYDRAULICS. 179 
 
 And A--d+2t 2 ==d'+^dt + . (208) 
 
 Deducting (207) from (208), and decomposing the 
 difference into factors, 
 
 A A=ftd,*l+-. (209) 
 
 Farther, the pipe has been regarded seamless, 
 whereas, in most cases, water pipes especially the 
 larger class are constructed with rivets and laps, or 
 bands, whose weight must be added to that of W of 
 (206). 
 
 Let n= this weight, expressed in form of percent- 
 
 age; thus: 
 
 W/-W (l+n). (210) 
 
 Let W ;/ =the entire weight of pipe, including cor- 
 rections shown by (209) and (210). 
 
 Then as fid, t only, as respects area of cross section 
 of shell, is involved in the value of W of (206), will 
 
 (211) 
 
 If the pipe is very thin, the factor -1 1 + -? r may be 
 omitted; if seamless (l-\-n), will be omitted. 
 
 TO FIND THE MEAN HYDRAULIC PRESSURE BELOW THE 
 LEVEL OF THE OUTLET. 
 
 Rule 45. Find in square feet, from the levelling 
 
180 PRACTICAL HYDRAULICS. 
 
 data, the area embraced, as represented in Fig. 20, 
 within the boundary of the line of pipe, O G C (axis 
 of pipe), and the horizontal line, O C, drawn from the 
 outlet, O, and intersecting the pipe line in C. Divide 
 this area by the length of the line, O C, in feet, the 
 quotient will be the mean hydrostatic ordinate in feet, 
 represented by OA. The length of this ordinate, 
 multiplied by the decimal, .484, will be the mean 
 hydrostatic pressure in pounds sought. 
 Rule 45 corresponds to Eq. (189). 
 
 TO FIND THE HYDRAULIC HEAD, SUCH THAT THE 
 WEIGHT OF THE MATERIAL EMPLOYED IN THE CON- 
 STRUCTION OF THE PIPE CARRYING A GIVEN QUANTITY 
 OF WATER, SHALL BE A MINIMUM. 
 
 Rule 46. Divide four times the length of the hy- 
 drostatic ordinate, as found according to Rule 45, by 3; 
 the quotient will be the hydraulic head, CI (Fig. 20), 
 required. 
 
 Rule 46 corresponds to Eq. (198). 
 
 Remark. In Fig. 20, I represents the point at 
 which the pressure in the pipe begins. This point de- 
 termined, that of I ; will readily be found, from the 
 levelling data, and will require to have an additional 
 head sufficient only to discharge the given quantity of 
 water at I. 
 
PRACTICAL HYDRAULICS. 181 
 
 TO FIND THE MEAN ORDINATE, E F, FlG. 20, IN- 
 VOLVING BOTH THE HYDRAULIC HEAD, I, AND THE 
 HYDROSTATIC ORDINATE, 'A O. 
 
 47, Divide the sum of the hydraulic head, 
 C I, and twice the hydrostatic ordinate by 2. 
 Rule 47 corresponds to Eq. (190). 
 
 TO FIND THE MEAN PRESSURE PER SQUARE INCH IN 
 POUNDS FOR THE ENTIRE PIPE. 
 
 Rule 48. Multiply the mean hydrostatic ordinate, 
 O A, Fig. 20, in feet by the decimal .7234. 
 Rule 48 corresponds to Eq. (203). 
 
 To FIND THE MINIMUM DIAMETER. 
 
 Rule 49. The minimum diameter is equal to .5965 
 times the fifth root of the quotient arising from divid- 
 ing the product of the coefficient of resistance, the 
 length of the pipe and the square of the discharge per 
 second, by the hight of the hydrostatic ordinate, O A, 
 Fig. 20, in feet. 
 
 Rule 49 corresponds to Eq. (204). 
 
182 PRACTICAL HYDRAULICS. 
 
 TO FIND THE MINIMUM MEAN THICKNESS. 
 
 Rule 50. Multiply the fifth root of the product of 
 the coefficient of resistance, the length of the pipe, 
 the square of the discharge per second, and the fourth 
 powfjr of the mean hydrostatic ordinate, O A, Fig. 20, 
 by the quotient arising from dividing 2.5908 by the 
 modulus of the working load or of safety, as shall be 
 required. 
 
 Rule 50 corresponds to Eq. (205). 
 
 TO FIND THE MINIMUM WEIGHT. 
 
 Rule 51. Case 1. The pipe being very thin, mul- 
 tiply the fifth root of the product of the square of 
 the coefficient of resistance, the seventh power of the 
 length of the pipe, the fourth power of the discharge 
 per second, and the cube of the mean hydrostatic or- 
 dinate, A, Fig. 20, by the quotient arising from 
 dividing .4046 times the weight of a cubic foot of the 
 material in the pipe by the modulus of the material. 
 
 Case 2. The pipe being thick as y\ of an inch or 
 more, and seamless, multiply the result obtained, ac- 
 cording to Case 1, by 1 (unit), increased by the quo- 
 tient arising from dividing the thickness of the shell 
 by the inner diameter. 
 
PRACTICAL HYDRAULICS. 188 
 
 Case 3. The pipe being thick, as T 3 ^ of an inch or 
 more, and constructed with rivets and laps or bands, 
 multiply the result obtained, according to Case 2, by 
 1 (unit), increased by their relative weight to that of 
 the pipe. 
 
 Rule 51 corresponds to Eq. (211). 
 
 The moduli with respect to the strength, working 
 load, and safety are given in Table 23. 
 
 The modulus of working load, as shown in Ex. 90, 
 is &=17,549 pounds. 
 
 Unless the iron is extra in quality, the modulus 
 ought to be less, as &=14,000. 
 
 The weight of a cubic foot of iron is usually esti- 
 mated at 480 pounds. 
 
184 
 
 TEACTICAL HYDRAULICS. 
 
 TABLE 24. 
 
 Number, Thickness and Weight of One Square Foot of 
 Sheet Iron. 
 
 BIRMINGHAM GAUGE. 
 
 AMERICAN GAUGB. HASWELL 
 
 No. 
 
 Thi'k 
 in. 
 
 Lbs. 
 
 No 
 
 Thi'k 
 in. 
 
 Lbs. 
 
 No. 
 
 Thi'k 
 in. 
 
 Lbs. 
 
 No. 
 
 Thick 
 in. 
 
 Lbs. 
 
 0000 
 
 .454 
 
 18.35 
 
 17 
 
 .058 
 
 2.34 
 
 0000 
 
 46 
 
 18.63 
 
 19 
 
 .036 
 
 1.45 
 
 000 
 
 .425 
 
 17.18 
 
 18 
 
 .049 
 
 1.98 
 
 000 
 
 .41 
 
 16.58 
 
 20 
 
 .032 
 
 1.29 
 
 00 
 
 .38 
 
 15.36 
 
 19 
 
 .042 
 
 1.70 
 
 00 
 
 .365 
 
 14.77 
 
 21 
 
 .028 
 
 1.15 
 
 
 
 .34 
 
 13.74 
 
 20 
 
 .035 
 
 1.42 
 
 
 
 .325 
 
 13.15 
 
 22 
 
 .025 
 
 1.03 
 
 1 
 
 .3 
 
 12.13 
 
 21 
 
 .032 
 
 1.29 
 
 1 
 
 .289 
 
 11.70 
 
 23 
 
 .023 
 
 .913 
 
 r 
 
 .284 
 
 11.48 
 
 22 
 
 .028 
 
 1.13 
 
 2 
 
 .258 
 
 10.43 
 
 24 
 
 .020 
 
 .814 
 
 c 
 
 t! 
 
 .259 
 
 10.47 
 
 23 
 
 .025 
 
 1.01 i 
 
 3 .229 
 
 9.29 
 
 25 
 
 .018 
 
 .724 
 
 4 
 
 .238 
 
 9.62 
 
 24 
 
 .022 
 
 .889 
 
 ^ 
 
 .204 
 
 8.27 
 
 26 
 
 .016 
 
 .644 
 
 
 
 .22 
 
 8.89 
 
 25 
 
 .02 
 
 .808 
 
 5 
 
 .182 
 
 7.37 
 
 27 
 
 .014 
 
 .574 
 
 e 
 
 .203 
 
 8.21 
 
 26 
 
 .018 
 
 .723 
 
 6 
 
 .162 
 
 6.56 
 
 28 
 
 .013 
 
 .511 
 
 7 
 
 .18 
 
 7.28 
 
 27 
 
 .016 
 
 .647 
 
 7 
 
 .144 
 
 5.84 
 
 29 
 
 .011 
 
 .455 
 
 8 
 
 .165 
 
 6.67 
 
 28 
 
 .014 
 
 .566 
 
 8 
 
 .128 
 
 5.20 
 
 30 
 
 .010 
 
 .405 
 
 9 
 
 .148 
 
 5.98 
 
 29 
 
 .013 
 
 .525 
 
 9 
 
 .114 
 
 4.63 
 
 31 
 
 .009 
 
 .360 
 
 10 
 
 .134 
 
 5.42 
 
 30 
 
 .012 
 
 .485 
 
 10 
 
 .102 
 
 4.13 
 
 32 
 
 .008 
 
 .321 
 
 11 
 
 .12 
 
 4.85 
 
 31 
 
 .010 
 
 .404 
 
 11 
 
 .091 
 
 3.67 
 
 33 
 
 .007 
 
 .286 
 
 12 
 
 .109 
 
 4.41 
 
 32 
 
 .009 
 
 .364! 
 
 12 
 
 .081 
 
 3.27 
 
 34 
 
 .0063 
 
 .254 
 
 13 
 
 .095 
 
 3.84 
 
 33 
 
 .008 
 
 .323 
 
 13 
 
 .072 
 
 2.92 
 
 35 
 
 .0056 
 
 .226 
 
 14 
 
 .083 
 
 3.36 
 
 34 
 
 .007 
 
 .283 
 
 14 
 
 .064 
 
 2.59 
 
 36 
 
 .005 
 
 .202 
 
 15 
 
 .072 
 
 2.91 
 
 35 
 
 .005 
 
 .202 
 
 15 
 
 .057 
 
 2.31 
 
 37 
 
 .0045 
 
 .180 
 
 16 
 
 .065 
 
 2.63 
 
 36 
 
 .004 
 
 .162 
 
 16 
 
 .051 
 
 2.05 
 
 38 
 
 .004 
 
 .159 
 
 
 
 
 
 
 
 17 
 
 .045 
 
 1.33 
 
 39 
 
 .0035 
 
 .142 
 
 
 
 
 
 
 
 18 
 
 .040 
 
 1.63 
 
 40 
 
 .0031 
 
 .127 
 
 Ex. 91. The following data from a sheet-iron in- 
 verted siphon being given, viz: 
 Length of pipe, 128.5 miles. 
 
 ^Elevations with respect to sea level: 
 <( Point of inlet, 1300 feet. 
 I Point of outlet, 350 feet. 
 
PRACTICAL HYDRAULICS. 185 
 
 Mean hydrostatic ordinate, as O A, Fig. 20, 305.5 
 feet; discharge of water per second, 37.57 cubic feet; 
 modulus of safety of the iron, 14,000 pounds; weight 
 of iron per cubic foot, 485 pounds; allowance for bands, 
 laps and rivets, 15 per cent; cost of laid pipe per 
 pound, 10 cents. Required, the minimum diameter, 
 thickness of shell, weight and cost of the siphon ? 
 Required, also, the diameter, thickness of shell, weight 
 and cost of the siphon, if 950 feet, the full hydraulic 
 head, be employed ? 
 
 Cal. 1st. The given hydrostatic ordinate is 305.5 
 feet. 
 
 By Rule 46, corresponding to Eq. (198), 305.5X4-^- 
 3=407.34 hydraulic head; 407.34-f-128.5=3.17 feet 
 fall per mile. 
 
 By Table 17, it is seen that for 3.17 feet fall per 
 mile, the pipe carrying 37.57 cubic feet per second 
 will approximate 48 inches in diameter, and that the 
 corresponding velocity is 3.20 feet per second. 
 
 By Table 16, for a velocity oi 3 feet in a 48-inch 
 pipe, the coefficient of resistance is=.0038. 
 
 By Rule 49, corresponding to Eq. (204), 
 
 . 5965 (^H2<i^|-y<^jj<l3_7,ii)2j 1 5- = 389 8 f eet = 
 46.776 inches, minimum diameter (internal). Ans. 
 
 By Rule 50, corresponding to Eq. (205), 
 
 { 2 f |||.(.0038xl28.5X5280x(305.5) 4 }*=.3694 
 inches in thickness. Ans. 
 
 By Rule 51, corresponding to Eq. (211), 
 
 oV- { (-0038) 2 (128.5 X 5280) 7 (37. 57) 4 (305. 5) 3 p 
 
186 PEACTICAL HYDBAULICS. 
 
 .15 = 143790750 pounds, minimum 
 
 weight. Ans. 
 
 Whence at lOc. per pound: Cost=$14,379,075.00. 
 Ans. 
 
 CW. 2d. 1300 350=950 feet total head; 950-^- 
 128.5-=7.392 feet fall per mile. 
 
 By Table 17, the diameter of pipe having 7.392 feet 
 fall per mile, and discharge 37.57 cubic feet of 
 water per second is 40 inches. 
 
 By Rule 47, corresponding to Eq. (190), (305. 5 X 
 2+ 950)+- 2=780.5 feet mean ordinate for the entire 
 pipe. 
 
 By Rule 43, corresponding to Eq. (187), 780.5 X 
 .434=338.737 mean ordinate in pounds for the entire 
 pipe. 
 
 By Rule 44, corresponding to Eq. (188), 338.737 
 X20-^ 14000=. 4839 inches thickness of pipe; 40X 
 3.1416-^-12=10.472 feet circumference of pipe; 
 10.472 X 128. 5 X5280X 485 X. 4839-*- 12 = 138954840 
 pounds weight of pipe, assumed seamless, and esti- 
 mated for the internal diameter. 
 
 By Rule 51, corresponding to Eq. (211), cases 2 and 
 3, 138954840(1 + -^^) XL 15 = 161,747,880 pounds 
 weight of pipe, employing the full head of 950 
 feet. Ans. 
 
 Whence, at lOc. per pound: Cost-$16,174,788.00. 
 
 Ans. 
 
 The difference in these results, viz., $16,174,788.00 
 
 $14,379,075. 00=$1,795,713.00, which amounts to 
 
PRACTICAL HYDRAULICS' * - 187 
 
 a saving of over 11 per cent by the application of the 
 principle hereinbefore demonstrated with respect to 
 the minimum weight and minimum cost of an in- 
 verted siphon. 
 
 There will, in fact, be a greater saving in practice, 
 arising from a less length of pipe under pressure, in 
 case of employing the smaller head. 
 
 FLOW OF WATER IN OPEN CHANNELS AND NATURAL 
 STREAMS. 
 
 The flow of water in open channels and natural 
 streams is subject to the same laws which govern its 
 flow in pipes. The force producing motion in the 
 water, and overcoming the resistances of the water 
 way, is that of gravity applied to an inclined plane. 
 A greater variety of forms, with respect to cross sec- 
 tion of streams, is presented in open channels and 
 natural streams than in pipes, thereby changing to a 
 greater extent the relations between the perimeters 
 and the areas of the cross sections of the former, than 
 of the latter. Thus, in the case of pipes, the "hy- 
 draulic mean radius" has been shown, uniformly, equal 
 to one-fourth of the diameter, while in open channels 
 their mean depths vary indefinitely. 
 
 The "mean depth" of an open channel or natural 
 stream is the ratio of the perimeter to the area of the 
 cross section of the stream. For the most part in hy- 
 
188 * PEACTICAL HYDRAULICS. 
 
 draulic computations, that portion of the perimeter 
 which bounds the bottom and sides of this area, 
 termed the "wet perimeter," is employed. 
 
 The "air perimeter," whose value does not often ex- 
 ceed one-tenth of an equal length of the wet perime- 
 ter, unless strong winds or other disturbing influences 
 obtain, is considered when great accuracy is required : 
 
 Let a =. area of cross section of stream; 
 
 >=_wet perimeter; 
 
 m pSiir perimeter; 
 
 m= coefficient of air perimeter; 
 
 T= hydraulic mean depth. 
 
 Thenr=-- (212) 
 
 P 
 
 (213) 
 
 p-f mp 
 
 Other things being equal, the greater the ratio of 
 the perimeter to the area of the cross section of a 
 stream of water, the less will be the resistance to flow. 
 
 The forms of cross sections, generally applied to 
 water ways, are rectangular and trapezoidal. 
 
 FORM OF RECTANGLE OF MAXIMUM CARRYING CAPA- 
 CITY. 
 
 Let p= perimeter (omitting air perimeter) ; 
 
 o;=hight of rectangle; 
 
 Then p 2# width of rectangle. 
 
PRACTICAL HYDRAULICS. 189 
 
 2~=r, maximum. (21 4) 
 P 
 
 Differentiating (214), x=, hight; (215) 
 
 $^3a>M w idth. (216) 
 
 FORM OF TRAPEZOID OF REGULAR FIGURE OF MAX- 
 IMUM CARRYING CAPACITY. 
 
 In Fig. 21, let =BAE, angle of slope of bank; 
 g=a side; then hight=f sin t. Mean width f -j-| 
 cos -t: 
 
 ^ 2 (sin + sin t cos t)r maximum. (2>7) 
 
 Differentiating (217), observing that sin 2 =1 cos 2 2, 
 cosM + ^-W (218) 
 
 Reducing (218), cos *=i=.5. (219) 
 
 By table natural sines, =60. (220) 
 
 Of the regular figures, the semi-circle consisting of 
 an infinite number of sides, so that at any point cos t 
 =l=r, offers the least resistance to flow. 
 
 By equations (215) and (216), it is seen that the 
 form of a rectangle, offering the least resistance to 
 flow, has its base or width equal to twice its hight; 
 and by equations (219) and (220), it is seen that of the 
 
190 PRACTICAL HYDRAULICS. 
 
 regular figures, the trapezoid whose angle of slope is 
 60, in other words the semi-hexagon, offers the least 
 resistance to flow. 
 
 THE ANGLE OF SLOPE AND THE AREA BEING GIVEN TO 
 DETERMINE THE MOST APPROPRIATE FORM OF A CANAL. 
 
 B 
 
 iie. 21. 
 
 Let in Fig. 21, p=A. B + B C + CD perimeter; b= 
 BC=bottom; Wangle A; n = cot t; a area; xd= 
 B E, depth of canal. Then 
 
 ix- (221) 
 
 ')i ; (222) 
 
 2 ) ; (223) 
 
 a=(b+nx)x-, (224) 
 
 - . ,^^^r^ 
 
 whence b=~ (225) 
 
 x 
 
 Substituting value of b of (225) in (223) and divid- 
 ing then both members by a, 
 
 minimum. (226) 
 
 X CL ^ 
 
 Differentiating (226) and reducing, 
 
 (227) 
 
PRACTICAL HYDRAULICS . 
 
 Substituting" the values of TI cot - 
 
 COS t 
 
 sin t 
 
 " 
 
 sin* 
 
 cos 
 
 
 From (225) b=x cot . 
 
 191 
 
 and 
 
 (228) 
 (229) 
 
 TABLE 25. 
 
 Dimensions of the most suitable forms of Canals, corres- 
 ponding to different angles of slopes, and to a given 
 area of cross section. 
 
 Angle of 
 Slope =t. 
 
 Ratio of 
 Perp. to 
 Base. 
 
 Relative 
 Slope. 
 n. 
 
 Depth. 
 d' 
 
 v 
 
 Bottom 
 Width. 
 b 
 
 ft 
 
 nd 
 
 ft 
 
 Top 
 Width. 
 
 b+2nd 
 
 T/a 
 
 Perimeter. 
 P 
 
 l/ 
 
 90 00' 
 
 lonO 
 
 .0 
 
 0.707 
 
 1.414 
 
 .0 
 
 1.414 
 
 2.828 
 
 78 41' 
 
 Son 1 
 
 0.200 
 
 0.734 
 
 1.217 
 
 0.147 
 
 .510 
 
 2.713 
 
 75 58' 
 
 4onl 
 
 0.250 
 
 0.734 
 
 1.161 
 
 0.186 
 
 .533 
 
 2.692 
 
 71 34' 
 
 3onl 
 
 0.333 
 
 0.752 
 
 1.079 
 
 0.251 
 
 .580 
 
 2.656 
 
 03 26' 
 
 2onl 
 
 0.500 
 
 0.759 
 
 0.938 
 
 0.379 
 
 .697 
 
 2.635 
 
 60 00' 
 
 26 on 15 
 
 0.577 
 
 0.760 
 
 0.877 
 
 0.439 
 
 .755 
 
 2.632 
 
 56 19' 
 
 3 on 2 
 
 0.667 
 
 0.759 
 
 0.812 
 
 0.506 
 
 .824 
 
 2635 
 
 53 8' 
 
 4 on 3 
 
 0.750 
 
 0.757 
 
 0.753 
 
 0.568 
 
 1.892 
 
 2.645 
 
 51 20' 
 
 5 on 4 
 
 0.800 
 
 0.753 
 
 0.724 
 
 0.603 
 
 1.960 
 
 2.654 
 
 45 00' 
 
 lonl 
 
 1.000 
 
 0.740 
 
 0.613 
 
 0.740 
 
 2.092 
 
 2.704 
 
 40 00' 
 
 21 on 25 
 
 1.192 
 
 0.722 
 
 0.525 
 
 0.860 
 
 2.246 
 
 2.771 
 
 36 52' 
 
 3 on 4 
 
 1.333 
 
 0.707 
 
 0.471 
 
 0.943 
 
 2.557 
 
 2.828 
 
 35 00' 
 
 7 on 10 
 
 1.402 
 
 0.697 
 
 0.439 
 
 0.995 
 
 2.430 
 
 2.870 
 
 33 41' 
 
 ?on3 
 
 1.500 
 
 0689 
 
 0.418 
 
 1.034 
 
 2.465 
 
 2.989 
 
 30 00' 
 
 23 on 40 
 
 1.732 
 
 0.664 
 
 0.356 
 
 1.150 
 
 2.656 
 
 3012 
 
 26 34' 
 
 Ion 2 
 
 2.000 
 
 0.636 
 
 0.300 
 
 1.272 
 
 2.844 
 
 3.144 
 
 21 48' 
 
 2on5 
 
 2.500 
 
 0.589 
 
 0.228 
 
 1.471 
 
 3.170 
 
 3.397 
 
 18 26' 
 
 Ion 3 
 
 3.000 
 
 0.548 
 
 0.188 
 
 1.645 
 
 3.478 
 
 3.646 
 
 14 2' 
 
 Ion4 
 
 4.000 
 
 0.485 
 
 0.119 
 
 1.941 
 
 4.001 
 
 4.121 
 
 11 19' 
 
 Ion 5 
 
 5.000 
 
 0.441 
 
 0.062 
 
 2.205 
 
 4.472 
 
 4.519 
 
 semi-cir. 
 
 
 
 0.798 
 
 
 
 1.596 
 
 2.507 
 
 
 
 
 
 
 
 
 
192 PEACTICAL HYDRAULICS. 
 
 TO FIND THE DIMENSIONS OF THE MOST SUITABLE 
 FOEM OF CANAL, WHEN THE ANGLE OF SLOPE OF THE 
 BANKS AND THE AEEA OF CEOSS SECTION AEE GIVEN. 
 
 Rule 52. Employing Table 25, multiply the square 
 root of the given area of cross section by the num- 
 ber in the table, which is opposite the given angle of 
 slope of bank, and in the column of the denomination 
 of the dimension sought, as "depth," "bottom width," 
 "top width," "perimeter" (wet). 
 
 Ex. 92. What dimensions must be given to the 
 cross section of a canal trapezoid of regular form 
 whose discharge of 64 cubic feet, with a velocity of 4 
 feet per second, is a maximum? 
 
 Gal By Eq. (220), it is shown that the angle of 
 slope 60, when the carrying capacity of a regular 
 trapezoidal canal is a maximum. 
 
 64-=- 4 =16 square feet, area of cross section. 
 
 (16)^ = 4 square root of area of cross section. 
 By Table 25, Rule 52, opposite 60, the angle of 
 
 slope in "depth" column, find .760; that is, -^--=.760; 
 
 or 
 
 But, as shown above, the square root of the area o? 
 cross section is=4 feet; hence, d, = .760X4 = 3. 01 
 feet, depth of canal. Ans. 
 
 Farther, opposite 60, in "bottom width" column, 
 
PRACTICAL HYDRAULICS. 193 
 
 find .877; that is, -7-- =.877; or 6=.877i/t, but 'as 
 
 1/66 
 
 shown above ]/c6=4; hence, &=. 877X4=3.508 feet, 
 bottom width. Ans. 
 
 Farther, opposite 60 find in " top width col- 
 umn" 1.755 feet; that is, b + 2nd === 1.755 feet; or b + 
 
 "]/ (A> 
 
 _2?id=l. 755X4=7.020 feet, top with. Ans. 
 
 Farther, opposite 60 in "perimeter column," 
 find 2.632; that is, p=2. 632X4=10.528 feet peri- 
 meter. Ans. 
 
 Ex. 93. The following data for a canal in loose 
 earth being given, viz : Discharge of water, 50 cubic 
 feet per second; velocity of flow, 2 feet per second ; 
 angle of slope, 26 34' equivalent to 1-2; it is re- 
 quired to determine the dimensions of the most appro- 
 priate form of cross section of the canal ? 
 
 Cal. 50-^-2=25 square feet, area of cross section; 
 l/ ~ =1/25=5 square root of area of cross section. 
 
 By Table 25, opposite 26 34', the given angle of 
 slope (l'-2), in depth column, find .636; that is 
 
 4= .636; or c/ y =.636 i/a. 
 
 I a 
 
 Substituting value of j/a=5; in this, d / =. 636x5= 
 3.18 feet, depth. Ans. 
 
 Farther, in column of ''bottom width," find 6= 
 .300 i/a, or by substituting -5 for /, Z>=. 300X5=1.5 
 feet, bottom width. Ans. 
 
 Farther, in column of "top width," find />X 2 H <l ( 
 
194 PRACTICAL HYDRAULICS. 
 
 2.844 i/o, or by substituting 5 for *\/'a, b -}-2nd= 
 2.844X5=14.22 feet, top width. Ans. 
 
 P. 
 
 p 
 
 /;=3.144X5=15.72 feet, perimeter. Ans. 
 
 Farther, in "perimeter" column, find 4==3.144, oi- 
 l/a 
 
 FORMULAS FOR THE FLOW OF WATER IN OPEN 
 
 STREAMS. 
 
 Fquation (130) is an expression for uniform velocity 
 of water in open streams, as well as in pipes. In case 
 the air perimeter of an open stream be considered, the 
 
 value of r .= - of (213) must be substituted for 
 
 r in (130); but in case the air perimeter be omitted, 
 equation (130) for the velocity of water in pipes re- 
 mains unchanged for the velocity of water in open 
 streams, that is: 
 
 Iii the factor - i- 2 ' c/ is a variable coefficient, 
 
 ( c f ) 
 
 whose value depends upon experiment. This simple 
 equation covers all. cases in practice, providing the 
 value of Cf be known. 
 
 Of the numerous formulas in use for finding the 
 velocity of water in open streams, the following from 
 Kutter, taken in connection with his table of coeffi- 
 
PRACTICAL HYDRAULICS. 195 
 
 cients for roughness of stream bed, seems entitled to 
 the foremost position, on account of its directness and 
 wide range of application. 
 
 KUTTER'S FORMULA. 
 
 
 In (231), v, T and s, respectively, denote velocity, 
 hydraulic mean depth, and sine of slope, and n repre- 
 sents the coefficient of roughness of the stream bed. 
 
 Eq. (231) may be somewhat simplified by putting 
 
 ; (232) 
 
 When 
 
 The first factor of the second member of Eq. 
 (231) is equal to the first factor of the second mem- 
 ber of Eq. (230), and being deduced from a wide 
 range of experiments with great care and masterly 
 skill, furnishes the best means known to the present 
 writer of determining, in the absence of actual expe- 
 
 riment in a special case, the value of -j Let c k 
 
 =this expression from Kutter's formula; then 
 
 v=c k (rs)*. (234) 
 
196 riUCTICAL HYDRAULICS. 
 
 TABLE 26. 
 
 Coefficients (n) for Roughness of Stream Beds. Compiled 
 from Kutter, Jackson and Fanning. 
 
 72,= .009 well planed timber in perfect order. 
 
 n .010 cement, glazed, coated material. 
 
 % = .01 2 unplaned timber in flumes. 
 
 7i=i.0l3 brickwork, cast and wrought-iron. 
 
 7i = .015 canvas lining, rectangular flumes with bat- 
 tens .5 inch apart. 
 
 ?i = .017 rubble, also, earth in highly regular cases. 
 
 7i=.020 coarse rubble, set dry, in bad condition, 
 very firm regular gravel, and flumes with batcens 2 
 inches apart. 
 
 n .0225, dry, coarse rubble in bad order; earth 
 canals and channels above average. 
 
 7i .0250 earth, canals and channels in good order. 
 
 7i .0275 earth, canals and channels below average. 
 
 7i=.030 earth, canals and channels in bad order. 
 
 7i=.035 rivers and canals in bad order, overgrown 
 with vegetation, and strewn with stones. 
 
 -^=.070 rivers in earth, with stones and weeds in 
 great quantities. 
 
 The principal elements, besides the force of gravity 
 involved in the determination of the velocity of an 
 open stream of water, are the hydraulic mean depth, 
 the sine of slope, and the degree of roughness of . the 
 stream bed. Now, if these relations be not changed, 
 it is evident that the velocity of the water will neither 
 
PRACTICAL HYDRAULICS. 197 
 
 be increased nor diminished by varying the form and 
 magnitude of the sectional area of the stream. 
 
 Illustrative of this proposition: 
 
 1st. The hydraulic mean radius or mean depth of 
 a circular pipe, in which d=diameter, is: 
 
 p 4tnd 4* 
 
 2d. The hydraulic mean depth of a V flume ( u vee 
 flume"), right angled at the bottom, in which <i=the 
 slant hight or width of side, 
 
 a d * d f9W\ 
 
 r= p=2X2d^ (236) 
 
 3d. The hydraulic mean depth of a rectangular 
 me or 
 
 depth is, 
 
 flume or canal, in which <i=the width, and ~= 
 
 d* d QP7X 
 
 = " (237) 
 
 4th. To determine a trapezoidal flume or canal, 
 
 in which the mean width and the angle of slope 
 
 of the sides are given, such that the hydraulic mean 
 
 depth shall be equal to one fourth of the mean width. 
 
 Let demean width. 
 
 x= width of side or slant hight. 
 
 2/=bottom width. 
 
 t angle of slope of sides.' 
 
198 PRACTICAL HYDRAULICS. 
 
 Then 
 
 
 x sin 2= depth. 
 
 (338) 
 
 
 xcost + y=d. 
 
 (239) 
 
 
 dx sin d 
 
 (240) 
 
 2x + y 4 ' 
 
 Whence: a 
 
 d 
 
 (241) 
 
 cos t -f- 4 sin 2 ' 
 
 Anrl -?/ rl J 
 
 4 sin t 2 ) 
 
 ^24^ 
 
 Case 1st. Put =45. 
 
 By Table of natural sines and cosines, sin 45 = 
 .70711; cos 45=.70711. 
 
 Substituting values of sin t and cos t in (241) and 
 (242), 
 
 aj=.6512 d. (243) 
 
 2/=.5395d. (244) 
 
 Deptha sin =.4605 d. (245) 
 
 a .4605 cT = d 
 
 Case 2nd. Put *==60. 
 
 By Table of natural sines and -cosines, sin 60 9 - 
 .86603; cos 60=. 50000. 
 
 Substituting values of sin 60 and cos 60 in (241) 
 and (-242), 
 
 ^=.5091 d. (247) 
 
 2/=.7454 d. (248) 
 
 Depth x sin =.4409 cl. (249) 
 
 p .4409cf d 
 
 And ^- (250) - 
 
PRACTICAL HYDRAULICS. 199 
 
 By giving different values to t in (241) and (242), 
 the form of sectional area of a trapezoidal flume or 
 canal may be varied indefinitely without change of 
 
 the hydraulic mean depth, ^=T* 
 
 The forms given in which =45 and =60, seem 
 to occur most frequently in practice. 
 
 Now, as under like conditions, excepting the forms 
 of stream beds, the velocity of water in flumes or 
 canals is equal to its velocity in pipes of equal hy- 
 draulic mean radii or depths. Table 17 giving the 
 velocity of flow in pipes, applies equally well to this 
 class of flumes or canals. 
 
 TO FIND THE VELOCITY OF WATER IN A FLUME OR 
 CANAL WHOSE HYDRAULIC MEAN DEPTH IS EQUAL TO 
 THAT OF A PIPE OF GIVEN DIAMETER. 
 
 Rule 53. Case 1st The flume being V shaped, 
 that is, quadrant in form (usually termed "vee flume)," 
 find opposite the given sine of slope or fall per mile, 
 in Table 17, the velocity of flow due a pipe whose 
 diameter is equal to the side, width or slant hight of 
 the flume. The quantity so found will be the ve- 
 locity sought. 
 
 Case 2d. The flume or canal being either rectangu- 
 lar or trapezoidal, find, opposite the given sine of slope 
 or fall per mile in Table 17, the velocity of flow due 
 
200 PRACTICAL HYDRAULICS. 
 
 a pipe whose diameter is equal to the mean width of 
 the flume or canal. The quantity so found will be 
 the velocity sought. 
 
 TO FIND THE FLOW OF WATER IN CUBIC FEET IN A 
 FLUME OR CANAL WHOSE HYDRAULIC MEAN DEPTH IS 
 EQUAL TO THAT OF A PIPE OF GIVEN DIAMETER. 
 
 Rule 54. Case 1st. The flume being V shaped, 
 quadrant in form, a " vee flume," find the velocity as 
 directed in Rule 53, arid multiply the velocity so found 
 by one half the square of the side width or slant- 
 depth of the ratio. 
 
 Case 2nd. The flume or canal being rectangular, 
 multiply the velocity found as directed in Rule 53, by 
 one half the square of the width of the water way. 
 
 Case 3d. The flume or canal being trapezoidal, if 
 the angle of the slope of the sides is equal to 45, mul- 
 tiply the velocity found as directed in Rule 53, by 
 .4605 times the square of the mean width; if the 
 angle of slope of the' sides is equal to 60, multiply 
 the velocity by .4409 times the square of the mean 
 width of the water way, and, in general, if the angle 
 of slope of the sides be equal to t, multiply the veloc- 
 ity found as directed in Rule 53, by the product of 
 the square of the mean width, and the ratio of the 
 mean width to the vertical depth of the water in the 
 flume or canal. 
 
PRACTICAL HYDRAULICS. 201 
 
 Rule 55. Another method of finding the flow of a 
 given flume is to multiply the discharge of a circular 
 pipe, Table 17, of equal fall per mile, and equal hy- 
 draulic mean depth, by the ratio of the sectional area 
 of the pipe to the sectional area of the given flume; 
 that is, if the flume is either quadrant in form, a "V 
 flume," or if it is rectangular, having its width equal 
 to twice its depth, multiply by .637; if it is trape- 
 zoidal, multiply by . 586 when the angle of slope of the 
 side is 45, and by .561 when it is 60. 
 
 COEFFICIENT OF ROUGHNESS INVOLVED IN -TABLE 17. 
 
 The measure of the degree of roughness of the inner 
 surfaces of the pipes for which the flow of water un- 
 der pressure has been computed in Table 17, closely 
 approximates .011 when referred to Kutter's scale of 
 coefficients (Table 26 of the present work) : that is 
 
 71=^.011. 
 
 Ex. 94. The fall per mile being 15.84 feet, what 
 will be the velocity and discharge per second of water 
 in a " V flume," whose side width or slant depth of 
 water is 2.5 feet, and the coefficient of roughness of 
 inner surfaces equal to .011 ? 
 
 Cat. With respect to velocity by Rule 53, in 
 Table 17, opposite the given fall 15.84, in velocity 
 column of 2.5 feet diameter, find 5.27 feet. Ans. 
 
202 PRACTICAL HYDRAULICS. 
 
 Calculation with respect to discharge: One-half the 
 square of side, 2.5X2.5-^2=3.125 square feet. 
 
 By Rule 51, 5.27X3.125=16.47 cubic feet. 4ns. 
 
 Or, by Rule 55, find, by Table 17, the discharge 
 of a pipe 2.5 diameter, for the given fall, 25.87 cubic 
 feet. 
 
 25.87X-637=16.48 cubic teet.Ans. 
 
 Ex. 95. The fall being 5.28 feet per mile, what 
 will be the discharge of water flowing in a flume 
 whose mean width is 8 feet, angle of side slope 45, 
 and coefficient of roughness, w=.011. 
 
 Cat. By Table 17, for a fall of 5.28 feet per mile, 
 and 8-foot pipe, the discharge per second is 350.5 
 cubic feet. Then by Rule 55, 
 
 350.5 X- 586=205. 39 cubic feet. Ans. 
 
 Remark. With respect to the flow of water in 
 pipes not differing largely in size, it may be assumed 
 without material error in practice, that the velocities 
 are proportionate to the respective diameters. 
 
 If greater accuracy be required, recourse to (133) 
 will need be had . 
 
 Ex. 96. The fall being 2.64 feet per mile, what is 
 the discharge per second in a flume 9 feet wide, 4.5 
 feet deep, and the coefficient of roughness, w=.011? 
 
 Gal By Table 17, the velocities for a fall of 2.64 
 feet per mile, due pipes 8 and 10 feet diameter 
 
PKACTICtL HYDKAULICS- 203 
 
 =^ 1 -5-=9 feet are 4.88, and 5.84 feet 
 =^-^-J-^-^=5.36 feet, velocity per second. 
 
 Whence, 5.36x9x9-5-2= 217.08 cubic feet. Ans. 
 
 K utter's Formula for the Flow of Water in 
 in Open Streams. 
 
 Among the most eminent experimeiitists in hy- 
 draulics, during the present half century, have been 
 D'Arcy and Bazin, Humphreys and Abbot, and Kut- 
 ter. Prior to this time, the science of hydraulics was 
 largely speculative and incoherent. The older hy- 
 draulicians had determined on meager data certain 
 laws which they erroneously held susceptible of gen 
 eral application. 
 
 D'Arcy and Bazin having collected a large amount 
 of experimental data, deduced therefrom and published 
 in 1835 and 1865, a formula better adapted than any 
 preceding for finding the flow of water in open streams 
 and pipes of medium size. The report of Humphreys 
 and Abbot on the ' 'Physics and Hydraulics" of the Mis- 
 sissippi river, published in 1861, is very justly esteemed 
 by engineers, first in importance, as to the extent, ac- 
 curacy, and value of its contributions to experimental 
 hydraulics. Their formulas, however, deduced from 
 their experiments, besides being quite complex and 
 
204 PRACTICAL HYDRAULICS. 
 
 tedious of application, give results too low for the flow 
 of water in small and medium sized streams. Thus, 
 between the formulas of D'Arcy and Bazin, and those 
 of Humphreys and Abbot, there existed a wide hiatus, 
 till it was effectually closed up in 1870 by the intro- 
 duction of Kutter's formula. 
 
 The mode in biief of Herr Kutter, in accomplishing 
 this difficult and laborious task, is substantially as 
 follows: 
 
 1st.; To divide the great mass of the observed and 
 trustworthy results at his command, appertaining to 
 the flow of water in open streams, into twelve classes, 
 arranged as shown in Table 26, with reference to the 
 degree of roughness of the stream-beds. 
 
 2d. To adopt, on careful comparison of various 
 formulas for the velocity of water under the imposed 
 conditions, the following formula of Chezy as the 
 basis : 
 
 v=c(rs)z, (251) 
 
 noting that c is variable. 
 
 3d. For the determination of the values of c, that 
 shall, when substituted with the given values of r; the 
 hydraulic mean depth, and of s, the sine of slope in 
 the Chezy formula (251), yield results respectively 
 corresponding to those determined by observation, he 
 makes extensive experiments with several trial formu- 
 las, among which is that of D'Arcy and Bazin, hitherto 
 considered in our discussion of the flow of water in 
 pipes. 
 
PRACTICAL HYDRAULICS. 205 
 
 Of these trial formulas, the following finally 
 adopted is one devised by himself, which is not only 
 more simple in form than that of D'Arcy and Bazin, 
 but yields results nearer in accord with those observed: 
 
 a 
 c= V. (252) 
 
 This formula, however, is faulty, in that it is limited 
 in application. 
 
 Thus by inversion, it is shown to be an equation t<5 
 
 a straight line whose abscissa= ^-, whereas the plot- 
 ted results of observation indicate a curve, and further 
 show that of is dependent upon the value of b" in a 
 word, that a' and V are variables, and not constants, 
 as at first assumed. 
 
 4th. To generalize Eq. (252) the variable terms 
 z for ci and x for 6' are substituted in it; whence, 
 
 (253) 
 
 5th. "After much examination and further com- 
 parison," (Kutter's words) he puts, 
 
 z=a + ^-; (254) 
 
 xn z l=a n. (255) 
 
206 PRACTICAL HYDRAULICS. 
 
 Substituting these values of z and x in (253), we 
 have 
 
 (256) 
 
 Equation (256) is found, however, not suited to the 
 extremes of inclination of water surface, nor to the 
 extreme limits of sectional area. 
 
 6th. To meet these requirements, in other words 
 to render the formula applicable to all cases whatever, 
 Kutter noting that "when r= infinity, c wi\l=z, and 
 the coefficients z will have their values represented by 
 an hyperbolic curve," makes in Eq. (253), 
 
 z=& + ~, (257) 
 
 s 
 
 in which A denotes the semi-axis of an hyperbola, m 
 the tangent of the inclination of its asym totes with 
 
 the axis of abscissa, and (s representing the sine of 
 slope) the abscissae; whence, 
 
 (259) 
 
 Substituting the values of z and x of Eqs. (258) and 
 (259) in (253), there results the equation in its general 
 form for the value of the coefficient c, viz: 
 
PRACTICAL HYDRAULICS. 207 
 
 m 
 
 n s /n^AA 
 
 . c= (260) 
 
 Substituting the value of c of Eq. (260) in (251), 
 there results the general formula of Kutter for the ve- 
 locity of water in open streams, viz: 
 
 (261) 
 1 I 
 4] 
 
 Combining (257) and (254), 
 
 
 Making s infinite in Eq. (262), 
 
 A=a+-. (263) 
 
 To determine the relation of c to 71, in its simplest 
 form, let 
 
 1=1, (264) 
 
 in which 1, as found by trial. 
 
 Substituting this value of I, and of -7 in Eq. (256), 
 and reducing, we obtain the relation sought, 
 
 -=n : (265) 
 
 c 
 
208 PRACTICAL HYDRAULICS. 
 
 which, as determined for the Mississippi river, is 
 
 w=-~.027. (266) 
 
 c 
 
 The value -of A in an hyperbola, coinciding with the 
 curve formed by plotting observed results, is found 
 to be: 
 
 A=60. (267) 
 
 Substituting the values of A=60 of (267), 1=1 of 
 (264), and ^=.027 of (266) in (263), transposing and 
 reducing, 
 
 a=A -=60 37=23. (268) 
 
 To find the value of tangent m, let an extreme case 
 be taken, in which the values, as determined from the 
 plotted curve, are: 
 
 8=0.00000363; (269) 
 
 0=487. (270) 
 
 Substituting these values, together with that of A== 
 60 of (267) in (257), transposing and reducing, 
 
 m=(z A)s=(487-60)X0.00000363=0.00155. (271) 
 
 Subsituting these constant values of a=23 of (268), 
 2=1 of (264), and w=0.00155 in (261), there results 
 for metrical measures: 
 
 H- nnniV, ::>M*- ( 272 ) 
 
PRACTICAL HYDRAULICS. 209 
 
 To reduce metrical measures employed in Eq. (272) 
 to those of a different system, let e denote the ratio of 
 the former to the latter, noting that n and s, repre- 
 senting ratios, are not affected by the reduction. 
 
 Substitute ?=23 + --h > ; (273) 
 
 . Ti S 
 
 (274 ) 
 
 (275) 
 
 Let z, x', r and v represent respectively the terms 
 to which z,x, r and v, of the metrical system are to be 
 
 z x' r , v' 
 
 reduced, then will z , x~ , r=~ and v=-. - 
 e e e e 
 
 Substituting these values of z } x, r and v in (275) 
 
 (276) 
 
 Multiplying in (276), both numerator and denomi- 
 nator within the parenthesis by e 4 ; also, multiplying 
 both sides of the equation by e, 
 
210 PRACTICAL HYDRAULICS. 
 
 z x' 
 
 Substituting the values of ~=e^z, and -fe^tx. in 
 
 e-2 62 
 
 (277). 
 
 With respect to the Kutter formula, equation (278) 
 is general for the reduction of the measures employed 
 in it to those of another system. An inspection shows 
 that by multiplying z and x, by e* (the square root of 
 the ratio of the different measures), each term of 
 the equation [1 (unit) being common,] will be the de- 
 nomination sought. 
 
 To render the equation in terms of English feet, 
 
 Let e=3.281, the number of feet in a meter. (279) 
 
 Substituting the value of e*=1.811 in (278) after 
 restoring the values of z and x of (273) and (274), 
 and omitting the accents with respect to v and r', 
 
 
 Equation (280) for the purposes of application may 
 be somewhat simplified in the following manner. 
 
 Putting the numerator inclosed in brackets under 
 the following form : 
 
PRACTICAL HYDRAULICS. 211 
 
 Numerator^- 41.6 + ' + 1 - 811 > ( 281 ) 
 
 and putting x=^l.6+n. (282) 
 
 S x 
 
 Substituting x for its value in (280), and then 
 multiplying both numerator and denominator by r% 
 there results: 
 
 = _ . s I. (283) 
 
 n 
 
 APPLICATION OF THE KUTTER FORMULA AS REN- 
 DERED BY EQS. (282) AND (283). 
 
 Ex. 97. In a rectangular flume or canal four feet 
 wide, two feet deep, the fall per mile is 4.752 feet 
 (equivalent to a sine of slope, s=.0009), and the co- 
 efficient of roughness of whose bed is 7i=.025. What 
 is the velocity of flow per second? 
 
 Gal. Area of cross section, 4X2=8; wet peri- 
 meter, 4+2+28. 
 
 Hydraulic mean depth, 8-^-8=1. 
 
 Square root hydraulic mean depth. r i =/i=l. 
 
 Square root of sine of slope, s i =i/.^=.03. 
 
 Substituting the given values of s=.0009, and n 
 .025 in Eq. (282), 
 
 . 025=1418. 
 
212 PRACTICAL HYDRAULICS. 
 
 Substituting the value of ?i=.025, and the values 
 as found of ^=1.118, s*=.03, and r*=l in Eq. (283), 
 
 L.118 + 1.81K 
 1.118 + 1 ) X - 8 ' 
 
 Reducing, ^=4orx .03=1.658 feet. Ans. 
 
 Ex. 98. In a rectangular flume or canal four feet 
 wide, two feet deep, the fall per mile is 4.752 (equiva- 
 lent to a sine of slope s .0009), and the coefficient of 
 roughness, of whose bed is n .012, what is the ve- 
 locity of flow per second? 
 
 Cat. It will be noted that Ex. 98 differs from Ex. 
 97, as relates to coefficient of roughness of the bed 
 only. 
 
 Substituting the given value of ?i=.012, the values 
 as found of =.536, s*=.03, and r*=l in Eq. (283), 
 
 Reducing, v=83.33 j X .03 = 3.82 feet. Ans. 
 
 Ex. 99.- It is required to construct a rectangular 
 canal, whose fall per mile shall be 2.112 feet (equiva- 
 lent to sine s=.0004), and coefficient of roughness of 
 bed 71=. 025, what must be its depth and width, for 
 it to discharge 108.2 cubic feet per second? 
 
 Gal. Let q a v = 108.2 cubic feet, the discharge 
 per second. ' (a) 
 
PRACTICAL HYDRAULICS. 213 
 
 Substitute the value of v of (283) in Eq. (a), 
 
 arxK + l.Sllx . M 
 
 q-= ( L_ \s*. (b) 
 
 n\ x~t~i* ) 
 
 Arranging terms of Eq. (6) with respect to r, 
 
 * (c \ 
 
 Substituting the values of s and n in (282), 
 
 In a rectangular canal, whose depth is d, and 
 width 2d, the hydraulic mean depth is: 
 
 d 
 
 area, a=2d 2 . (/) 
 
 Substituting the values of a? 1.216, s^ .02, ^= 
 
 025, r=|, a=2cZ 2 , and 5=108.2 in Eq. (c), and ob- 
 
 erving that d*=d' t 
 
 cZ* 31.59 di=54.33. 
 
214 PRACTICAL HYDRAULICS. 
 
 Solving Eq. (g) by Hutton's method for the resolu- 
 tion of equations of the higher order: 
 
 
 
 
 
 31.59 
 
 54.33 
 
 2 
 
 4 
 
 8 
 
 16 
 
 32. 
 
 .82 
 
 9 
 
 g 
 
 24 
 
 64 
 
 
 
 
 
 
 
 .41 
 
 53.51 
 
 4 
 
 12 
 
 32 
 
 80 
 
 160. 
 
 43.06 
 
 2 
 
 12 
 
 48 
 
 160 
 
 54.89 
 
 
 
 
 
 
 
 
 10 4"> 
 
 6 
 
 24 
 
 80 
 
 240 
 
 215.3 
 
 8.65 
 
 2 
 
 16 
 
 80 
 
 34.5 
 
 62.3 
 
 
 
 
 
 
 
 
 1QA 
 
 8 
 
 40 
 
 160 
 
 274.5 
 
 277.6 
 
 1.79 
 
 2 
 
 20 
 
 12.5 
 
 37.1 
 
 10.7 
 
 
 
 10 
 
 60 
 
 172.5 
 
 311.6 
 
 288.3 
 
 
 2 
 
 2.4 
 
 13. 
 
 39.8 
 
 10.8 
 
 
 12 
 
 62.4 
 
 185.5 
 
 351.4 
 
 299.1 
 
 
 
 2.5 
 
 13.5 
 
 6.4 
 
 
 
 
 64.9 
 
 199.0 
 
 357.8 
 
 
 
 
 2.5 
 
 14. 
 
 
 
 
 
 67.4 
 
 213. 
 
 
 
 
 
 2.6 
 
 2. 
 
 
 
 
 
 70.0 
 
 215. 
 
 
 
 
 
 2.6 
 
 
 
 
 
 2.236 
 
 72.6 
 
 Thus #=2.236. (h) 
 
 Squaring, d= 5 feet depth; whence 2d = 10 feet 
 width. Ans. 
 
 Ex. 100. In a rectangular canal two feet deep, 
 four feet wide, the fall per mile is 4.752 feet, equiva- 
 lent to sine of slope s=.0009, and the observed ve- 
 
PRACTICAL HYDRAULICS. 215 
 
 locity per second 2.578 feet, what is the value of the 
 coefficient n for the roughness of the bed? 
 
 Gal Let m=.41.6+^?i. (a) 
 
 s 
 
 Substituting m of (a) for its value in (282), 
 
 x=mn. (b) 
 
 Substituting mn for x in (283), and arranging 
 terms with respect to n, 
 
 /v r* r shn\ 
 n*-f(- -lnp=1.8llr> 3 (c) 
 
 V V 771 / 
 
 Substituting the value of s=.0009 in Eq. (a), 
 
 44.7. (d) 
 
 From the given data, the hydraulic mean depth is 
 
 found to be: 
 
 r=4X2-*-(4-f2 + 2)=l. (e) 
 
 Substituting in Eq. (c), the values of ??i=44.7 of 
 Eq. (d), r=l, r-=l, s s =.03, and the given value of 
 
 v=2.578 feet. 
 
 7i 2 +. 01072^=. 000471. (/) 
 
 Completing, square ^ 2 +J 1 +(.00536) 2 =,. 0004997. (^J 
 
 Extracting root and transposing, n =.017. Ans. 
 
 Ex. 101. In a rectangular canal two feet deep, 
 four feet wide, the coefficient of roughness of the bed 
 is7i=.012, the observed velocity of flow 4.946 feet 
 per second, what is the sine of slope? 
 
216 
 
 PRACTICAL HYDRAULICS. 
 
 Cal. Arranging terms of Eq. (280), with respect 
 to s, 
 
 s /41.6 n z v4-nv rK 2 /- .00281?" 
 
 r-f-1 
 
 7it;rK | / . 00 
 .811r/ 88 + Ul.6 w r 
 
 __ 
 + l.SlTr 
 
 (") 
 
 Substituting the given values of t; 4.946, n= 
 .012, and the value of .r=l, r' 1, as found in the 
 preceding example, in Eq. (a): 
 
 si .03852 si -f- .00001459 s*=. 0000008663 . (I) 
 
 Solving Eq. (b) by Hutton's method for the resolu- 
 tion of cubic equations: 
 
 .03872 
 
 - .03852 
 .03 
 
 + .0000146 
 - .0002556 
 
 + .0000008663 
 - .000007230 
 
 - .00852 
 .03 
 
 -.0002410 
 .0006444 
 
 .C000080963 
 .OU00070336 
 
 .02148 
 .03 
 
 .0004034 
 .0004758 
 
 .0000010647 
 .0000010306 
 
 .05148 
 .008 
 
 .0008792 
 .0005398 
 
 .0000000341 
 .0000000308 
 
 .05948 
 .008 
 
 .0014190 
 .0000533 
 
 33 
 
 .06748 
 .008 
 
 .0014723 
 .0000538 
 
 .07548 
 .0007 
 
 .0015261 
 .0000015 
 
 .07618 
 .0007 
 
 .0015276 
 
 .07688 
 .0007 
 
 .07758 
 
PRACTICAL HYDEAULICS. 217 
 
 Thus 8*=. 03872. (c) 
 
 Squaring, s=.00l5, sine of slope. Ans. 
 
 Remark. It will be observed that Eq. (c) obtained 
 in the solution of Ex. 98, Eq. (c) in that of Ex. 99, 
 and Eq. (a) in that of Ex. (100), are general and ap- 
 plicable for the solution respectively of all similar 
 problems. 
 
 Table 27 has been computed by the author of the 
 present work direct from Eqs. (282) and (283), for the 
 velocity of water in open streams differing in regime 
 and slope, and varying from the size of a small ditch 
 to that of the Mississippi river. The computation has 
 been made for four different values of n, to-wit: n= 
 .012, ?i=.0l7, n= .025, and w=.035. 
 
 (For explanation of the values of n, see Table 26.) 
 
 Thus, the hydraulic mean depth being r=-~ =1, and 
 
 the fall per mile F=5.28 feet, or sine of slope s=.001, 
 the velocity per second for the respective values of n, 
 as shown by Table 27, are: 
 
 When %=.012, the velocity is v= 4.028 feet; 
 7i=.0l7, the velocity is v=2.720 feet; 
 71,=. 025, the velocity is ?; 1.755 feet; 
 7i=. 035, the velocity is #=1.190 feet. 
 
 In the several headings of the table, F represents in 
 feet the fall per mile, and s the equivalent sine of 
 slope. 
 
 Table 28 has been prepared to be used in connection 
 
218 PEACTICAL HYDRAULICS. 
 
 with Table 27. In the trapezoidal forms of canal 
 beete, considered -in this table, the bottom and sides are 
 equal each to each in the same cross section. 
 
 Table 29 has been computed to facilitate in finding 
 the wet perimeter of the bed of a trapezoidal canal. 
 
 Let 6=bottom width; 
 
 d depth of water depth of bank; 
 
 m=ratio of depth to base of bank; 
 
 m d=base of bank; 
 
 y=slope of bank; 
 
 jp=wet perimeter. 
 
 Then y=d (1 + m 2 )*; (284) 
 
 p=2d (1 + m 2 ) * + &. (285) 
 
 The computed values of the base and bank slope 
 for a unit depth are ari&nged under their respective 
 headings. 
 
PRACTICAL HYDRAULICS. 
 
 219 
 
 TABLE 27. 
 
 Flow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O12. 
 
 if s * 
 
 ^"l 
 
 ^">T 
 
 w^J |f 
 
 21? 
 
 " II || 
 
 ^|y 
 
 ^?^ 
 
 ^IT 
 
 J^I 
 
 !! 
 
 g. 2. g 
 
 ||i 
 
 lllg 
 
 go* of" 
 
 I ||E 
 
 i-2- 
 
 a? 2 
 
 I S |H 
 
 3~$ 
 
 VJ ai** 
 
 *.2j 
 
 4S 
 
 4fp.S 
 
 <PF 
 
 vrp^ 
 
 frsa 
 
 .25 
 .3 
 
 .0856 
 .1016 
 
 .2627 
 .3064 
 
 .4126 
 .4780 
 
 .6250 
 .7204 
 
 .7864 
 .9046^ 
 
 .9206 
 1.058 
 
 1.038 
 1.192 
 
 1.144 
 1.313 
 
 .4 
 
 .1329 
 
 .3894 
 
 .6004 
 
 .8989 
 
 1.134 
 
 1.312 
 
 1.477 
 
 1.612 
 
 .5 
 
 .1635 
 
 .4676 
 
 .7146 
 
 1.062 
 
 1.325 
 
 I.t45 
 
 1.739 
 
 1.912 
 
 .6 
 
 .1925 
 
 .5416 
 
 .8213 
 
 1.214 
 
 1.513 
 
 1.762 
 
 1.981 
 
 2.178 
 
 .7 
 
 .2226 
 
 .6123 
 
 .9227 
 
 1.359 
 
 1.690 
 
 1.966 
 
 2.209 
 
 2.428 
 
 .8 
 
 .2508 
 
 .6804 
 
 1.020 
 
 1.495 
 
 1.856 
 
 2.159 
 
 2.425 
 
 2.664 
 
 .9 
 
 .2796 
 
 .7*57 
 
 .111 
 
 1.625 
 
 2.016 
 
 2.343 
 
 2.631 
 
 2.889 
 
 1. 
 1.26 
 
 .3073 
 .3752 
 
 .8090 
 .9493 
 
 .201 
 .411 
 
 1.761 
 
 2.044 
 
 2.170 
 2.527 
 
 2.520 
 2.932 
 
 2.828 
 3.287 
 
 3.034 
 3.608 
 
 1.5 
 
 .4408 
 
 1.100 
 
 .606 
 
 2.316 
 
 2.858 
 
 3.312 
 
 3.711 
 
 4.071 
 
 2. 
 
 .5665 
 
 1.359 
 
 .960 
 
 2.806 
 
 3.453 
 
 3.995 
 
 4.474 
 
 4.905 
 
 2.5 
 
 .6366 
 
 1.596 
 
 2.281 
 
 3.247 
 
 3.987 
 
 4.606 
 
 5.158 
 
 5.652 
 
 3. 
 
 .8017 
 
 1.815 
 
 2.576 
 
 3.651 
 
 4.477 
 
 5.171 
 
 5.783 
 
 6.335 
 
 3.5 
 
 .9127 
 
 2.021 
 
 2.852 
 
 4.121 
 
 4.930 
 
 5.691 
 
 6.362 
 
 6.967 
 
 4. 
 
 .9968 
 
 2.216 
 
 3.111 
 
 4.380 
 
 5.356 
 
 6.189 
 
 6.905 
 
 7.561 
 
 4.5 
 
 .124 
 
 2.402 
 
 3.357 
 
 4.714 
 
 5.759 
 
 6.640 
 
 7.419 
 
 8.122 
 
 5. 
 
 .226 
 
 2.579 
 
 3.590 
 
 5.030 
 
 6.141 
 
 7.077 
 
 7.822 
 
 8.653 
 
 5.5 
 
 .324 
 
 2.741 
 
 3.814 
 
 5.333 
 
 6.506 
 
 7.495 
 
 8.370 
 
 9.160 
 
 6. 
 
 .421 
 
 2.977 
 
 4.029 
 
 5.624 
 
 6.856 
 
 7.896 
 
 8.615 
 
 9.647 
 
 6.5 
 
 .516 
 
 3.135 
 
 4.236 
 
 5.903 
 
 7.192 
 
 8.281 
 
 9.244 
 
 10.11 
 
 7. 
 
 .608 
 
 3.290 
 
 4.435 
 
 6.171 
 
 7.515 
 
 8.651 
 
 9.655 
 
 10.56 
 
 7.5 
 
 .699 
 
 3.443 
 
 4.629 
 
 6.433 
 
 7.831 
 
 9.012 
 
 10.06 
 
 11.00 
 
 8. 
 
 .783 
 
 3.590 
 
 4.817 
 
 6.686 
 
 8.134 
 
 9.359 
 
 10.44 
 
 11.42 
 
 8.5 
 
 .875 
 
 3.656 
 
 4.999 
 
 6.931 
 
 8.430 
 
 9.695 
 
 10.81 
 
 11.83 
 
 9. 
 
 .961 
 
 3.792 
 
 5 TJB 
 
 7.169 
 
 8.715 
 
 10.02 
 
 11.18 
 
 12.26 
 
 9.5 
 
 .046 
 
 3.926 
 
 5 3TO 
 
 7.402 
 
 8.996 
 
 10.34 
 
 11.54 
 
 12.61 
 
 10. 
 
 2.128 
 
 4.056 
 
 5!518 
 
 7.632 
 
 9.268 
 
 10.66 
 
 11.88 
 
 12.99 
 
 11. 
 
 2.292 
 
 4.309 
 
 5.846 
 
 8.067 
 
 9.798 
 
 H.25 
 
 12.55 
 
 13.73 
 
 12. 
 
 2.450 
 
 4.551 
 
 6.157 
 
 8.485 
 
 10.29 
 
 11.83 
 
 13.19 
 
 14.42 
 
 13. 
 
 2.604 
 
 4.785 
 
 6.458 
 
 8.889 
 
 10.78 
 
 12.39 
 
 13.81 
 
 15.09 
 
 14. 
 
 2.754 
 
 5.010 
 
 6.748 
 
 9.273 
 
 11.25 
 
 12.92 
 
 14.40 
 
 15.74 
 
 15. 
 
 2.900 
 
 5.228 
 
 7.029 
 
 9.652 
 
 11.70 
 
 13.43 
 
 14.97 
 
 16.36 
 
 16. 
 
 3.044 
 
 5.439 
 
 7.299 
 
 10.01 
 
 12.13 
 
 13.93 
 
 15.52 
 
 16.97 
 
 17. 
 
 3.184 
 
 5.645 
 
 7.563 
 
 10.37 
 
 12.56 
 
 U.42 
 
 16.06 
 
 17.55 
 
 18. 
 
 3.328 
 
 5.846 
 
 7.820' 
 
 10.71 
 
 12.97 
 
 4.89 
 
 16.58 
 
 18.13 
 
 19. 
 
 3.457 
 
 6.166 
 
 8.069 
 
 11.04 
 
 13.37 
 
 5.35 
 
 17.09 
 
 18.68 
 
 f- 
 
 3.590 
 
 6.361 
 
 8.313 
 
 11.37 
 
 13.76 
 
 5.79 
 
 17.59 
 
 19.22 
 
 . 
 
 3.720 
 
 6.551 
 
 8.551 
 
 11.63 
 
 14.14 
 
 6.23 
 
 18.07 
 
 19.75 
 
 22 % - 
 
 3.848 
 
 6.737 
 
 8.783 
 
 12.00 
 
 14.51 
 
 6.65 
 
 18.54 
 
 20.26 
 
 23. 
 
 3.974 
 
 6.918 
 
 9.010 
 
 
 14.87 
 
 7.06 
 
 19.01 
 
 
 24. 
 
 4.098 
 
 7.096 
 
 9.233 
 
 
 15.23 
 
 7.47 
 
 19.46 
 
 
 25. 
 
 4.220 
 
 7.271 
 
 9.451 
 
 12.83 
 
 15.58 
 
 7.87 
 
 19.95 
 
 
 50. 
 
 6.843 
 
 10.89 ' 
 
 13.95 
 
 
 
 
 
 
 100. 
 
 10.79 
 
 16.07 
 
 20.36 
 
 
 
 
 -" 
 
 
220 
 
 PRACTICAL HYDRAULICS, 
 
 TABLE 27. 
 
 Flow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O12. 
 
 ^ flfffl 
 
 
 
 - 
 
 
 
 
 *D 
 
 
 II 0-0. 
 
 II 03 
 
 w 
 
 o ' *" 
 
 $J 
 
 lisS 
 
 l||b 
 
 *$ 
 
 ^ ^ O o 
 
 *. 2. "o to 
 
 .25 
 .3 
 
 1.241 
 1.423 
 
 1.330 
 1.526 
 
 1.415 
 1.623 
 
 1.494 
 1.713 
 
 1.841 
 2.110 
 
 2.132 
 2.443 
 
 3.391 
 
 3.884 
 
 4.919 
 5.504 
 
 .4 
 
 1.761 
 
 1.898 
 
 2.007 
 
 2.118 
 
 2.606 
 
 3.017 
 
 4.762 
 
 6.790 
 
 .5 
 
 2.072 
 
 2.220 
 
 2.359 
 
 2.489 
 
 3.134 
 
 3.544 
 
 5.625 
 
 7.968 
 
 .6 
 
 2.358 
 
 2.526 
 
 2.685 
 
 2.833 
 
 3.403 
 
 4.029 
 
 6.388 
 
 9.053 
 
 .7 
 
 2.629 
 
 2.815 
 
 2.991 
 
 3.154 
 
 3.877 
 
 4.485 
 
 7.280 
 
 10.07 
 
 .8 
 
 2.883 
 
 3.088 
 
 3.257 
 
 3.459 
 
 4.253 
 
 4.919 
 
 7.794 
 
 11.03 
 
 .9 
 
 3.127 
 
 3.348 
 
 3.556 
 
 3.750 
 
 4.605 
 
 5.326 
 
 8.385 
 
 11.95 
 
 1.0 
 
 3.360 
 
 3.596 
 
 3.820 
 
 4.028 
 
 4.946 
 
 5.719 
 
 9.064 
 
 12.83 
 
 1.25 
 
 3.902 
 
 4.176 
 
 4.435 
 
 4.676 
 
 5.738 
 
 6.633 
 
 10.50 
 
 14.87 
 
 1.5 
 
 4.403 
 
 4.710 
 
 5.001 
 
 5.272 
 
 6.468 
 
 7.475 
 
 11.81 
 
 16.74 
 
 2. 
 
 5.302 
 
 5.671 
 
 6.018 
 
 6.344 
 
 7.778 
 
 8.986 
 
 14.22 
 
 20.11 
 
 2.5 
 
 6.107 
 
 6.531 
 
 6.851 
 
 7.304 
 
 8.951 
 
 10.33 
 
 16.35 
 
 23.13 
 
 3. 
 
 6.843 
 
 7.317 
 
 7.674 
 
 8.180 
 
 10.02 
 
 11.84 
 
 18.30 
 
 25.88 
 
 3.5 
 
 7.525 
 
 8.044 
 
 8.436 
 
 8.992 
 
 11.01 
 
 12.71 
 
 20.10 
 
 28.43 
 
 4. 
 
 8.165 
 
 8.727 
 
 9.151 
 
 9.153 
 
 11.94 
 
 13.78 
 
 21.79 
 
 30.81 
 
 4.5 
 
 8.769 
 
 9.372 
 
 9.826 
 
 10.47 
 
 12.82 
 
 14.80 
 
 23.39 
 
 33.07 
 
 5. 
 
 9.341 
 
 9.983 
 
 10.46 
 
 11.15 
 
 13.65 
 
 15.75 
 
 24.90 
 
 35.20 
 
 5.5 
 
 9.888 
 
 10.56 
 
 11.20 
 
 11.80 
 
 14.44 
 
 16.67 
 
 26.34 
 
 37.24 
 
 6. 
 
 10.41 
 
 11.13 
 
 11.79 
 
 12.44 
 
 15.21 
 
 17.55 
 
 27.73 
 
 39.19 
 
 6.5 
 
 10.91 
 
 11.66 
 
 12.36 
 
 13.02 
 
 15.94 
 
 18.39 
 
 29.05 
 
 41.07 
 
 7. 
 
 11.40 
 
 12.17 
 
 12.91 
 
 13.60 
 
 16.64 
 
 19.20 
 
 30.32 
 
 42.86 
 
 7.5 
 
 11.87 
 
 12.68 
 
 13.44 
 
 14.16 
 
 17.32 
 
 19.99 
 
 31.57 
 
 44.62 
 
 8. 
 
 12.32 
 
 13.16 
 
 13.96 
 
 14.70 
 
 17.98 
 
 20.75 
 
 32.77 
 
 46.31 
 
 8.5 
 
 12.76 
 
 13.63 
 
 14.45 
 
 15.23 
 
 18.62 
 
 21.43 
 
 33.93 
 
 
 
 9. 
 
 13.19 
 
 14.09 
 
 14.94 
 
 15.73 
 
 19.24 
 
 22.20 
 
 35.05 
 
 
 9.5 
 
 13.61 
 
 14.54 
 
 15.41 
 
 16.23 
 
 19.85 
 
 22.90 
 
 
 
 10. 
 
 14.02 
 
 14.97 
 
 15.87 
 
 16.71 
 
 20.44 
 
 23.58 
 
 
 
 11. 
 
 14.81 
 
 15.81 
 
 16.76 
 
 17.64 
 
 21.58 
 
 
 
 
 12. 
 
 15.56 
 
 16.61 
 
 17.61 
 
 18.54 
 
 22.67 
 
 
 
 
 
 13. 
 
 16.28 
 
 17.38 
 
 18.42 
 
 19.41 
 
 
 
 
 
 14. 
 
 16.97 
 
 18.13 
 
 19.21 
 
 20.28 
 
 
 
 
 
 15. 
 
 17.64 
 
 18.84 
 
 19.97 
 
 
 
 
 
 
 16. 
 
 18.29 
 
 19.53 
 
 20.71 
 
 
 
 
 
 
 17. 
 
 18.93 
 
 28.21 
 
 
 
 
 
 
 
 18. 
 
 19.54 
 
 
 
 
 
 
 
 
 19. 
 
 20.14 
 
 
 
 
 
 
 
 
PRACTICAL HYDRAULICS. 
 
 221 
 
 TABLE 27. 
 
 Flow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O17. 
 
 Hydraulic 
 Mean Depth, 
 T== _a_ 
 P 
 
 ,*">? 
 
 lid 
 
 32% 
 
 F=.264. 
 =. 00005. 
 Velocity. 
 Feet. 
 
 <!<*, w 
 
 ?!! . 
 
 HP 
 
 <*%, *i 
 
 stiJ- 
 
 g.oo 
 
 <}co Hj 
 
 gfsfbr' 
 
 ^? 
 
 TJ <L II II 
 
 rt> o^ to 
 
 S-2.2M 
 
 ' ?Z$ 
 
 *21? 
 
 |8^ 
 
 ^s? 
 
 <J TJ 
 
 nan II 
 ggbs* 
 
 ~$f$ 
 
 .25 
 
 .0567 
 
 .1704 
 
 .2615 
 
 .3956 
 
 .5004 
 
 .5844 
 
 .6597 
 
 .7288 
 
 .3 
 
 .0675 
 
 .2000 
 
 .3051 
 
 .4595 
 
 .5800 
 
 .6769 
 
 .7635 
 
 .8412 
 
 .4 
 
 .0887 
 
 .2568 
 
 .3877 
 
 .5796 
 
 .7296 
 
 .8499 
 
 .9577 
 
 .054 
 
 .5 
 
 .1095 
 
 .3110 
 
 .4657 
 
 .6922 
 
 .8691 
 
 .011 
 
 1.138 
 
 .2f6 
 
 .6 
 
 .1301 
 
 .3628 
 
 .5272 
 
 .7979 
 
 .9997 
 
 .162 
 
 1.307 
 
 .438 
 
 .7 
 
 .1503 
 
 .4127 
 
 .6101 
 
 .8985 
 
 1.123 
 
 .305 
 
 1.467 
 
 .612 
 
 .8 
 
 .1702 
 
 .4612 
 
 .6778 
 
 .9951 
 
 1.239 
 
 .441 
 
 1.619 
 
 .781 
 
 .9 
 
 .1892 
 
 .5079 
 
 .7423 
 
 1 086 
 
 1.355 
 
 .572 
 
 1.766 
 
 1.986 
 
 1. 
 
 .2093 
 
 .5535 
 
 .8(166 
 
 1.176 
 
 1.465 
 
 .698 
 
 1.907 
 
 2.095 
 
 1.25 
 
 .2569 
 
 .6625 
 
 .9565 
 
 1.386 
 
 1.722 
 
 .989 
 
 2.237 
 
 2.456 
 
 1.5 
 
 .3034 
 
 .7659 
 
 1.097 
 
 1.581 
 
 1.957 
 
 2.269 
 
 2.544 
 
 2.792 
 
 2.0 
 
 .3934 
 
 .9581 
 
 1.356 
 
 1.939 
 
 2.398 
 
 2.768 
 
 3.109 
 
 3.402 
 
 2.5 
 
 .4802 
 
 1.136 
 
 1.592 
 
 2.338 
 
 2.793 
 
 3.221 
 
 3.605 
 
 3.952 
 
 3. 
 
 .5629 
 
 1.302 
 
 1.812 
 
 2.563 
 
 3.157 
 
 3.808 
 
 4.069 
 
 4.458 
 
 3.5 
 
 .6457 
 
 1.460 
 
 2.018 
 
 2.843 
 
 3.496 
 
 4.025 
 
 4.506 
 
 4.929 
 
 4. 
 
 .7253 
 
 1.609 
 
 2.213 
 
 3.100 
 
 3.816 
 
 4.390 
 
 4.906 
 
 5.372 
 
 4.5 
 
 .8030 
 
 1.753 
 
 2.455 
 
 3.358 
 
 4.119 
 
 4.735 
 
 5.291 
 
 5.792 
 
 5. 
 
 .8590 
 
 1.890 
 
 2.576 
 
 3.594 
 
 4.407 
 
 5.064 
 
 5.657 
 
 6.191 
 
 5.5 
 
 .9533 
 
 2.023 
 
 2.745 
 
 3.824 
 
 4.683 
 
 5.379 
 
 6.007 
 
 6.574 
 
 6. 
 
 1.026 
 
 2.101 
 
 2.910 
 
 4.044 
 
 4.943 
 
 5.682 
 
 6.344 
 
 6.941 
 
 6.5 
 
 1.098 
 
 2.275 
 
 3.C68 
 
 4.256 
 
 5.204 
 
 5.973 
 
 6.667 
 
 7.134 
 
 7. 
 
 1.168 
 
 2.394 
 
 3.220 
 
 4.460 
 
 5.450 
 
 6.253 
 
 6.978 
 
 7.633 
 
 7.5 
 
 1.238 
 
 2.512 
 
 3.361 
 
 4.659 
 
 5.690 
 
 6.526 
 
 7.282 
 
 7.964 
 
 8. 
 
 1.307 
 
 2.582 
 
 3.433 
 
 4.851 
 
 5.921 
 
 6.790 
 
 7.575 
 
 8.280 
 
 8.5 
 
 1.373 
 
 2.740 
 
 3.570 
 
 5.038 
 
 6.147 
 
 7.(i47 
 
 7.861 
 
 8.595 
 
 9. 
 
 1.439 
 
 2.845 
 
 3.790 
 
 5.220 
 
 6.365 
 
 7.296 
 
 8.137 
 
 8.896 
 
 9.5 
 
 1.505 
 
 2.951 
 
 3.923 
 
 5.397 
 
 6.579 
 
 7.540 
 
 8.407 
 
 9.191 
 
 10. 
 
 1.569 
 
 3.055 
 
 4.054 
 
 5.444 
 
 6.788 
 
 7.777 
 
 8.673 
 
 9.479 
 
 11. 
 
 1.696 
 
 3.256 
 
 4.307 
 
 6.044 
 
 7.192 
 
 8.256 
 
 9.182 
 
 10.03 
 
 12. 
 
 1.820 
 
 3.519 
 
 4.539 
 
 6.226 
 
 7.572 
 
 8.878 
 
 9.669 
 
 10.56 
 
 13. 
 
 1.939 
 
 3.638 
 
 4.783 
 
 6.537 
 
 7.950 
 
 9.100 
 
 10.14 
 
 11.08 
 
 14. 
 
 2.059 
 
 3.819 
 
 5.009 
 
 6.680 
 
 8.309 
 
 9.487 
 
 10.59 
 
 11.57 
 
 15. 
 
 2.175 
 
 3.994 
 
 5.257 
 
 7.125 
 
 8.654 
 
 9.901 
 
 11.03 
 
 12.05 
 
 16. 
 
 2.289 
 
 4.165 
 
 5.438 
 
 7.404 
 
 8.990 
 
 10.28 
 
 11.45 
 
 12.51 
 
 17. 
 
 2.401 
 
 4.331 
 
 5.644 
 
 7.675 
 
 9.317 
 
 10.65 
 
 11.86 
 
 12.96 
 
 18. 
 
 2.511 
 
 4.493 
 
 5.846 
 
 7.940 
 
 9.635 
 
 11.01 
 
 12.26 
 
 13.39 
 
 19. 
 
 2.619 
 
 4.651 
 
 6.181 
 
 8.196 
 
 9.943 
 
 11.36 
 
 12.65 . 
 
 13.82 
 
 20. 
 
 2.726 
 
 4.806 
 
 6.231 
 
 8.448 
 
 10.24 
 
 11.71 
 
 13.03 
 
 14.23 
 
 21. 
 
 2.836 
 
 4.957 
 
 6.417 
 
 8.694 
 
 10.54 
 
 12.04 
 
 13.40 
 
 14.64 
 
 22. 
 
 2.935 
 
 5.096 
 
 6.598 
 
 8.933 
 
 10.82 
 
 12.37 
 
 13.77 
 
 15.03 
 
 23. 
 
 3.036 
 
 5.249 
 
 6.777 
 
 9.273 
 
 11.11 
 
 12.69 
 
 14.12 
 
 15.42 
 
 24. 
 
 3.137 
 
 5.391 
 
 6.952 
 
 9.397 
 
 11.38 
 
 13. CO 
 
 14.47 
 
 15.80 
 
 25. 
 
 3.237 
 
 5.531 
 
 7.123 
 
 9.622 
 
 11.65 
 
 13.31 
 
 14.81 
 
 16.17 
 
 50. 
 
 6.402 
 
 8.439 
 
 10.67 
 
 14.27 
 
 17.22 
 
 19.64 
 
 21.83 
 
 
 100. 
 
 8.766 
 
 12.64 
 
 15.75 
 
 9.0.90 
 
 
 
 
 
222 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 27. 
 
 FJow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O17. 
 
 Hydraulic 
 Mean Depth, 
 va 
 
 F=3.696 
 8=,0007. 
 Velocity. 
 Feet. 
 
 <!F 
 
 *Tj.JI II 
 
 *glr 
 
 !8^ 
 r *8g 
 
 ,ITT 
 
 as -ui 
 
 *ll 
 
 < ^ 
 
 Its 
 
 ' < W.tO 
 
 <!^ 
 
 *4 . II II 
 
 G> O ' M 
 
 r fPs 
 
 *j|f ? 
 
 ! S-88 
 - * f* 
 
 ^00 *J 
 
 II- 1 i 
 
 g. Q. o to 
 
 ,fr- 
 
 .25 
 .3 
 .4 
 .5 
 .6 
 .7 
 .8 
 .9 
 1. 
 1.25 
 1.5 
 2. 
 2.5 
 3. 
 3.5 
 4. 
 1.5 
 5. 
 5.5 
 6. 
 8.5 
 7. 
 7.5 
 B. 
 3 5 
 
 .7893 
 .9128 
 1.143 
 1.358 
 1.558 
 1.748 
 1.928 
 2.102 
 2.268 
 2.658 
 3.020 
 3.678 
 4.271 
 4.817 
 5.324 
 5.802 
 6.255 
 6.684 
 7.096 
 7.491 
 7.871 
 8.212 
 8.594 
 8.938 
 9 166 
 
 .8463 
 .9785 
 1.200 
 1.455 
 1.708 
 1.871 
 2.064 
 2.249 
 2.427 
 2.654 
 3.231 
 3.931 
 4.567 
 5.149 
 5.823 
 6.200 
 6.683 
 7.141 
 7.583 
 8.003 
 8.408 
 8.797 
 9.178 
 9.545 
 9 902 
 
 .9002 
 1.039 
 1.303 
 1.546 
 1.773 
 1.988 
 2.193 
 2.445 
 2.578 
 3.020 
 3.430 
 4.174 
 4.844 
 5.462 
 6.036 
 6.526 
 7.086 
 7.572 
 8.036 
 8.483 
 8.912 
 9.324 
 9.683 
 10.12 
 10 49 
 
 .9508 
 1.074 
 1.375 
 1.632 
 1.872 
 2.098 
 2.314 
 2.521 
 2.720 
 3.185 
 3.618 
 4.401 
 5.108 
 5.758 
 6.362 
 6.931 
 7.469 
 7.979 
 8.469 
 8.7?5 
 9.390 
 9.824 
 10.25 
 10.66 
 11 05 
 
 1.172 
 1.354 
 1.693 
 2.009 
 2.302 
 2.580 
 2.844 
 3.098 
 3.344 
 3.911 
 4.543 
 5.347 
 6.260 
 7.053 
 7.790 
 8.484 
 9.141 
 9^164 
 1TO6 
 10.93 
 11.48 
 12.01 
 12.53 
 13.01 
 13 51 
 
 1.358 
 1.568 
 1.961 
 2.325 
 2.664 
 2.985 
 3.290 
 3.568 
 3.863 
 4.521 
 5.130 
 6.540 
 7.231 
 8.145 
 8.995 
 9.794 
 10.95 
 11.26 
 11.95 
 32.61 
 13.25 
 13.86 
 14.45 
 15.03 
 15 23 
 
 2.160 
 2.494 
 3.116 
 
 3.eaL 
 
 4.2HF 
 
 5 '219 
 5.680 
 6.124 
 7.163 
 8.143 
 9. 869 
 11.44 
 12.88 
 14.22 
 15.48 
 16.67 
 17.81 
 18.88 
 19.93 
 20.93 
 21.88 
 22.82 
 23.73 
 24 61 
 
 3.C61 
 3.534 
 4.414 
 5.230 
 5.9S8 
 6.707 
 7.474 
 8.042 
 8.669 
 10.14 
 11.44 
 13.96 
 16.18 
 18.22 
 20.11 
 21.89 
 23.57 
 25.17 
 26.70 
 28.17 
 29.48 
 30.94 
 32.26 
 33.54 
 
 9 
 
 9 597 
 
 10 25 
 
 10 86 
 
 11 44 
 
 13 98 
 
 16 13 
 
 25.46 
 
 
 9 5 
 
 9 914 
 
 10 59 
 
 11 22 
 
 11 82 
 
 14 44 
 
 16 66 
 
 
 
 ) 
 
 10 22 
 
 10 91 
 
 11 57 
 
 12 18 
 
 14 89 
 
 17.18 
 
 
 
 
 10 82 
 
 11 55 
 
 12 24 
 
 12^93 
 
 15 76 
 
 18 18 
 
 
 
 2 
 
 11 39 
 
 12 16 
 
 12 89 
 
 13*67 
 
 16 58 
 
 19.12 
 
 
 
 
 11 95 
 
 12 75 
 
 13 51 
 
 14 23 
 
 17 38 
 
 20 00 
 
 
 
 1 
 
 12 48 
 
 13 32 
 
 14 11 
 
 14 86 
 
 18 10 
 
 
 
 
 > ' 
 
 13 99 
 
 13 86 
 
 14 69 
 
 15 47 
 
 18.89 
 
 
 
 
 5' *. 
 
 13 49 
 
 14 36 
 
 15 25 
 
 16 06 
 
 19 61 
 
 
 
 
 f 
 
 13 97 
 
 14 91 
 
 15 79 
 
 16 64 
 
 20.31 
 
 
 
 
 j '*. 
 
 14 44 
 
 15 41 
 
 16 32 
 
 17 19 
 
 
 
 
 
 | 
 
 14 89 
 
 15 90 
 
 16 80 
 
 17 73 
 
 
 
 
 
 o 
 
 15 34 
 
 16 33 
 
 17 34 
 
 18 26 
 
 
 
 
 
 1 
 
 15 78 
 
 16 82 
 
 17 83 
 
 18 78 
 
 
 
 
 
 2 
 
 16 20 
 
 17.29 
 
 18.31 
 
 19.28 
 
 
 
 
 
 3 
 
 16 62 
 
 17 73 
 
 18 78 
 
 19 77 
 
 
 
 
 
 1 
 
 1 7 03 
 
 18 17 
 
 19 24 
 
 
 
 
 
 
 5. 
 
 17.43 
 
 18.59 
 
 19.69 
 
 
 
 
 
 
PRACTICAL HYDRAULICS. 
 
 223 
 
 TABLE 27. 
 
 Flow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O25. 
 
 Hydraulic 
 Mean Depth, 
 
 r=-- 
 
 p 
 
 tl? 
 
 *. 
 
 *?S5 
 
 v& 
 
 2.J 
 
 NfSB 
 
 <!!!*> 
 
 ifi 1 
 
 *|ip 
 
 <<*> 3 
 *j2j II 
 
 w 
 
 -3*> *4 
 
 iHi 
 
 <*-?; 01 
 
 ' ^ ^^ 
 
 ^11 ? 
 
 bcj 0, N .|| 
 n o - 
 
 vT.^K 
 
 <^*a 
 igg & fl II 
 
 a o g 
 
 *4pS 
 
 *2T? 
 
 Is-S* 
 
 r 4i 
 
 .25 
 
 .0365 
 
 .1022 
 
 .1584 
 
 .2394 
 
 .3031 
 
 .3528 
 
 ....3978 
 
 .4387 
 
 .3 
 
 .0435 
 
 .1206 
 
 .1862 
 
 .2802 
 
 .3519 
 
 .4142 
 
 .4642 
 
 .5117 
 
 .4 
 
 .0624 
 
 .1564 
 
 .2394 
 
 .3580 
 
 .4483 
 
 .5238 
 
 -.5899 
 
 .6498 
 
 .5 
 
 .0712 
 
 .1909 
 
 .2901 
 
 .4316 
 
 .5393 
 
 .6296 
 
 -.7086 
 
 .7800 
 
 .6 
 
 .0848 
 
 .2243 
 
 .3388 
 
 .5017 
 
 .6258 
 
 .7296 
 
 ,8208 
 
 .9027 
 
 .7 
 
 .0982 
 
 .2567 
 
 .3858 
 
 .5691 
 
 .7089 
 
 .8256 
 
 .9282 
 
 1.020 
 
 .8 
 
 .1115 
 
 .2884 
 
 .4311 
 
 .6341 
 
 .7887 
 
 .9180 
 
 1.031 
 
 1.134 
 
 .9 
 
 .1247 
 
 .3192 
 
 .4761 
 
 .6966 
 
 .8653 
 
 1.006 
 
 1.130 
 
 1.243 
 
 1. 
 
 .1378 
 
 .3494 
 
 '.5183 
 
 .7576 
 
 .9400 
 
 1.093 
 
 1.226 
 
 1.348 
 
 1.25 
 
 .1662 
 
 .4223 
 
 .6211 
 
 .9025 
 
 1.117 
 
 1.297 
 
 1.457 ' 
 
 1.598 
 
 1.5 
 
 .2017 
 
 .4921 
 
 .7186 
 
 1.039 
 
 1.283 
 
 1.488 
 
 1.669 
 
 1<832 
 
 2. 
 
 .2636 
 
 .6239 
 
 .9006 
 
 1.291 
 
 1.590 
 
 1.841 
 
 2.06? 
 
 2.264 
 
 2.5 
 
 .3235 
 
 .7134 
 
 1.079 
 
 1.514 
 
 1.872 
 
 2.164 
 
 2.422 
 
 2.655 
 
 3. 
 
 .3826 
 
 .8641 
 
 1.227 
 
 1.739 
 
 2.132 
 
 2.464 
 
 2.755 
 
 3.020 
 
 3.5 
 
 .4399 
 
 .9758 
 
 1.377 
 
 1.943 
 
 2.303 
 
 ;2.746 
 
 3.068 
 
 3.361 
 
 4. 
 
 .4965 
 
 1.083 
 
 1.519 
 
 2.136 
 
 2.610 
 
 3.010 
 
 3.371 
 
 3.684 
 
 4.5 
 
 .5518 
 
 1.214 
 
 1.656 
 
 2.321 
 
 2.832 
 
 3.264 
 
 3.645 
 
 3.993 
 
 5. 
 
 .6062 
 
 1.290 
 
 1.7S6 
 
 2.496 
 
 3.045 
 
 3.506 
 
 3.921 
 
 4.279 
 
 5.5 
 
 .6600 
 
 1.382 
 
 1.913 
 
 2.666 
 
 3.247 
 
 3.738 
 
 4.161 
 
 4.566 
 
 6. 
 
 .7128 
 
 1.476 
 
 2.034 
 
 2.831 
 
 3.443 
 
 3.962 
 
 4.420 
 
 4.838 
 
 6.5 
 
 .7650 
 
 1.567 
 
 2.153 
 
 2.988 
 
 3.634 
 
 4.180 
 
 4.662 
 
 5.100 
 
 7. 
 
 .8165 
 
 1.655 
 
 2.267 
 
 3.141 
 
 3.815 
 
 4.388 
 
 4.893 
 
 5.352 
 
 7.5 
 
 .8671 
 
 1.741 
 
 2.379 
 
 3.291 
 
 3.996 
 
 4.592 
 
 5.121 
 
 5.599 
 
 8. 
 
 .9202 
 
 1.826 
 
 2.488 
 
 3.433 
 
 4.169 
 
 4.788 
 
 5.340 
 
 5.840 
 
 8.5 
 
 .9607 
 
 1.909 
 
 2.594 
 
 3.575 
 
 4.337 
 
 4.982 
 
 5.552 
 
 6.070 
 
 9. 
 
 .016 
 
 1.990 
 
 2.697 
 
 3.712 
 
 4.500 
 
 5.168 
 
 5.760 
 
 6.295 
 
 9.5 
 
 .065 
 
 2.069 
 
 2:790 
 
 3.846 
 
 4.661 
 
 5.352 
 
 5.961 
 
 6.518 
 
 10. 
 
 .113 
 
 2.146 
 
 2.898 
 
 3.978 
 
 4.818 
 
 5.530 
 
 6.160 
 
 6.734 
 
 11. 
 
 .208 
 
 2.353 
 
 3.091 
 
 4.357 
 
 5.101 
 
 6.056 , 
 
 6.568 
 
 7.131 
 
 12. 
 
 .299 
 
 2.444 
 
 3.276 
 
 4.491 
 
 5.413 
 
 6.208 
 
 6.911 
 
 7.552 
 
 1:5. 
 
 .392 
 
 2.583 
 
 3.455 
 
 4.714 
 
 5.695 
 
 6.530 
 
 7.267 
 
 7.939 
 
 14. 
 
 .482 
 
 2.724 
 
 3.629 
 
 4.941 
 
 5.967 
 
 6.838 
 
 7.612 
 
 8.311 
 
 15. 
 
 1.569 
 
 2.793 
 
 3.797 
 
 5.162 
 
 6.228 
 
 7.136 
 
 7.942 
 
 8.671 
 
 16. 
 
 1.658 
 
 2.988 
 
 3.960 
 
 5.375 
 
 6.484 
 
 7.426 
 
 8.262 
 
 9.021 
 
 17. 
 
 1.742 
 
 3.115 
 
 4.127 . 
 
 5.585 
 
 6.732 
 
 7.708 
 
 8.575 
 
 9.362 
 
 18. 
 
 1.828 
 
 3.241 
 
 4.278 
 
 5.788 
 
 6.975 
 
 7.986 
 
 8.879 
 
 9.695 
 
 19. 
 
 1.915 
 
 3.362 
 
 4.419 
 
 5.985 
 
 7.209 
 
 8.272 
 
 9.186 
 
 10.02 
 
 20. 
 
 1.992 
 
 3.480 
 
 4.514 
 
 6.178 
 
 7.439 
 
 8.532 
 
 9.488 
 
 10.33 
 
 21. 
 
 2.074 
 
 3.598 
 
 4.719 
 
 6.368 
 
 7.647 
 
 8.770 
 
 9.749 
 
 10.68 
 
 22. 
 
 2.155 
 
 3.712 
 
 4.862 
 
 6.553 
 
 7.884 
 
 9.020 
 
 10.02 
 
 10.95 
 
 23. 
 
 2.234 
 
 3.824 
 
 5.000 
 
 6.734 
 
 8.018 
 
 9.264 
 
 10.30 
 
 11.24 
 
 24. 
 
 2.312 
 
 3.934 
 
 5.150 
 
 6.911 
 
 8.310 
 
 9.502 
 
 10.56 
 
 11.52 
 
 25. 
 
 2.390 
 
 4.052 
 
 5.270 
 
 ;7.085 
 
 8.516 
 
 9.738 
 
 10.82 
 
 11.81 
 
 50. 
 
 4.127 
 
 6.325 
 
 8.064 
 
 10.70 
 
 12.80 
 
 14.61 
 
 16.21 
 
 17.67 
 
 100. 
 
 6.916 
 
 9.673 
 
 12.00 
 
 15.89 
 
 18.94... 
 
 21.58 
 
 
 
224 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 27. 
 
 Plow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O25. 
 
 -If 
 
 -tjdJI ll_ 
 
 <<?? "9 
 o' 1 
 
 j GO *^ 
 
 lll^ 
 
 $11 
 
 <<*, "3 
 Kl'l 
 
 nil 
 
 nil 
 
 * ls l 
 
 
 ?s 
 
 
 ^ ' 
 
 
 9. * 
 
 << . * 
 
 ' C^po 
 
 .25 
 
 .4759 
 
 .5116 
 
 .5457 
 
 .5765 
 
 .7072 
 
 .8197 
 
 1.301 
 
 1.847 
 
 .3 
 
 .5551 
 
 .5951 
 
 .6330 
 
 .6847 
 
 .8238 
 
 .9544 
 
 1.515 
 
 2.150 
 
 .4 
 
 .7045 
 
 .7548 
 
 .8028 
 
 .8491 
 
 .044 
 
 1.209 
 
 1.918 
 
 2.722 
 
 .5 
 
 .8318 
 
 .8986 
 
 .9630 
 
 1.019 
 
 .254 
 
 1.450 
 
 2.299 
 
 3.259 
 
 .6 
 
 .9787 
 
 1.048 
 
 1.114 
 
 1.177 
 
 .446 
 
 1.674 
 
 2.655 
 
 3.765 
 
 .7 
 
 1.106 
 
 1.184 
 
 1.259 
 
 1.330 
 
 .633 
 
 1.890 
 
 2.996 
 
 4.248 
 
 .8 
 
 1.228 
 
 1.315 
 
 1.398 
 
 1.477 
 
 .813 
 
 2.097 
 
 3.324 
 
 4.711 
 
 .9 
 
 1.345 
 
 1.440 
 
 1.530 
 
 1.616 
 
 .984 
 
 2.295 
 
 3.637 
 
 5.1-4 
 
 1. 
 
 1.459 
 
 1.564 
 
 1.664 
 
 1.755 
 
 2.151 
 
 2.4S8 
 
 3.941 
 
 5.585 
 
 1.25 
 
 1.729 
 
 1.855 
 
 1.980 
 
 2.092 
 
 2.545 
 
 2.943 
 
 4.660 
 
 6.600 
 
 1.5 
 
 1.981 
 
 2.127 
 
 2.265 
 
 2.383 
 
 2.914 
 
 3.368 
 
 5.333 
 
 7.551 
 
 2. 
 
 2.446 
 
 2.621 
 
 2.787 
 
 2.934 
 
 3.590 
 
 4.143 
 
 6.418 
 
 9.290 
 
 2.5 
 
 2.804 
 
 3.035 
 
 3.261 
 
 3.434 
 
 4.206 
 
 4.861 
 
 8.693 
 
 10.90 
 
 3. 
 
 3.262 
 
 3.487 
 
 3.702 
 
 3.899 
 
 4.775 
 
 5.514 
 
 8.719 
 
 12.31 
 
 3.5 
 
 3.630 
 
 3.880 
 
 4.113 
 
 4.237 
 
 5.310 
 
 6.131 
 
 9.694 
 
 13.71 
 
 4. 
 
 3.979 
 
 4.24S 
 
 4.503 
 
 4.746 
 
 5.817 
 
 6.712 
 
 10.61 
 
 15.04 
 
 4.5 
 
 4.310 
 
 4.599 
 
 4.872 
 
 5.139 
 
 6.297 
 
 7.272 
 
 11.49 
 
 16.24 
 
 5. 
 
 4.625 
 
 4.936 
 
 5.223 
 
 5.509 
 
 6.754 
 
 7.795 
 
 12.32 
 
 17.41 
 
 5.5 
 
 4.929 
 
 5.255 
 
 5.559 
 
 5.866 
 
 7.192 
 
 8.300 
 
 13.11 
 
 18.53 
 
 6. 
 
 5.220 
 
 5.566 
 
 5.886 
 
 6.211 
 
 7.614 
 
 8.78$ 
 
 13.89 
 
 19.62 
 
 6.5 
 
 5.503 
 
 5.864 
 
 6.198 
 
 6.543 
 
 8.025 
 
 9.257 
 
 14.62 
 
 20.68 
 
 7. 
 
 5.773 
 
 6.152 
 
 6.501 
 
 6.862 
 
 8.416 
 
 9.709 
 
 15.33 
 
 21.67 
 
 7.5 
 
 6.640 
 
 6.432 
 
 6.795 
 
 7.178 
 
 8.803 
 
 10.15 
 
 16.03 
 
 22.65 
 
 8. 
 
 6.297 
 
 6.703 
 
 7.080 
 
 7.485 
 
 9.194 
 
 10.59 
 
 16.71 
 
 23.60 
 
 8.5 
 
 6.545 
 
 6.966 
 
 7.356 
 
 7.773 
 
 9.527 
 
 10.91 
 
 17.16 
 
 24.13 
 
 9. 
 
 6.789 
 
 7.224 
 
 7.626 
 
 8.054 
 
 9.884 
 
 11.40 
 
 17.96 
 
 25.42 
 
 9.5 
 
 7.001 
 
 7.461 
 
 7.890 
 
 8 . 332 
 
 10 23 
 
 11 79 
 
 18 . 62 
 
 
 10. 
 
 7.260 
 
 7.721 
 
 8.148 
 
 8.601 
 
 10 .'53 
 
 12! 18 
 
 19.24 
 
 
 11. 
 
 7.948 
 
 8.454 
 
 8.919 
 
 9.306 
 
 11.21 
 
 12.89 
 
 20.39 
 
 
 
 12. 
 
 8.141 
 
 8.655 
 
 9.126 
 
 9.642 
 
 11.83 
 
 13.64 
 
 21.49 
 
 
 13. 
 
 8.556 
 
 9.090 
 
 9.585 
 
 10.13 
 
 K.43 
 
 14.33 
 
 22.62 
 
 
 14. 
 
 8.937 
 
 9.503 
 
 10.03 
 
 10.59 
 
 13! 02 
 
 15 . or 
 
 
 
 15. 
 
 9.318 
 
 9.982 
 
 10.46 
 
 11.05 
 
 13.57 
 
 15.64 
 
 
 
 16. 
 
 9.723 
 
 10.32 
 
 10.87 
 
 11.49 
 
 14.11 
 
 16.27 
 
 
 
 17. 
 
 10.09 
 
 10.71 
 
 11.28 
 
 11.92 
 
 14.64 
 
 16.88 
 
 
 
 18. 
 
 10.44 
 
 11.08 
 
 11.67 
 
 12.33 
 
 15.16 
 
 17.47 
 
 
 
 19. 
 
 10.79 
 
 11.45 
 
 12.05 
 
 12.75 
 
 15.66 
 
 18.05 
 
 
 
 20. 
 
 11.10 
 
 11.81 
 
 12.43 
 
 13.14 
 
 16 . 15 
 
 18.61 
 
 
 
 21. 
 
 11.46 
 
 12.16 
 
 12.80 
 
 13.53 
 
 16.61 
 
 19.16 
 
 
 
 22. 
 
 11.78 
 
 12.48 
 
 13.15 
 
 13.91 
 
 17.09 
 
 19.70 
 
 
 
 23. 
 
 12.10 
 
 12.82 
 
 13.50 
 
 14.27 
 
 17.55 
 
 20.22 
 
 
 
 24. 
 
 12.41 
 
 13.15 
 
 13.84 
 
 14.53 
 
 17.99 
 
 
 
 
 25. 
 
 12.72 
 
 13.48 
 
 14.18 
 
 15.00 
 
 18.43 
 
 
 
 
 50 ! 
 
 19 '.01 
 
 2o! 13 
 
 
 
 
 
 
 
PRACTICAL HYDRAULICS. 
 
 225 
 
 TABLE 27. 
 
 Flow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O35. 
 
 *H 
 
 <1* NJ 
 
 ^ 
 
 ~ 
 
 ~^ 
 
 ^ 
 
 ^*>*l 
 
 ~~ 
 
 ~ 
 
 ** ^ ^ 
 
 
 ^y, ft* J' || 
 
 ^2 ii if 
 
 *T\ 3- 11 il 
 
 hel L 11 11 
 
 w ^L 11 II 
 
 i-ri <' 1! 11 
 
 *2-l.|. 
 
 *tl! 
 
 ?|l| 
 
 *ffi 
 
 So gin 
 
 -tii 
 
 ^ili 
 
 *f|| 
 
 S 4IP 
 
 
 j?^' 
 
 
 
 
 
 
 
 
 
 .25 
 
 .0251 
 
 .0674 
 
 .1033 
 
 .1551 
 
 .1947 
 
 .2279 
 
 .2565 
 
 .2832 
 
 .3 
 
 .0300 
 
 .0799 
 
 .1219 
 
 .1825 
 
 .2288 
 
 .2676 
 
 .3011 
 
 .3322 
 
 .4 
 
 .0397 
 
 .1040 
 
 .1545 
 
 .23^3 
 
 .2944 
 
 .3439 
 
 .3867 
 
 .4215 
 
 .5 
 
 .0493 
 
 .1280 
 
 .1929 
 
 .2860 
 
 .3571 
 
 .4166 
 
 .4682 
 
 .5160 
 
 .6 
 
 .0588 
 
 .1546 
 
 .2266 
 
 .3347 
 
 .4171 
 
 .4862 
 
 .5462 
 
 .6015 
 
 .7 
 
 .0683 
 
 .1737 
 
 .2594 
 
 .3817 
 
 .4752 
 
 .5547 
 
 .6215 
 
 .6841 
 
 .8 
 
 .0771 
 
 .1959 
 
 .2944 
 
 .4274 
 
 . 5313 
 
 .6184 
 
 .6941 
 
 .7628 
 
 .9 
 
 .0870 
 
 .2177 
 
 .3225 
 
 .4718 
 
 .5858 
 
 .6815 
 
 .7648 
 
 .8412 
 
 1. 
 
 .0963 
 
 .2391 
 
 .3555 
 
 .5152 
 
 .6389 
 
 .7429 
 
 .8334 
 
 .9165 
 
 1.25 
 
 .1192 
 
 .2911 
 
 .4267 
 
 .6212 
 
 .76ri3 
 
 .8898 
 
 .9976 
 
 1.096 
 
 1.5 
 
 .1418 
 
 .3413 
 
 .4972 
 
 .7521 
 
 .8872 
 
 1.029 
 
 1.153 
 
 1.266 
 
 2. 
 
 .2160 
 
 .4372 
 
 .6303 
 
 .9035 
 
 1.112 
 
 1.288 
 
 1.442 
 
 1.583 
 
 2.5 
 
 .2298 
 
 .5283 
 
 . 7542 
 
 1.076 
 
 1.322 
 
 1 . 528 
 
 1.711 
 
 1>75 
 
 3. 
 
 .2727 
 
 .6297 
 
 .8734 
 
 1.296 
 
 1.518 
 
 1.753 
 
 1.961 
 
 2.100 
 
 3.5 
 
 .3147 
 
 .6991 
 
 .9861 
 
 1.392 
 
 1.704 
 
 1.966 
 
 2.198 
 
 2.407 
 
 4. 
 
 .3562 
 
 .7800 
 
 1.094 
 
 1.539 
 
 1.881 
 
 2.169 
 
 2.423 
 
 2.653 
 
 4.5 
 
 .3971 
 
 .8583 
 
 1.198 
 
 1.6SO 
 
 2.050 
 
 2.363 
 
 2.638 
 
 2.888 
 
 5. 
 
 .4374 
 
 .9343 
 
 1.296 
 
 1.815 
 
 2.213 
 
 2.543 
 
 2.846 
 
 3.113 
 
 5.5 
 
 .4774 
 
 1.008 
 
 1.396 
 
 1.946 
 
 2.370 
 
 2.728 
 
 3.045 
 
 3.331 
 
 6. 
 
 .5168 
 
 1.081 
 
 1.490 
 
 2.075 
 
 2.464 
 
 2.902 
 
 3.238 
 
 3.540 
 
 6.5 
 
 .5558 
 
 1.151 
 
 1.583 
 
 2.195 
 
 2.669 
 
 3.069 
 
 3.425 
 
 3.744 
 
 7. 
 
 . 5944 
 
 .219 
 
 1.671 
 
 2.314 
 
 2.811 
 
 3.232 
 
 3.605 
 
 3.940 
 
 7.5 
 
 .6327 
 
 .287 
 
 1.759 
 
 2.431 
 
 2.971 
 
 3.391 
 
 3.782 
 
 4.132 
 
 8. 
 
 .6706 
 
 .353 
 
 1.802 
 
 2.544 
 
 3.0S6 
 
 3.545 
 
 3.953 
 
 4.319 
 
 8.5 
 
 .7080 
 
 .418 
 
 1.927 
 
 2.655 
 
 3.218 
 
 3.695 
 
 4.120 
 
 4.501 
 
 9. 
 
 .7454 
 
 .482 
 
 2.009 
 
 2.763 
 
 3.347 
 
 3.843 
 
 4.284 
 
 4.678 
 
 9.5 
 
 .7823 
 
 .514 
 
 2.089 
 
 2.868 
 
 3.473 
 
 3.986 
 
 4.444 
 
 4.851 
 
 10. 
 
 .8189 
 
 .605 
 
 2.163 
 
 2.972 
 
 3.597 
 
 4.127 
 
 4.600 
 
 5.022 
 
 11. 
 
 .8911 
 
 .726 
 
 2.315 
 
 3.173 
 
 3.837 
 
 4.401 
 
 4.904 
 
 5.351 
 
 12. 
 
 .9625 
 
 .842 
 
 2.463 
 
 3.368 
 
 4.068 
 
 4.663 
 
 5.195 
 
 5.668 
 
 13. 
 
 1.033 
 
 .958 
 
 2.612 
 
 3.556 
 
 4.204 
 
 4.917 
 
 5.477 
 
 5.974 
 
 14. 
 
 1.102 
 
 2.063 
 
 2.751 
 
 3.738 
 
 4.508 
 
 5.163 
 
 5.750 
 
 6.271 
 
 15. 
 
 1.171 
 
 2.174 
 
 2.886 
 
 3.914 
 
 4.717 
 
 5.401 
 
 6.014 
 
 6.557 
 
 16. 
 
 1.238 
 
 2.279 
 
 3.017 
 
 4.086 
 
 4.921 
 
 5.633 
 
 6.271 
 
 6.836 
 
 17. 
 
 1.305 
 
 2.382 
 
 3.146 
 
 4.253 
 
 5.120 
 
 5.858 
 
 6.500 
 
 7.108 
 
 18. 
 
 1.372 
 
 2.483 
 
 3.271 
 
 4. 489 
 
 5.314 
 
 6.079 
 
 6.765 
 
 7.373 
 
 19. 
 
 1.437 
 
 2.582 
 
 3.394 
 
 4.576 
 
 5.502 
 
 6.293 
 
 7.003 
 
 7.631 
 
 20. 
 
 1.501 
 
 2.679 
 
 3.514 
 
 4.732 
 
 5.688 
 
 6.503 
 
 7.235 
 
 7.883 
 
 21. 
 
 1.569 
 
 2.774 
 
 3.632 
 
 4.885 
 
 5.869 
 
 6.709 
 
 7.463 
 
 8.131 
 
 22. 
 
 1.629 
 
 2.867 
 
 3.747 
 
 5.033 
 
 6.046 
 
 6.909 
 
 7.6S2 
 
 8.372 
 
 23. 
 
 1.692 
 
 2.951 
 
 3.861 
 
 5.180 
 
 6.228 
 
 7.105 
 
 7.903 
 
 8.607 
 
 24. 
 
 1.754 
 
 3.049 
 
 3.972 
 
 5.324 
 
 6.398 
 
 7.299 
 
 8.118 
 
 8.840 
 
 25. 
 
 1.816 
 
 3.139 
 
 4.0S1 
 
 5.465 
 
 6.556 
 
 7.488 
 
 8.327 
 
 9. 068 
 
 50. 
 
 3.223 
 
 5.038 
 
 6.385 
 
 8.420 
 
 10.04 
 
 11.44 
 
 12.70 
 
 13.81 
 
 100. 
 
 5.825 
 
 7.878 
 
 9.991 
 
 12.71 
 
 1 .10 
 
 17.15 
 
 19.03 
 
 20.68 
 
226 
 
 PRACTICAL HYDRAULICS. 
 
 TABLE 27. 
 
 Flow of Water per Second in Open Streams, the Coefficient 
 of Roughness of whose beds is n=.O35. 
 
 IT * 
 
 ** **t 
 
 ^ . jl Ij 
 
 ||?l 
 
 M 
 
 ill? 
 
 |f 
 
 fill 
 
 nil 
 
 *j2.^f | 
 
 $. 2. o to ' 
 
 *|*|| 
 
 Ur^i 
 
 irsp 
 
 
 
 NjB. 
 
 & p 
 
 t. - 
 
 <" ^ bo 
 
 .25 
 
 .3071 
 
 .3294 
 
 .3503 
 
 .3700 
 
 .4562 
 
 .5163 
 
 .8401 
 
 1.190 
 
 .3 
 
 .3603 
 
 .3365 
 
 .4108 
 
 .4339 
 
 . 5345 
 
 .6191 
 
 .9843 
 
 1.395 
 
 .4 
 
 .4621 
 
 .4954 
 
 .5266 
 
 .5561 
 
 .6346 
 
 .7927 
 
 1.259 
 
 1.784 
 
 .5 
 
 .5590 
 
 .5992 
 
 .6368 
 
 .6723 
 
 .8273 
 
 .9577 
 
 1.521 
 
 2.155 
 
 .6 
 
 .6517 
 
 .6983 
 
 .7419 
 
 .7832 
 
 .9634 
 
 1.115 v 
 
 1.770 
 
 2.507 
 
 .7 
 
 . 7408 
 
 .7937 
 
 .8241 
 
 .8901 
 
 1.094 
 
 1.266 
 
 2.010 
 
 2.846 
 
 .8 
 
 .8270 
 
 .8858 
 
 .9411 
 
 .9932 
 
 1.221 
 
 1.412 
 
 2.240 
 
 3.173 
 
 .9 
 
 .9107 
 
 .97f>3 
 
 1.036 
 
 1.097 
 
 1.343 
 
 1.555 
 
 2.464 
 
 3.488 
 
 1. 
 
 .9920 
 
 1.062 
 
 1.128 
 
 1.190 
 
 1.482 
 
 1.691 
 
 2.682 
 
 3.797 
 
 1.25 
 
 1.186 
 
 1.270 
 
 1.343 
 
 1.422 
 
 1.734 
 
 2.019 
 
 3.200 
 
 4.529 
 
 1.5 
 
 1.370 
 
 1.468 
 
 1.556 
 
 1.64/1 
 
 2.615 
 
 2.329 
 
 3.6S9 
 
 5.221 
 
 2. 
 
 1.672 
 
 1.330 
 
 1.943 
 
 2.049 
 
 2.512 
 
 2.903 
 
 4.596 
 
 6.502 
 
 2.5 
 
 2.026 
 
 2.167 
 
 2.299 
 
 2.424 
 
 2.972 
 
 3.433 
 
 5.430 
 
 7.632 
 
 3. 
 
 2.393 
 
 2.482 
 
 2.643 
 
 2.650 
 
 3.400 
 
 3.927 
 
 6.201 
 
 8.783 
 
 3.5 
 
 2.600 
 
 2.779 
 
 2.947 
 
 3.107 
 
 3.804 
 
 4.393 
 
 6.945 
 
 9.820 
 
 4. 
 
 2.865 
 
 3.061 
 
 3.247 
 
 3.422 
 
 4.188 
 
 4.836 
 
 7.642 
 
 10.80 
 
 4.5 
 
 3.118 
 
 3.331 
 
 3.543 
 
 3.723 
 
 4.556 
 
 5.270 
 
 8.307 
 
 11.74 
 
 5. 
 
 3.361 
 
 3.590 
 
 3.807 
 
 4.011 
 
 4.907 
 
 5.663 
 
 8.945 
 
 12.65 
 
 5.5 
 
 3.594 
 
 3.840 
 
 4.070 
 
 4.239 
 
 5.246 
 
 5.928 
 
 9.559 
 
 13.57 
 
 6. 
 
 3.820 
 
 4.081 
 
 4.325 
 
 4.557 
 
 5.573 
 
 6.429 
 
 10.15 
 
 14.35 
 
 6.5 
 
 4.039 
 
 4.314 
 
 4.572 
 
 4.817 
 
 5.889 
 
 6.793 
 
 10.72 
 
 15.16 
 
 7. 
 
 4.250 
 
 4.539 
 
 4.810 
 
 5.067 
 
 6.194 
 
 7.145 
 
 11.28 
 
 15.94 
 
 7.5 
 
 4.459 
 
 4.759 
 
 5.044 
 
 5.313 
 
 6.493 
 
 7.490 
 
 11.81 
 
 16.70 
 
 8. 
 
 4.657 
 
 4.973 
 
 5.269 
 
 5.551 
 
 6.783 
 
 7.823 
 
 12.34 
 
 17.44 
 
 8.5 
 
 4.853 
 
 5.181 
 
 5.490 
 
 5.783 
 
 7.065 
 
 8.148 
 
 12.85 
 
 18.16 
 
 9. 
 
 5.044 
 
 5.372 
 
 5.705 
 
 6.008 
 
 7.340 
 
 8.475 
 
 13.05 
 
 18.87 
 
 9.5 
 
 5.231 
 
 5.584 
 
 5.916 
 
 6.230 
 
 7.610 
 
 8.775 
 
 13.84 
 
 19.56 
 
 10. 
 
 5.413 
 
 5.778 
 
 6.121 
 
 6.447 
 
 7.855 
 
 9.079 
 
 14.32 
 
 20.23 
 
 11. 
 
 5.781 
 
 6.155 
 
 6.521 
 
 6.8B7 
 
 8.385 
 
 9. 668 
 
 15.24 
 
 
 12. 
 
 6.108 
 
 6.504 
 
 6.904 
 
 7.269 
 
 8 . 674 
 
 10.47 
 
 16.13 
 
 
 13. 
 
 6.438 
 
 6.740 
 
 7.227 
 
 7.661 
 
 9.538 
 
 10.78 
 
 17.03 
 
 
 14. 
 
 6.757 
 
 7.209 
 
 7.635 
 
 8.033 
 
 9.811 
 
 11.30 
 
 17.80 
 
 
 15. 
 
 7.064 
 
 7.537 
 
 7.982 
 
 8.403 
 
 10.25 
 
 11.82 
 
 18.62 
 
 
 16. 
 
 7.365 
 
 7.856 
 
 8.320 
 
 8.759 
 
 10.69 
 
 12.31 
 
 19.40 
 
 
 17. 
 
 7.657 
 
 8.167 
 
 8.643 
 
 9.104 
 
 11.11 
 
 12.80 
 
 20.16 
 
 
 18. 
 
 7.942 
 
 8.471 
 
 8.970 
 
 9.442 
 
 11.52 
 
 13.27 
 
 
 
 19. 
 
 8.031 
 
 8.763 
 
 9.231 
 
 9.769 
 
 11.92 
 
 13.71 
 
 
 
 20. 
 
 8.489 
 
 8.849 
 
 9.58fl 
 
 10.09 
 
 12.30 
 
 14.18 
 
 
 
 21. 
 
 8.755 
 
 9 125 
 
 9.886 
 
 10.4* 
 
 12.69 
 
 14.61 
 
 
 
 22. 
 
 9.015 
 
 9.614 
 
 10.18 
 
 10.71 
 
 13.06 
 
 15.05 
 
 
 
 23. 
 
 9.263 
 
 9.884 
 
 10.46 
 
 11.01 
 
 13.43 
 
 15.47 
 
 
 
 24. 
 
 9.518 
 
 10.15 
 
 10.74 
 
 11.31 
 
 13.78 
 
 15.88 
 
 
 
 25. 
 
 9.763 
 
 10.41 
 
 11.02 
 
 11.60 
 
 14.14 
 
 16.28 
 
 
 
 50. 
 
 14.86 
 
 15.84 
 
 16.75 
 
 17.62 
 
 21.43 
 
 
 
 
PEACTICAL HYDRAULICS. 
 
 227 
 
 TABLE 28. 
 
 Dimensions of Water Ways Corresponding to Their Given 
 Hydraulic Mean Depths. 
 
 M a Square. 
 
 Rectangle. Triangle. 
 
 Semi-circle. 
 
 If 
 
 
 
 
 
 V "7 
 
 \ 7 
 
 -TI 
 
 II 5- 
 
 1:1 
 
 
 112 
 
 \oy 
 
 V_x 
 
 II J 
 
 bide. 
 
 Area. 
 
 D'ptfc 
 
 Width 
 
 Area. 
 
 Side. 
 
 Area. 
 
 Diam'tr 
 
 Arei. 
 
 i 
 
 Feet. 
 
 Sq. Ft. 
 
 Ft. 
 
 Ft. 
 
 Sq. Ft. 
 
 Feet. 
 
 Sq. Ft. 
 
 Feet. 
 
 Sq. Ft. 
 
 .25 
 
 .75 
 
 .563 
 
 
 .5 
 
 1.0 
 
 .5 
 
 1. 
 
 .5 
 
 1. 
 
 .393 
 
 .3 
 
 .9 
 
 .81 
 
 
 .(i 
 
 1.2 
 
 .72 
 
 1.2 
 
 .72 
 
 1.2 
 
 .565 
 
 .4 
 
 1.2 
 
 1.44 
 
 
 .8 
 
 1.6 
 
 1.28 
 
 1.6 
 
 1.28 
 
 1.6 
 
 1.005 
 
 .5 
 
 1.5 
 
 2.25 
 
 1 
 
 .0 
 
 2.0 
 
 2.0 
 
 2.0 
 
 2.00 
 
 2. 
 
 1.571 
 
 .6 
 
 1.8 
 
 3.24 
 
 1 
 
 .'2 
 
 2.4 
 
 2.88 
 
 .2.4 
 
 2.88 
 
 2.4 
 
 2.262 
 
 .7 
 
 2.1 
 
 4.41 
 
 ] 
 
 .\ 
 
 2.8 
 
 3.92 
 
 2.'8 
 
 3.92 
 
 2.8 
 
 3.079 
 
 .8 
 
 2.4 
 
 5.76 
 
 ] 
 
 .(i 
 
 3.2 
 
 5.12 
 
 3.2 
 
 5.12 
 
 3.2 
 
 4.021 
 
 .9 
 
 2.7 
 
 7.29 
 
 ] 
 
 .S 
 
 3.6 
 
 '6.48 
 
 3.6 
 
 6.48 
 
 3.6 
 
 5.089 
 
 1. 
 
 3. 
 
 9.00 
 
 2.0 
 
 4.0 
 
 8.00 
 
 4. 
 
 8.00 
 
 4. 
 
 6.283 
 
 1.25 
 
 3.75 
 
 14.06 
 
 2.5 
 
 5.0 
 
 12.5 
 
 5. 
 
 12.5 
 
 5. 
 
 9.818 
 
 1.5 
 
 4.5 
 
 20.25 
 
 3.0 
 
 6.0 
 
 18.0 
 
 6. 
 
 18.0 
 
 6. 
 
 14.137 
 
 2. 
 
 6. 
 
 36. 
 
 4.0 
 
 8.0 
 
 32.0 
 
 8. 
 
 32.0 
 
 8. 
 
 25.133 
 
 2.5 
 
 7.5 
 
 56.25 
 
 5:0 
 
 10.0 
 
 50.0 
 
 10. 
 
 50.0 
 
 10. 
 
 39.27 
 
 3. 
 
 9. 
 
 81. 
 
 6.0 
 
 12.0 
 
 72.0 
 
 12. 
 
 72.0 
 
 12. 
 
 56.549 
 
 3.5 
 
 10.5 
 
 110.25 
 
 7.0 
 
 14.0 
 
 98.0 
 
 14. 
 
 98. 
 
 14. 
 
 76.696 
 
 4. 
 
 12. 
 
 144.00 
 
 * 
 
 
 16.0 
 
 128. 
 
 10. 
 
 128. 
 
 16. 
 
 100.53 
 
 4.5 
 
 13.5 
 
 182.25 
 
 9. 
 
 18.0 
 
 162. 
 
 IS. 
 
 162. 
 
 18. 
 
 127.23 
 
 5. 
 
 15. 
 
 225.00 
 
 10. 
 
 20.0 
 
 200. 
 
 20. 
 
 200. 
 
 20. 
 
 157.08 
 
 5.5 
 
 16.5 
 
 272.25 
 
 11. 
 
 22.0 
 
 242. 
 
 22. 
 
 242. 
 
 22. 
 
 190.07 
 
 6. 
 
 18. 
 
 324.00 
 
 12. 
 
 24.0 
 
 288. 
 
 24. 
 
 288. 
 
 24^ 
 
 226.2 
 
 6.5 
 
 19.5 
 
 380.25 
 
 13. 
 
 26.0 
 
 338. 
 
 26. 
 
 338. 
 
 26. 
 
 265.5 
 
 7. 
 
 21. 
 
 441.00 
 
 14. 
 
 28.0 
 
 392. 
 
 28. 
 
 392. 
 
 28. 
 
 307.0 
 
 7.5 
 
 22.5 
 
 506.25 
 
 15. 
 
 30.0 
 
 450. 
 
 30. 
 
 450. 
 
 30. 
 
 353.4 
 
 8. 
 
 24. 
 
 576.00 
 
 16. 
 
 32.0 
 
 512. 
 
 32. 
 
 512. 
 
 32. 
 
 402.1 
 
 9. 
 
 27. 
 
 729.00 
 
 18. 
 
 36.0 
 
 648. 
 
 36. 
 
 648. 
 
 36. 
 
 508.9 
 
 10. 
 
 30. 
 
 900.00 
 
 20. 
 
 40.0 
 
 800. 
 
 40. 
 
 800. 
 
 40. 
 
 628.3 
 
228 
 
 PEACTICAL HYDKAULICS. 
 
 1 
 
 00 
 
 l\ 
 
 II 
 
 Hydraulic 
 Mean Depth, 
 
 r- r- IH r- r- <M co Tj i co J> t a e> o -H < 
 
 i S O5 00 kO t- CO CO 00 00 Tt< S-l 'O * L^ * 1^ -* i-( OO 
 1H '~ ( rtrHrH5-]^M^COCO-*lO< 
 
 r- i- i- i- i- <M eo rj us CD t-. oo as o r-i 01 co * in co oo 
 
PRACTICAL HYDKAULICS. 
 
 229 
 
 TABLE. 29. 
 
 Relations of Depth, Base and Slope of a Bank of a Trap- 
 ezoidal Canal. 
 
 fatio of 
 Depth 
 
 1 
 
 m 
 
 1 x m 2 )* 
 
 Ratio of 
 Deoth 
 
 1 
 
 m 
 
 (1+m 2 )* 
 
 to Base. 
 
 Depth. 
 
 Base. 
 
 S'ope. 
 
 to Base. 
 
 Depth. 
 
 Base. 
 
 Slope. 
 
 5:1 
 
 1 
 
 .2000 
 
 .0198 
 
 19:4 
 
 1 
 
 .2105 
 
 1.0219 
 
 24:5 
 
 1 
 
 .2083 
 
 .0215 
 
 9:2 
 
 1 
 
 .2222 
 
 1.0244 
 
 23:5 
 
 1 
 
 .2174 
 
 .0234 
 
 17:4 
 
 1 
 
 .2353 
 
 1.0272 
 
 22:5 
 
 1 
 
 .2273 
 
 .0255 
 
 15:4 
 
 1 
 
 .2667 
 
 1.0349 
 
 21:5 
 
 1 
 
 .2381 
 
 .0280 
 
 7:2 
 
 1 
 
 .2856 
 
 1.0400 
 
 4:1 
 
 1 
 
 .2500 
 
 .0306 
 
 13:4 
 
 1 
 
 .3077 
 
 1.0463 
 
 19:5 
 
 1 
 
 .2632 
 
 .0364 
 
 11:4 
 
 1 
 
 .3636 
 
 1.0641 
 
 18:5 
 
 1 
 
 .2778 
 
 .0379 
 
 5:2 
 
 1 
 
 .4000 
 
 1.0770 
 
 17:5 
 
 1 
 
 .2941 
 
 .0423 , 
 
 9:4 
 
 1 
 
 .4444 
 
 1.0940 
 
 10:5 
 
 1 
 
 .3125 
 
 .0477 
 
 7:4 
 
 1 
 
 .5714 
 
 1.1517 
 
 3:1 
 
 1 
 
 .3333 
 
 .0541 
 
 3:2 
 
 1 
 
 .6667 
 
 1 . 2046 
 
 14:5 
 
 1 
 
 .3571 
 
 .0619 
 
 5:4 
 
 1 
 
 .8000 
 
 1.2806 
 
 13:5 
 
 1 
 
 .3846 
 
 .0714 
 
 3:4 
 
 1 
 
 1.3333 
 
 1.6667 
 
 12:5 
 
 1 
 
 .4167 
 
 .0833 
 
 1:2 
 
 1 
 
 2.0000 
 
 2.2361 
 
 11:5 
 
 1 
 
 .4545 
 
 .0985 
 
 1:4 
 
 1 
 
 4.0000 
 
 4.1231 
 
 2:1 
 
 1 
 
 .5000 
 
 .1180 
 
 14:3 
 
 1 
 
 .2143- 
 
 1-0227 
 
 9:5 
 
 1 
 
 .5556 
 
 .1439 
 
 13:3 
 
 1 
 
 .2308 
 
 1.0263 
 
 8:5 
 
 1 
 
 .6250 
 
 .1793 
 
 11:3 
 
 1 
 
 .2727 
 
 1.0365 
 
 7:5 
 
 1 
 
 .7143 
 
 .2575 
 
 10:3 
 
 1 
 
 .3000 
 
 1.0440 
 
 6:5 
 
 1 
 
 .8333 
 
 .3017 
 
 8:3 
 
 1 
 
 .3750 
 
 1.0680 
 
 1:1 
 
 1 
 
 l.oro 
 
 .4142 
 
 7:3 
 
 1 
 
 .4286 
 
 1.0880 
 
 4:5 
 
 1 
 
 1.250 
 
 .6008 
 
 5:3 
 
 1 
 
 .6000 
 
 1.1662 
 
 3:5 
 
 1 
 
 1.667 
 
 .9436 
 
 4:3 
 
 1 
 
 .7500 
 
 1.2500 
 
 2:5 
 
 1 
 
 2.500 
 
 2.6926 
 
 2:3 
 
 1 
 
 1.5000 
 
 1.8028 
 
 1:5 
 
 1 
 
 5.000 
 
 5.0990 
 
 1:3 
 
 1 
 
 3.0000 
 
 3.1622 
 
 5:24 
 
 1 
 
 4.8000 
 
 4.9031 
 
 4:19 
 
 1 
 
 4.7500 
 
 4.8541 
 
 5:23 
 
 1 
 
 4.6000 
 
 4.7074 
 
 4:18 
 
 1 
 
 4.5000 
 
 4.6098 
 
 5:22 
 
 1 
 
 4.4000 
 
 4.5122 
 
 4:17 
 
 1 
 
 4.2500 
 
 4.3660 
 
 5:21 
 
 1 
 
 4.2000 
 
 4.3174 
 
 4:15 
 
 1 
 
 3.7500 
 
 3.8810 
 
 5:19 
 
 1 
 
 3.8000 
 
 3.9295 
 
 4:14 
 
 1 
 
 3.5000 
 
 3.6401 
 
 5:18 
 
 1 
 
 3.6000 
 
 3.7363 
 
 4:13 
 
 1 
 
 3.2500 
 
 3.4004 
 
 5:17 
 
 1 
 
 3.4000 
 
 3.5440 
 
 4:11 
 
 1 
 
 2.7500 
 
 2.9262 
 
 5:16 
 
 1 
 
 3.2000 
 
 3.3526 
 
 4:9 
 
 1 
 
 2.2500 
 
 2.4622 
 
 5:14 
 
 1 
 
 2.8000 
 
 2.9732 
 
 4:7 
 
 1 
 
 1.7500 
 
 2.0156 
 
 5:13 
 
 1 
 
 2.6000 
 
 2.7851 
 
 4:6 
 
 1 
 
 1.5000 
 
 1.8028 
 
 5:12 
 
 1 
 
 2.4000 
 
 2.6000 
 
 3:14 
 
 1 
 
 4.6667 
 
 4.7726 
 
 5:11 
 
 1 
 
 2.2000 
 
 2.4161 
 
 3:13 
 
 1 
 
 4.3333 
 
 4.4472 
 
 5:9 
 
 1 
 
 1.8000 
 
 2.0591 
 
 3:11 
 
 1 
 
 3.6666 
 
 3.8006 
 
 5:8 
 
 1 
 
 1.6000 
 
 
 3:10* 
 
 1 
 
 3.3333 
 
 3.4801 
 
 5:7 
 
 1 
 
 1.4000 
 
 l!7205 
 
 3:8 
 
 1 
 
 2.6667 
 
 2.8480 
 
 5:6 
 
 1 
 
 1.2000 
 
 1.5621 
 
 3:7 
 
 1 
 
 2.3333 
 
 2.5386 
 
230 PRACTICAL HYDRAULICS. 
 
 PRACTICAL APPLICATION OF TABLES 27, 28 AND 29, 
 
 TO DETERMINE THE VELOCITY AND DISCHARGE OF AN 
 OPEN STREAM OF WATER. 
 
 Rule 56. From the given dimensions of the stream 
 find by Table 28 or Table 29, according to the condi- 
 tions, the hydraulic mean depth. Turn then to Table 
 27 for the given coefficient n, for roughness of bed. 
 In this table, opposite the ''hydraulic mean depth" as 
 determined, in the column headed by the given fall 
 per mile, will be found the velocity sought. Multiply 
 the velocity so found by the area of the cross section 
 of the stream for the discharge. 
 
 Ex. 102. The side of a square flume of unplaned 
 plank is 3 feet, the fall per mile-4.752 feet (s=.0009), 
 and the coefficient of roughness of its bed, ?i=.0l2. 
 (See Table 26.) What, per second, is the velocity, 
 and what the discharge? 
 
 Col. In Table 28, find in "side" column for a 
 square flume or canal. 3 feet; opposite which, in "area" 
 column, is found 9 square feet, and opposite which, in 
 hydraulic mean depth column, is found 1. Turning 
 now to Table 27, computed for coefficient of rough- 
 ness of bed, 7i^=.012, find in hydraulic mean depth 
 column 1; opposite which, in velocity column, for the 
 given fall, F=4.752, (s=.0009) will be found the ve- 
 locity sought, viz. : . 
 
 v= 3. 82 feet. 
 
PRACTICAL HYDRAULICS. 231 
 
 Then g .=3. 82x9= 34. 38 cubic feet. Ana. 
 
 Ex. 103. The depth of a rectangular canal "being 
 5 feet, the width 10 feet, the fall per mile 2.64 feet, 
 F=2.64, s=.0005, and the coefficient of roughness 
 of the bed n-=.0l7 (see Table 26), what will be 
 the velocity of the water, and what the discharge in 
 cubic feet? 
 
 Cal. Find in Table 28, under ' 'rectangle" in 
 "depth" and "width" columns, the given depth and 
 width 5 and 10 feet; opposite which, in "area" col- 
 umn, will be found 50 square feet; and in the "hy- 
 draulic mean depth column," will be found 2.5. 
 
 Turning now to Table 27, for n=.0~Ll, find in "hy- 
 draulic mean depth column" 2.5; opposite which, in 
 velocity column for the given fall, F=2.64, will be 
 found the velocity sought, viz. : 
 
 v---= 3.605 feet. 
 
 Then g=3. 605X50=180.25 cubic feet. Ana. 
 
 Ex. 104. The side (wetted) of a V (vee) flume of 
 unplaned plank, being 2.4 feet, the fall per mile 26.4 
 feet, and the coefficient for roughness of the bed, ti= 
 .012 (see Table 26), what is the velocity of the 
 water per second, and what the discharge in cubic 
 feet? 
 
 Col. Find in Table 28, under triangle in side col- 
 umn, the given side 2.4 feet; opposite which, in area 
 column, is found 2.88 square feet, and in ' 'hydraulic 
 mean depth column," .6. 
 
 Turning now to Table 27, for 7i^=.0l2, find in "hy- 
 
232 PEACTICAL HYDRAULICS. 
 
 draulic mean depth" column, .6; opposite which, in ve- 
 locity . column for the given fall, F=26.4, will be 
 found the velocity sought, viz.: 
 
 v=6. 338 feet. 
 
 Then =6.338X 2.88=18.25 cubic feet. Ans. 
 
 Ex. 105. The depth of a regular semi-hexagonal 
 canal in earth, being 3 feet, the fall per mile F=5.28 
 feet (s=.00l), and the coefficient for roughness of bed 
 n.Q2o (see Table 26), what is the velocity of 
 the water and what the discharge? 
 
 Col. Find in Table 28, under semi-hexagon in 
 depth column, 3 feet, opposite which, in area column, 
 is found 15.588 square feet, and in hydraulic mean 
 depth column 1.5. 
 
 Turning now to Table 27, for n=.025, find in hy- 
 draulic mean depth column 1.5, opposite which, in 
 velocity column for the given fall, F=5.28 feet, will 
 be found the velocity sought, viz. : 
 
 w=2. 383 feet. 
 
 Then 0=15. 588X2. 383=37.15 cubic feeb. Ans. 
 
 Ex. 106. In a trapezoidal canal (bottom-slope of 
 bank), the angle of slope of bank is 45, the depth 
 4.394 feet, the fall per mile F=7.92 feet, and the co- 
 efficient for roughness of bed n= .01 7 (see Table 26); 
 what is the velocity and what the discharge of water 
 per second? 
 
 Gal. Find in Table 28, under trapezoid, with bank 
 slope of 45, in depth column, 4.394 feet, the given 
 
PEACTICAL HYDRAULICS. 23b 
 
 depth ; opposite which, in area column, is found 
 46.606 square feet, and in hydraulic mean depth col- 
 umn, 2.5. 
 
 Turning to Table 27, find in hydraulic mean depth 
 column 2.5, opposite which, in velocity column for 
 the given fall, F=7.92 feet, is found the velocity 
 
 sought, viz. : 
 
 v=6.26 feet. 
 
 Then 5=46.606x6.26=291.75 cubic feet. Ans. 
 
 Ex. 107. In a trapezoidal canal, in which the bot- 
 tom is equal a side, and the ratio of the depth to the 
 base of bank is as 2:1, the depth of water is 2.591 
 feet, the fall per mile F=10.56 feet (s=.002) and the 
 coefficient for roughness of bed, 7i=.025 (see Table 26), 
 what, per second, is the velocity and what the dis- 
 charge? 
 
 Col. Find in Table 28, under trapezoid 2*. 1, in 
 depth column, the given depth 2.591 feet; opposite 
 which in area column is found 10.86 square feet, and 
 in hydraulic mean depth column 1.25. 
 
 Turning now to Table 27 for n= .025 find in hy- 
 draulic mean depth column 1.25; opposite which in 
 velocity column for the given fall F=10.56 is found 
 the velocity sought, viz. : 
 
 v= 2.943 feet. 
 
 Then 5=10.86X2.943=31.96 ciibic feet. Ans. 
 
 Ex. 108. In a trapezoid canal, in which the bot- 
 tom is equal to a side, and the ratio of depth to the 
 base of the bank is as 1:2, the depth of water" is 
 
234 PRACTICAL HYDRAULICS. 
 
 5.543 feet, the fall per mile 1.056 feet, and the coeffi- 
 cient for the roughness of the bed, 7i=.035 (see 
 Table 26), what is the velocity, and what the dis- 
 charge per second ? - 
 
 Cat. Find in Table 28, under trapezoid 1-2, in 
 depth column the given depth 5.543 feet; opposite 
 which in area column is found 130.04 square feet, 
 and hydraulic mean depth column 3.5. Turning now 
 to Table 27, find in hydraulic mean depth column 3.5; 
 opposite which in velocity column for the given fall, 
 F=1.056, will be found the velocity sought, viz. : 
 
 v= 1.392 feet. 
 
 Then 9-l30.04Xl.392-=18l.02 cubic feet. Ans. 
 
 Ex. 109. The diameter of a semi-circular canal 
 being 8 feet, the fall per mile 26.4 feet, and the coeffi- 
 cient for roughness of bed 7i=.0l2, (see Table 26), 
 what is the velocity per second and what the dis- 
 charge ? 
 
 'Gal. Find in Table 28, under semi-circle, in diam- 
 eter column 8 feet, the given diameter; opposite which 
 in area column is found 25.133 square feet, and in 
 hydraulic mean depth column, 2. 
 
 Turning now to Table 27, findln hydraulic mean 
 depth column 2; opposite which in velocity column 
 for the fijiven fall, F=26.4 feet, is found the velocity 
 sought, viz. : 
 
 <y--=14.22 feet. 
 
 Then =25.133X14.22 357.39 cubic feet. Ans. 
 'Remark. It will be observed that Table 27, is 
 
PBACTIGJLL HYDBAULICSi 235 
 
 equally well adapted to finding the flow of water in 
 circular pipes as in semi-circular canals. For the hy- 
 draulic mean depth is the same in each. Thus in Ex- 
 ample 108, were it required to determine the velocity 
 of water in a circular pipe running full, the only 
 change required in the calculation would be to double 
 the area, whence would occur a corresponding change 
 in the result, as follows: 
 
 357.39X2=714.78 cubic feet. 
 
 In case of foul pipes it will be better to employ 
 Table 27, rather than Table 17, computed for clean 
 iron pipes, the coefficient of roughness for whose walls 
 as shown is 71^.011. 
 
 Ex. 110. The observed data of a canal are as fol- 
 lows: 
 
 Width of bottom, 15 feet. 
 
 Depth of water, 4.5 feet. 
 
 Ratio of depth to base, 2:5. 
 
 Fall per mile, 3.168 feet. 
 
 Coefficient of roughness of bed .017. 
 
 What is the velocity of flow per second, and what 
 the discharge ? 
 
 Gal. Find in Table 29 the given ratio of depth to 
 base, viz.: 2:5; opposite which in columns of depth, 
 base and slope, computed for a depth of unity are 
 found 1, 2.5 and 2.6926. Multiplying each by the 
 given depth, there results 4.5, 11.25 and 12.1167, the 
 depth, base and slope of bank. The wet perimeter is 
 
236 PEACTICAL HYDRAULICS. 
 
 equal to the sum of the bottom and twice the slope of 
 bank: 
 
 p= 15 + 12.1167X2=39.2334 feet. 
 
 The mean width of the stream is equal to the sum 
 of the bottom and of the base of the bank: 
 
 ^=15 + 11.25=26.25. 
 
 The area of cross -section of stream is equal to the 
 product of the mean width and depth : 
 
 a=26.25X 4.5 118.125 square feet. 
 
 The hydraulic mean depth is equal to the quotient 
 arising from dividing the area of cross-section by the 
 wet perimeter: 
 
 r= JL=H8.125-*- 39.2334 -=3.01. 
 
 Turning now to Table 27 for the given coefficient 
 for roughness of bed, 7i=.0l7, find in hydraulic mean 
 depth column 3, the nearest approximate to that de- 
 termined, viz.: 3.01; opposite which in velocity column 
 for the given fall per mile, F=3.168, is found the 
 velocity sought, viz.: 
 
 v=-4.458 feet. 
 
 Then g=-118.125X 4.458 = 526.6 cubic feet. Am. 
 Ex. 111. The data for the Sacramento river being 
 as follows, viz.: 
 
 Fall per mile .528 feet. 
 
 Depth, 25 feet. 
 
 Width of bottom, 453.33. 
 
 Ratio of depth to base of bank, 5'- 12. 
 
PEACTICAL HYDRAULICS. 237 
 
 Coefficient of roughness of bed, .025. 
 
 What will be the velocity per second, and what the 
 discharge ? 
 
 Gal. Find in Table 29 the given ratio, 5 : 12, oppo- 
 site which are found 1, 2.4 and 2.6, multiplying each 
 by the given depth, 25 feet, there results 25, 60 and 
 65, the deptli, base and slope. 
 
 Then wet perimeter =453.33 + 65X2= 583.33 feet. 
 
 Mean width=453.33 + 60=513.33. 
 
 Sectional area-513.33 + 25^12,833.25 square feet, 
 
 Hydraulic mean depth=12,833. 25-^ 583.33=22. 
 
 By Table 27, the velocity for the hydraulic mean 
 depth, 22 in velocity column for the given fall of .528 
 feet is: 
 
 ?;=4.862 feet. 
 
 Whence, 2=12,833.25X4.862=62395.26 cubic 
 feet. Ans. 
 
 INTERPOLATION. 
 
 For the purposes of interpolation where the ex- 
 tremes are not far apart, in Table 27, the velocities, 
 without any considerable error in practice may be 
 assumed proportionate, either to the sines of slope on 
 one hand, or to the hydraulic mean depths on the 
 other. 
 
 Ex. 112. The coefficient of roughness of bed being 
 % .012, the hydraulic mean depth .5, the extreme 
 sines of slope s=.0015, and 8= .005, what, in a regu 
 
238 PRACTICAL HYDRAULICS. 
 
 lar arithmetical series will be the six interpolated 
 velocities ? 
 
 Cal.By Table 27, for 7^.012, and for the given 
 hydraulic mean depths the velocities due the given 
 slopes are 3.134 and 5.625 feet per second. Then 
 (5.625 3.134)-^7=- .356 common difference. 
 
 Whence, the interpolated velocities will be 3.490, 
 3.846, 4.202, 4.558, 4.914 and 5.270 feet. Ans. 
 
 Ex. 113. The coefficient of roughness of bed being 
 7i=.012, the given sine of slope s=.0007, the ex- 
 tremes of hydraulic mean depths 1 and 2, what, in a 
 regular arithmetical series, will be the three inter- 
 polated velocities? 
 
 Cal.By Table 27, for ^=.012, the velocities due 
 the given hydraulic mean depths 1 and 2, are 3.360 
 and 5.302 feet. Then (5.302 3.360)-^4=.4855 com- 
 mon difference; whence the interpolated velocities will 
 be 3.845, 4.330, and 4.816 feet. Ans. 
 
 MEAN VELOCITY. 
 
 To find the mean velocity of an open stream of 
 water, various devices, as tight tin tubes, loaded each 
 at one end, so as to float vertically in still water, and as 
 nearly so in streams as the current will permit the 
 Pitot tube, patent logs, etc., are employed. Wooden 
 floats, of nearly the specific gravity of water, are, 
 however, mostly used in common practice. By these 
 
PRACTICAL HYDRAULICS. 239 
 
 the surface velocity is taken; thence the mean ve- 
 locity is determined by calculation based on experi- 
 mental data. 
 
 Having cliosen a straight section of a stream free 
 as possible, to be found from eddies, roughness and 
 foulness of bottom, and having divided the stream 
 into sections parallel with its course, cast into it 
 wooden floats, note the time taken by a float in 
 each section to pass through a given distance, as 100 
 feet, and thence determine the surface sectional ve- 
 locity per second. Divide the sum of the several ve- 
 locities of the floats by their numbers; the result will 
 be the mean surface velocity of the stream per second ; 
 whence, the mean velocity for the mean depth of the 
 entire cross section can be calculated as above stated. 
 
 CENTRAL SURFACE AND CORRESPONDING MEAN 
 VELOCITY. 
 
 Having made se\$eral trials with floats, as above de- 
 scribed, let the greatest velocity, well established by 
 any one of them, be taken as the central surface ve- 
 locity. For the central surface velocities, from five- 
 tenths ( .5) of a foot to six (6) feet, the corresponding 
 mean velocities have, by the aid of experimental data, 
 and the following empirical formula of Prony, viz. : 
 
 ' <*> 
 
240 
 
 PRACTICAL HYDEAULICS. 
 
 been computed, and the results arranged in Table 30. 
 In Eq. (286), v denotes the mean velocity, and V 
 the central surface velocity of a stream of water. 
 
 TABLE 30. 
 
 Central Surface and Corresponding Mean Velocities of 
 Streams. 
 
 Central Surface 
 Velocity. 
 Feet. 
 
 Mean 
 Velocity. 
 Feet. 
 
 Central Surface 
 Velocity. 
 Feet. 
 
 Mean 
 Velocity. 
 Fett. 
 
 .5 
 
 .382 
 
 3.5 
 
 2.852 
 
 1.0 
 
 .774 
 
 4.0 
 
 3.284 
 
 1.5 
 
 1.174 
 
 4.5 
 
 3.721 
 
 2.0 
 
 1.584 
 
 5.0 
 
 4.165 
 
 2.5 
 
 2.000 
 
 5.5 
 
 4.609 
 
 3.0 
 
 2.424 
 
 6.0 
 
 5.058 
 
 Ex. 114. What is the quantity of flow in a stream 
 in which the cross section is 50 square feet and the 
 central surface velocity 3 feet per second? 
 
 Gal. In Table 30, opposite the given central sur- 
 face velocity 3 feet, find in mean velocity column 
 2.424 feet. Then 
 
 2.424X50=121.2 cubic feet per second. Ans. 
 
 Rough Approximate. A rough approximate is 
 readily found by taking one-half the product of the 
 surface width, central depth, and central velocity of 
 a stream. 
 
 This rough approximate rule is based on the as- 
 
PRACTICAL HYDRAULICS. 241 
 
 sumption that the cross section of stream is parabolic, 
 and the mean velocity equal to three-fourths (.75) of 
 the central surface velocity. 
 
 Ex. 115. The surface width of a stream is 25 feet, 
 the central depth 4 feet, and the central surface ve- 
 locity 1.5 feet, what is the flow per second? 
 
 Gal 25X4X1.5-^-2=75 cubic feet. Ans. 
 
 QUANTITY OF WATER REQUIRED FOR VARIOUS MINING 
 PURPOSES. 
 
 Hydraulic Mining. Hydraulic mining, properly, 
 comprises all classes of mining in which the metallic 
 substance sought is separated from its earthy mass or. 
 matrix by means of water. The term, however, as 
 employed in California, is, for the most part, restricted 
 to that class of mining in which a stream of water is 
 projected under great pressure from a nozzle against 
 a deep gravel deposit or earthy formation for the pur- 
 poses of disintegrating the mass, thence freeing the 
 gold and carrying off the debris. The relation of the 
 quantity of water employed to that of material re- 
 moved varies in different mines and in different parts 
 of the same mine. 
 
 Duty of an Inch of Water. This phrase involves in 
 its meaning the work of disintegration; but as the 
 projecting head is variable from 50 to 350 feet and 
 upward, the phrase seems to refer chiefly to that por- 
 tion of the work performed by the water in carrying 
 off the debris in a sluice, the grade of which is usually 
 
242 PBACTICAL HYDRAULICS. 
 
 6 inches per 12 feet. Experience shows that the duty 
 of a 24-hour miner's inch, under a 7-inch head, equiva- 
 lent, as shown by Table 8, to 2,230 cubic- feet flow in 
 twenty-four hours, is in the lower portions of certain 
 mines as follows: 
 
 Cubic Yards. 
 
 North Bloomfield Mine 3.5 
 
 Milton Mine 2.4 
 
 Excelsior Mine 2.0 
 
 Gold Run Mine 3.5 
 
 In the upper portions of the same mines the duty 
 of the miner's inch was much greater, say 5 to 10 
 cubic yards. 
 
 Thus, at the Gold Run mine, for six years to No- 
 vember 1, 1881, 4,389,791 cubic yards were worked 
 with 1,124,367 miner's inches of water; whence the 
 duty of per inch was 3.9 cubic yards. 
 
 In the State Engineer's report to the legislature of 
 the State of California, 1880, the estimated inch duty 
 is as follows, viz.: 
 
 Cubic Yards. 
 
 Yuba River Mines 3.5 
 
 Bear River Mines 3.0 
 
 American River Mines 4.5 
 
 One instance is brought to the attention of the 
 writer showing the inch duty to have been 19 cubic 
 yards. 
 
 Drift Mining. Drift mining consists in excavating 
 the lower material of a gravel mine by hand, raising 
 it through a shaft to the surface, or carrying it by 
 wheelbarrows and cars through a tunnel to a dump, 
 whence it is shoveled or piped into a sluice to be freed 
 of its gold, and thereupon carried off as debris. A 
 
PRACTICAL HYDRAULICS. 243 
 
 the larger cobble, bowlders and barren blocks of rock 
 are usually left in a drift mine, and the material 
 broken smaller than in hydraulic mining, a corres- 
 pondingly less quantity of water is required for work- 
 ing a cubic yard. 
 
 The duty of a 24-hour inch (2,230 cubic feet) in 
 this class of mining varies according to the character 
 of the gravel, whether hard and cemented, clayey or 
 sandy, from 3 to 20 cubic yards. 
 
 Quartz Mining. The contents of one ton of quartz, 
 in its normal condition in the lode, is estimated at 
 13 cubic feet, and at 20 cubic feet when the quartz is 
 broken, as it usually comes from the mine. Adopting 
 the lode measurement it is seen that a cubic yard of 
 quartz is 27-*- 13=2.08 tons nearly. 
 
 Experience shows that the duty of a miner's inch is 
 as follows: 
 
 Duty of a miner's inch (under 4-inch pressure) in 
 the reduction and amalgamation of silver ores in a 
 "stamp silver mill," Nevada, 3.25 cubic yards or 6.76 
 tons; in the reduction and amalgamation by riffles, or 
 copper plate, in "stamp gold mill," California, 5.T8 
 cubic yards or 12 tons. 
 
 Duty of miner's inch (under 7-inch pressure) in the 
 former case (silver) 4.3 cubic yards, or 8.93 tons; in 
 the latter case (gold) 6.65 cubic yards, or 15.88 tons. 
 
 The volume of water to that of ore is, in working 
 silver ores, Nevada, 19.5 to 1; in working gold ores, 
 California, 11.1 to 1; in working copper ores, Lake 
 Superior, 20 to 1. 
 
244 PRACTICAL HYDRAULICS. 
 
 QUANTITY OF WATER REQUIRED FOR PURPOSES OF 
 IRRIGATION. 
 
 As the area of land is usually expressed in denomi- 
 nation of acres, a convenient unit of measure for irri- 
 gating purposes is that quantity of water which will 
 cover one acre one inch deep. This quantity is 3,630 
 cubic feet. 
 
 The total depth of irrigation, as practiced in Cali- 
 fornia, varies for different soils and products from two 
 to five feet. 
 
 Ex. 116. It is proposed to irrigate 1000 acres of 
 land, 50 inches in depth, in 100 days, by means of a 
 canal whose fall per mile is to be 1.056 feet (s=.0002), 
 coefficient of roughness of bed 7i=.0l7, bottom width 
 equal to slant width of side, and ratio of depth to base 
 of bank as 1 2, what will be the dimensions of the 
 canal? 
 
 CaL 3630X50-^-100=:1815 cubic feet; 1815 X 
 1000 1815000 cubic feet per day; 1815000^86400- 
 21.007 cubic feet per second. 
 
 Assume, by way of trial, the hydraulic mean depth 
 to be 1.25. Then, in Table 27, for =.017, the ve- 
 locity for hydraulic mean depth 1.25 is 1.386 feet per 
 second; whence, 21.007-^- 1.386=15.16 square feet, 
 area of cross section of canal. 
 
 By Table 28, for hydraulic mean depth 1.25, the 
 sectional area is 16.59 square feet, under trapezoid 1 ' 2. 
 This approximate is sufficiently near to meet the re- 
 quirements of practice. The dimensions of the canal 
 
PRACTICAL HYDBAULICS. 245 
 
 then, as per Table 28, are: side =bottom 4.426 feet; 
 depth=1.980 feet. Ans. 
 
 If greater accuracy be required, proceed as in the 
 solution of Ex. 99. 
 
 Ex. 117. How many acres can be irrigated 40 
 inches in depth in 75 days, by means of a semi- 
 hexagonal canal five feet deep, the fall per mile being 
 1.584 feet (s=.0003), and the coefficient for roughness 
 of bed being n = .025? 
 
 Cal. By Table 28, it is seen that the hydraulic 
 mean depth and the area of a semi-hexagon five feet 
 deep, are respectively: 2.5 feet and 43.301 square feet. 
 
 By Table 27, for w=.025, fall per mile 1.584 feet, 
 the velocity corresponding to hydraulic mean depth 
 2.5 is 1.872 feet per second. 
 
 Then 43.301X1-872=81. 059472 cubic feet per sec- 
 ond; 81.059472X86400X75=525265378.5 cubic feet; 
 3,630X40=145200 cubic feet per acre; 525265378.5 
 -*- 145200 -3617.5 acres. Ans. 
 
 MEASUREMENT OF THE POWER OF WATER AS A MOTOR. 
 
 The unit in the measurement of power is a foot- 
 pound that is, the amount of energy necessary to 
 raise one pound weight vertically through a distance 
 of one foot. On the other hand, one pound falling by 
 the force of gravity through a distance of one foot, 
 generates a foot-pound. 
 
 The amount of energy required to raise one pound 
 vertically 550 feet, is equal to the amount of energy 
 
246 PRACTICAL HYDRAULICS. 
 
 necessary to raise 550 pounds vertically one foot in 
 bight. 
 
 This amount .of energy rendered in one second is 
 termed a horse-power that is, 550 foot-pounds ren- 
 dered in one second, is the value of a horse power in 
 mechanics. 
 
 The weight of a cubic foot of fresh water is esti- 
 mated in practice at 62.5 pounds. 
 
 Ex. 118. How many horse-power will 10 cubic 
 feet of water, applied to an overshot water wheel, 40 
 feet diameter, render, the efficiency of the wheel being 
 75 per cent, and one foot being allowed for clearance? 
 
 Gal. 40 1=39 feet, effective head; 62.5X10X39 
 X .75H-550---33.24 horse-power. Ans. 
 
 TABLE 30. 
 
 Limiting Velocities in Open Streams. Jackson's Hydraulic 
 
 Manual. 
 
 Feet per Second. 
 
 For the worst or most sandy soil 2.5 
 
 For sandy soil generally 2 . 75 
 
 For ordinary loam 3 . 
 
 For firm gravel and hard soil 4 . 
 
 For brick work, ashlar, or rubble in cement ....5.5 to 7.5 
 
 For hard, sound, stratified rock 10 . 
 
 For very hard homogeneous rock 14 . or 15 . 
 
 Limits usual for canals 1 . to 4 . 
 
 Limits for irrigating channels , 1 . to 3 . 
 
 Limits for sewers and brick conduits 1 . to 4 . 5 
 
 Limits for self -cleans ing sewers and drainage pipes 2.5 to 4.5 
 
 Remark. The importance of the data, given in 
 Table 30, will be seen at a glance. 
 
 Thus, if a velocity exceeding 2.5 feet per second be 
 given a stream in very sandy soil, destruction by eros- 
 
I 
 
 PRACTICAL HYDRAULICS. 247 
 
 ion of the bed of the canal will ensue; while on the 
 other hand, if the velocity shall not exceed one foot 
 per second at first, the canal will be liable to become 
 choked up by the growth of vegetation. 
 
 FLOOD-FLOW OF STREAMS. 
 
 Various formulse have been devised for the flow of 
 streams in times of floods. These empirical formulae 
 are, at best, but rough approximates to the true flow. 
 
 The following formula, given by Fanning, is an ex- 
 pression for the " recorded flood measurement of 
 American streams" in New England and the Middle 
 States: 
 
 Q=200 M*, (287) 
 
 in which Q denotes the number of cubic feet discharge 
 per second, and M the area of water shed in square 
 miles. 
 
 As California is more mountainous than New Eng- 
 land and the Middle States, and fully as subject to 
 heavy downfall of rain in the mountainous regions, it 
 is not improbable that the flood discharge here will 
 exceed that indicated by formula (287). Let it hence 
 be accepted until otherwise determined. 
 
 Ex. 119. The water-shed of the main Sacramento 
 river contains twenty-four thousand seven hundred 
 and eight square miles. What will be its flood dis- 
 charge per second? 
 
 . (24,708)1x200=915647 cubic feet. Ans. 
 
248 PEACTICAL HYDRAULICS. 
 
 TABLE 32. 
 
 Miscellanies. 
 
 1 cubic foot of distilled water (U. S. standard), ba- 
 rometer 30 inches, 39.83 Fahr. =62.3793 Ibs. 
 
 1 cubic foot of distilled water (British standard), ba- 
 rometer 30 inches, 62 Fahr.---62.32l Ibs, 
 
 1 cubic foot of distilled water (U. S. standard) = 
 7.48052 gallons. 
 
 1 cubic inch of distilled water (U. S. standard) = 
 0.0361 Ibs. 
 
 1 gallon (U. S. standard) --231 cubic inches= 
 0.133681 cubic feet 8.3389 pounds water. 
 
 1 gallon, imperial (British standard) =277. 12,3 cu- 
 bic inches=- 0.160372 cubic feet= 10 Ibs. water. 
 
 1 gallon (N. Y. statute measure), barometer 30 
 inches, 39.83 Fahr.=221.184 cubic inches=8 Ibs. 
 water. 
 
 1 pound avoirdupois=16 ounces=7000 grains (U. 
 S. standard)=27.7015 cubic inches. 
 
 1 pound Troy=l pound apothecary =12 ounces= 
 5760 grains. 
 
 1 ounce avoirdupois=437.5 grains. 
 
 1 ounce Troy=l ounce apothecary =480 grains. 
 
 1 chain=100 links=4 rods=66 feet 792 inches. 
 80 chains=l statute mile=320 rods==1760 yards= 
 5280 feet 63, 360 inches. 
 
 1 geographical, nautical or sea-mile = 6,086. 5 feet in 
 longitude; and 6,076.5 feet in latitude. 
 
 1 league (English) =3 nautical miles. 
 
 1 metre=3. 2808992=3.281 in practice. 
 
 1 square metre=l centiare=10.7643 square feet. 
 
 1 are=100 square metres=1076.43 square feet. 
 
 1 cubic metre=l stare= 35.3166 cubic feet. 
 
 1 vara=2.75 feet. 
 
PRACTICAL HYDRAULICS. 249 
 
 1 legua (Mexican) =5000 varas linear= 13,750 feet 
 =2.60417 miles. 
 
 100 vara lot=100 varas square -=75625 square 
 feet 1.73611 acres. 
 
 1 legua, Mexican (of land) 6.7817 square miles 
 4340.27778 acres. 
 
 1 acre=4 roods 10 square chains 160 square 
 rods 43560 square feet. 
 
 1 section=l square mile 640 'acres. 
 
 1 township=36 sections=6 miles square= 36 square 
 miles. 
 
 1 cubic yard=27 cubic feet=16,656 cubic inches. 
 
 1 hundredweight (British) =8 stone=112 pounds. 
 
 1 ton (long ton), commercial- 20 hundredweight 
 2240 pounds. 
 
 1 ton (short ton), U. S.=2000 pounds. 
 
 1 quintal^ 100 pounds. 
 
 1 fathom=6 feet; 1 cable length 120 fathoms. 
 
 1 point -fg of an inch. 
 
 1 line=6 points^^ of an inch. 
 
 12 inches=l foot; 3 feet = l yard. 
 
 5J yards=l rod. 
 
 1 foot board measure =1 foot square and 1 inch 
 thick. 
 
 12 feet board measure=l cubic foot. 
 
 1 foot-pound- work required to raise one pound 
 vertically one foot. 
 
 1 second foot-pound=work required to raise one 
 pound vertically one foot in one second of time. 
 
 1 minute foot-poundwork required to raise one 
 pound vertically one foot in one minute. 
 
 1 (one degree), centigrade =1.8 (degrees), Fah- 
 renheit. 
 
 1 barometric inch column of mercury, with one 
 square inch base and one inch high. 
 
250 PKACTICAL HYDRAULICS. 
 
 Atmospheric pressure per square inch=14.7 pounds 
 =30 barometric inches nearly, at 39.83 Fahr. 
 
 1 ounce Troy, gold, 1000 fine=$20.67l8. 
 
 1 ounce Troy, gold coin, U. S., 900 fine=$l8.6046. 
 
 1 pound avoirdupois, gold coin, U. S., 900 fine= 
 $271.375. 
 
 1 ounce Troy, silver, 1000 fine-=$l. 29293. 
 1 ounce Troy, silver, U. S., 900 fine=$l.l63636. 
 1 pound avoirdupois, silver coin, U. S., 900 fine= 
 $16.96969. 
 
 1 dollar, U. S. gold coin=23.22 grains gold + 2. 58 
 grains copper 25.8 grains. 
 
 1 dollar, U. S. silver coin=37l.25 grains silver + 
 41.25 grains copper=412.5 grains. 
 
 1 pound sterling 1 so vereign=l 13.001 grains gold 
 4-10.273 grains copper =; 12 3. 274 grains weight, fine- 
 ness 22 carats= 916.6667. 
 
 1 grain gold, 1000 fine=- $.0430663 mint value. 
 
 1 grain silver, 1000 fine=-$. 0026936 mint value. 
 
 1 gramme gold, 1000 fine $.6646142 mint value. 
 
 1 gramme silver, 1000 fine $. 0415686 mint value. 
 
 1 cubic foot air^. 0806726 pounds --=564.7082 gr's. 
 
 1 pound of air at 39.83 = 12.387 cubic feet by vol. 
 
 1 cubic foot hydrogen^. 005042 pounds=35.2743 
 grains. 
 
 25 cubic feet of sand=l ton. 
 
 18 cubic feet of earth =l'ton. 
 
 17 cubic feet of clay=-l ton. 
 
 13 cubic feet of quartz, unbroken in lode = l ton. 
 
 18 cubic feet of gravel or earth, before digging 
 27 cubic feet when dug. 
 
 20 cubic feet of quartz, broken (of ordinary fineness 
 coming from the lode) 1 ton, contract measurement. 
 
 1 horse-power (H. P.)=550 second foot-pounds= 
 33000 minute foot-pounds. 
 
 OF THE 
 
 WNIVER 
 
HOSKIN'S NEW HYDRAULIC GIANT. 
 
 Of all the Hydraulic Nozzles or Giants made, no other has been so universally 
 adopted or so satisfactory in results. It is no exaggeration to say that it is now 
 almost exclusively used in all mining countries, and its great superiority every- 
 where acknowledged, so that the Hoskin Giant has come to be regarded as the 
 standard for this class of appliances. Several improvements have recently been 
 made which greatly increase its efficiency, as well as its security and convenience. 
 
 Horizontal and vertical motions are now made with only one joint, and this 
 so protected as to be durable and easily kept in order. The Nozzle Butt is at- 
 tached to the pipe in such a way as to counteract the downward movement com- 
 mon to this class of machines when working under great pressure. The pipe is 
 balanced by matching the notch in its flange with corresponding one in the 
 flange of the elbow. 
 
 This Machine is fully secured by Letters Patent in this, 
 as well as other mining countries, and full protection guar- 
 anteed to all purchasers. 
 
 HOSKIN'S & PERKINS' PATENT DEFLECTORS. 
 
 Special attention is called to the many advantages of these Deflectors, and to 
 the fact that they are adapted to no other Giants. By means of a hand-lever the 
 stream can be deflected to any desired angle without moving the body of the 
 machine, and it is moved so easily that any child can operate it. This is the 
 most valuable invention that has ever been applied to this class of machines, and 
 gives the Hoskin Machine great pre-eminence over all others. 
 
 ROMAN'S SAFETY PRESSURE EQUALIZER. 
 
 A simple and most effective device for equalizing the strain on hydraulic 
 pipes, and compensating for*all variations of pressure. It relieves the direct 
 shock upon the pipes, and prevents the collapse or disastrous breaks so common 
 from this cause. It is an indispensable and universally recognized necessity for 
 all hydraulic operations. One required for each line of pipe. 
 FOR CIRCULAR AND PRICE LIST. Address 
 
 PACIFIC IRON WORKS, 
 
 127 First Street, 1OO N. Clinton St., 145 Broadway, 
 
 SAN FRANCISCO. CHICAGO. NEW YORK. 
 
JOSHUA HENDY MACHINE WORKS, 
 
 (INCORPORATED SEPTEMBER 29, 1882.) 
 
 Nos. 39 to 51 Fremont St., San Francisco, Cal. 
 
 MANUFACTURERS OF- 
 
 Hydraulic Gravel Elevators 
 
 HYDRAULIC GIANTS, 
 
 HYDRAULIC MINING PIPE, 
 QUARTZ and SAW-MILL MACHINERY, 
 
 NEW and SECOND-HAND BOILERS, ENGINES 
 
 AND MACHINERY OF EVERY DESCRIPTION. 
 
 AGENTS FOR THE SALE OF 
 
 "Cummer" Engines, from Cleveland, Ohio,* 
 
 Porter Manufacturing Co.'s Engines and Boilers, 
 
 "Baker" Rotary Pressure Blowers, 
 "Wilbraham" Rotary Piston Pumps, 
 
 " Boggs & Clarke " Centrifugal Pumps, 
 
 The Volker & Felthousen Manuf'g Co.'s 
 
 Buffalo Duplex Steam Pumps, 
 
 P. Blaisdell & Co.'s Machinists' Tools. 
 
1.64453