:-NRLF SB 77 5 3D > * WA^ A./!? $MI .. OifiiP* REESE LIBRARY UNIVERSITY OF CALIFORNIA. Received Accessions No. ^/ 5<- Shelf No. Jb HYDRAULIC TABLES, COEFFICIENTS, AND lOBMULffi, FOR FINDING THE DISCHARGE OF WATER FKOM ORIFICES, NOTCHES, WEIRS, PIPES, AND RIVERS. BY JOHN NEVILLE, CIVIL ENGINEEB, M.E.I.A., COUNTY 6UBVEYOB OF LOUTH AND OF THE COUNTY OF THE TOWN OF DBOOHEDA. WITH EXTENSIVE ADDITIONS, NEW FORMULA, TABLES, AND GENERAL INFORMATION ON RAIN-FALL CATCHMENT-BASINS, DRAINAGE, SEWERAGE, WATER SUPPLY FOR TOWNS AND MILL POWER, " It ought to be more generally known, that theory is nothing more than the conclusions of reason from numerous and accurately observed phenomena, and the deductions of the laws which connect causes with effects ; that practice is the application of those general truths and principles to the common affairs and purposes of life ; and that science is the recorded experience and discoveries of mankind, or, as it has been well defined, ' the know- ledge of many, orderly and methodically digested, and arranged, so as to become attainable by one.'" AMERICAN QUABTEBLY LONDON: JOHN WEALE, 59, HIGH HOLBORN. 1860-1. TO MAJOR-GENERAL SIR THOMAS AISKEW LARCOM, K.C.B., LL.D., F.R.S., M.B.I.A., ETC., OF THE ROYAL ENGINEERS, UNDER SECRETARY OF STATE FOR IRELAND, THIS WORK IS INSCRIBED BY THE AUTHOR. CONTENTS. Introduction to the Second Edition * *; &.* ;'-;..**< .-*> Introduction to the First Edition -*. ' * ~ 4.-; - #T- *1 SECTION I. Application and Use of the Tables, Formulae, &c. Extra Horse-power required in Pumping Engines from Friction in the Pipes Table of Heads due to Friction and of Discharges f . ~ w SECTION II. ' Formulae for the Velocity and Discharge from Orifices, Weirs, and Notches Coefficients of Ve- locity, Contraction, and Discharge Practical Remarks on the Use of the Formulae ; ^. . -.# i f*-j. SECTION III. Experimental Results and Formulse Co- efficients of Discharge for Orifices, Notches, and Weirs SECTION IV. Variations in the Coefficients from the Posi- tion of the Orifice General and Partial Contraction Velocity of Approach Various Practical Formulae for the Discharge over Weirs and Notches Central and Mean Velocities . . . . ' . ". '" .' *~"l SECTION V. Submerged Orifices and Weirs Contracted River Channels . j 4 * . ,, . .. SECTION VI. Short Tubes, Mouth-pieces, and Approaches Alteration in the Coefficients from Friction by in- creasing the Length Coefficients of Discharge for Simple and Compound Short Tubes Shoots SECTION VII. Lateral Contact of the Water and Tube Atmospheric Pressure Head measured to the Dis- charging Orifice Coefficient of Resistance Formula for the Discharge from a Short Tube Diaphragms Oblique Junctions Formula for the Time of the Sur- face smking a given depth Lock Chambers *' : ' SECTION VIII. Flow of Water in Uniform Channels Mean Velocity Mean Radii and Hydraulic Mean Depths Border Train Hydraulic 1 nclination Ef- fects of Friction Formulse for calculating the Mean Velocity Application of the Formulae and Tables to the Solutions of three useful Problems PAGES v. to xii. 1 to!4 1643 4463 6395 96139 140150 150165 166181 162234 11 CONTENTS. SECTION IX. Best Forms of the Channel Regimen Velocity Equally Discharging Channels . . . 235253 SECTION X. Effects of Enlargements and Contractions Backwater Weir Case Long and Short Weirs . 254 271 SECTION XI. Bends and Curves Branch Pipes Dif- ferent Losses of Head General Equation for finding the Velocity Hydrostatic and Hydraulic Pressure Piezometer Catchment Basins Rain-fall per annum 271 289 SECTION XII. Rain-fall Catchment Basins Discharge into Channels Discharge from Sewers Loss from Evaporation, &p. . . . ; , . . 289315 SECTION XIII. Water Supply for Towns -Strength of Pipes Sewerage Estimates and Cost Thorough Drain- ageArterial Drainage ^ > . . + . , p. 315364 SECTION XIV. Water Power Dynamometer, or Friction Brake Overshot, Breast, and Undershot Vertical Wheels Turbines and Horizontal Wheels Hydraulic $am Water-pressure Engine Work of Flour Mills . 364399 TABLES NOT EMBODIED IN THE TEXT. TABLE I. Coefficients of Discharge from Square and Dif- ferently proportioned Rectangular Lateral Orifices in Thin Vertical Plates . . ... . . 400401 TABLE II. For finding the Velocities from the Altitudes and the Altitudes from the Velocities .... 402 411 TABLE III. Square Roots for finding the Effects of the Velocity of Approach, when the Orifice is small in pro- portion to the Head. Also, for finding the Increase in the Discharge from an Increase of Head. (See pp. 101 and 107.) ......... 412413 TABLE IV. For finding the discharge through Rectan- gular Orifices j in which n =-^. Also, for finding the Effects of the Velocity of Approach to Weirs, and the Depression on the Crest. (See pp.101 and 107.) . . 414 417 TABLE V. Coefficients of Discharge for different Ratios of the Channel to the Orifice . . 418423 CONTENTS. Ill TABLE VI. The Discharge over Weirs or Notches of One Foot in Length, in Cubic Feet per Minute. (See pp. IQQ and 111 to 127.) 424429 TABLE VII. For finding the Mean Velocity from the MaximumVelocity'at the surface, in Mill-races, Streams, and Rivers with Uniform Channels ; and the Maximum Velocity from the Mean Velocity. (See p. 184.) y 430 TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. (See p. 195) ..."...,'' 431443 TABLE IX. For finding the Discharge in Cubic Feet, per Minute, when the Diameter of a Pipe or Orifice, and the Velocity of Discharge, are known, and vice versd . 444445 TABLE X. For finding the Depths on Weirs of different Lengths, the Quantity discharged over each being sup- posed constant. (See p. 271.) 446 TABLE XI. Relative Dimensions of Equally-Discharging Trapezoidal Channels, with Slopes from to 1, up to 2 to 1. (See pp. 249 and 252.) ..... 447 TABLE XII. Discharges from the Primary Channel in the first column of Table XI. (See p. 253.) ; . 448449 TABLE XIII. The Square Roots of the Fifth Powers of Numbers for finding the Diameter of a Pipe, or Dimen- sions of a Channel from the Discharge, or the reverse ; showing the relative Discharging Powers of Pipes of different Diameters, and of any similar Channels what- ever, closed or open. (See pp, 31, 42, 228, &c.) . . |j 450 TABLE XIV. English and French Measures . . . 451454 TABLE XV. Weight, Specific Gravity, Ultimate Strength and Elasticity of Various Materials . . . 655 COBRECTION. 2 2 For o = 3 c dN / 2# h, p. 120, line 18, read o =g- i -r T = cTfTK = *36. We must, therefore, multiply channel 2-75 2-75 by -857, which gives 2*36 for the ratio of the mean velocities in the orifice and in the channel approaching it. With this new value of the ratio of the channel to the orifice, we find, as before, the value of the co- efficient from TABLE V. to be -651. The remarks throughout the work, with the auxiliary tables, will be found of much use in determining the coefficients for different ratios of the channel to the orifice, notch, or weir, and the corrections suited to each. If in this example we were considering, other things being the same, the alteration in the coeffi- cient for a notch, or weir, it would be found from the Table, column 4, to be *672 instead of *645 found in column 3, for an orifice sunk some depth 24 THE DISCHARGE OF WATER FROM below the surface. For the corrections suited to mean and central velocity, and to the nature of the approaches, we must refer to the body of this work and to the auxiliary tables therein at the end of SECTION IY. EXAMPLE 12. What is the discharge aver a weir 50 feet long ; the circumstances of the overfall, crest, and approaches, being such that the coefficient of discharge is '617, when the head measured from the water in the weir basin, 6 feet above the crest, is 17i inches? TABLE VI. will give the discharge in cubic feet per minute, over each foot in length of weir, for various depths up to 6 feet. It is divided into two parts ; the first for " greater coefficients," viz. -667 to -617; and the second for "lesser co- efficients," viz. -606 to -518. The coefficient assumed being -617, we find the discharge over 1 foot in length, with a head of 17J inches, to be 348-799 cubic feet per minute ; hence the required discharge is 50 x 348-799 = 17439-95 cubic feet. The determination of the coefficient suited to the circumstances of the overfall, crest, approaches, and approaching section, will be found discussed else- where through this work. The valuable Table de- rived from Mr. BlackwelPs experiments will also be of use ; but the heads being taken at a much greater distance back from the crest than is generally usual, the coefficients taken from it for heads greater than 5 or 6 inches, will be found under the true ones for heads measured immediately at or about 6 feet, above the crest. For heads measured on the crest, the small Table of coefficients in SECTION III., applicable to the purpose, will be of use. OEIFICES, WEIRS, PIPES, AND RIVERS. 25 EXAMPLE 13. What is the mean velocity in a large channel, when the maximum velocity along the central line of the surface is 31 inches per second? TABLE VII. gives 25-89 inches for the required velocity, and for smaller channels 24*86 inches. In order to find the mean velocity at the surface from the maximum central velocity, the latter must be multiplied by 914. The velocity at the surface is best found by means of a floating hollow ball, which just rises out of the water, The velocity at a given depth is best found by means of two hollow balls connected with a link, the lower being made heavier than the upper, and both so weighted by the admission of a certain quan- tity of water that they shall float along the current, the upper one being in advance but nearly vertical over the other. The velocity of both will then be the velocity at half the depth between them. The velocity at the surface, found by means of a single ball, being also found, the velocity lost at the half depth is had by subtracting the common velocity due to the linked balls from that of the single ball at the surface. The velocity at any given depth is then easily found by a simple proportion ; but the result will be most accurate when the given depth is nearly half the distance between the balls, which distance can never exceed the depth of the channel. Pitot's tube, Woltmann's tachometer, the hydrometric pendu- lum, the rheometer, and several other hydrometers, have been used for finding the velocity ; but these instruments require certain corrections suited to each separate instrument, as well as kind of instrument, 26 THE DISCHAKGE OF WATEK FROM and are not so correct or simple, for measuring the velocity in open channels, as a ball and linked balls. EXAMPLE 14. What is the discharge from a river having a surface inclination of 18 inches per mile, or 1 in 3520, 40 feet wide, with nearly vertical banks, and 3 feet deep ? The area is 40 x 3 = 120 feet, and the border 40 + 2 x 3 = 46 feet ; therefore the 120 hydraulic mean depth is -TTT = 2-61 feet zz 2 feet 7-3 inches*. With this and the inclination we find from TABLE VIII. 28-27 + 2-75 X ^ = 28-87 inches per second =: 28-87 x 5 =z 144-35 feet per minute for the mean velocity ; hence we get 144-35 x 120 = 17,322 cubic feet per minute for the required dis- charge. For channels with sloping banks we have only to divide the border, which is always known, into the area for the hydraulic mean depth, with which, and the surface inclination, we can always find the velocity by TABLE VIII., and thence the discharge. Unless the banks of rivers be protected by stone pavement or otherwise, the slopes will not continue permanent ; it is therefore almost useless to give the discharges for channels of particular widths and side slopes, When the mean velocity is once known, the remaining calculations are those of mere mensuration, and they should be made separately. This example may also be solved, practically, by means of TABLES XI. and XII. A channel 40 x 3 * For greater hydraulic depths than 144 inches, the extent of the TABLE, divide hy 9, and find the corresponding velocity. This multiplied by 3 will be the velocity sought. ORIFICES, WEIKS, PIPES, AND RITE has the same conveying power as one 70 x XL, which latter, TABLE XII. discharges with a fall of 18 inches in the mile, 17,157 feet; or about one per cent, less than that previously found. EXAMPLE 15. The diameter of a very long pipe is li inch, and the rate of inclination , or ivhole length of the pipe divided by the whole fall, is 1 in 71^ ; what is the discharge in cubic feet per minute ? The 1*5 hydraulic mean depth, or mean radius, is -j- = -375 3 inches = ^ inch. Consequently we find from TABLE VIII. the velocity in inches per second equal to 25-09 - 1-92 x ^ = 25-09 - -29 = 24-80. The discharge in cubic feet per minute for a IJ-inch pipe is now found most readily by means of TABLE IX., as follows : Inches. Cubic feet. For a velocity of 20-0 the discharge is 1-227 4-0 -245 8 -049 ,, 34-8 1-521 Whence the discharge in cubic feet per minute is 1-521. For short pipes, of 100 or 200 feet in length, and under, the height due to the velocity and orifice of entry must be deducted from the whole height to find the proper hydraulic inclination, and also the height due to bends, curves, cocks, slides, and erogation. The neglect of these corrections has led some writers into mistakes in applying certain formulae, and in test- 28 THE DISCHAEGE OF WATER FEOM ing them by experimental results obtained with short pipes. We shall now apply the TABLES to the deter- mination of the discharge from short pipes, and compare the results with experiment, referring gener- ally to equation (153) and the remarks preceding it for a correct and direct solution. EXAMPLE 16. What is the discharge in cubic feet per minute from a pipe WO feet long, with a fall or head of 35 inches to the lower end, when the diameter is 1 J inch ? Find also the discharge from pipes 80 feet t 60 feet, 40 feet, and 20 feet, of the same diameter and having the same head. If the water be admitted by a stop-cock at the upper end, the coefficient due to the orifice of entry will probably be about -75 or less, -815 being that for a clear entry to a short cylindrical tube. The approximate inclination is 100 x 12 or i= 1 in 34*3 ; but as a portion of the fall must be absorbed by the velocity and orifice of entry, we may assume for the present that the inclination is 1 in 35. With this inclination and the mean radius IL- 3 -jr zz g inches, we find the mean velocity from TABLE VIII. to be 38-06 inches. Now when the coefficient due to the orifice of entry and velocity is -75, we find from TABLE II. the head due to this velocity to be 3| inches nearly, whence 35 - 3f = 31f = 31-625 100 x 12 inches is the height due to friction, and ^^ o J.*D^jO equals 1 in 37-9, the inclination, very nearly. With this new inclination we find, as before, from TABLE VIII. the mean velocity of discharge to be now 36-35 ORIFICES, WEIRS, PIPES, AND RIVERS. 29 inches ; and by repeating the operation we shall find the velocity to any degree of accuracy in accord- ance with the table, and the shorter the pipe is, the oftener must it be repeated. The height due to 36-35 inches taken from TABLE II. as before, with a coeffi- cient of -750, is 3^ = 3-125 inches. The corrected fall due to the friction is now 35 - 3-125 == 31-875, and 01.07- equal 1 in 37-6, the corrected inclina- tion. With this inclination we find the corrected velocity to be now 36-53 inches per second. It is not necessary to repeat the operation again. The discharge determined from TABLE IX. is as follows : Inches. Cubic feet. For a velocity of 30-00 the discharge is 1-841 6-00 -368 50 -031 03 -002 36-53 The experimental discharge found by Mr. Provis was 2-264 cubic feet per minute in one experiment, and 2-285 in another. The discharge from the shorter pipes may be found in a similar manner, and we place the results alongside the experimental ones given in the work referred to below* in the following short table : * " Transactions of the Institution of Civil Engineers," vol. ii. p. 203. " Experiments on the Flow of Water through small Pipes." By W. A. Provis. The small Tables in SECTIONS VI. and VIII. of this edition give at once the coefficient to be mul- tiplied by^^n, or 8-v/lf, to find the velocity when the ratio of the diameter to the length of the pipe is known. They will be found of great advantage in calculating directly the velocity from short pipes. For long pipes, see the TABLE pp. 42 and 43. 30 THE DISCHARGE OF WATER FROM EXPERIMENTAL AND CALCULATED DISCHARGES FROM SHORT PIPES. 1, 4 1 .a !* A o ^o fa 1 .a 1 III II jji II 11 w I'l if If 3 w w 100 35 2-275 37-082 3* 31| 37-6 36-53 2-242 80 35 2.500 40-750 3f 3H 30-8 41-18 2-521 60 35 2-874 46-846 5 30 24-0 48-02 2-946 40 35 3-504 57-115 n 27i 17-5 58-50 3-590 20 35 4.528 73-801 12* 221 10-7 78-61 4-824 The velocities in the fourth column have been cal- culated by the writer from the observed quantities discharged, from which the height due to the orifice of entry and velocity in column 5 is determined, and thence the quantities in the other columns as above shown. The differences between the experimental and calculated results are not large, and had we used a lesser coefficient than 8 750 for calculating the re- duction of head due to the velocity, stop-cock, and orifice of entry, say -715, the calculated results, and those in all of Mr. Provis's experiments in the work referred to, would be nearly identical.* EXAMPLE 17. It is proposed to supply a reservoir near the town of Drogheda with water by a long pipe, having an inclination of 1 in 480, the daily supply to be 80,000 cubic feet ; what must the diameter of * In a late work, " Researches in Hydraulics," the author is led into a series of mistakes as to the accuracy of Dtf Buat's and several other formulae, from neglecting to take into consideration the head due to the velocity and orifice of entry when testing them by the experiments above referred to. ORIFICES, WEIRS, PIPES, AND RIVERS. 31 the pipe be ? The discharge per minute must be * =56* cubic feet, nearly. Assume a pipe whose "mean radius" is 1 inch, or diameter 4 inches, and the velocity per second found from TABLE VIII. will be 14*41 inches. We then have from TABLE IX., Inches. Cubic feet. For a velocity of 10-00 a discharge of 4-363 4-00 1-745 40 -175 01 -004 14-41 6-287 The discharge from a pipe 4 inches in diameter would be therefore 6-287 cubic feet per minute. We then have 4* : d$ : : 6'287 : 56, or 1 : rf* : : -196 : 56 : : 1 : 286; therefore d^ =. 286, and d =. 9-61 inches, nearly, as may be found from TABLE XIII., &c. This is nearly the required diameter. It is to be observed that the diameters thus found will not always agree exactly with those found from Du Buat's or other formulas, nor with each other, because the discharges are not strictly as d^ ; but in practice the difference is imma- terial, and the approximative value thus found can be easily corrected. If we assumed a pipe whose diameter is 1, the operation would have been more simple ; for the velocity would then be, TABLE VIII., at the given inclination, 6-4 inches ; and the discharge 175 cubic feet, TABLE IX. Hence we get d% * Hydraulic Tables, Weale, 1854, give at once this discharge for a pipe between 9 and 10 inches diameter, also the TABLE, p. 42. 32 THE DISCHARGE OF WATER FROM = 320, and, therefore, TABLE XIII., d 10 inches nearly, which differs about half-an-inch from the former value, 9-6 inches, found by assuming a pipe of 4 inches to calculate from. It is necessary to understand that different results must be expected, in working from practical formulae, for different operations. When once an approximative value is obtained, it can be easily corrected to any required degree of accuracy. Again the velocity in inches per second, from a cylin- drical pipe 6 inches in diameter, is nearly equal to the discharge in cubic feet per minute ; and as 6^ m 88*2, we have 88*2^ : d^ : : the velocity in inches per second from a 6-inch pipe : the discharge per minute from a pipe whose diameter is d. Hence this pro- portion would enable us to find, very nearly, the discharge from the diameter and fall ; or the diameter from the discharge and fall by finding the velocity only, due to a 6-inch pipe. See TABLE pp. 42 and 43. EXAMPLE 18. The area of a channel is 50 square feet, and the border 20*6 feet ; the surface has an in- clination of 4 inches in a mile ; what is the mean 50 velocity of discharge? ^g = 2*427 feet = 29-124 inches is the hydraulic mean depth ; and we get from Table VIII. 12-03 - -- - = 12-03 - 19 = 11*84 inches per second for the required velo- city. Though this velocity will be found under the true value for straight clear channels, it will yet be more correct for ordinary river courses, with bends OKIFICES, WEIES, PIPES, AND KIVERS. 33 and turns, of the dimensions given, than the velocity found from equation (114). For a straight clear channel of these dimensions, Watt found the mean velocity to be 13-5 to 14 inches; that is to say, 17 at top, 10 at bottom, and 14 in the middle, Our formula v = 140 (r sfi 11 (r sft gives v = 1*143 feet, or nearly a mean of these two. EXAMPLE 19. A pipe 5 inches in diameter, 14,637 feet in length, has a fall of 44 feet ; what is the dis- charge in cubic feet per minute ? The inclination is 1 i r* Q IT K -jj = 332*7, and mean radius j = 1J. We then find from TABLE VIII. the velocity equal to 19-81 + L \*. 5 4 - = 19-81 + -16 = 19-97, or 20 inches per second very nearly ; and by TABLE IX. the discharge in cubic feet per minute is, as before found to be, 13-635. The TABLE, p. 42, gives, by inspection, 13-6 feet. EXAMPLE 20. What is the velocity of discharge from a pipe or culvert 4 feet in diameter, having a fall of 1 foot to a mile ? Here s =: goftQ' and r = 1 foot. We then find the velocity of discharge from TABLE VIII. to be 14-09 inches, equal to 1-174 feet per second. By calculating from the different formulae referred to below, we shall find the velocities, when r s = -0001894, and \/Ts = -01376, as follows. Velocity in feet. Eeduction of Du Buat's formula .... equation (81) 1-174 Girard's do. (Canals with aquatic plants and very slow velocities) (86) -621 Prony's do. (Canals) .... (88) 1*201 D 34 THE DISCHAKGE OF WATER FROM Velocity in feet. Seduction of Prony's formula (Pipes) . . equation (90) 1-257 Prony's do. (Pipes and Canals) . (92) 1-229 Eytelwein's do. (Kivers) ... (94) 1-200 Eytelwein's do. (Kivers) ... (96) 1-285 Eytelwein's do. (Pipes) ... (98) 1-364 Eytelwein's do. (Pipes) ... (99) 1-350 Dr. Young's do. . . .>. . (104)1-120 *D'Aubuisson's do. (Pipes) . . (109)1-259 *D'Aubuisson's do. (Kivers) . . (111) 1-199 ,, The writer's do. (Clear straight Channels with small velocities) (114) 1-268 Weishaeh's do. (Pipes) ... (119) 1-285 The author's, for Pipes and Kivers 1-295 We have calculated this example from the several formulae above referred to, whether for pipes or rivers in order that the results may be more readily com- pared. The formula from which the velocities and tables for the discharges of rivers are usually cal- culated is, for measures in feet, v 94-17 an( ^ i n the second lv* If we wish to substitute the fall per mile for the hydraulic inclination, the first of these will again become h = -* = Is for the loss per mile ; and in the second case, h = Y^. = i s f OT the loss per mile in feet. If the velocity were so low as about 1 foot per second, then v = 90 \/ r s, and we should find , lv j -- If for the inclination we substitute the fall per mile, this will become h = -* = Is for the loss per mile in feet. The loss of head varies in the same pipe with the velocity, and must be calculated differently, for small and for high velocities, when using the common for- mulae. The TABLE near the end of SECTION VIII. will ?* ^ always give the correct value of rs, and thence *=~' In addition to the loss of head arising from fric 1 tion, losses also o'ccur from straight or curved bends, from diaphragms, from junctions, and from the ori- fices of entry and discharge ; these must be deter- mined separately for each case, as is shown hereafter, and added together and to the loss arising from fric- tion, and the sum to the height the water is to be raised, before the full or total head for determining the power of an engine can be accurately known. OEIFICES, WEIES, PIPES, AND KIVEES. 41 The TABLE on the next two pages will be found of great practical utility in solving all questions con- nected with water-pipes and sewers discharging fully- diluted sewage. In using it we can interpolate, by inspection, for intermediate diameters or inclinations. For greater diameters, divide those given by 4, and multiply the corresponding velocity found in the table by 2, and the corresponding discharge in the table by 32. If the object be to find the size of the channel, divide greater given velocities by 2, and multiply the diameters or inclinations found from the table by 4 ; also divide greater discharges by 32, and multiply the diameters found from the table by 4. The small auxiliary table, p. 43, embodied in the larger one, is of great use in making allowance for the velocity and orifice of entry in short pipes, before finding the head due to friction. The table also gives the different diameters and inclinations which, taken together, give the same velocity or discharge ; and it enables us, from inspection, to select that relation of diameter to declivity which is best suited for other engineering aspects of the question. Taken in con- nexion with TABLES VIII., XL, XII., and XIII., this table completes the means of finding, by inspection, the dimensions, inclinations, velocities, and discharges of every class of water-channel or sewage-conduit required in engineering practice. TABLE XIV. gives the comparative values of Eng- lish and French measures ; and TABLE XV. gives the weight, specific gravity, and ultimate strength and elasticity of various materials with which the engineer has to operate. TABLE for finding, very nearly, the velocity and discharge from Cylindrical Water Pipes or Sewers, when the diameter and fall are given. Any two of the four quantities, the velocity, discharge, diameter, and fall or inclination, being given, the others can be found in THE TABLE from inspection. sift -w o 5 a A * p,. flj* Mean by- 1 draulic incli-l nation of 1 pipe or sewerj The VELOCITY IN INCHES PER SECOND is given in the first horizontal line for each inclination or fall ; and the DISCHARGE IN CUBIC FEET PER MINDTE in the next following one. lin. 2 in. 1 3 in. diameter diameter | diameter 4 in. diameter 5 in. diameter 6 in. diameter 7 in. diameter 8 in. diameter 9 in. diameter 10 in. diameter One in 1 5280 1-7 2-5 3-2 3-8 4-2 4.7 5-1 5-5 5-9 6-2 05 27 79 1-6 29 4-6 6-8 9-6 129 169 2 2640 2-5 3-8 4-7 5-6 6-3 6-9 7-5 8-1 8-6 9-1 07 41 1-2 2-4 4-3 68 101 14-2 191 24-9 3 1760 3-1 4-7 5-9 7-0 7-8 8-7 9'4 10-2 10-8 11-5 08 51 1-4 30 53 85 126 17-7 239 31-2 4 1320 3-6 5-5 6-9 8-2 9-2 10-2 ii-l 11-9 12-7 13-4 10 60 1-7 3-6 63 100 148 20-8 280 36-7 5 1056 4-1 6-2 7-9 9'3 10-4 11-6 12-5 135 14-4 15-2 11 68 19 4-0 7-1 11-3 16-8 236 31-7 41-5 6 880 4-6 6-9 8*7 10-3 11-5 12-8 13-9 15-0 15-9 16-9 12 76 2-2 45 79 126 186 261 351 460 7 754 5-0 7-5 9-5 11-2 1-2-6 14-0 15-1 16-3 17-4 18-4 14 82 2-3 49 86 13-7 20-2 285 383 502 8 660 5-4 8-1 10-2 12-0 13-5 15-0 16-3 17-6 18-7 20-0 15 89 2-5 53 92 14-8 21-8 306 41-3 541 9 587 5-7 8-7 11-0 12-9 14-5 16-1 17-4 18-8 20-0 21-2 16 95 27 56 99 15-8 233 328 441 57-7 10 528 6-1 9-2 116 13-7 15-4 17-1 18-5 19-9 21-2 22-5 17 1-00 29 60 92 167 24-7 348 46-8 61-3 11 480 6-4 9-7 12-3 14-4 16-2 18-0 19-5 21-0 22-4 23-7 17 11 3-0 6-3 111 17-7 261 36-7 49-4 64-7 12 440 6-7 10-2 12-9 15-2 17-1 18-9 20-5 22-1 23-5 24-9 18 11 32 6-6 11-6 186 27-4 386 519 679 13-2 400 7-1 10-8 13*6 16-0 18-0 20-0 21-7 23-3 24-8 26-3 19 1-2 33 69 123 196 28-9 40-7 548 71-7 15-1 350 7-7 11-6 14-7 17-2 19-4 21-6 23-4 25-2 26-8 28-4 21 1-3 36 75 33-2 21-2 31-2 439 591 77-4 17-6 300 8-4 12-7 16-0 18-8 21-2 23-5 25-5 27-5 29-2 31-0 23 1-4 3-9 82 14-4 231 34-1 40-8 646 845 21-1 250 9-4 14-1 17-8 20-9 23-5 26-1 28-3 30-5 32-5 34-4 26 15 4-4 92 160 25-7 378 533 71-7 93-8 26-4 200 10-6 16-0 20-2 23-8 26-8 29-7 32-2 34-7 36-9 39-1 29 1-7 5-0 104 18-2 29-2 431 60-6 816 1067 35-2 150 12-5 19-0 23-9 28-1 31-6 35-1 38-1 41-0 43-7 46-3 34 21 59 123 21-6 345 509 71-6 96-4 126-2 52-8 100 15-9 24-1 30-4 35-7 40-1 44-6 48-3 52-1 55-4 58-7 43 2-6 7-4 15-6 27-3 438 64-6 90-9 122-3 1601 587 90 16-9 25-6 32-3 38-0 42-7 47-4 51-4 55-4 58-9 62-5 46 28 79 166 29-1 46-6 687 96-7 130-2 170-3 C6- 80 18-1 27-5 34-6 40-7 45-8 509 55-2 59-4 63-2 67'0 49 30 8-5 17-8 312 499 73-7 103-7 1396 1827 754 70 19-6 30-0 37-5 44-1 49-6 55-1 59-7 64-4 68-4 72-5 53 32 9-2 192 338 54-1 79-8 112-3 1512 197-7 88- 60 21-5 32-6 41-1 48-3 54-4 60-4 65-5 70-6 75-1 79-5 59 36 101 211 371 59-3 87-5 123-2 165-8 2169 105-6 50 24-0 36-4 45-8 53-9 60-7 67-4 73-1 78-7 83-7 88-7 65 4-0 112 235 413 662 97-6 137-4 1849 242-0 132- 40 27-4 41-6 52-5 61-7 69-4 77-1 83-6 90-1 95-8 101-5 75 4-5 128 26-9 47-3 75-7 111-7 157-2 211-6 2768 176 30 32-6 49-5 62-5 73-4 82-6 91-7 99-5 107-2 1140 120-8 89 5-4 153 320 563 90-0 1329 1871 251-8 3294 212-2 25 36-4 55-3 69-8 8-2-0 92-2 102-4 lii-l 119-7 127-3 134-9 99 6-0 171 35-8 62-9 ICO 6 148-4 2089 2812 367-9 264-1 20 41-7 63-3 79-9 93-8 105-6 117-3 127-2 137-0 145-7 154-4 114 69 196 409 720 1151 1699 2392 3219 421-2 352- 15 49-6 75-3 95-0 111-7 125-6 1396 151-3 163-1 173-4 183-8 1-35 8-2 23-3 48-3 856 1370 202-2 284-6 383-1 1501-2 528 10 63-3 96-0 1212 142-4 160-2 178-0 192-9 207-9 221-1 1234-3 173 105 29-7 621 109-2 174-7 257-8 3629 4885 6390 TABLE for finding, very nearly, the velocity and discharge from Cylindrical Water Pipes or Sewers, when the diameter and fall are given. Any two of the four quantities, the velocity, discharge, diameter, or inclination being given, the others can fee found in THE TABLE from inspection. il Mean hy- 1 draulic incli-l nation of 1 pipe or sewer| The VELOCITT IN INCHES PER SECOND is given in the first horizontal line for each inclination or fall ; and the DISCHARGE IN COBIC FEET PER MINUTE in the next following one. 12 in. 1 14 in. diameter) diameter 16 in. diameter 18 in. diameter 20 in. i 22 in. diameter diameter 24 in. 26 in. diameter| diameter 28 in. diameter 30 in. diameter One in i 5280 6-8 7-4 8-0 8-5 8-9 9-4 9-8 10-3 10-7 ll'l 26-9 40- 56- 75- 98- 124- 155- 189- 228- 271- 2 2640 10-1 10-9 11-7 12-5 13-2 13-9 14-5 15 1 15-7 16-3 40 58 82 110 144 183 228 278 336 400 3 1760 13 14 15 16 17 17 18 19 20 20 50 73 103 138 180 229 285 349 421 500 4 1320 15 16 17 18 19 20 21 22 23 24 58 86 120 163 212 269 335 410 494 588 5 1056 17 18 20 21 22 23 24 25 26 27 66 97 136 184 240 305 380 464 560 665 6 880 19 20 22 23 24 26 27 28 29 30 73 108 151 203 265 337 420 514 620 737 7 754 20 2-2 24 25 27 28 29 30 32 33 80 118 165 222 289 368 458 560 676 804 8 660 22 24 25 27 29 30 31 33 34 35 86 127 177 239 312 397 493 604 728 866 9 587 23 25 28 29 30 32 33 35 36 38 92 135 192 255 333 424 527 645 778 925 10 528 25 27 29 31 32 34 36 37 39 40 97 144 201 271 354 450 560 685 826 982 11 480 26 28 30 32 34 36 38 39 41 42 103 152 213 286 373 475 590 723 871 1036 12 440 27 30 32 34 36 38 39 41 43 44 108 159 223 300 392 498 620 759 916 1089 13-2 400 29 31 34 36 38 40 42 43 45 47 114 168 236 317 414 526 655 801 9i 36 1149 15-1 350 31 34 36 39 41 43 45 47 49 50 123 181 254 342 446 568 707 865 1043 1240 17-6 300 34 37 40 42 45 47 49 51 53 55 134 198 278 374 487 620 772 944 1139 1354 21-1 250 38 41 44 47 50 52 55 57 59 61 149 219 308 415 541 688 857 1048 1264 1504 26-4 200 43 47 50 53 56 59 62 65 67 70 169 249 350 472 616 783 975 1192 1438 1710 35-2 150 51 55 59 63 67 70 73 76 79 82 200 294 414 558 728 925 1152 1409 1700 2021 52-8 100 65 70 75 80 85 89 93 97 101 105 254 374 526 708 923 1174 1462 1788 2157 2565 58*7 90 69 74 80 85 90 95 99 Sop Velocit ies in inches, oer 270 398 559 753 982 1250 1556 c >t> second, due to heads. 2 , ' and for measures in feet, * The velocities for different heights are given in the column number 1, TABLE II. ORIFICES, WEIES, PIPES, AND RIVERS. 45 v = 8-025 v/ h, and h = 64-403 If v be in feet, and h in inches, then v = 2-317 h = = -01553 v*. = -1864 v\* COEFFICIENT OF VELOCITY. Let the vessel A B c D, Fig. 1, be filled with water to the level E F : then it has been found, by experi- ment, that the velocity of discharge through a small orifice o, in a thin plate, at the distance of half the diameter outside it, in the vena-contracta, will be very nearly that due to a heavy body falling freely from the height A, of the surface of the water E F, above the centre of the orifice. The velocity of discharge * The force of gravity increases with the latitude, and decreases with the altitude above the level of the sea, but not to any con- siderable extent. If A be the latitude, and h the altitude, in feet, above the mean sea level, then we may, generally, take g = 32-17 (1 -0029 cos 2A) x (I ), in which E, the radius of the earth at the given latitude is equal to 20887600 ( 1 + -0016 cos 2X). 46 THE DISCHARGE OF WATER FROM determined by the equation v v/2 g h, for falling bodies, is, therefore, called the " theoretical velocity'' If we now put v d for the actual mean velocity of dis- charge in the vena-contracta, and c v for its ratio to the theoretical velocity v, we shall get v d zz c v v ; and by substituting for v, its value (2.) v, c v is termed "the coefficient of velocity ;" its numerical value, at about half the diameter from the orifice, is about *974 ; and, consequently, t? d = -974 v/2 g h. This for measures in inches becomes v d = 27-077 and for measures in feet v* = 7-816 The orifice o, is termed an horizontal orifice in Fig. 1, and in Pig 2 a vertical or lateral orifice. When * The velocities for different heights calculated from this formula, are given in the column numbered 2, TABLE II. It has been latterly asserted in a Blue Book that theoretically v d = f V 2 g h. It is not necessary here to combat this error, which confounds the discharge with its velocity, and a single practical fact, applicable only to a thin plate, with a theoretical principle. The experimental^ discharge approximates to f */ % g h multiplied by the area of the orifice ; but the theoretical velocity \/ 2 g h always approximates to the experimental velocity, or *974 */ 2 gh, obtained immediately outside the orifice in the vena-contracta. It would be unnecessary to allude to this theory here if it were not supported and put forward by three eminent engineers whose authority may mislead others. Vide p. 4. Brief observations of Messrs. Bidder, Hawksley, and Bazalgette on the answers of the Government Referees on the METROPOLITAN MAIN DEAINAGE, or- dered by the House of Commons to be printed 13th July, 1858. OEIFICES, WEIRS, PIPES, AND RIVERS. 47 small, each is found to have practically the same velocity of discharge, when the centres of the con- tracted sections are at the same depth, h, below the surface ; but when lateral orifices are large, or rather deep, the velocity at the centre is not, even prac- tically, the mean velocity ; and in thick plates and modified forms of adjutage, the mean velocities are found to vary. VENA-CONTRACTA AND CONTRACTION. It has been found that the diameter of a column issuing from a circular orifice in a thin plate, is con- tracted to very nearly eight-tenths of the whole diameter at the distance of the radius from it, and that at this distance the contraction is greatest. The ratio of the diameter of the orifice to that of the contracted vein, vena-contracta, is not always found constant by the same or different experimentalists. Newton makes it 1 : -841, J md ' Before, that of the j. 707 I areas as 1 : ) (7156 (622 1 : -6432 1 : -64 656 Poleni 1 : f -846 \ -788 Borda 1 : 802 Michellotti 1 : 8 Bossut 1 : f -81 {818 Du Buat 1 : 816 Venturi 1 : 798 Eytelwein 1 : 8 Bayer 1 : 7854 ,,1: '667 1: '637 1: -64 1 : '617 Bayer's value for the contraction has been deter- mined on the hypothesis, that the velocities of the particles of water as they approach the orifice from all sides, are inversely as the squares of their 48 THE DISCHARGE OF WATER FROM distances from its centre ; and the calculations made of the discharge from circular, square, and rectan- gular orifices, on this hypothesis, coincide pretty closely with experiments. FORM OF THE CONTRACTED VEIN. Let o R zz d, Fig. 3, be the diameter of an orifice ; then at the distance R s zz -^ the contraction is found to be greatest ; we shall assume the contracted di- ameter o r zz '7854 d. If we suppose the fluid column between o R and o r to be so reduced, that the curve lines R r and o o shall become arcs of circles, then it is easy to show from the properties of the circle, that the ra- dius c r must be equal to 1-22^. The mean velocity in the orifice, OR, is to that in the vena-contracta, o r, as *617 : 1 ; and the mouth piece, R r o o, Fig. 4, in which o p zz \ o R, and or zz -7854 x o R, will give for the velocity of discharge at o r, the vena-contracta, v, zz -974 c-- Fig. 4 zz 7-816 in feet very nearly. In speaking of the velo- city of discharge from orifices in thin plates, we always assume it to be the velocity in the vena-contracta, and not that in the orifice itself, ORIFICES, WEIRS, PIPES, AND RIVERS. 49 which varies with the coefficient of discharge, unless in TABLE II., where the mean velocity in the latter, as representing c d v/ 2 g h, is also given. COEFFICIENTS OF CONTRACTION AND DISCHARGE. If we put A for the area of the orifice o R, Pig. 3, and c c x A for that of the contracted section at o r, then c c is called the "coefficient of contraction." The velocity of discharge v^ is equal to c v \/2 gh, equa- tion (2). If we multiply this by the area of the contracted section c c x A, we shall get for the dis- charge D =i c v x c c x A N/ 2 g h* It is evident A \/ 2 g h would be the discharge if there were no contraction and no change of velocity due to the height h; c v x c c is therefore equal to the coefficient of discharge. If we call the latter c d , we shall have the equation (3.) Ct = c v xc c , and hence we perceive that the "coefficient of discharge" is equal to the product of the coefficients of velocity and contraction. In the foregoing expression for the discharge D, h must be so taken, that the velocity at that depth shall be the mean velocity in the orifice A. In full prismatic tubes the coefficients of velocity and discharge are equal to each other. * The expression c v c c */2gh=:c d */2gh is the coefficient of the area A, and, consequently, represents the mean velocity in the orifice ; the coefficient of which is, therefore, equal to c d . The values of the velocity c d \/2 g h t for different heights and coefficients, are given in TABLE II. 50 THE DISCHARGE OF WATER FROM MEAN AND CENTRAL VELOCITY. In order to find the mean velocity of discharge from an orifice, it is, in the first instance, necessary to determine the velocity due to each point in its surface, and the discharge itself; after which, the mean velocity is found by simply dividing the area of the orifice into the discharge. The velocity due to the height of water at the centre of a circular, square, or rectangular orifice, is not strictly the mean velocity, nor is the latter in these, or other figures, that at the centre of gravity. When, how- ever, an orifice is small in proportion to its depth in the water, the velocity of efflux determined for the centre approaches very closely to the mean velocity ; and, indeed, at depths exceeding four times the depth of the orifice, the error in assuming the mean velocity to be that at the centre of the orifice is so small as to be of little or no practical consequence, and for lesser depths it never exceeds 6 per cent. It is, there- fore, for greater simplicity, the practice to determine the velocity from the depth h of the centre of the orifice, unless in weirs or notches ; and the coeffi- cients of discharge and velocity in the following pages have been calculated from experiments on this assumption, unless it shall be otherwise stated. DISCHARGES THROUGH ORIFICES OF DIFFERENT FORMS IN THIN PLATES. The orifices which we have to deal with in practice are square, rectangular, or circular; and sometimes, perhaps, triangular or quadrangular in form. It will ORIFICES, WEIKS, PIPES, AND RIVERS. 51 be necessary to give here only the theoretical ex- pressions for the discharge and velocity for each kind of form, but as the demonstrations are unsuited to our present purposes we shall omit them. TRAPEZOIDAL ORIFICES WITH TWO HORIZONTAL SIDES. Put d for the vertical depth of an orifice, h t for the altitude of pressure at top, above the upper side, and h b for the altitude at bottom, above the lower side, we then get ^b ^t = d. Let us also represent the top or upper side of the orifice A or c, Fig. 5, by / t , and the lower or bottom lt + /b side by / b , and put = /. Now, when l t = / b , the trapezoid becomes a paral- lelogram whose length is I and depth d ; and putting h for the depth to the centre of gravity, we get the equation *t + 2 = ^ 2 = * The general expression for the discharge, D, through a trapezoidal orifice, A, is then B3 52 THE DISCHARGE OF WATER FROM (4.) D^v2 X in which c d is the coefficient of discharge ; and when the smaller side is uppermost as at c, (5.) D= PARALLELOGRAMIC AND RECTANGULAR ORIFICES. When / t z= 4 = I, the orifice becomes a parallelo- gram, or a rectangle, B, and we have for the discharge (6.) D= NOTCHES. When the upper sides of the orifices A, B, and c, rise to the surface as at A O , B O , and c , h t becomes nothing, and we get, as h b = d, for the trapezoidal notch A with the larger side up, (7.) D = c d N/2^x 3"" 2 for the trapezoidal notch, c , with the smaller side up, (8.) D = C the same in form, but not in value, as the preceding equation ; and for a parallelogramic or rectangular notch B , ORIFICES, WEIRS, PIPES, AND RIVEES. 53 (9.) D = c, v/^7 X -Id = - cj N/ 2 ff . o o It is easy to perceive that the forms of equations (4) and (5), and also of equations (7) and (8), are iden- tical. The values for the discharge in equations (6) and (9) are equally applicable, whether the form of the orifice be a parallelogram or a rectangle, the only difference being in the value of the coefficient of discharge, c d , which becomes slightly modified for each form of orifice. TRIANGULAR ORIFICES WITH HORIZONTAL BASES, AND RECTILINEAL ORIFICES IN GENERAL. When the length of the lower side, 4 = 0, the orifice becomes a triangle, D, Fig. 6, with the base upwards. Fig. 6 In this case, equation (4) becomes (10.) .. which gives the discharge through the triangular orifice, D. When / t = 0, in equation (5), the orifice becomes a triangle, F, with the base downwards ; in this case, we find for the value of the discharge, 54 THE DISCHARGE OF WATER FROM (11.) r, = c dN /2^xf />(*!-! X**-** As any triangular orifice whatever can be divided into two others by a line of division through one of the angles parallel to the horizon ; and as the dis- charge from the triangular orifice D or F is the same as for any other on the same base and between the same parallels, we can easily find, by such a divi- sion, the discharge from any triangle not having one side parallel to the horizon, and thence the discharge from any rectilineal figure whatever by dividing it into triangles. If the triangle F be raised so that the base shall be on the same level with the upper side of the trian- gular orifice D ; if, also, the bases be equal, and also the depths, we shall find, by adding equations (10) and (11), and making the necessary changes indicated by the diagram, (12.) D = c d s2 X {+- 2 X for the discharge from a parallelogram E with one diagonal horizontal. Now this is the same as the discharge from any quadrilateral figure whatever, having the same horizontal diagonal, and also having the upper and lower angles on the same parallels, or at the same depths, as those of the parallelogram. If the orifices D, F, and E rise to the surface of the water, as at D O , E O , F O , we shall then have for the dis- charge from the notch D O , which for a right angled triangle becomes ORIFICES, WEIRS, PIPES, AND RIVERS. 55 : D = C d N X <. For the discharge from the notch F O , (14.) D = c dV /2^ x ^^: and for the discharge through the notch E O , (15.) D=c d v/2^x ] ^{4~2^} = c d v/2^x -9752/^1 When the parallelogram E O becomes a square / = 2 d, and hence, (16.) D=c dV / 2^X-9752/*x v/J = c d v/2^X'34478 l\ The foregoing equations will enable us to find an expression for the discharge from any rectilineal orifice whatever, as it can be divided into triangles, the discharge from each of which can be determined as already shown in the remark following equation (11.) The examples which we have given will be found to comprehend every form of rectilineal orifice which occurs in practice ; but for the greater number of orifices, sunk to any depth below the surface, the * In the Civil Engineer and Architect's Journal, 1858, p. 370, it is stated that Professor Thompson, Belfast College, gave at the British Association in Leeds for a right angled triangle, for discharges of from 2 to 10 cubic feet per minute, the expression Q = '317 H"ST, in which Q is the quantity in cuhic feet per minute, and H the head in inches. Now the ahove equation for a coeffi- cient of -617 becomes, for inch measures, D = 17-153 x T % d 2 = _5 9-15 d2; or by multiplying by 60, and dividing by 1728, to reduce the discharge to feet per minute, we get D = -317 d^, identically the same as Professor Thompson derived from his experiments. All sections of a triangular notch are similar tri- angles, and hence the advantage of a triangular-notch-gauge, where it can be used, as, probably, the coefficient remains constant. Professor Thompson, I believe, first drew attention to this. 56 THE DISCHAEGE OF WATEE FROM discharge will be found with sufficient accuracy by multiplying the area by the velocity due to the centre. CIRCULAR AND SEMICIRCULAR ORIFICES. The discharge through circular and semicircular orifices in thin plates can only be represented by means of infinite series. Let us represent by s 1 the sum of the series Fig.7 -\ \x / v Y \ f 1 \ = H e^^-gjF^=^ m _f\ 1357 9\/l 1 3 5\/^ ^2 4 6 o 10 12 2 4 6 o / Let us also represent by s 2 the sum of the series _M(i.IV . (I.I.* 3*1416( ^2 * 3^ A ^~ V2 ' 4 * 6 2*4*6*8 1(K ^3 5 then the discharge from the semicircle G, Fig. 7, with the diameter upwards and horizontal, is (17.) D = c d \/%g~h x 3-1416 r 8 (^ + s 2 ). And the discharge from the semicircle i, with the diameter downwards and horizontal, is (18.) D = c d \/2gh x 3-1416 7-2 (s 1 .s 2 ). If we put A for the area, we shall also have for the discharge from a circle H, OKIFICES, WEIES, PIPES, AND EIVEKS. 57 (19.) v = c (i \2g/i X In each of these three equations (17), (18), and (19), h is the depth of the centre of the circumference below the surface, and r the radius. When the orifices rise to the surface, we have for the discharge from a semicircular notch G O , with the diameter horizontal and at the surface, (20.) D = c d v2#^ X '9586 r 2 = c d v2#r X '6103 A ; when the circumference of the semicircle is at the surface, and the diameter horizontal, as at I , when the horizontal diameter of the semicircle is uppermost, and at the depth r below the surface, (22.) D = c d v/27rx 1-8667 r^^v/^^x 1-1884 A; and when the circumference of the entire circle is at the surface, as at H O , (23.) D = c d v/2<7>x 3-0171 r 2 = c d \/2^r x '9604A. If we desire to reduce equations (20), (21), and (22), to others in which the depth h of the centre of gravity from the surface is contained, we have only to substitute ^ for r in equation (20), and we shall get, for the discharge from a semicircle with the diameter at the surface, (24.) D = c d \/2gh X '0367 A : also, by substituting 7/yrinr for r in equation (21), we get, for the discharge from a semicircle when the circumference is at the surface and the diameter horizontal, (25.) D = c d V Zgh X -9653 A ; 58 THE DISCHARGE OF WATER FROM and when the horizontal diameter is uppermost, and at the depth r below the surface r = i .4044 an ^ (26.) D = c d v/2#A x -9957 A. As A stands for the area of the particular orifice in each of the preceding expressions for the dis- charge, it must be taken of double the value, in equation (23) for instance where it stands for the area of a circle, that it is in equations (20), (21), or (23), where it represents only the area of a semicircle. MEAN VELOCITY. The mean velocity is easily found by dividing the area into the discharge per second given in the pre- ceding equations. For instance, the mean velocity in the example represented in equation (9), is equal , which is had by dividing the area Id into the discharge ; and in like manner the mean velocity in equation (23) is -9604 c d \/ 2g r. PRACTICAL REMARKS ON THE DISCHARGE FROM CIRCULAR ORIFICES. It has been shown, equation (19), that, for the discharge from a circle, we have x 2 A*! in which h is the depth of the centre, A the area, and s l the sum of the series I 1 -/' 1 l \f l -r 1 l 3 5 \( l l 3 V- II Vl'TAi'lJF UTi'lJUTOF ORIFICES, WEIRS, PIPES, AND RIVERS. 59 and it has also been shown, equation (23), that, when the circumference touches the surface, this value becomes D = c d v/2^r x '9604 A. Now when h is very large compared with r, it is easy to perceive that 2 ^ = 1, and hence (27.) D = c d \/%gh x A. As this is the formula commonly used for finding the discharge, it is clear, if the coefficient c d remain con- stant, that the result obtained from it for D would be too large. The differences, however, for depths greater than three times the diameter, or 6 r, are practically of no importance ; for, by calculating the values of the discharge at different depths, we shall find, when h = r, that D = c d (28.) ;? 3 r 2' 7 r X '960 A ; X '978 A; X -985 A; X '989 A; X '992 A ; X '996 A; X -998 A; X '9987 A ; X '9991 A. These results show very clearly that, for circular orifices, the common expression for the discharge c d N/2^^ x A is abundantly correct for all depths h h h = 3r, 4r, 5r, )> D D D = Cd = C d = C d 60 THE DISCHARGE OF WATER FROM exceeding three times the diameter, and that for lesser depths the extreme error cannot exceed four per cent, in reduction of the quantity found by this formula. We shall show, hereafter, when discussing the value of c d , that from the sinking of the surface, and perhaps other causes, the discharge at lesser depths is even larger than that exhibited by the expression c d \/2 g h, x A, the value of the coefficient of discharge, c d , being found to increase as the depths h decrease. In fact, the sides of the orifice, the rounding of the arrises, and the depth and position with reference to the sides of the vessel, and surface of the water, are of far greater practical importance than extreme accuracy in the mathematical formula. PRACTICAL REMARKS ON THE DISCHARGE FROM RECTANGULAR ORIFICES. It has been shown, equation (6), that the discharge from rectangular orifices, with two sides parallel to the horizon or surface of the water, is expressed by the equation D = c d x \ v/2~> x I (h\ - hj }, o in which / is the horizontal length of the orifice, A b the depth of water on the lower, and h t the depth on the upper, side. As it is desirable in practice to change this form into a more simple one, in which the height h of the centre and depth d of the orifice only shall be included, we then have h b == h + and ORIFICES, WEIRS, PIPES, AND RIVERS. 61 h t = h --- By substituting these values of A b and 2 k t in the foregoing equations, and developing the result into a series, the terms of which, after the third, may be neglected, and putting A. for the area I d, we shall find, _ ( d* ^ (29.) D = c d v/2 g h x A|l ^g-^ [very nearly. We have therefore for the accurate theoretical dis- charge (30.) D = d -- for the approximate discharge D = c v2 X A , jl and for the discharge by the common formula D = c d v2 X A. When the head (h) is large compared with (d) the height of the orifice, each of the three last equations gives the same value for the discharge ; but as the common expression c d \/2 g h X A is the most simple ; and as the greatest possible error in using it for lesser depths does not exceed six per cent., viz. when the orifice rises to the surface and becomes a notch, it is evidently that formula best suited for practical pur- poses. The following table and equations will show more clearly the differences in the results as obtained from the true, the approximate, and the common for- mulce, applied to " lesser " heads ; and they will also explain, to some extent, why " coefficients " deter- mined from the common formula, and that used by Poncelet and Lesbros, should decrease as the orifice approaches the surface. 62 THE DISCHARGE OF WATER FROM (31.) d 2> 5d D= 596 2 635 635 668 M 577 3 606 606 606 572 4 593 593 593 593 593 Means 613 613 632 593 585 The most valuable series of experiments of which we are possessed are those made at Metz, by Poncelet and Lesbros. They were made with orifices 8 inches wide, nearly, and of different vertical dimen- sions placed at various depths down to 10 feet. The discrepancies as to any general law in the relation of the different values of the coefficient of discharge c d 72 THE DISCHARGE OF WATER FROM to the size and depth of the orifice in the preceding experiments, have been remedied 'to a great extent by these. They give an increase of the coefficients for the smaller and very oblong orifices as they approach the surface, and a decrease under the same circum- stances in those for the larger square and oblong orifices. There are a few depths where maximum and minimum values are obtained : we use the terms "maximum and minimum values" for those which are greater in the one case and less in the other than the coefficients immediately before and after them, and not as being numerically the greatest or least values in the column. We have marked with a *, in the arrangement of the coefficients, TABLE I,, these maximum and minimum values. The heads given in this table were measured to the upper side of the orifices, and by adding half the depth (d) to any particular head, we shall obtain the head at the centre. As a perceptible sinking of the surface takes place in heads less than from five to three times the depth of the orifice, the coefficients are arranged in pairs, the first column containing the coefficients for heads measured from the still water surface some distance back from the orifice, and the second ob- tained when the lesser heads, measured directly at the orifice, were used. A very considerable increase in the value of the coefficients for very oblong and shallow small orifices, may be perceived as they ap- proach the surface, and the mean value for all rectilinear orifices at considerable depths, seems to approach to *605 or *606. OEIFICES, WEIKS, PIPES, AND RIVERS. 73 We have shown, equation (29), that the discharge is approximately, in which expression d is the depth of the orifice, and h the head at its centre. Now it is to be observed, that it is not the value of c d simply, which is given in TABLE I., but the value of c d x 2 g h, equation (29). The coefficients in the table are, therefore, less than the coefficients of discharge, strictly so called, by a c d? quantity equal to Q ^ , g . The value of this expression is in general very small, and it is easy to perceive from the first of the expressions in equation (31), p. 62, that it can never exceed 4-2 per cent., or more correctly 0417 in unity. If we wish to know the discharge from an orifice 4 inches square = 4" x 4", with its centre 4 feet below the surface, which is equivalent to a head of 3 feet 10 inches at the upper side, we find from the table the value of c d jl ,A = 601 ; hence we shall get D = -601 x A v/2# h = -601 x - x 8'025 x 2 = 601 x i x 16-05 = - x 9-646 = 1-072 9 9 cubic feet per second. In the absence of any expe- riments with larger orifices, we must, when they occur, use the coefficients given in this table ; and, in order to do so with judgment, it is only necessary to observe the relations of the sides and heads. For example, if the size of an orifice be 16"x4", we must 74 THE DISCHARGE OF WATER FROM seek for the coefficient in that column where the ratio of sides is as four to one, and if the head at the upper side be five times the length of the orifice, we shall find the coefficient *626, which in this case is the same for depths measured behind, or at the orifice. For lesser orifices, the results obtained from the experiments of Michelotti and Bossut, pages 67 and 68, are most applicable; and also the coefficients of Rennie, page 71. It is almost needless to observe, that all these coefficients are only applicable to orifices in thin plates, or those having the outside ar- rises chamfered as in Fig. 8. Very lit- tle dependence can be placed on calculations of the quantities of water discharged from other orifices, unless where the coefficients have been already obtained by experiment or correct inference for them. If the inner arris next the water be rounded, the coefficient will be increased. NOTCHES AND WEIRS. We have already given some coefficients, pages 69 and 70, derived from the experiments of Du Buat, Brindley and Smeaton, and Poncelet and Lesbros, for finding the discharge over notches in the sides of large vessels ; and it does not appear that there is any difference of importance between these and those for orifices sunk some depth below the surface, when the proper formula for finding the discharge for each ORIFICES, WEIRS, PIPES, AND RIVERS. 75 is used. If we compare Poncelet and Lesbros' co- efficients for notches, page 70, with those for an orifice at the surface, TABLE L, we perceive little practical difference in the results, the head being measured back from the orifice, unless in the very shallow depths, and where the ratio of the length to the depth exceeds five to one. The depths being in these examples less than an inch, it is probable that the larger coefficients found for the orifice at the surface, arise from the upper edge attracting the fluid to it and lessening the effects of vertical con- traction, as well as from less lateral contraction. In- deed, the results obtained from experiments with very shallow weirs, or notches, have not been at all uniform, and at small depths the discharge must proportionably be more affected by movements of the air and external circumstances than when the depths are considerable. We shall see that in Mr. Blackwell's experiments the coefficient obtained for depths of 1 and 2 inches was -676 for a thin plate 3 feet long, while for a thin plate 10 feet long it in- creased up to -805. The experiments of Castel, with weirs up to about 30 inches long, and with variable heads of from 1 to 8 inches, lead to the coefficient -497 for notches ex- tending over one-fourth of the side of a reservoir ; and to the coefficient -664 when they extend for the whole width. For lesser widths than one-fourth, the coefficients decrease down to -584 ; and for those extending between one-third of, and the whole width, they increase from -600 to -665 and -680. Bidone finds c d = -620, and Eytelwein c d = -635. It will be 76 THE DISCHARGE OF WATER FROM perceived from these and the foregoing results, that the third place of decimals in the value of c d , and even sometimes the second, is very uncertain ; that the coefficient varies with the head and ratio of the notch to the side in which it is placed ; and we shall soon show that the form and size of the weir, weir- basin, and approaches, still further modify its value. When the sides and edge of a notch increase in thickness, or are extended into a shoot, the coeffi- cients are found to reduce very considerably ; and for small heads, to an extent beyond what the in- crease of resistance, from friction alone, indicates. Poncelet and Lesbros found, for orifices, that the addition of a horizontal shoot, 21 inches long, re- duced the coefficient from *604 to '601, with a head of 4 feet ; but for a head of only 4J inches, the coefficient fell from -572 to -483, the orifice being 8" x 8". For notches 8 inches wide, with a hori- zontal shoot 9 feet 10 inches long, the coefficient fell from -582 to -479, for a head of 8 inches ; and from 622 to -340, for a head of only 1 inch. Castel found also, for a notch 8 inches wide with a shoot 8 inches long attached and inclined at an angle 4 18', that the mean coefficient for heads from 2 to 4J inches was only -527. Little dependence can be placed on ex- perimental results obtained for shoots which partake of the nature of short pipes, and should be treated in like manner to find the discharge.* We have obtained the following table of coefficients from some experiments made by Mr. Ballard, on the river Severn, near Worcester, " with a weir 2 feet * Trait6 Hydraulique, par D'Aubuisson, pp. 46, 94 et 95. OKIFICES, WEIRS, PIPES, AND RIVERS. 77 COEFFICIENTS FOR SHORT WEIRS OVER BOARDS. Heads measured on the crest. Depths in inches. Coefficients. Depths in inches. Coefficients. Depths in inches. Coefficients. 1 762 3 801 5 733 i| 662 8J 765 H 713 ii 673 3* 748 i 735 u 692 31 740 5| 729 2 684 4 759 6 727 | 702 4* 731 7 716 ^ 756 4i 744 8 726 8* 786 4f 745 Mean 732 long, formed by a board standing perpendicularly across a trough."* The heads or depths were here measured on the weir, and hence the coefficients are larger than those found from heads measured back to the surface of still water. Experiments made at Chew-Magna, in Somerset- shire, by Messrs. Blackwell and Simpson, in 1850f, give the following coefficients. COEFFICIENTS DERIVED FROM THE EXPERIMENTS OF BLACKWELL AND SIMPSON. Heads in inches. Coefficients. Heads in inches. Coefficients. Heads in inches. Coefficients. Ito I 591 44 743 6 749 Ito 11 626 41 760 6 3 748 16 16 16 23 to 21 16 682 4 I 741 6fg to 6i 747 24 665 41 16 750 61 5 16 772 2 32 670 725 7 21 717 35 32 21 665 5 780 8 802 g29 653 51 781 8 to 8 13 737 32 16 16 215 654 5L 3 749 8 16 750 16 32 16 725 itoS 751 9 781 4 745 ^ 728 Mean 723 * Civil Engineer and Architect's Journal for 1851, p. 647. f Civil Engineer and Architect's Journal for 1851, pp. 642 and 645. 78 THE DISCHARGE OF WATER FROM " The overfall bar was a cast-iron plate 2 inches thick, with a square top." The length of the over- fall was 10 feet. The heads were measured from still water at the side of the reservoir, and at some distance up in it. The area of the reservoir was 21 statute perches, of an irregular figure, and nearly 4 feet deep on an average. It was supplied from an upper reservoir, by a pipe 2 feet in diameter and of 19 feet fall ; the distance between the supply and the weir was about 100 feet. The width of the reservoir as it approached the overfall was about 50 feet, and the plan and section, Fig. 9, of the weir and overfall in connection with it, will give a fair idea of the circumstances attending the experiments. For heads over 5 inches the velocity of approach to the weir was "perceptible to the eye," though its amount was not determined. We perceive that the coefficient (derived from two experiments) for a depth of 8 inches is *802, while the coefficient (derived from three experiments) for a depth of 7J inches is ORIFICES, WEIRS, PIPES, AND RIVERS. 79 717, and for depths from 8 to 8H inches the mean coefficient is -743 : as all the attendant circumstances appear the same, these discrepancies and others must arise from the circumstances of the case : perhaps the supply, and, consequently, the velocity of ap- proach, was increased while making one set of ex- periments, without affecting the still water near the side where the heads appear to have been taken. By comparing the results with those obtained by one of the same experimenters, Mr. Black well, on the Kennet and Avon Canal, we shall immediately per- ceive that the velocity of approach, and every circumstance which tends to alter and modify it, has a very important effect on the amount of the discharge, and, consequently, on the coefficient. The experiments made by Mr. Blackwell, on the Kennet and Avon Canal, in 1850*, afford very valuable instruction, as the form and width of the crest were varied, and brought to agree more closely with actual weirs in rivers than the thin plates or boards of earlier experimenters. We have calculated and ar- ranged the coefficients in the following table from these experiments. The variations in the values for different widths of crest, other circumstances being the same, are very considerable ; and the differences in the coefficients, at depths of 5 inches and under, for thin plates and crests 2 inches wide, are greater than mere friction can account for ; and greater also than the differences at the same depths between the coefficients for crests 2 inches thick, and 3 feet long. * Civil Engineer and Architect's Journal, 1851, p. 642. 80 THE DISCHARGE OF WATER FROM When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients are marked with a *. The effect of the converging wing-boards is very strongly marked. NOTE. Francis' experiments give a coefficient of '565 for a level crest 3 feet wide, and a head slope of 3| to 1, see p. 121. Crests 3 feet wide. 00 ||5 t- O . O . CO b- CO OS .rH .'I* O 1 ', '. * in i-H OS 00 CO 00 CO O 00 fr- * rH rH \ CO OO O co-^ oo ^i ^ ^p { !i * * # * * Gx t?* t^ t>- O O b* CS OS m O OS * 00 CO CO ^ Tj< O ^ ^ ^ -^ ll l^-COOSO ,rHt .00 CO CO CO O . CO CT 1 ^ * OS TtlJOOT^ OO * ^ fe 15 CO rHCO .COrHCJ T^H -rH CO CO ^^ * rH OS OS OOOrflO O^-rH bo Si CO C7* O5 CO ^ -IIOAJ989I 9qj Ut 'saqout ut spB9H rHdCO^OCOJ^OOCSOcMT* rH rH rH OEIFICES, WEIRS, PIPES, AND RIVERS. 81 The plan and section, Fig. 10, will give a fair idea of the approach to, and nature of the overfall made The dotted lines on Plan show the sub- merged masonry ap- pearing at C in Section. use of in these experiments. The area of the reservoir was 2 A. In. 3 OP., and the head was mea- sured from the surface of the still water in it, which remained unchanged between the beginning and end of each experiment. The width of the approach A B from the reservoir was about 32 feet ; the width at a b about 13 feet, below which the waterway widened suddenly, and again narrowed to the length of the overfall. The depth in front of the dam appears to have been about 3 feet ; the depth on the dam, next the overfall, about 2 feet ; and the depth on the sunk masonry in the channel of approach, about 18 inches. Altogether, the circumstances were such as to in- crease the amount of resistances between the reservoir, from which the head was measured, and the overfall, particularly for the larger heads, and we accordingly 82 THE DISCHARGE OF WATER FROM see that the coefficients become less for heads over six inches, with a few exceptions. The measure- ments of the quantities discharged appear to have been made very accurately, yet the discharges per second, with the same head and same length of over- fall, sometimes vary ; for instance, with the plank 2 inches thick and 10 feet long, the discharge per second for 4 inches head varies from 6-098 cubic feet to 6-491 cubic feet, or by about one-sixteenth of the whole quantity. Most of the results, however, are means from several experiments. The quantities discharged varied from one-tenth of a cubic foot to 22 cubic feet per second, and the duration of the experiments from 24 to 420 seconds. If we compare the coefficients for a plank 10 feet long and 2 inches thick in the foregoing table with those for the same overfall at Chew-Magna, we shall immediately per- ceive how much the form of the approaches affects the discharge. Indeed, were the area of the reser- voir at Chew-Magna even larger than that for the Kennet and Avon experiments, it would be found, notwithstanding, that the coefficients in the former would still continue the larger, though not fully as large as those found under the particular cir- cumstances.* * There is a very important omission in all the preceding experiments on weirs and notches. In Fig. 10, for instance, it would have been necessary to obtain the heads at A B and a b in each experiment, above the crest, and also the head on and a few feet above the crest itself. These are, perhaps, best calculated by means of the observed velocity of approach. They would indicate the resistances at the different passages of approach, and enable us to calculate the coefficients correctly, and thereby render them more generally applicable to practical purposes. The coefficients ORIFICES, WEIRS, PIPES, AND RIVERS. 83 The following table gives the mean results of 88 experiments made by Francis, at the Lower Lock, Lowell, Massachusetts, in 1852. The duration of each of these experiments varied from 180 to 822 seconds. The coefficients in column 10 have been 1 2 3 4 5 6 7 8 9 10 1 it il >* gi fr* 11 iff .s| ^ 03 multiplier ormula in i PH . -4j 2 a o o -*^* ^ r< ** T 1 S Q) tM .2 p< Si \ Observed n over weir i Observed di cubical fe cond. Observed ve! proach in second. 5-^ A Values of formula it | + o Values of th c in the columnS. Correspondi the coefflc charrge c< 1 9-997 1-55 62-6 78 1-56 1-56 62-6 3-32 621 2 9-997 1-24 45-6 59 1-25 1-25 45-4 3-33 623 3 9-997 1-00 33-4 44 1-00 1-00 32-5 8-32 621 4 7-997 1-01 26-8 36 1-02 1-02 26-3 3-36 628 5 9-997 1-05 36- 97 1-06 1-06 35-8 3-35 626 6 9-995 0-98 32-6 54 0-99 .98 32-4 3-34 624 7 9-995 1-00 33-5 55 1-01 1-00 33-3 3-33 623 8 9-997 0-80 23-5 33 80 80 23-4 3-32 621 9 9-997 0-82 25- 75 83 83 24-8 3-34 624 10 9-995 0-80 23-9 40 80 80 23-8 3-34 624 11 9-997 0-62 16-2 23 62 62 16-0 3-33 623 12 9-997 0-65 17-5 53 65 65 17-2 3-33 623 13 7-997 0-68 14-6 45 68 68 14-5 3-34 623 calculated by ourselves, and the other results con- densed from the large table given in Francis' Book.* in the two previous tables are not as valuable as they otherwise would be from this omission. The level of still water near the banks is below that of the moving water in the current, therefore, heads measured from still water must give larger coefficients than if taken from the centre of the current. This may account, to some extent, for the larger coefficients in the first table, but apart from this, the short contracted channel immediately above the water-fall, Fig. 9, must increase the coefficients. .* Lowell Hydraulic Experiments. New York, 1855. G3 84 THE DISCHAEGE OF WATEE FEOM The heads given in the 6th column are those which would give the observed discharge from the formula o As we have also equation (39) we must, therefore, have the values of which are given in column 6. The values of h" in column 8 are those which would be found by resolving the equation D=O(/+ -Ink")*!* n being the number of end contractions, and c a multiplier varying from 3*32 to 3-36. 2 In this table the theoretical head ~ zz -0155 vjdue to *& the velocity of approach has been used and does not exceed -02 of a foot. We are of opinion, however, that the head is much greater, and should be taken v 2 "2 ^o~~ = '04 vl or thereabouts. This would reduce c d x &g the values of the coefficient of discharge c d in the 10th column. The differences between A, ti, and h" in columns 3, 6, and 7 are here, practically, of little moment, and the value of c d in column 10 would be nearly the same derived from either. The crest of the weir experimented upon was 1 inch thick. The weir measuring 10 feet x 13 inches x 1 inch, the top was rounded off at both arrises, leaving the central horizontal portion one quarter of an inch wide. The general result of these experiments verifies the OEIFICES, WEIKS, PIPES, AND KIVEES. 85 ordinary coefficient for notches in thin plates from 617 to -628 for the value of c d . Professor Thomson's experiments with right- angled triangular notches, in thin plates, give a mean coefficient of -617. Vide Note p. 55. HEAD, AND FROM WHENCE MEASURED. By referring to TABLE I., we shall see that there is a difference in the coefficients as obtained from heads measured on or above the orifice. This dif- ference is greater in notches, or weirs, than in orifices sunk below the surface ; and when the crest of a weir is of some width, the depths upon it vary. In the Kennet and Avon experiments, the heads mea- sured from the surface of the water in the reservoir, and the depths at the "outer edge" (by which we understand the lower edge) of the crest were as follows : DIFFEEENCE OF HEADS MEASUEED ON AND ABOVE WEIES. I Heads on crests Heads on II 2 inches thick. crests 3 feet wide. a -9 I I ao'3 l! Jfc !fe |1 fi bJDOJ V 1 eo 1 D 11 w 11- 03 W ll || "S-2.9 1 1 .. 7 16 .. i 4 r 16 5 16 2 .. i 1 16 3 . . ll? U to 1J 4 3to2g 3J U If u .. .. li 5 3* 3 r 2 i 14 .. 14 6 41 4 2| .. 'i j'*:;; 24 2| 2i 7 .. ft 2 .. 8 61 'J ;>". ; : r 31 34 9 grV .. .. ....... .. 4i N .. 10 4 I 86 THE DISCHARGE OF WATER FROM No intermediate heads are given, but those registered point out very clearly the great differences which often exist between the heads measured on a weir, or notch, and those measured from the still water above it ; and how the form of the weir itself, as well as the nature of the approaches, alters the depth pass- ing over. On a crest 2 feet wide, with 14| inches depth on the upper edge, we have found that the depth on the lower edge is reduced to 11J inches, or as 1*26 to 1. The head taken from 3 to 20 feet above the crest, where the plane of the approaching water surface becomes curved, is that in general which is best suited for finding the discharge by means of the common coefficients, but a correct section of the channel and water-line, showing the different depths upon and for some distance above the crest, is neces- sary in all experiments for determining accurately by calculation the value of the coefficient of dis- charge c d . Du Buat, finding the theoretical expression for the discharge through an orifice of half the depth h, equation (6) to agree pretty closely with his experiments, seems to have assumed that the head h is reduced to ^ in passing over. This is a reduction, however, which never takes place unless with a wide crest and at its lower edge, or where the head h is measured at a ORIFICES, WEIRS, PIPES, AND RIVERS. 87 considerable distance above the weir, and when a loss of head due to the distance and obstructions in channel takes place. When there is a clear weir basin immediately above the weir, we have found that, putting h for the head measured from the sur- face in the weir basin, and h w for the depth on the upper edge of the weir, that (32.) h A w for measures in feet, and (33.) A A w = - for measures in inches. The comparative values of h and h w depend, however, a good deal on the par- ticular circumstances of the case. Dr. Robinson found* Am 1-111 A w , when h was about 5 inches. The expressions we have given are founded on the hypothesis, that h A w is as the velocity of discharge, or as the \/A nearly. For small depths, there is a practical difficulty in measuring with sufficient accu- racy the relative values of h and A w . Unless for very small heads the sinking will be found in general to vary from to -, and in practice it will always be useful to observe the depths on the weir as well as the heads for some distances (and particularly where the widths contract) above it. In order to convey to our readers a more definite idea of the differences between the coefficients for heads measured at the weir, or notch, and at some distance above it, we shall assume the difference of A w A w .,1 A A w the heads A A w zz ; then , , =r, and-zz r ? * Proceedings of the Eoyal Irish Academy, vol. iv. p. 212. 00 THE DISCHARGE OF WATER FROM hence h = r *~ h w and A ff = ^_ k. r r + J- Now the discharge may be considered as that which would take place through an orifice whose depth is h w with a head over the upper edge equal to A ^ w zz-^; hence from equation (6) the discharge is equal to and substituting for h% its value ( r ' A W V, we shall \ r ' find the value of (34.) D = As the value of the discharge would be expressed by 2 X s if the head h h^ were neglected, it is evident the coefficient is increased, under the circumstances, from c d to or, more correctly, the common formula has to be multiplied by (l + i)* (!)*, to find the true dis- charge, and the value of this expression for different values of - = n w m be found in TABLE IV. If we suppose that * z. ^w ., 1 1 h-hv = JQ, then - =j-g = n ; and we find from the table(l + i) l ~( -)* = 1-1221. ORIFICES, WEIRS, PIPES, AND RIVERS. 89 Now if we take the value of c d for the full head h to be -628, we shall find 1-1221 x '628 =-705, rejecting the latter figures, for the coefficient when the head is 1 2 measured at the orifice ; and if - = JQ n, we should find in the same manner the new coefficient to be 1-2251 x '628 = -769 nearly. The increase of the coefficients determined, page 77, from Mr. Ballard's experiments is, therefore, evident from principle, as the heads were taken at the notch ; and it is also pretty clear that, in order to determine the true dis- charge^ the heads both on, at, and above a weir should be taken. Most of the discrepancies in the coefficients determined from experiment have arisen from imper- fect and limited observations of the facts. Amongst these the velocity of approach should never be neglected by observers, as its effect on the discharge is often considerable in increasing the quantity. The effect of the form of the weir and approaches is scarcely ever sufficiently considered by professional men. Most of the discussions which arose with reference to the gaugings on the Metropolitan MAIN DRAINAGE QUESTION would have been obviated if the calculators, or engineers, had taken into account the different circumstances attendant on it, instead of applying generally a formula suited to a particular case, namely, a thin crest, a small notch, and a large body of water immediately above it ; and applied a correct formula for finding the effects of the velocity of approach. The two following tables have been reduced to English measures of feet, from Boileau's experiments ; 90 THE DISCHARGE OF WATER FROM they show the relation of the head to the depth on the crest at the upper arris. The coefficient for the head h being known, we may, from our equation (34), calculate that due to h & on the weir. TABLE showing the ratio of the head, h, to the depth., h^, on a Plank Weir of the full width of the Channel, immediately at the upper edge, or j, see equation (33), when the sheet of water is free after passing over, with air under it. Head h in feet. Values of the head h divided by the thickness of the sheet of water passing over the weir immediately at the upper edge ; average r- = F = 1-2 between heads of 3 and 14 inches. Height of weir in feet, 8&. Height of weir in feet, 1-07'. Height, of weir in feet, 1-33'. Height of weir in feet, Ml*. 1 1-339 1-285 13 1-282 1-320 1-250 16 1-260 1-285 1-228 20 1-234 1-243 1-249 1-214 23 1-223 1-232 1-231 1-205 26 1-216 1-232 1-223 1-200 3 1-212 1-228 1-218 1-199 33 1-210 1-225 1-217 1-199 39 1-206 1-221 1-112 1-197 46 1-202 1-216 1-206 53 1-199 1-201 . . 59 1-196 1-195 .. 66 1-192 1-191 82 1-186 .. 99 1-184 . . *''"*. '" 1-15 1-182 *W' C If we were to use the head h w instead of h, to cal- culate the discharge, when j- zi 1*2, then a coefficient /i w of -628 for the head h would become -769 for the head A w in equation (34) : for - = '2, and, therefore, TABLElV.,-628x(l'2)*-(-2)* = -628x1-2251 =-769. ORIFICES, WEIRS, PIPES, AND RIVERS. 91 TABLE showing the ratio j , equation (33), when the sheet of water ' l Vf passing over is in contact with the crest and with the water im- mediately below a Plank Weir. h Values of i for different heights of weirs and for different "w Head h in feet. heads : mean value for heads between 3 and 14 inches, equal 1=1-25. Height of weir in feet, 1-07'. Height of weir in feet, 1-1'. Height of weir in feet, 1-38'. 43 1-283 46 1-275 1-291 49 1-256 1-266 1-281 53 1-250 1-258 1-271 59 1-236 1-245 1-254 66 1-225 1-232 1-241 73 1-216 1-223 79 1-208 1-216 86 1-202 1-208 92 1-198 1-203 99 ,{!;* - 1-198 .:. If we were to use the head h w instead of h to cal- culate the discharge, when y- = 1-25, then a coeffi- cient of -628 for the head h would become -799 for the head h w in equation (34) : for - = -25 ; and, therefore, the value of c d {(l + -)* - (-)*J,TABLE IV., is -628 x (1-25)* - (-25)*-= -628 x 1'2725 = -799: and so on we may calculate the value of the coeffi- cient to be applied to the depth h w on the weir, for any other ratios between h and h w by means of equation (34). Boileau made some valuable experiments at Metz, 92 THE DISCHARGE OF WATER FROM which were published in 1854. They give the fol- lowing results for vertical plank weirs extending from side to side of the channel, when the water passed over without adhering to the crest : Height of weir over bot- torn of channel in feet. Head above Mean coefficient. 3- -2 to 1-6 -645 1-3 -16 to -5 -622 6 -15 to -25 -625 When the water passing over was joined to the crest, and no air between the sheet passing over and the water below the weir, the experiments gave Height of weir over bot- -^ , , torn of channel in feet. Mean coefficient. 2- 1- to 1-6 -694 1-3 -6 to 1-8 -690 6 -36 to 1-3 -675 When the plank weir leant up-stream 4 inches to a foot, the mean value of c d was -620, the height of weir being 1-5 foot, and with heads from -23 to *5 foot. When its crest was rounded to a semi-cylin- der, the coefficient was, with a head of % 26 foot, '696, and with a head of -52 foot, *843 ; the water adhering to the crest. With a head of -6 foot the coefficient was -867, and with a head of -85 foot, '840, when the water passed over without air between it and the water below the crest. The following tables give the experimental and reduced coefficients for vertical plank weirs of different heights, and with different heads, when the water passes over in a full sheet, and also when it is joined to the crest and lower water. Also for plank weirs suitable for sluices, leaning up-stream with a slope of one-third horizontal to one vertical. ORIFICES, WEIRS, PIPES, AND RIVERS. 93 < F*4f>4iH* > *'**rHC9C$CQO3eOCQOQCQHf'^4i^4<)QtQ<3f^S3 CDCOCOCDCOCDCDCDCDCDCDCDCDCDCDCDCDCOCOCOCDCD CO CO CD CD e ^ CD CD CD CD fn CD " ^^ CD ef "> cf ^ CD tn CD CD ert ef ^ ff^ fff ~> CD COCCCOc^O>C > ?CTCC^^TH-t*-^OOCDt-COCDlOlOO'OcO CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD T-i COCOCOCOC?O?COCC'CO^fl--H-^'OOCOCOCOCDCDCO CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD tft CD C\?CO-H CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD COt~ CD CD COCDOOCDCOCOCDCDCDcD H/* H^ *O ^O CD CD CD CD CD CD CD CD 1^^- i>* t^- GO CDCDCOCOCOCOCOCOCOcDCOCOCDCDCOCO ot-i>QOcscocoioooaoc?ci-i^t^ t^DjT i rHr-'23 the crest into the back-water J feet feet feet feet feet feet feet feet 66 -83 1-00 1-16 1-32 1-48 1'65 2-00 31 -38 -45 -51 -59 -66 92 which shows that the head was drowned (noyee) when the depth of the lower channel below the crest of the weir was less than 24 times the head passing over, taking a general average. ORIFICES, WEIRS, PIPES, AND RIVERS. 95 TABLE of Experimental Coefficients for Plank Weirs leaning up- stream, when the crest has the down-stream arris rounded to a quadrant; and when the crest is cylindrical and projecting up- stream in the form of a knob. Head h in Plank weir leaning np-sti earns one-third to oni- ; the lower arris of crest rounded off to a quadrant of a circle with a radius the full thickness of the plank. Plank weir leaning upwards one- third to one, the crest rounded and projecting in front beyond the plank, so as to be thicker than it. feet. Water free from curve of crest 13 foot thick. Water in contact with curve of crest -17 foot thick. Water in contact with curve of crest -3 foot thick. Water in contact with curve of crest -33 foot thick. 16 589 651 20 589 672 23 594 697 26 612 697 30 633 721 670 33 642 747 604 686 36 649 766 625 700 39 655 768 648 714 43 661 795 669 727 46 667 802 687 741 49 675 702 753 53 679 715 765 56 685 729 775 59 741 786 63 753 795 66 762 802 69 808 72 813 The effect of the form of the crest in increasing the coefficients is distinctly observable in this Table, although the weirs experimented on overhung the water above, between the crest and the bottom of the channel. We must protest against the notation adopted by Boileau and Morin, of giving only two-thirds of the coefficient of discharge, c d , for weirs, instead of the full and true value. The correct formula for the dis- charge from a weir, is D zz - lh \/2gh. Now they 3 assume a coefficient due to an incorrect formula D zz lh \/2gh, which reduces c d to - c d to give the same 3 final results. This leads also to an unnecessary dis- tinction between the coefficients of orifices at the surface, or notches, and orifices sunk to some depth, which, practically, have the same, or nearly the same, general value. 96 THE DISCHARGE OF WATER FROM SECTION IY. VARIATIONS IN THE COEFFICIENTS FROM THE POSITION OF THE ORIFICE. GENERAL AND PARTIAL CONTRACTION. VELOCITY OF APPROACH. PRACTICAL FORMULAE FOR THE DISCHARGE OVER WEIRS AND NOTCHES. CENTRAL AND MEAN VELOCITIES. A glance at TABLE I. will show us that the coeffi- cients increase as the orifices approach the surface, to a certain depth dependent on the ratio of the sides, and that this increase increases with the ratio of the length to the depth : some experimenters have found the increase to continue uninterrupted for all orifices up to the surface, but this seems to hold only for depths taken at or near the orifice when it is square or nearly so : it has also been found that the co- efficient increases as the orifice approaches to the sides or bottom of a vessel: as the contraction becomes imperfect the coefficient increases. These facts probably arise from the velocity of approach being more direct and concentrated under the re- spective circumstances. The lateral orifices A, B, c, D, E, F, G, H, i, and K, Fig. 11, have coefficients dif- fering more or less from each other. The coefficient for A is found to be larger than either of those for B, c, E, or D ; that for G or K larger than that for H or i ; that for H larger than that for i ; and that for F, where the contraction is general, least of all. The contraction of the fluid on entering the orifice F removed from the bottom and sides is com- plete; it is termed, therefore, " general contraction;" that at the orifices A, E, G, H, i, K, and D, is inter- fered with by the sides ; it is therefore incomplete, ORIFICES, WEIRS, PIPES, AND RIVERS. 97 and termed "partial contraction'' The increase in the coefficients for the same-sized orifices at the same Fig-.ll mean depths may be assumed as proportionate to the length of the perimeter at which the contraction is partial, or from which the lateral flow is shut off ; for example, the increase for the orifice G is to that for H as cd -{-de : de; and in the same manner the increase for G is to that for E as cd + de : c d. If we put n for the ratio of the contracted portion c d e to the entire perimeter, and, as before, c d for the coefficient of general contraction, we shall find the coefficient of partial contraction to be equal to (35.) c d + *09 n = c d -f- *1 n nearly, for rectangular orifices. The value of the second term *09 n is derived from various experiments. If we assume *617 for the mean value of c d , we may change the expression into the form (1 + *146 n) c d . When n=%, this becomes 1-036 c d ; when n = |, it becomes T073 c d ; and when n f , contraction is prevented for three-fourths of the perimeter, and the coefficient for partial contraction becomes l'L09c d . The form which we have given equation (35) is, however, the simplest ; but the value of n must not exceed f . If in this case c d =. -617, the coefficient H 98 THE DISCHARGE OF WATER FROM for partial contraction becomes '617 + -09 x f = 617 + -067 =z -684, Bidone's experiments give for the coefficient of partial contraction (1 + -152n)c d ; and Weisbach's (1 + -132 w) c d . VARIATION IN THE COEFFICIENTS FROM THE EFFECTS OF THE VELOCITY OF APPROACH. Heretofore we have generally supposed the water in the vessel to be almost still, its surface level un- changed, and the vessel consequently large compared with the area of the orifice. When the water flows to the orifice with a perceptible velocity, the contracted vein and the discharge are both found to be in- creased, other circumstances being the same. If the area of the vessel or channel in front exceed thirty times that of the orifice, the discharge will not be perceptibly increased by the induced velocity in the conduit ; but for lesser areas of the approaching channel corrections due to the velocity of approach become necessary. It is clear that this velocity may arise from either a surface inclination in the channel, an increase of head, or a small channel of approach. We get equation (6) for the discharge from a rectangular orifice A, Fig. 12, of the length /, with a head measured from still water in which A b and k t are measured to the surface at some distance back from the orifice, as shown in the section. The water here, however, must move along the channel towards the orifice with considerable velocity. If A be the area of the orifice, and c the area of the OEIFICES, WEIRS, PIPES, AND RIVERS. 99 channel, we may suppose with tolerable accuracy that Fig. 12 this velocity is equal to -v , in which v represents c the mean velocity in the orifice. If we also represent by v & the velocity of approach, we get the equation (36.) '; : ,f;':'; : * = x... and consequently the theoretical height (h a ) due to it is (37.) h & = ^ x - = '0155 o 2 c in feet measures.* The height A a may be considered as an increase of head, converting A b into h b + h & , and h t into h t + h & . The discharge therefore now becomes (38.) D = which, for notches or weirs, is reduced to * When the approaching velocity passes through the orifice without contraction, it is evident that the head h & required to produce that A s v 2 velocity, in the orifice with contraction, must be h & = ^ x <> - 2 -^ case equation (40) becomes instead of h & = ^2 x cf~> In like manner we must have h & = ^ = c"a x = '04 vl in feet measures when v & is the velocity of approach and c d = -617. H3 100 THE DISCHARGE OF WATER FROM (39.) D =1 c^^Tg { (Ab + Aj'r _ At} * as h t then vanishes. As D is also equa to A x v , equation (37) may be changed into D 2 1 D 2 (40.) h a X - - -0155 -gin feet measures. zg If this value for h & be substituted in equations (38) and (39), the resulting equations will be of a high order and do not admit of a direct solution ; and in (38) and (39), as they stand, h & involves implicitly the value of D, which we are seeking for. By find- ing at first an approximate value for the velocity of approach, the height A a due to it can be easily found, equation (37) ; this height, substituted in equation (38) or (39), will give a closer value of D, from which again a more correct value of h a can be de- termined ; and by repeating the operation the values of D and h & can be had to any degree of accuracy. In general the values found at the second operation will be sufficiently correct for all practical purposes. It has been already observed that, for orifices, it is advisable to find the discharge from a formula in which only one head, that at the centre, is made use of ; and though TABLE IV., as we shall show, enables us to calculate the discharge with facility from either formula, it will be of use to reduce equation (38) to * The formula for the discharge over weirs, taking into account the velocity of approach, D = 2-95 c d l*J h -f -115 vj, given by D'Aubuisson, Traite Hydraulique, seconde Edition, pp. 78 et 95, and adopted by some English writers and engineers, is incor- rect in principle. In feet measures it becomes D = 5-35c d ZfeX V h + -03494 t?J, which form, with alterations in the numerals and measures, was used for calculating discharges of sewers during the METROPOLITAN MAIN DRAINAGE discussion. ORIFICES, WEIES, PIPES, AND EIVERS. 101 a form in which only the head (h) at the centre is used. The error in so doing can never exceed six per cent., even at small depths, equation (31), and this is more than balanced by the observed increase in the coefficients for smaller heads. The formula for the discharge from an orifice, h, being the head at the centre, is and when the additional head h a due to the velocity of approach is considered, which may be changed into (41.) D= Equation (39), for notches, may be also changed to the form (42.) D= this is similar in every way to the equation (43.) ^ for the discharge from a rectangular orifice whose depth is d, with the head h t , at the upper edge. TABLE III. contains the values of j 1 + v- a | in equation (41), and TABLE IV. the values of in equation (42), or the similar expression in (43), T- or -> being put equal to n; and we perceive that the effect of the velocity of approach is such as to in- crease the coefficient from c d to c d { 1 + j- & } for orifices ft i 102 THE DISCHARGE OF WATEE FROM sunk some distance below the surface, and into for weirs when h & is the height due to the velocity of approach, h the depth of the centre of the orifice, and A b the head on the weir. A few examples, show- ing the application of the formulas (41), (42), and (43), and the application of TABLES I., II., IIL, and IV. to them, will be of use. We shall suppose, for the present, the velocity of approach v & to be given, and no extra head be required to maintain it through v 2 the orifice : in other words when h= = '017^ in feet measures nearly. EXAMPLE I. A rectangular orifice, 12 inches wide by 4 inches deep, has its centre placed 4 feet below the surface, and the water approaches the head with a velocity of 28 inches per second ; what is the dis- charge ? For an orifice of the given proportions, and sunk to a depth nearly four times its length, we shall find from TABLE I. *616 +'627 /jn-i i c d -T 621 nearly. As the coefficient of velocity, equation (2), for water flowing in a channel is about -956, we shall find, column No. 3, TABLE II. the height A a =l|izl-125 inch nearly, corresponding to the velocity 28 inches. Equation (41), now becomes D=12 X4>/2^X -621 f OEIFICES, WEIES, PIPES, AND KIVEES. 103 We also find v/2# h = 192 -6 inches when h = 48 inches, TABLE II. ; therefore D=12x4xl92-6X'62lfl+ }* * 48 ' =9244-8x'621{l + -0234p=9244-8x -621x1-0116, (as {1-0234}>=1-0116 from TABLE III.) = 9244-8 x 628 nearly = 5 805 -1 cubic inches=3-36 cubic feet per second. Or thus: The value of -621 x (1-0234) being found equal -628, D=AX-628v/2#x48. Now for the coefficient -628, and h = 48 inches, TABLE II. gives us -628v/2^ x 48 =120-96 inches; hence we get D=12 x 4x120-96=5806-08 cubic inches=3'36 cubic feet, the same as before, the difference -38 in the cubic inches being of no practical value. If we v 2 find h & from the formula h & = n 2 =2*6 inches. then r we shall get i>=3-41 cubic feet nearly. If the centre of the orifice were within 1 foot of the surface, the effect of the velocity of approach would be much greater ; for then Cd x x + * = (from TABLE L) . 623 = (from TABLE III.) -623 x 1-047 = -652 instead of 628. In this case the discharge is D = 12 x 4 x -652 X 12 = 12 x 4 x -652x96-3 (from TABLE II.) = 12 X 4 x 62-8 = 3014-4 cubic inches = 1-744 cubic feet per second. Or we may find the value of 652 v/2# h directly from TABLE II. thus : The value of -628 y/2^ x 12 = 60-48 -628 The value of -666 \/1g x 12 = 64-14 -652 38 ~T~ 3-66 ::~24: 2-31. Hence -652 \/2^h = 60-48 + 2-31 = 62-79, and the 104 THE DISCHAEGE OF WATER FROM discharge zz 12 x 4 x 62-79 x 3013-92 cubic inches zz 1-744 cubic feet per second, the same as before. v 2 If we take A a zz f~ 2=: 2 -6 inches, we shall find DZZ Agc and A zz 7 x - n^ 3 hence D zz - x 7 x ~^g X - X 628{(l-047) l -(-047) f }. 3 8 3 v / The value of (1-047)* - (-047)* will be found from TABLE IV. equal to 1-0612 ; the value of v 2# x ^ will be found from TABLE II. equal to 6-552, viz. by dividing the velocity 78-630, to be found opposite 8 inches, by 12 ; hence Dzz-x7x-X 6-552 x '628 X 1'0612 3 3 zz x 7 X 4-368 x -628 x 1'0612 OKIFICES, WEIKS, PIPES, AND RIVERS. 105 = - x 7 x 4-368 x -666 nearly 3 = - x 7 x 2-909 -1 X 1-939 3 = 13-573 cubic feet per second = 814-38 cubic feet per minute. Or thus: From TABLE VI, we find, when the coefficient is -628, the discharge from a weir 1 foot long, with a head of 8 inches, to be 109-731 cubic feet per minute. The discharge for a weir 7 feet long, when -r = *047 is therefore 109-731 x 7 x 1-0612 = 815-12 cubic feet per minute. The difference between this value and that before found, 814-38 cubic feet is immaterial, and has arisen from not continuing all the products to a suffi- v * cientnumberof places of decimals. If A a :n 2 =. -87 &g c& inch, then D zz 14-51 cubic feet per second nearly. We have, in equations (36) and (37), pointed out the relations between the channel, orifice, velocity of approach, and velocity in the orifice, viz. . = X t^andA. = X =, in which A. = (neglecting, for the present, the coefficient of velocity in passing through the orifice). As v is the actual v velocity in the orifice, must be the theoretical velocity due to the head h + h^ and therefore v* vl v\ ;, and h =. -3 ; hence ** 4A**=? C C d A V & A 106 THE DISCHARGE OF WATER FROM We have hence (44.) i a - C ' A " \ / j - Q *2 Q ? fl/ C "" Cft A substituting this value in equations (41) and (42), there results (45.) D = A v^AXCajl +^T or D = A < X in which m = -, for the discharge from an orifice at some depth, and for the discharge from a weir, (46.) ^A The two last equations give the discharge when the o ratio of the channel to the orifice - = m is known, and also when the whole quantity of water passing through the orifice, that due to the velocity of approach as well as that due to the pressure, suffers a contrac- tion whose coefficient is c d . When h & zz ^~ "9 ^ C is when the velocity of approach v & passes through the orifice without contraction, we shall get h. vl A 2 1 h ~ vl - vl -" c 2 - A 2 ^ m 2 - 1' consequently, in this case, equation (45) becomes (46o.) D rz A x/2^1 X c d x (l + i^if > ' T ^ and equation (46) in like manner changes into OEIFICES, WEIES, PIPES, AND E1VEES. 107 The last members of these two equations are the same as the like members in (45) and (46), when c d? within the brackets = 1 ; consequently we shall easily find their values for the coefficient 1 in the last page of TABLE Y., for the respective values of c 'h \ m == - and also for those of ^ = 2 _ , When c d zz 1, equation (45) may be changed into Igh \\ This is the equation of Daniel Bernoulli, and only a particular case of the one we have given. 2 2 1 If we put n = gfc&^ the values of (l + gjtn^l)* no o *| and of {l + m ^ } - { m 2 C l_ ^2}^ respectively, can be easily had from TABLES III. and IV. We have, however, calculated TABLE V. for different ratios of the channel to the orifice, and for different values of the coefficient of discharge. This table gives at once the values of as new coefficients, and the corresponding value of T~* or y- zz a o4f h> h* nP cl h i A S * When T 5 = -5 T- sa ^ - -3 we shall have in EXAMPLE II. /& Wl * X O^ - * A -* = -11 and (l + ~* - ( 1-133, TABLE IV., (or 108 THE DISCHARGE OF WATER FROM It is equally applicable, therefore, to equations (41) and (42) as to equations (45) and (46). For in- stance, we find here at once the value of 628 {(1-047)1 --(-047)*} in EXAMPLE II., p. 104, equal to -666, as T- a i= *047, and the next value to it for the coefficient -628, in the table, is *046, opposite to which we find -666, the new coefficient sought. The sectional area of the channel in this case, as appears from the first column, must be about three times that of the weir or notch. TABLE V. is calculated from coefficients, c d , in still water, which vary from -550 to 1. Those from '606 to -650, and the mean value -628 are most suited for application in practice. When the channel is equal to the orifice, the supply must equal the discharge, and for open channels, with the mean coefficient 628, we find, accordingly, from the table, the new coefficient 1*002 for weirs ; or 1 very nearly as it should be. We also find, in the same case, viz. when A =z c, and c d zz -628, that for short tubes, Fig. 13, the re- sulting new coefficient be- comes *807. This, as we shall afterwards see, agrees very closely with the experimental results. When the coefficients in still water are less than -628, or more correctly -62725, the orifice, according to our TABLE V. for the coefficient 1.) Hence in this case *628 x 1-133 = -712 the new coefficient suited to the velocity of v 2 approach. Here of course h & = ^^ (see Note p. 99). OEIFICES, WEIRS, PIPES, AND EIVEES. 109 formula, cannot equal the channel unless other re- sistances take place as from friction in tubes longer than one and a half or two diameters, or in wide crested weirs ; and for greater coefficients the junc- tion of the short tube with the vessel must be rounded, Fig. 14, on one or Fig. 14 more sides ; and in weirs or notches the approaches must slope from the crest and ends to the bottom and sides, and the overfall be sudden. The converging form of the approaches must, however, increase the velocity of approach ; and therefore v a is greater than - x V Q when c is measured between c r o and R o, Fig. 14, to find the discharge, or new coefficient of an orifice placed at r o. As the coefficients in TABLE V. are suited for orifices at the end of short cylindrical or prismatic tubes at right angles to the sides or bottom of a cistern, a correction is required when the junction is rounded off as at nor o, Fig. 14. When the chan- nel is equal to the orifice, the new coefficient in equation (45) becomes The velocity in the short tube Fig. 14 is to that in ( 1 ) the short tube Fig. 13 as 1 to c d y^ 2 nearly, or vl C d ; for the mean value c d zz -628, as 1 to -807. Now, as - is assumed equal to - in the cylindrical or pris- J10 THE DISCHARGE OF WATER FROM matic tube, Fig. 13, - - z=- in the tube Fig. 14 A V & with the rounded junction, for v & becomes TOTI hence, in order to find the discharge from orifices at the end of the short tube, Fig. 14, we have only to multiply the /t numbers representing the ratio - in the first column, ^L TABLE V., by -807, or more generally by c d jj-^ 2} , and find the coefficient opposite to the product. Thus if c d =; *628, we find, when - zi l,c d j-j -- 2} A. N JL c/d ) zz -807 in the table. If, again, we suppose - z= 3, A. then 3 x '807 2-421, the value of - for the tube Fig. 14, and opposite this value of -, taken in column 1, we shall find -651 for the new coefficient. For the cylindrical or prismatic tube, Fig. 13, the new coeffi- cient would be only -642. When the head h & is how- v 2 ever equal to ^ * c z the results must be modified accordingly (see Note p. 99).* * Professor Rankine gives the value of the coefficient of dis- charge, or contraction, for varying values of A and c at a diaphragm in a pipe by the formula 618 When- = o, c d = 1; and when - = 1, c d = -618 ; as it should be c c very nearly for an orifice in a thin plate, to which only, and to A, in the short tube, Fig. 14, the formula is suited (see SECTION X). ORIFICES, WEIRS, PIPES, AND RIVERS. Ill PRACTICAL FORMULA FOR THE DISCHARGE OVER WEIRS. In order to reduce the preceding formulae for weirs and notches to some of the forms in common use, with definite combined numerical coefficients, by substituting 8*025 for v/2^, equation (39) be* comes for feet measures (A.) D a =5-35 c d / {&+ A>-A a *}, and for inch measures, as \f%g = 27*8, the dis- charge, taken also in cubic feet, becomes (B.) D a = -01072 c d I {(h b - A a )t - A>* } When the length I is taken in feet and the depth in inches, we shall have (0.) D a =-1287c d /{(A,^AJ*-A>*}- The three last equations being for seconds of time, we shall get, when the time is taken in minutes for feet measures, the discharge in cubic feet ; (D.) D a -321 c d for inch measures (E.) D. = -6433c and for lengths (I) in feet and depths in inches , : (F.) * = W*Cil{(h, + h$-hJ}. (j If The latter equation, when the coefficient of discharge, c d , is taken at -614 becomes ,D a = 4-74 / {(A b + A a )-A fc *}, and (G.) JDn:4-74/^, when the velocity of ap- ( proach vanishes. For a coefficient of -617 112 THE DISCHARGE OF WATER FROM -^}' and (H.) -p = 4*76// when the velocity of ap- ( proach vanishes. For a coefficient of -623 a ' ' JD=:4'81/A with no perceptible approach. For a coefficient of -628 - A.*}' and a ^ ' JDzr4-85 /A with no perceptible approach. For a coefficient of -648 \ '* 1 D zz 5 I ti% with no perceptible approach. For a coefficient of f or -667 D a iz5-14 / {(A b + A a )^ - Aj}' and ' *' JDzz5*14 /At with no perceptible approach. For a coefficient of -712 = 5-5 I ((Ah-1-A.Fr-A.IV' and ' ' JDz=5.5 / 7^ with no perceptible approach. And finally for a coefficient of -81 /D a = 6-3 /{(AH+AJ* A^Vand (0.) D 6-3 / A when the velocity of approach vanishes. The theoretical value of k & in each of the foregoing equations is in terms of the velocity of approach v a * * h * - 27 in which 2 g must be taken equal to 64-403 for heads in feet, and equal to 772-84 for heads in inches. But it is evident that in order to produce the velocity per second v & passing through the notch with a ORIFICES, WEIRS, PIPES, AND RIVERS. 113 nearly still-water basin above it, that A a must be v l v l increased from its theoretical value ^- to 2 -?in * 9 C ^ L 9 which expression c d is the coefficient of discharge due to the particular notch, or weir, and its attendant circumstances ; whence we must take /P v * Theoretical head Now, unquestionably, the most general coefficient both for notches and submerged orifices, in thin plates, for gauging, whether triangular, rectangular, or circular, is -617, when the orifice or notch is small compared with the approaching channel ; whence for measures in feet A a zz -0408 v% and v a zz4'95 For measures in inches, A a = -0034 t and v & = 17-2 And for measures in which v & is expressed in feet per second, and 7 a in inches h & zz '49 vl 9 and v & zz 1*43 v/A a . By substituting these values of A a , found in terms of the approaching velocity, according to the standards used in the equations from (A) to (p) inclusive, and also in equation (H), we shall be enabled to find the proper discharge from a notch in a thin plate. The values of A a , equation (p), can be found at once in inches from the observed values of v & , to be also taken in inches/for coefficients varying from -584 to -974, by means of TABLE II. Thus, with a coefficient of -617, we shall find, for an approaching velocity of 36 inches per second, that h & becomes 4f = 4-4 inches nearly, while for a coefficient of -666, it is only 3|zz i 114 THE DISCHARGE OF WATER FROM 3-8 inches ; and for a coefficient of 1, the theoretical head is but If zzl-7 inches nearly. From the very nature of the case the approaching velocity must continue nearly unimpaired through the notch with but a very slight reduction arising from the viscidity of the water when it enters the aperture, and separates from the lateral fluid. But in order to give this unimpaired velocity by means of an extra head h & , it is evident that h & must be in- creased above the theoretical value by the amount due to the coefficient of discharge; or, as before v 2 v* stated. h a must be increased from ~ to -%- This * 9 c &^9 value of h a is, perhaps, something too large, owing to the reduction of v & at the moment it enters the notch and is acted upon by the overfall, drawing it away, as it were, from the lateral water above the crest. The numerical results of the respective formula from (A) to (o), inclusive, can be obtained by modify- ing the form as in equation (42) into (Q.) J in which D is the discharge found, when there is no velocity of approach, by the common form D = 5*35 x c d / h*, for which separate values are given in equa- tions from (H) to (o) inclusive ; and numerical values in TABLE VI. ; and l+ "" a multi l )lier suited to the velocity of approach, the values of which can be found from TABLE IV. Suppose, for OEIFICES, WEIKS, PIPES, AND KIVEES. 115 example, D =: 158-1 cubic feet per minute, A b 10 inches, and h a = 4 inches, which is that due to an approaching velocity of 3 feet per second with a coefficient of -648 ; then the multiplier becomes (l + -4)l_ -4! 1-4035, TABLE IV. Hence the dis- charge due to an approaching velocity of 3 feet is 158-1 x 1*4035 = 221-9 cubic feet, or an increase of about 40 per cent. Also, if the common formula were used, it is plain that the coefficient -648 should be increased to -648 x 1-4035, or to -909 nearly, which approximates within 10 per cent, of the theoretical value. Nothing can show more clearly the necessity for varying the coefficients when the ordinary formulae are used, even for a notch in a thin plate : for other notches the coefficients, even for still water above the crest, vary considerably. The form of the equation used by D'Aubuisson and several other writers is (R.) D a = in which c and c are numerical coefficients, and v a the velocity of approach. This form is incorrect in principle, although the values of c and c can be so taken as to give resulting values for D a approximately correct. For feet measures, and time in seconds, Professor Downing makes, after D'Aubuisson, p. 37 of his translation, D a = c d X 5-35 / \hl + -03494^. Doctor Robinson* gives for like measures and time, values varying from D a = 3-55 / H*| o 00 ? ^ 999 o o oo O O rH 9 5 it *5 V I- . CO TH OS o o oo I I 10 9^-3 1 00 O O l> CO CO *H 8 - >O TH 00 CO s W ^ *. s +3 rH rH l> H co -S J rH rH 1-1 *lf 1 H I "I 's 2 10 9 -rf rH OS b * J 5 S* % rH (M O 00 00 t> GO lO p|'S> X 03 1 0< 00 CO OS 9 ^J agg, P> ^. g H 1 1 I t CO CO 1Q g J rH CO OS (N ^ ,H CO w ^ ^ a ^ H < ^* -73 .a TO 00 rH < r^ W d 5 o . s^ t 1 CO 10 l CO OS 1 OQ ^^ O *f) (9b *o . H ^ ? o "S CO CO Ol d !E^ H^ 4 ec * < c* ^ ** fi ,^i ?S "d i CO CO rH o t* os ct S S cp l^l GO '+3 P CO O *H CO ^ **"* **^ v 00 00 ^* 00 10 Pi j* Q O S s rH rH rH rH J rS.S ** V 3 s 5 i lO CO O b b co 00 1 ^ 2 rH d ^c^l - ^3 >O iO O ^ 8 . ?^^ S d O CO 3 CO >o cc ^ ?S CO S CO CO 55 &2 ^ w >o o op I> l> -.8 1 T *? ib >b -^ ig the Metropolitan Sewage D TROACH IN FEET PER SECOND. nanner of obtaining the expe f to -737 nearly. scriptions and Formulae. g 1 o 1 1 velocities of stream in \ t per second . . . j OJ O ^>_, O) 5 v 5 y fl 5 ^ 4 f fJLii : 4-8 \/A 3 H- -1875 Ji v* & ) lal quantities measured . 2S of the coefficients, #, \ the formula = x \/h* + -8 A 2 v V obtained from the actual an titles discharged . ./ , ' * A w t ?|j 8 r2 .S fi S c^ H c < < o < o ^ ORIFICES, WEIRS, PIPES, AND RIVERS. 119 uncertain ; but as the equation D a = 5'5\/hl +'81% hi appears to have been framed by Mr. Taylor, to express special experiments made for Mr. Simpson, in which the quantities varied from 5 to 152 cubic feet per minute, and for heads on a four-foot weir varying from 1 inch to 8 inches,* we must conclude the coefficient for heads measured from still water above the crest in those experiments suited to the form of the weir used, and its attendant circum- stances, is -712. The equations (39) and those from (A) to (o) may be easily changed into forms in which only the depth h b) the velocity of approach, and the coefficient of velocity (in this case equal to that of discharge) c d are introduced. It is, however, only necessary here to reduce the general form (A) p. Ill, for feet measures, which becomes, after, substituting for h & its value c 2 x* 2 tf an( * ma ki n g some reductions, and for time in minutes the discharge is R91 (T.) D a = -- l{ (64-4 C\h, + tfl* - } in which v & still continues the velocity in feet per second, as determined from observation. These for- * Vide p. 22, Letter dated 16th August, 1858, from the Government Referees to the Eight Hon. Lord John Manners, on the subject of the Metropolitan Main Drainage. 120 THE DISCHARGE OF WATER FROM mulge may be again reduced to many others. If we take h b in inches (T) becomes (U.) D a = Mr. Pole, in a letter to Mr. Simpson and Captain Gallon, already referred to, gives the special value, D a = 1-06 / {(3 ^+^)t _<;*}, which corresponds very closely with the experiments made for Mr. Simpson. If we assume c d = -712, which also closely corresponds with those experi- ments, our equation (U) becomes for them D a = 1-225 / {(2-72 h, + !)* J} ; but the amount of the discharge must always depend on the coefficient c d , equation (U) suited to the special circumstances of the case under consideration. The form of equation for the discharge proposed by Mr. Boy den * includes the effects of the end con- tractions : it is D =z c {/ bnh^h^ in which c zz f c d \/%gh, n the number of end con- tractions, / the length of the weir, h^ the head measured from the surface of the water above the curvature of approach, and b a coefficient due to the nature of the end contractions. The mean nume- rical exponent of this formula, derived by Francis from his experiments, is for feet measures, per second, Dzz3-33 (I - * Francis's Lowell Hydraulic Experiments, p. 74. t Ibid, p. 119. OKIFICES, WEIRS, PIPES, AND RIVERS. 121 but the value of c varied from 3-303 to 3-3617. These results give corresponding values of c d = -617 to -628, and when c = 3-33, c d = -623. The experi- mental results compared with this formula have been referred to at p. 83. Francis's Lowell experiments on a wooden dam 10 feet long, level and 3 feet wide at the crest, with a head slope of 3i to 1 in a channel 10 feet wide, give, for heads between 6 and 20 inches, a mean coefficient of -563 or -565. This for feet measures would give for the discharge per second Dm 3-02Ai For greater depths, on this width of crest, the dis- charge would probably rise as high as 3-lA^or 3-3^. The section of the dam was the same as that erected by the Essex Company across the Merrimack Kiver, at Lawrence, Massachusetts. See, also, table of coefficients, p. 80. In equation (13), pp. 54 and 55, we have given a general expression for the value of D through a tri- angular notch. Professor Thomson, of the Queen's College, Belfast, in a paper read at the British Asso- ciation at Leeds in 1858, says : " The ordinary rectangular notches, accurately ex- perimented on as they have been, at great cost and with high scientific skill, in various countries, with the view of determining the necessary formulas and coefficients for their application in practice, are for many purposes suitable and convenient. They are, however, but ill adapted for the measurement of very variable quantities of water, such as commonly occur to the engineer to be gauged in rivers and streams. 122 THE DISCHARGE OF WATER FROM If the rectangular notch is to be made wide enough to allow the water to pass in flood times, it must be so wide that for long periods, in moderately dry weather, the water flows so shallow over its crest, that its indications cannot be relied on. To remove, in some degree, this objection, gauges for rivers or streams are sometimes formed, in the best engineer- ing practice, with a small rectangular notch cut down below the general level of the crest of a large rectan* gular notch. If now, instead of one depression being made for dry weather, we use a crest wide enough for use in floods, we conceive of a large number of de- pressions extending so as to give the crest the appearance of a set of steps of stairs, and if we conceive the number of such steps to become in- finitely great, we are led at once to the conception of the triangular instead of the rectangular notch. The principle of the triangular notch being thus arrived at, it becomes evident there is no necessity for having one side of the notch vertical, and the other slanting; but that, as may in many cases prove more convenient, both sides may be made slanting, and their slopes may be alike. It is then to be observed, that by the use of the triangular notch, with proper formulas and coefficients derivable by due union of theory and experiments, quantities of running water from the smallest to the largest may be accurately gauged by their flow through the same notch. The reason of this is obvious, from consider- ing that in the triangular notch, when the quantity flowing is very small, the flow is confined to a small space admitting of accurate measurement ; and that ORIFICES, WEIRS, PIPES, AND RIVERS. 123 the space for the flow of water increases as the quantity to be measured increases, but still continues such as to admit of accurate measurement. " Further, the ordinary rectangular notch, when ap- plied for the gauging of rivers, is subject to a serious objection from the difficulty or impossibility of pro- perly taking into account the influence of the bottom of the river on the flow of the water to the notch* If it were practicable to dam up the river so deep that the water would flow through the notch as if coming from a reservoir of still water, the diffi- culty would not arise. This, however, can seldom be done in practice, and although the bottom of the river may be so far below the crest as to produce but little effect on the flow of the water when the quantity flowing is small, yet when the quantity becomes great, the velocity of approach comes to have a very material influence on the flow of the water, but an influence which is usually difficult, if not impracticable to ascertain with satisfactory ac- curacy. In the notches now proposed of a trian- gular form, the influence of the bottom may be rendered definite, and such as to affect alike (or at least by some law that may be readily determined by experiment) the flow of the water when very small, or when very great, in the same notch. The method by which I propose that this may be effected consists in carrying out a floor, starting exactly from the vertex of the notch, and extending both up-stream and latterally, so as to form a bottom to the channel of approach, which will both be smooth and will serve as the lower bounding surface 124 THE DISCHARGE OF WATER FROM of a passage of approach unchanging in form while increasing in magnitude, at the places at least which are adjacent to the vertex of the notch. The floor may be either perfectly level, or may consist of two planes, whose intersection would start from the vertex of the notch> and would pass up-stream per- pendicularly to the direction of the weir board ; the two planes slanting upwards from their intersection more gently than the sides of the notch. The level floor, although theoretically not quite so perfect as the floor of two planes, would probably for most practical purposes prove the more convenient ar- rangement. " With reference to the use of the floor it may be said, in short, that by a due arrangement of the notch and the floor a discharge orifice and channel of approach may be produced, of which (the upper surface of the water being considered as the top of the channel and orifice) the form will be un- changed or but little changed, with variations of the quantity flowing ; very much less certainly than is the case with rectangular notches. " Whatever may be the result in this respect, the main object must be to obtain, for a moderate number of triangular notches of different forms, and both with and without floors at the passage of approach, the necessary coefficients for the va- rious forms of notches and approaches selected, and for various depths in any one of them, so as to allow of water being gauged for practical pur- poses, when in future convenient, by means of similarly formed notches and approaches. The util- ORIFICES, WEIRS, PIPES, AND RIVERS. 125 ity of the proposed system of gauging it is to be particularly observed, will not depend upon a per- fectly close agreement of the theory described with the experiments, because a table of experimental coefficients for various depths, or an empirical for- mula slightly modified from the theoretical one, will serve all purposes. "To one evident simplification in the proposed system of gauging, as compared with that by rect- angular notches, I would here advert, namely, that in the proposed system the quantity flowing comes to be a function of only one variable namely, the measured head of water while in the rectangular notches it is a function of at least two variables, namely, the head of water, and the horizontal width of the notch ; and is commonly also a function of a third variable very difficult to be taken into account, namely, the depth from the crest of the notch down to the bottom 'of the channel of approach, which depth must vary in its influence with all the varying ratios between it and the other two quantities of which the flow is a function. " The proposed system of gauging also gives facil- ities for taking another element into account which often arises in practice namely, the influence of back water on the flow of the water in the gauge, when, as frequently occurs in rivers, it is found impracticable to dam the river up sufficiently to give it a clear overfall free from the back or tail water. For any given ratio of the height of the tail water above the vertex of the notch to the height of head water above the vertex of the notch, I would an- 126 THE DISCHARGE OF WATER FROM ticipate that the quantities flowing would still be approximately at least, proportional to the \ power of the head, as before ; and a set of coefficients would have to be determined experimentally for different ratios of the height of the head water to the height of the tail water above the vertex of the notch. " I have got some preliminary experiments made on a right-angled notch in a vertical plane surface, the sides of the notch making angles of 45 with the horizon, and the flow being from a deep and wide pool of quiet water, and the water thus approaching the notch uninfluenced by any floor or bottom. The principal set of experiments as yet made were on quantities of water varying from about 2 to 10 cubic feet per minute ; and the depths or heads of the water varied from 2 inches to 4 inches in the right-angled notch. From these experiments I derive the formula Q = 0-317 H* where Q is the quantity of water in cubic feet per minute, and H the head as measured vertically in inches from the still water level of the pool down to the vertex of the notch. This formula is submitted at present temporarily as being accurate enough for use for ordinary practical purposes for the measure- ment of water by notches similar to the one experi- mented on, and for quantities of water limited to nearly the same range as those in the experiments ; but as being, of course, subject to amendment by more perfect experiments extending through a wider range of quantities of water." In the first edition of this book we gave the gene- ral form of the equation for the discharge through ORIFICES, WEIRS, PIPES, AND RIVERS. 127 triangular notches, and also showed the general appli- cation of the coefficients -617 to -628 for all forms of orifices and notches in thin plates. '617, as shown in note p. 55, gives a result identical with the prac- tical results of Professor Thomson's experiments. The great advantage of the triangular notch for gauging is, that the sections for all depths flowing over are similar triangles, and therefore the coeffi- cient probably remains constant, or nearly so, not only for one but for all species of triangles, when the depth at the open is not very little indeed in propor- tion to the width flowing over at the surface. The disadvantage of the proposed triangular form of depression, if permanent in a dam, would be that the angular point should be at a lower level than the top of a horizontal crest to maintain the same level, above, of the water during floods ; and therefore the power of the water and head would be reduced at the period when most required for mill-power or navigation purposes ; that is, during dry weather. For drainage purposes the winter level or that du- ring floods, must evidently be kept down, unless when the banks are steep, and along rapids ; but these remarks do not apply to dams erected across inillraces or streams where the banks are, generally, considerably above floods. These remarks refer to occasions for permanent gauging to find the relations of evaporation, absorption, and discharge in given catchment areas. In notch gauging to determine the useful effect of water engines, rectangular forms in thin plates have the coefficients already well deter- mined, and the calculations are easy. 128 THE DISCHAKGE OF WATEK FKOM DIFFERENT EFFECTS OF CENTRAL AND MEAN VELOCITIES. There is, however, another element to be taken into consideration, and which we shall have to refer to more particularly hereafter ; it is this, that the central velocity, directly facing the orifice, is also the maximum velocity in the tube, and not the mean velocity. The ratio of these velocities is 1 : -835 /-i nearly; hence, in the example, p. 110, where - 3, A we get 3 x *835 = 2-505 for the value of - in column A 1, TABLE Y., opposite to which we shall find -649, the coefficient for an orifice of one-third of the section of the tube when cylindrical or prismatic, Fig. 13; and 3 x '835 x '807 = 2-02 nearly, oppo- site to which we shall get -661 for the coefficient when the orifice is at the end of the short tube, Fig. 14, with a rounded junction. We have, there- c c* fore, - x *835 equal to the new value of for A A finding the discharge from orifices at the end of /i cylindrical or prismatic tubes, and - x '835 x '807 A C C z= - x '67 nearly for the new value of J - when A A finding the discharge from orifices at the end of a short tube with a rounded junction. The ratio of a mean velocity in the tube to that facing the orifice cannot be less than *835 to 1, and varies up to 1 to 1 ; the first ratio obtaining when the orifice is pretty small compared with the sec- ORIFICES, WEIRS, PIPES, AND RIVERS. 129 Fig 15 D tion of the tube, and the other when they are equal. If we suppose the curve D c, whose abscissae (A b) repre- sent the ratio of the orifice to the section of the tube, and whose ordinates (be) repre- sent the ratio of the mean velocity in the tube to that facing the orifice, to be a parabola, we shall find the following values : Katio of the orifice to the channel, or values of AB 1 2 3 4 5 6 7 8 9 1-0 Values of dc. 165 163 158 150 139 124 106 084 059 031 000 Ratio of the mean velocity of approach in a tube or channel to that directly opposite the orifice, or values of b c 835 837 842 850 861 876 894 916 941 969 1-000 These values of b c are to be multiplied by the cor- /- responding ratio - in order to find a new value, opposite to which will be found, in the table, the coefficient for orifices at the ends of short prismatic K 130 THE DISCHARGE OF WATER FROM or cylindrical tubes ; and this new value again mul- tiplied by -807, or more generally by c d (T 3}*, will /~i give another new value of -, opposite to which, in A the table, will be found the coefficient for orifices at the ends of short tubes with rounded junctions. EXAMPLE III. What shall be the discharge from an orifice A, Fig. 16, 2 feet long by 1 foot deep, when the 117 '945 12 value of ' is 3, and A the depth of the centre of A I foot 6 inches below the surface? We have D t zz 2 x 1 X (TABLE II.) = 2 x 9-829 x 19-658 cubic feet per second for the theoretical discharge. From the table on last page the coefficient for the mean velocity, /i facing the ori&ce, is about -86 ; hence - x *86 = 3 x A. 86 zz 2-58. If we take the coefficient from TABLE L, we shall find it (opposite to 2, the ratio of the length of the orifice to its depth) to be -617 ; and, for this coefficient, opposite to 2-58, in TABLE V., or the next number to it, we find the required coefficient -636 ; hence the discharge is -636 x 19-658 = 12-502 cubic feet per second. If we assume the coefficient in still water to be -628, then we shall obtain the new co- * See p. 106, with reference to the modifications of equations * (45) and (46) into (45 a) and (46a) suited to & a ORIFICES, WEIRS, PIPES, AND RIVERS. 131 efficient '647, and the discharge would be '647 x 19-658 = 12-719 cubic feet. If the junction of the tube with the cistern be rounded, as shown by the dotted lines, we have to multiply 2*58 by *807, which / gives 2-08 for the new value of -, opposite which we A. shall find, in TABLE V., when the first coefficient is 628, the new coefficient -659; and the discharge in this case would be -659 x 19-658 zz 12-955 cubic feet per second. It is not necessary to take out the coefficient of mean velocity facing the orifice to more than two places of decimals. For gauge notches in thin plates placed in streams and millraces, Fig. 17, the mean coefficient 628, for still water, may be assumed ; thence the new /-! coefficient suited to the ratio - may be found, as in A the first portion of EXAMPLE III. We shall leave the working out of the results when h a is taken equal to the student. EXAMPLE IV. What shall be the discharge through the aperture A, equal ^feet by I foot, when the channel is to the orifice as 3-375 to 1, and the depth of the centre is 1 *2 5 foot below the surface, taken at about 3 feet above the orifice? Here the coefficient of the approaching velocity is -85 nearly, whence the new /-( value of - is 3-375 x -85 = 2-87 ; and as C A = -628, K3 132 THE DISCHARGE OF WATER FROM we shall get from TABLE V. the new coefficient '644. Hence D = 2 x 1 x x -644 (TABLE !!.)= 2 x 8*972 x -644 12 V = 17-944 x -644 = 11*556 cubic feet per second. Weisbach finds the discharge, by an empirical formula, to be 11*31 cubic feet. If the coefficient be sought in TABLE I., we shall find it -617 nearly, from which, in TABLE V., we shall find the new coefficient to be *632 : hence 17-944 x *632 = 11-341 cubic feet per second. If the coefficient *6225 were used, we should find the new coefficient equals *638, and the discharge 11-468 cubic feet. Or thus: The ratio of the head at the upper edge to the depth of q the orifice is = *75, and from TABLE IY. we find 12 (1*75)* (-75)* = 1-6655. Assuming the coefficient to be *644, we find from TABLE VI. the discharge per minute over a weir 12 inches deep and 1 foot long to 208-650 + 205'119 on/^oo^ i i j be - - = 206*884 cubic feet nearly; and 2 as the length of the orifice is 2 feet, we have 2 X 206*884 X 1-6655 nn AC . n , . , ,. , - = 11*482 cubic feet per second, which 60 is the correct theoretical discharge for the coefficient 644, and less than the approximate result, 11*556 cubic feet above found, by only a very small dif- ference. The velocity of approach in this example must be derived from the surface inclination of the stream. The working out of this example and the v 2 increase of the discharge when h & = a 2 will afford practice to the student. ORIFICES, WEIRS, PIPES, AND RIVERS.' For notches or Poncelet weirs the approachin velocity is a maximum at or near the surface. If the central velocity at the surface facing the notch be 1, the mean velocity from side to side will be -914. We may therefore assume the variation of the central to the mean velocity to be from 1 to '914; and hence the ratio of the mean velocity at the surface of the channel to that facing the notch or weir cannot be less than -914 to 1, and varies up to 1 to 1 ; the first ratio obtaining when the notch or weir occupies a very small portion of the side or width of the channel, and the other when the weir extends for the whole width. Following the same mode of calculation as at p. 129, Fig. 15, we shall find as follows : Ratio of the Values of Values of width of the notch J l. ~ to the width of _. a 6 > _.*' C > thechaunel. ]?lg. 15. Fig. 15. -086 -914 1 -085 -915 2 -083 -917 3 -078 -922 4 -072 -928 5 -064 -936 6 -055 -945 7 -044 -956 8 -031 -969 9 -016 -984 1-0 -000 1-000 These values of b c are to be used as before in /i order to find the value of -, opposite to which in the At 134 THE DISCHARGE OF WATER FROM tables, and under the heading for weirs, will be found the new coefficient. EXAMPLE V. The length of a weir is 10 feet; the width of the approaching channel is 20 feet; the head, measured about 6 feet above the weir, is 9 inches ; and the depth of the channel 3 feet: what is the discharge ? Assuming the circumstances of the overfall to be such that the coefficient of discharge for heads, measured from still water in a deep weir basin or reservoir, will be -61 ?, we shall find from TABLE VI. the discharge to be 128-642 x 10 = 128642 cubic feet per minute; but from the smallness of the channel the water approaches the weir with some velocity, and - = = 8. We have also the width A 10 X ^ of the channel equal to twice the width of the weir, and hence (small table, p. 133,) 8 x -936 = 7-488 for P the new value of -. From TABLE V. we now find the A. *fi24 new coefficient - = -623, and hence the dis- 2 charge is - - = 1298-93 cubic feet per minute. Or thus : As the theoretical discharge, TABLE VI., is 2084-96 cubic feet, we get 2054-96 x -623 = 1298-93, the same as before. In this example, however, the mean velocity approaching the overfall bears to the mean velocity in the channel a greater ratio than 1 : -936, as, though the head is pretty large in pro- portion to the depth of the channel, the ratio of the sections - = - is small. We shall therefore be more C 8 correct by finding the multiplier from the small table, ORIFICES, WEIRS, PIPES, AND RIVERS. 135 /-i p. 129. By doing so the new value of - is 8 x -838 A = 6*704. From this and the coefficient *617 we shall find, as before from TABLE V., the new coefficient to be -627 ; hence we get 2084-96 x -627 = 1307-27 cubic feet per minute for the discharge. The foregoing solution takes for granted that the velocity of approach is subject to contraction before arriving at the overfall or in passing through it ; now, as this reduces the mean velocity of approach from 1 to -784, TABLE V., when the coefficient for heads in still water is '617, we have to multiply the ri value of- = 6-704, last found, by -784, and we get A. 6-704 x -784 = 5-26 for the value - due to this cor- A. rection, from which we find the corresponding co- efficient in TABLE V. to be -629, and hence the cor- rected discharge is 2084-96 x -629 = 1311-44 cubic /-i feet. It is to be borne in mind that the value of - in A. TABLE Y. is simply an approximate value for the ratio -of the velocity in the channel facing the orifice to the velocity in the orifice itself ; and the corrections applied in the foregoing examples were for the pur- pose of finding this ratio of velocity more correctly r\ than the simple expression -gives it. The following auxiliary table will enable us to find the correction, and thence the new coefficient, with facility. Thus, if the channel be five times the size of the orifice, and a loss in the approaching velocity takes place equal to that in a short cylindrical tube, we get 136 THE DISCHAKGE OF WATEE FKOM AUXILIARY TABLE, TO BE USED WITH TABLE V. FOR MORE NEARLY FINDING THE COEFFICIENT OF DISCHARGE NEARLY SUITED TO EQUATIONS (45 a) AND (46 a). Eatio of the orifice 1 to the channel, or J l Multipliers due to I the difference of I the central and 1 mean velocity only.| Multipliers for finding the new values of-^ in TABLE V., when the water approaches and passes through the orifice, without contraction or loss of velocity. Coefflc*' 639 Coeffic*' 628 Coefflc 1 617 Coeffic*' 606 Coeffic* 1 595 Coeffic*' 584 Coeffic 1 573 835 69 67 65 64 62 60 58 1 837 70 68 66 64 62 60 59 2 842 70 68 66 64 62 61 59 3 850 71 69 67 65 63 61 59 4 861 72 69 68 66 64 62 60 5 876 73 71 69 67 65 63 61 6 894 74 72 70 68 66 64 62 7 916 76 74 72 70 68 66 64 8 941 78 76 74 72 70 68 66 9 969 81 78 76 74 72 70 68 1-0 1-000 831 807 784 762 740 719 699 5 x '842 zz 4-210 for the new value of -, opposite to which, in TABLE V., will be found the coefficient sought. If the coefficient for still water be '606, we shall find it to be *6 12 for orifices and '623 for weirs. But when the water approaches without loss of velocity, we find from the auxiliary table '64 for the multiplier instead of -842, and consequently the new value of -becomes A 5 x '64 zz 3-2, from which we shall find *617 to be the new coefficient for orifices and -636 for weirs. The auxiliary table is calculated by multiplying the numbers in the second column (see third column, table, p. 129) by the value of c d x J - 3-2! , which will be found from TABLE V., for the different values of c d in the table, viz. ORIFICES, WEIRS, PIPES, AND RIVERS. 137 639, -628, -617, -606, -595, -584., and -573, to be 831, -807, -784, -762, -740, -719, and -699 respec- tively, as given in the top and bottom lines of figures. c 2 When -3-^2- in equations (45) and (46) is equal to , 2 _^j in equations (45a) and (46), then c d zz 1, and c d (l + ^ - 2 } in equation (45) is equal to ^ vn c d j 1 1 H -- 2 1 I i n equation (45a) ; and c d \(l + in e( l uatio11 ( 46 ) is e( l ual to l +^^=Tl""^=r ^ equation (460); and therefore the coefficient found from TABLE V. for c d = 1 will give the multiplier for d , outside the brackets, in (45o) and (46a), to find the new coeffi- cients. Thus in the last example m = 5, and hence TABLE V. for c d = 1, we find { 1 +-^r l f = 1-021 and {(1 + ^f - fcj^r-i)*} - 1-OB5. Hence 1-021 x '606 = -619 nearly ; and 1-055 x '606 = -639 nearly, the new coefficients found from the other method being *617 and -636, the difference by both methods being of no great practical importance. It is necessary to observe, that in equations (45), (46), (45a), and (46o), the head due to the velocity of supply or approach, A a , must be extra to the head k, and no part of it, and that as is indicated by the equations m can never be so small as unity. These equations are not, therefore, strictly applicable to orifices in the short tubes, Fig. 15 and Fig. 16, al- 138 THE DISCHARGE OF WATER FROM though they can be made practically so within definite limits. The initial value of c d itself varies consider- ably with the position and form of the orifice ; for a mean value of *707 it changes according to the rela- 707 tion of c and A into -7271 an( ^ f r a value of 618 for an orifice, central in a thin plate, Professor Bankine's formula,, p. 110, is applicable. In weirs at right angles to channels with parallel sides, the sectional area can never equal that of the channel unless it be measured at or above the point A, where the sinking of the overfall commences ; and unless also the bed c D and surface A B have the same inclination. In all open channels, as mill- races, streams, rivers, the supply is derived from the surface inclination of A B, and this inclination regulates itself to the discharging power of the over- fall. When the overfall and channel have the same width, and it is considerable, we have, as shall appear hereafter, 91 \/ h s for the mean velocity in the channel, where h is the depth in feet and s the rate of inclination of the surface A B. We have also - \/1gh for the theoretical velocity of discharge at 3 the overfall, of equal depth with the channel, and, when both velocities are equal, - \/!Tgh zz 5-35 \/~h = 91 \HTs ; 8 from which we find ORIFICES, WEIRS, PIPES, AND RIVERS. 139 ,= -=00348, the inclination of B A when the supply is equal to the theoretical discharge at the overfall. If the co- efficient at the overfall were *628, or, which is nearly the same thing, if a large and deep weir basin inter- vene between the weir and channel, Fig. 19, A a would be level, the velocity of approach would be destroyed, and we should have 5-35 X '628 \/~h = 3-36 %/T= 91 \Ths ; and thence the inclination of A B s = =-00136 734 very nearly. When we come to discuss the surface inclination of rivers, we shall see that the conditions here assumed and the resulting surface inclinations would involve a considerable loss of head. If the quantity discharged under both circumstances be the same, and h be the depth in the first case, Fig. 18, we shall then have the head in the latter case, Fig. 19, equal ( ) T h = 1-36 h very nearly, from which vJ'SG'' and the surface inclination the extent of the back- water may be found with sufficient accuracy. When, in Fig. 19, the inclination of A B exceeds , the head 734 at a must exceed the depth of the river above A. We must refer to pages further on, SECTION X, for some remarks on the backwater curve. 140 THE DISCHARGE OF WATER FROM SECTION V. SUBMERGED ORIFICES AND WEIRS. CONTRACTED RIVER CHANNELS. The available pres- sure at any point in the depth of the ori- fice A, Fig. 20, is equal to the difference of the pressures on each side. This difference is equal to the pressure due to the height k, between the water surfaces on each side of the orifice ; in this case, the velocity is (47.) ^ and the discharge (48.) D = / c? c d \/2 g h ; in which, as before, / is the length, and d the depth of the rectangular orifice A. When the orifice is partly submerged, as in Fig. 21, we may put ^ b h = d<2 for the submerged depth, and k~h t = d l} the remaining portion of the depth ; whence d l + d 2 = d is the entire depth. The dis- charge through the submerged depth d 2 iscjd 2 ^x \/ 2 g h, and the discharge through the upper portion d l is ORIFICES, WEIRS, PIPES, AND RIVERS. 141 whence the whole discharge assuming the coeffi- cient of discharge c d is the same for the upper and lower depths is (49.) D = c d /\/2^m 2 \/A-)- - (h? A t 2") L We may, however, equation (31), assume that very nearly, and hence (50.) D = Ash i + -=h -^ this equation may be changed into (51.) = c,ld. 2 In either of these forms the values of c d v 2 ff A, can be had from TABLE II., and the value of the dis- charge D thence easily found. When the water approaches the orifice with a determinate velocity, the height h & due to that velo- city can be found from TABLE II., and the discharge is then found by substituting h + A a and h t + A a for h and h t in the above equations. In the submerged weir, Fig. 22, A be- comes equal to d^ and h t =. ; the discharge, equation (49), then be- comes 142 THE DISCHARGE OF WATER FROM ( 52 When the water approaches with a velocity due to the height h & , then h becomes h + A a , h t = h &) and equation (49) becomes (53.) D= In the improvement of the navigation of rivers, it is sometimes necessary to construct weirs so as to raise the upper waters by a given depth, d^ The discharge D is in such cases previously known, or easily determined, and from the values of d^ and D we' can easily determine, equation (52), the value of (54.) 3 or, by taking the velocity of approach into account, (55.) d 2 =- * This value of d 2 must be the depth of the top of the weir below the original surface of the water, in order that this surface should be raised by a given depth, d lt When h & is small compared with d 2 , we may take = 2 _ x -in equation (55). 3 3 \/d l + A a EXAMPLE VI. A river whose width at the surface is 7 Q feet, whose hydraulic mean depth is k&feet, and whose cross sectional area is 325 feet, has a surface inclination of 1 foot per mile to what depth below, or height above the surface must a weir at right angles \i xv^V. rv^v^ ORIFICES, WEIRS, PIPES, AND RIVERS. 143 to the channel be raised, so that the depth of water immediately above it shall be increased by %\feet ? When the hydraulic mean depth is 4-4 feet, and the fall per mile 1 foot, we find from TABLE VIII. that the mean velocity of the river is 29-98 or 30 inches very nearly per second. The discharge is, therefore, 325x2izz812-5 cubic feet per second, or 48750 cubic feet per minute. Hence, = 696*4 cubic feet, must pass over each foot in length of the weir per minute. Assuming the coefficient c d -628 in the first instance, we find from TABLE VI. the head passing over a weir corresponding to this discharge to be 27-4 inches ; but as the head is to be increased by 3J feet, or 42 inches, it is clear that the weir must be perfect ; that is, have a clear overfall, and rise 42 27-4 = 14-6 inches over the original water surface. In order that the weir may be submerged, or imperfect, the head could not be increased by more than 27-4 inches. Let us, there- fore, assume in the example, that the increase shall be only 18 instead of 42 inches; the weir then becomes submerged, and we have, from equation (54), d 2 = u " u * _ -i x 18" (as 7 = 1 foot). 628 \/18"x2# 3 The value of the first part of this expression is found from TABLE VI. or TABLE II. equal to 696-4 696-4 = 07,1. ! = 1*88 feet = 22-56 m.; - X-X 370-341 370 ' 341 18 2 net hence 22-56 yzz 10-56 inches is the value of c? 2 ; 144 THE DISCHARGE OF WATER FROM that is, the submerged weir must be built within 10-56 inches of the surface to raise the head 18 inches above the former level. If, however, the velocity of approach be taken into account, we shall find this velocity equals ,o = 2 feet per second very nearly ; and the height, or value of A a , due to o this velocity, taken from TABLE II., is - = -75 inches nearly ; therefore, from equation (55), d = 696-4 2 (18-75)* -(-76)* 628 v/2^ x 18-75" s X \Sl8-T5 The value of x 696 ' 4 = (from TABLE VI.) 628 x/2#x 18-75" 696-4 696-4 = 1 . 84fflet = 22H)8ilL5 393-75 378 ' 81 1875 2 (18-75)* -(-75)* 2 2 (-75)* x ^- -i x 18 ' 75 -3 - o fiK = 12-5 --x ^ - 12-5- -1 =z 12-4. 3 4*33 Hence d 2 = 22-08 12-4 ~ 9*68 inches, or about 1 inch less than the value previously found from equa- tion (54). The mean coefficient of discharge was here assumed to be -628. Experiments on submerged weirs show that the value of c d varies from -5 up to *8, but as this coefficient would reduce the value of dj or not much more than half the previous value ; but this would only increase the whole height of the weir by 9-68 - 4-93 = 4-75 inches. As D = cJ\7Tg {(d, + h h] for a perfect weir with a free overfall, it is clear that when D is greater than -c^l \/'Yg {(^ + h^ hf}, the weir 3 is imperfect or submerged. For backwater curve see SECTION X. In the following table of coefficients from Lesbros* d 2 is measured from that point below the weir where its value is a minimum. On examining equa- tion (52), it will be seen that the equation D zz c d / (d l + d 2 ) v/2 g d l adopted by Lesbros is incorrect, and can only be safely used within the limits of his experiments. * Vide p. 84, deuxieme Edition, Hydraulique, par Arthur Morin. Paris, 1858. 146 THE DISCHARGE OF WATER FROM Values of d l pqy Values of the coefficient c , in the d formula D = c 1 (d + d\ d V i a/ X-V/20^. Values of d i Values of the coefficient c, in the d Oin ula D =- c I (d + d \ d \ l */ X*/20d~. d i+ d * 001 227 060 519 002 295 080 517 003 363 100 516 .- , 150 512 004 430 200 507 005 496 250 502 006 556 300 497 007 597 350 492 008 605 400 487 009 600 450 480 010 596 ^___ _^__ 015 580 500 474 020 570 550 466 025 557 600 '459 030 546 700 444 035 537 800 427 040 531 900 409 045 526 1-000 390 050 522 > The experimental values are those shown between the horizontal lines, the others above the upper ones, and below the lower ones, were deduced from calculations bj Lesbros. The true value of the discharge is expressed by the equation D zz c d / j - di + d 2 r X v/2 g d 1} and the values of c d in the above table are, therefore, too small, applied to the correct formula. When d^ = d 2 the table gives c d = -474. Now for weirs in which the sheet passing over is " drowned," the general value of the coefficient is about -67 ; this would give the coefficient for the lower portion d 2 , in the true formula, equal to -503, and a mean co- efficient c d in the correct formula (52) equal to -569 nearly. When d 2 = 200 d l9 the apparent limits of the experiments on the other side, then the mean value of c d zz *496 nearly in equation (52). These results would show that the coefficient due to the submerged depth d 2 , in the first and last experiments, ORIFICES, WEIRS, PIPES, AND RIVERS. 147 is equal to about *5 nearly, (but varies to *6 nearly in some of the middle experiments,) or thereabouts, and, therefore, equation (52) for submerged weirs, as the coefficient for the upper part d l is -67, would become (52A.) D = I X { '445 d, + -5 d 2 } x which for feet measures would become again (52B.) D = /X v/^ X {3-56 ^ + 44}* for the discharge in cubic feet per second over a submerged weir, Fig. 22. CONTRACTED RIVER CHANNELS. When the banks of a river, whose bed has a uniform inclination, approach each other, and con- tract the width of the channel in any way, as in Fig. 23, the water will rise in the channel above the contracted portion A, until the increased velocity of discharge compensates for the reduced cross section. If we put, as before, d for the increase of depth immediately above the contracted width, and d 2 for the previous depth of the channel, we shall find the quantity of water passing through the lower depth, d 2 , equal to cjd 2 \/2gd ly in which / is the width of the contracted channel at A, and the quantity of water overflowing through d l equal to - c d / d l \/2gd l ; 3 L3 148 THE DISCHARGE OF WATER FROM and hence the whole discharge through A is (56.) D = c d / v/^M (d 2 + * d\ \ O ' When our object is to find the width I of the con- tracted channel, so that the depth of water in the upper stretch shall be increased by a given depth d l9 we shall find (57.) /= When the velocity of approach is considerable, or when the height h & due to it becomes a large portion of d 1} its effect must not be neglected. In this case, as before, we find the discharge through the depth d 2 equal to c^ld 2 v2#(<^i + A a ) ; and the discharge through the depth d l equal to - c d / v/2^ {(d l + A a ) 7 Ay } ; and hence the whole discharge is (58,) = cj^{d,( from which we shall find (59.) /= If the projecting spur or jetty at A be itself sub- merged, these formulae must be extended ; the man- ner of doing so, however, presents no difficulty, as it is only necessary to find the discharges of the different sections according to the preceding formulae, and then add them together ; but the resulting for- mula so found is too complicated to be of much practical value. ORIFICES, WEIRS, PIPES, AND RIVERS. 149 HEADS ARISING FROM PIERS AND BACKWATER ABOVE BRIDGES. Equations (56), (57), (58), and (59), are applicable to cases of contraction of river channels caused by the construction of bridge-piers and abutments, when the width / is put for the sum of the openings between them. The value of the coefficient c d will depend on the peculiar circumstances of each case; we have seen that it rises from -5 to -7 in some cases of submerged weirs, and for cases of contracted chan- nels it rises sometimes as high as -8, particularly when they are analogous to those for the dis- charge through mouth-pieces and short tubes. When the heads of the piers are square to the chan- nel, the coefficient may be taken at about *6 ; when the angles of the cut-waters or sterlings are ob- tuse, it may be taken at about '7 ; and when curved and acute, at -8. With this coefficient, a head of 2| inches will give a velocity of very nearly 36 inches, or 3 feet per second ; but as a certain amount of loss takes place from the velocity of the tail-water being in general less than that through the arch, also from obstructions in the passage, and from square-headed and very short piers, the coefficient may be so small in some cases as -5, which would require a head of 6f inches to obtain the same velocity. This head is to the former as 54 to 21. The selection of the proper coefficient suited to any particular case is, therefore, a matter of the first importance in determining the effect of obstructions in river channels : we shall have to recur to this subject again, but it is necessary to observe here, that the form of the approaches, the 150 THE DISCHARGE OF WATER FROM length of the piers compared with the distance between them, or span, and the length and form of the obstruc- tion compared with the width of the channel, must be duly considered before the coefficient suited to the particular case can be fixed upon. Indeed, the coeffi- cients will always approximate towards those, given in the next section, for mouth-pieces, shoots, and short tubes similarly circumstanced. For some further remarks on contracted channels, see SECTION X. SECTION VI. SHORT TUBES, MOUTH-PIECES, AND APPROACHES. ALTERATION IN THE COEFFICIENTS FROM FRICTION BY INCREASING THE LENGTH. COEFFICIENTS OF DISCHARGE FOR SIMPLE AND COMPOUND SHORT TUBES. SHOOTS. The only orifices we have heretofore referred to were those in thin plates or planks, with a few inci- dental exceptions. It has been shown, page 48, Fig. 4, that a rounding off, next the water, of the mouth- piece increases the coefficient ; and when the curving Fig. 24. assumes the form of the vena-contracta, the coefficient increases to -986, or nearly unity. The discharge OKIFICES, WEIES, PIPES, AND EIVEKS. 151 from a short cylindrical tube A, Fig. 24, whose length is from one and a half to three times the diameter, is found to be very 'nearly an arithmetical mean between the theoretical discharge and the discharge through a circular orifice in a thin plate of the same diameter as the tube, or -814 nearly. If, however, the inner arris be rounded, or chamfered off in any way, the coefficient will increase until, in the tube B, Fig. 24, with a properly-rounded junction, it becomes unity very nearly. In the conical short tubes c and D the coefficients are found to vary according to some func- tion of the converging or diverging angles o, o, and according as we take the lesser or greater diameter to calculate from. When the length of the tube exceeds twice the diameter, the friction of the water against the sides may be taken into account. The following table, calculated by us, for a coeffi- cient of friction -00699, due to a discharging velocity of about eighteen inches per second, see SECTION VIII., shows the resistance arising from friction in pipes of different lengths in relation to the diameter, and will be found of considerable practical value. It will be perceived that the calculations are made for three different orifices of entry. First, when the arrises are rounded, as in B, Fig. 24, with a coefficient of 986 ; secondly, when the arrises are square, as in A, with a coefficient of -815 ; and, thirdly, when the pipe projects into the vessel, when the coefficient of entry becomes reduced to -715. The velocity is v = c d v/2#A, h being measured to the lower end of the tube. 152 THE DISCHAEGE OF WATER FEOM COEFFICIENTS FOE SHORT AND LONG TUBES. Number of diame- ters in the length of the pipe. Corresponding coefficients of discharge, showing the effects of friction. Number of diame- ters in the length of the pipe. Corresponding coefficients of discharge, showing the effects of friction. 2 diameters 986 814 715 650 diameters 228 225 223 5 936 779 690 700 220 217 215 10 884 747 668 750 213 211 209 15 840 720 649 800 206 205 203 20 801 695 630 850 201 199 197 25 767 673 615 900 195 193 192 30 737 653 598 950 190 189 187 35 711 634 584 1000 186 184 183 40 693 617 570 1100 177 '176 175 45 665 601 558 1200 170 169 168 50 646 586 546 1400 158 157 156 100 513 480 458 1600 148 147 146 150 439 418 403 1800 139 139 138 200 389 375 364 2000 132 132 131 250 354 345 334 2200 126 126 125 300 327 318 311 2400 120 120 120 350 304 297 292 2600 116 116 116 400 287 280 276 2800 112 112 112 450 271 266 262 3000 ,, 108 108 108 500 258 254 250 3200 105 105 104 550 247 243 240 3400 102 102 101 600 237 234 231 3600 099 099 099 We see from this table, that the effect of adding to the length of the pipe is greatest next the orifice of entry. The effect of a few diameters added to the length in long pipes is, practically, immaterial ; but in short pipes it is considerable. As for orifices in thin plates, so also for short tubes, the coefficients are found to vary according to the depth of the centre below the surface of the water, and to increase as the depths and diameter of the tube decrease. Poleni first remarked that the discharge through a short tube was greater than that OEIFICES, WEIRS, PIPES, AND RIVERS. 153 through a simple orifice, of the same diameter, in the proportion of 133 to 100, or as -617 to -821. CYLINDRICAL SHORT TUBES, A, FIG. 24. The experiments of Bossut, as reduced by Prony, give the following coefficients, at the corresponding depths, for a cylindrical tube A, Fig. 24, 1 inch in diameter and 2 inches long. The depths are given in COEFFICIENTS FOE SHORT TUBES, FROM Heads in feet. Coefficients. Heads in feet. Coefficients. Heads in feet. Coefficients. 1 818 6 806 11 805 2 807 7 806 12 804 3 807 8 805 13 804 4 807 9 805 14 804 5 806 10 805 15 -.803 Paris feet in the original, but the coefficients remain the same, practically, for depths in English feet. Yenturi's experiments give a coefficient *823 for a short tube A, 1 J inch in diameter and 4j inches long, at a depth of 2 feet 8J inches, the coefficient through an orifice in a thin plate of the same diameter and at the same depth being -622. We have calculated these coefficients from the original experiments. The mea- sures were in Paris feet and inches, from which the calculations were directly made ; and as the difference in the coefficient for small changes of depth or dimensions is immaterial or vanishes, as may be seen by the foregoing small table, and as 1 Paris inch or foot is equal to 1*0658 English inches or feet, the former measures exceed the latter by only 154 THE DISCHARGE OF WATER FROM about iVth. We may therefore assume that the coeffi- cient for any orifice, at any depth, is the same, whether the dimensions be in Paris or English feet or inches. This remark will be found generally useful in the con- sideration of the older continental experiments, and will prevent unnecessary reductions from one stand- ard to another where the coefficients only have to be considered. The mean value derived from the experiments of Michelotti, at depths from 3 to 20 feet, and with short tubes A fromi inch to 3 inches in width, isc d zz -814. Buff's experiments* give the following results for a tube A of an inch wide and A of an inch long, nearly. BUFF'S COEFFICIENTS FOE SMALL SHORT TUBES. Head in inches. Coefficient. Head in inches. Coefficient. Head in inches. Coefficient. 1* 855 6 840 23 829. ^ 861 14 840 32 826 The increase for smaller tubes and for lesser depths appears by comparing these results with the foregoing, and from the results in themselves, generally. Weis- bach's experiments give a mean value for c d zz -815, and for depths of from 9 to 24 inches the coefficients 843, -832, -821, -810 respectively, for tubes -, -, -, 10' 10' 10' and - of an inch wide, the length of each tube being three times the diameter. D'Aubuisson and CastePs * Annalen der Physik und Chemie von Poggendorff, 1839, Band 46, p. 243. ORIFICES, WEIRS, PIPES, AND RIVERS. 155 experiments with a tube -61 inch diameter and 1-57 inch long, give -829 for the coefficient at a depth of 10 feet. When a pipe projects into a cistern and has a sharp edge, the coefficient falls so low as -715. We have calculated the coefficients in the two fol- lowing short tables, from Rennie's experiments with glass orifices and tubes, Table 7, p. 435, Philosophical Transactions for 1831. The form of the orifices, or length of the shorter tubes is not stated, but it is probable from the result, that the arrises of the ends were in some way rounded off; it is stated they were " enlarged." Indeed, the discharges from the short tube or orifice of J inch diameter exceed the theo- retical ones in the proportion of 1-261 to l,and 1*346 to 1. These results could not have been derived from a simple cylindrical tube, but might have arisen from the arrises being more or less rounded at both ends, and the orifice partaking of the nature of a compound tube, which may be constructed, as we shall hereafter show, so as to increase the theoretical discharge from 1 up to 1-553. The resulting coefficients for the | COEFFICIENTS FOE SHORT TUBES, THE ENDS ENLARGED. Head in feet 4 inch diameter. inch diameter. finch diameter. linch diameter. 1 1-231 831 766 912 2 1-261 839 820 920 3 1-346 838 821 8SO 4 1-261 831 829 991 and | inch tubes, approach very closely to those obtained by other experimenters, but those for the inch tube are too high, unless the arris at the ends was also rounded. The coefficients derived from the 156 THE DISCHAEGE OF WATEE FEOM experiments with a cylindrical glass tube 1 foot long, as here given, are very variable ; like the others they COEFFICIENTS DERIVED FROM EXPERIMENTS WITH A GLASS TUBE ONE FOOT LONG. Heads in feet. iinch diameter. | inch diameter. 1 inch diameter. 1 inch diameter. 1 892 703 691 760 2 914 734 718 749 3 931 723 709 777 4 914 725 677 815 are, however, valuable, as exhibiting the uncertainty attending "experiments of this nature," and the ne- cessity for minutely observing and recording every circumstance which tends to alter and modify them. Indeed, for small tubes, a very slight difference in the measurement of the diameter must alter the result a good deal, particularly when it is recollected that measurements are seldom taken more closely than the sixteenth of an inch, unless in special cases. As the author, however, states, p. 433 of the work re- ferred to, that the " diameters of the tubes at their extremities were carefully enlarged to prevent wire edges from diminishing the sections;" this circum- stance alone must have modified the discharges, and would account for most of the differences. The coefficient for rectangular short tubes differs in no way materially from those given for cylindrical ones, and maybe taken on an average at -814 or -815. SHOKT TUBES WITH A ROUNDED MOUTH-PIECE, B, FIG. 24. When the junction of a short tube with a vessel takes the form of the contracted vein, Figs. 3 and 4, page 48, the mean value of the coefficient c d 956, OKIFICES, WEIRS, PIPES, AND RIVERS. 157 and the actual discharge is found to be from 93 to 99 per cent, of the theoretical discharge. Weisbach, for a tube 1| inch long and T 9 o inch diameter, rounded at the junction, found at 1 foot deep c d = -958, at 5 feet deep Q969, and at 10 feet deep c d 975. These experiments show an increase in the coefficients, in this particular case, for an increase of depth. Any other form of junction than that of the contracted vein, will reduce the discharge, and the coefficients will vary from -715 to -814, and to -986, according to the change in the junction from the cylindrical, pro- jecting into the vessel, to the square and properly curved forms. The coefficients derived from Ventures experiments will be given hereafter. SHORT CONICAL CONVERGENT TUBES, C, FIG. 24. The experiments of D'Aubuisson and Castel lead to the following coefficients of discharge and velocity* from a conically convergent tube c at a depth of 10 COEFFICIENTS FOB CONICAL CONVEEGENT TUBES. Converging angle o. Coefficient of discharge. Coefficient of velocity. Converging angle o. Coefficient of discharge. Coefficient of velocity. 1 858 858 14 943 964 2 873 873 16 937 970 3 908 908 18 931 971 4 910 909 20 922 971 5 920 916 22 917 973 6 925 923 26 904 975 80 931 933 30 895 976 lOo 937 950 40 869 980 12 942 955 50 844 985 * Traite d'Hydraulique, Paris, p. 60. 158 THE DISCHAEGE OF WATEK FROM feet. We have interpolated the original angles and coefficients so as to render the table more convenient to refer to, for practical purposes, than the original. The diameter of the tube at the smaller or discharging orifice in the experiments was *61 inches, and the length of the axis T57 inch; that is, the length was 2-6 times the smaller diameter of the tube. The coefficient became *829 for the cylindrical tube, i.e. when the angle at o was nothing. The angle of con- vergence o determines, from the proportions, the length of the inner and longer diameter of the tube. The coefficients of discharge increase up to '943 for an angle of 13 J or 14 degrees, after which they again decrease ; but the coefficients of velocity increase as the angle of convergence, o, increases from -829, when the angle is zero up to -985 for an angle of 50 degrees. When D is the discharge and A the area of the sec- tion, we have, as before shown, D =C A A \/^gh; but as, in conically convergent or divergent tubes, the inner and outer areas (or, as they may be called, the receiving and discharging sections) vary, it is clear that, the discharge being the same, and also the theo- retical velocity \/2 g h, the coefficient c d must vary inversely with the sectional area A, and that c d XA must be constant. For the coefficients tabulated, the sectional area to be used is that at the smaller or outside end of a convergent tube c, Fig. 24. For a short tube c, whose length is -92 inch, lesser diameter 1-21 inch, and greater diameter 1-5 inch, we have found, from Venturi's experiments, that c d 607 if the larger diameter be used in the calcu- OEIFICES, WEIES, PIPES, AND RIVERS. 159 lation, and c d =-934 when the lesser diameter is made use of, the discharge taking place under a pressure of 2 feet 8J inches. The earlier experiments of Poleni, when reduced, furnish us with the following coefficients : A tube 7-67 inches long, 2167 inches diameter at each end, gave c d zz-854; the like tube with the inner or receiving orifice increased to 2| inches, c d 903 ; increased to 3-5 inches, c d 898 ; increased to 5 inches, c d 888 ; and increased to 9*83 inches, c d 864. The depth or head was 21-33 inches, the discharging orifice 2-167 inches diameter, and the length 7-67 inches, in each case. In the conically divergent tube D, Fig. 24, the co- efficient of discharge is larger than for the same tube c, convergent, when the water fills both tubes, and the smaller sections, or those at the same distances from the centres o o, are made use of in the calcu- lations. A tube whose angle of convergence, o, is 5 nearly, with a head of from 1 to 10 feet, whose axial length is 3 J inches, smaller diameter 1 inch, and lar- ger diameter 1-3 inch, gives, when placed as at c, -921 for the coefficient ; but when placed as at D, the co- efficient increases to -948. In the first case the smaller area, used in both calculations, being the re- ceiving, and in the other the discharging, orifice. The coefficient of velocity is, however, larger for the tube c than for the tube D, and the discharging jet of water has a greater amplitude in falling. The effects of conically diverging tubes will, however, be better perceived from the experiments on compound short tubes. 160 THE DISCHARGE OF WATER FROM EFFECTS OF COMPOUND ADJUTAGES AND ADMISSION OF AIR INTO SHORT TUBES. If the tube A, Fig. 24, be pierced all round with small holes at the distance of about half its diameter from the reservoir, the discharge will be immediately reduced in the proportion of '814 to *617. Venturi found the reduction for a tube li inch diameter and 4i inches long, at a depth of 2 feet 10i inches, as 41 to 31, or as *823 to -622. As long as one hole re- mained open, the discharge continued at the same reduced rate ; but when the last hole was stopped, the discharge again increased to the original quantity. If a small hole be pierced in a tube 4 diameters long, at the distance of li or 2 diameters at farthest from the junction, the discharge will remain unaffected. This shows that the contraction in the cylindrical tube extends only a short distance from the junction, probably li or li diameter, including the whole cur- vature of the contraction. The contraction at the entrance into a tube from a reservoir accounts for the coefficients for a short tube A, Fig. 24, and the short tubes, diagrams 1 and 2, Fig. 25, being each the same decimal nearly, when Fig. 25. OR : or : : I : '8, or when or is not less than ORX '79, ORIFICES/WEIRS, PIPES, AND RIVERS. 161 O Tl and is at the distance of nearly - - from o R. The 2 form of the junction o o r R remaining as we have de- scribed it, the following coefficients will enable us to judge of the discharging powers of differently formed short mouth-pieces. They have been deduced and calculated by us, principally, from Venturi's ex- periments.* These coefficients show very clearly that any cal- culations from the mere head of water and size of the orifice, without taking into consideration the form of the discharging tube and its connection with the re- servoir, are very uncertain ; and that the discharge can only be correctly obtained when all the circum- stances of the case, including the form of the dis- charging orifice and its approaches, have been duly considered. .oil JfcfM 0%0 89MCfg 6rft prated ajj I {I When a tube similar to diagram 5, Fig. 25, has the junction o o r R rounded, as in Fig. 4, page 48, the outer extremity s t s T, such that s t or, ss = 9 st, and the diameter s T = 1-8 times the diameter ,9 , with a short central cylindrical piece o r s t between, the coefficient of discharge corresponding to the diameter or=:rs will increase to 1-493 or 1-555 ; that is, the 1-493 1.555 discharge is =: 2-4, or zz 2-5 times as much *622 *622 as through an orifice (whose diameter is o r) in a thin plate, and - - : srjK9 times as much as through a i * 822 ;sifj ."asifour T of s a bur, . ,.'i fl { B*6* j.,.. . .0 .o/C ni KB sniff ift/i \ * See Nicholson's translation of Venturi's Experimental IB quiries, published in the Tracts on Hydraulics, London, 1836. The coefficients in the table, next page, have been all calculated for the first time by us. M 162 THE DISCHARGE OF WATER FROM TABLE OF COEFFICIENTS FOR SHORT TUBES AND MOUTH-PIECES. Description of orifice, mouth-piece, or short tube. Coefficients for the diameter OR. Coefficients for the diameter or. 1. An orifice 1 inch diameter in a thin plate . . 2. A cylindrical tube 1 inch diameter and 4 ; inches long, A, Fig. 24 622 823 974 823 3. A short tube with a sharp end projecting intc the cistern 715 715 4. A cylindrical tube, B, Fig. 24, having the June tion rounded, as in Fig. 4, page 48 611 956 5. A short conical convergent mouth-piece, c, Fig 24, of the proportions of o o r R, Fig. 25 607 934 6. The like tube divergent, with the smaller dia meter at the junction with the reservoir length 3 inches, lesser diameter 1 inch, anc greater diameter 1*3 inch . 561 948 7. The tube, o o u v r R, diagram 2, Fig. 25, when o R = 1 inch, o r = 1-21 inch, u v 1-21 inch and ouc=rv= 2 inches, the cylindrical por- tion bein 01 shown bv dotted lines 600 923 8. The same tube when o u 11 inches 567 873 The same tube when o u 23 inches 531 817 9. The tube, oossitrn, diagram 2, Fig. 25, in which OR=S=ST = I inch, from o to s If inch, and ss = 3 inches, gives the same co- efficient as the cylindrical tube, result No. 2 (see No. 19) viz 823 1-266 0. The tube, diagram 1, Fig. $5, o R = l inch. . . 1. The same tube, having the spaces oso and r t R between the mouth-piece o o r n and the cylindrical tube o s T R open to the influx of 804 785 1-237 1-209 12. The double conical tube, o o s TTR, diagram 3, Fig. 25, when OR = ST = ! inch, or = l-21 inch, o o= '92 inches, and o s = 4-l inches . . 3. The like tube when, as in diagram 4, Fig. 25, o o T R o s T T and o o s = 1*84 inch . . . . 928 823 1-428 1-266 4. The like tube when, sT = l-46 inch, and os 2'17 inches 823 1-2C6 15. The like tube when ST = 3 inches, and os = 911 1-400 16. The like tube when os = 6| inches, and ST enlarged to 1*92 inch . . 1-020 1-569 17. The like tube when ST = 2| inches, and os 12i inches 1-215 1-855 18. A tube, diagram 5, Fig. 25, when o s = r f=3 inches, or=s=l-21 inch, and the tube o s T r the same as described in No. 12, viz. g T 1 1 inch and s s = 4'1 inches 895 1-377 9. The tube, diagram 2, Fig. 25, when s T is en- larged to 1-97 inch, and s s to 7 inches, the other dimensions remainin ' as in No 9. 945 1-454 20. When the junction of o s r t with s s T t, dia- gram 2, Fig. 25, is improved, the other parts 850 1-309 2 ^ Another experiment gives 847 1'303 ORIFICES, WEIRS, PIPES, AND RIVERS. 163 short cylindrical tube A, Fig. 24, whose diameter is also o r. Venturi was of opinion that this discharge con- tinued even when the central cylindrical portion orst was of considerable length ; but this was a mistake, as the maximum discharge is obtained when it is reduced so that o o r a and s s t T shall join, as in diagram 3, Fig. 25. We see from No. 16 of the fore- , 1-569 . 1-569 going coefficients that - = 2-52 and 7 =1-91 are, '622 "822 perhaps, nearer to the maximum results obtainable by comparing the discharge from a compound tube ooSTrR, diagram 3, Fig. 25, with those through an orifice in a thin plate, and through a short cylin- drical tube. When the form of Fig\26 the tube becomes curvilineal throughout, as in Fig. 26, s T = 1/8 o r and o s = 9 o r, the coefficient suited to the diameter o r will be 1*57 nearly, and the discharge will be - - =2 '5 2 times as *622 much as through an orifice o r in a thin plate. The whole of the preceding coefficients have been determined from circumstances in which the co- efficient for an orifice in a thin plate was -622, and for a short cylindrical tube *822 or -823. When the circumstances of head and approaches in the reser- voir are such as to increase or decrease those primary coefficients, the other coefficients for com- pound adjutages will have to be increased or de- creased proportionately. After examining the foregoing results, it appears sufficiently clear that the utmost effect produced by M3 164 THE DISCHABGE OF WATER FROM the formation of the compound mouth -piece o o s T r R, with the exception of No. 17, is simply a restoration of the loss effected by contraction in passing through the orifice o R in a thin plate, and that the coefficient 2 *5 applied to the contracted section at o r is simply equal to the theoretical discharge* or the coefficient unity, applied to the primary orifice o R ; for, as orifice o R : orifice o r : : 1 : '64, very nearly, when o o r R takes the form of the vena-contracta, and the coefficient of discharge for an orifice o r in a thin plate is -622, we get the theoretical discharge through the orifice o R, to the actual discharge through an orifice o r, so is 1 to -622 x '64, so is 1 : -39808 :: 1 : '4 very nearly ; and as -4 x 2'5 = 1, it is clear that the form of the tube oosflrR^ when it produces the foregoing effect, simply restores the loss caused by contraction in the vena-contracta* Venituri's sixteenth experiment, from which we have derived the coefficients in No. 17, gives the coefficient f%i5 m ior ine orifice d W E?^ 1i fMl f Hnmcai > ^SF thaf J greater discharge than the theoretical, through the receiving orifice, may be obtained. It is, however, observable that Venturi, in his seventh proposition, does not rely on this result, and Eytelwein's expe- riments do not give a larger coefficient than 2-5 applied to the contracted orifice o r, which, we have above shown, is equal to the theoretical discharge through o R. 1 SHOOTS. When the sides and under edge of an orifice or notch increasie in thickness, so as to be converted into a shoot or small channel, open at the top, the OEIFICES, WEIKS, PIPES, AND KIVERS. 165 coefficients reduce very considerably, and to some extent beyond what the increased resistance from friction, particularly for small depths, indicates. Poncelet and Lesbros* found for orifices 8" x 8", that the addition of a horizontal shoot 21 inches long reduced the coefficient from -604 to *601, with a head of about 4 feet ; but for a head of 4 J inches the coefficient fell from -572 to -483. )!i 'S)W' notches 8" wide, with the addition of a horizontal shoot P' 1,0" long, the coefficient fell from -582 to -479 for a head of 8"; and from -622 to -340 for a head of I". Castel also found for a notch 8" wide, with the addi- tion of a shoot 8" long, inclined 4 18', the mean co- efficient for heads from 2" to 4i", to be -527 nearly. The effects arising from ' friction alone will be per- ceived from the short table at the beginning of this section, p. 1521' ( TIJ > r]y ^ The orifice of entry into a shoot and its position with reference to the sides and bottom modify the discharge, the head remaining constant. Lesbrosf has given the coefficients suited to different positions : df "shootis both within and without a cistern, and from notches and submerged Orifices ; but, however valuable these ate in some respects, they are of little practical use to the engineer. The general principles which are involved in the modification of these coeffi- cients have, however, been already pointed out by us when discussing the effects 1 ' of ' ; fchfc 'position of ; toe orifice, and the addition of short tubes, on the dis- charge. Equation (74s. ), . 1 8$, ! -is here &ppllfcabte j * Trait6 d'Hydraulique, pp. 46 et 94. Morin's Hydrauliqne, deuxieme Edition, pp. 29 et 40. 166 THE DISCHAKGE OF WATEE FKOM SECTION YIL LATERAL CONTACT OF THE WATER AND TUBE. ATMOSPHERIC PRESSURE. HEAD MEASURED TO THE DISCHARGING ORI- FICE. COEFFICIENT OF RESISTANCE. FORMULA FOR THE DISCHARGE FROM A SHORT TUBE. DIAPHRAGMS. OB- LIQUE JUNCTIONS. FORMULA FOR THE TIME OF THE SURFACE SINKING A GIVEN DEPTH. LOCK CHAMBERS. The contracted vein o r is about 8 times the diameter o R; but it is found, notwith- standing, that water, in passing through a short tube of not less than 14 diameter in length, fills the whole of the discharging orifice s T. This is partly effected by the outflowing column of water carrying forward and exhausting the air between it and the tube, and by the external air then pressing on the column so as to enlarge its diameter and fill the whole .tube. When once the water approaches closely to the tube, or is caused to approach, it is attracted and adheres with some force to it. The water between the tube and the vena-contracta is, however, rather in a state of eddy than of forward motion, as appears from the experiments, with the tube, diagram 2, Fig. 25, giving the same discharge as the simple cylindrical tube. If the entrance be contracted by a diaphragm, as at o R, Fig. 27, the water will also generally fill the tube, if it be only sufficiently long. Short cylin- drical tubes do not fill when the discharge takes place in an exhausted receiver ; but even diverging tubes, OKIFICES, WEIRS, PIPES, AND RIVERS. 167 V, Fig. 24, will be filled, under atmospheric pressure, when the angle of divergence, o, does not exceed 7 or 8 degrees, and the length be not very great nor very short. When a tube is fitted to the bottom or side of a vessel, it is found that the discharge is that due to the head measured from the surface of the water to the lower or discharging extremity of the tube. It must, however, be sufficiently long, and not too long, to get filled throughout. G-uiglielmini first referred this effect to atmospheric pressure, but the first simple explanation is that given by Dr. Mathew Young, in the Transactions of the Royal Irish Academy, vol. vii., p. 56. Yenturi, also, in his fourth proposition, gives a demonstration. The values of the coefficients for short cylindrical tubes, which we have given p. 162, have been derived from experiment. Coefficients which agree pretty closely with them, and which are derived from the coefficients for the discharge through an orifice in a thin plate, may, however, be calculated as follows : Let c be the area of the approaching section, Fig. 27, A the area of the discharging short tube, and a the area of the orifice o R which admits the water from the vessel into the tube : also put, as before, h for the head measured from the surface of the water to the centre of the tube, and diaphragm o R ; v for the actual velocity of discharge at s T ; v & for the velocity of approach in the section c towards the diaphragm o R; and c c for the coefficient of contraction in passing from o R to o r ; then we have c X v & = A X v, the contracted section o r z= C Q x a, and consequently the velocity at the contracted section is equal to 168 THE DISCHAKGE OF WATEE FKOM = - Now a theoretical head equal to ac c ac c ,2 : i? - - is necessary to change the velocity v & into v by the action of gravity ; but as the water at the contracted A v section o r, moving with a velocity , strikes against ac c the water between it and T s, moving, from the nature of the case, with a slower velocity,* a certain loss of effect takes place from the impact. If this be, sup- posed, suddeD, then writers on mechanics have shown that a loss of head, equal to that due to the difference of the velocities,- - v, before and after the impact must take place. This loss of head is therefore equal to V^Au'.ii- \) y 2 -^2 * jnorli fttiw whence we must have the whole head, ' J()1 : ewoflo] 8J3 i-)li>h ,noi)oyay[Mj^i-'g^n:jfojlJ ad o Js (bO.| ^:-r j-jo) -ofifo^oftib from which we find for the velocity from a short tube, '-f< ;< J \\ f ' j- orlt Q _ ;i(M]iJ)fu^n 9jo o^inoo oil] Now, as v2 ^ h would be the velocity of discharge were there no resistances^ jss sustained, it^s ( _ l _ -|| evident that J ^ _ ^ , /_A_ Tx 2 [ becomes as it ( C 2 ' \ac c ) } f'K)il gniaajsq Sir EolJert Kane's translation of Riinhnan's book on Horizontal Water Wheels, p, 49frjjnoo Olli Jfi Ydioolyv OilJ ORIFICES, WEIRS, PIPES, AND RIVERS. 169 were a coefficient of velocity. When the diameter of the diaphragm o R becomes equal to the diameter s T of the tube, A zz a, and as the coefficient of velocity be- comes equal to the coefficient of discharge when there is no contraction, we get in such case this coefficient, which we shall also call c of, expressed by the formula oj fcaoji>i-- [ - g 1 s 1 J ' " 1 1 ^ + ( -- 1) ( '* i t - 111 J-ilOiJOiJj' i ! Mlkll 1 ) Y'UiHHODQJJ and when the approaching section c is very large compared with the area A, '.oitfAHoaM uo ^,oa-^.ao>:ATj4ifc3a uo TKaioi'a'iaoo (63.) '" '6'-of=\,,,i ,A' * "T ( -- 1) i 1 ',)lTAM8iaM If c c -z -64, we shall find from the last equation c of= -872 ; if C G -601, c o/z= -833 ; if c c = -617, c of= -847 ; and if ^ -62l;-^/ i !> 856. These results are in excess of those derived from experi- ment with cylindrical short tubes, perfectly square at the ends and of uniform bore. As some loss, how- ever, takes place in the eddy between o r and the tube, and from the friction at the sides, not taken into account in the above calculation, they will ac- count for the differences of not more than from 4 to * When the diaphragm is placed in a> tube of uniform bore, then c= A, and we shall get }l 0} oi/b Bestir io4i3i -663 x 2 ff = -949 1 2g> r " 949 * = ' 663 v ' whence we get v\ = -698 v\ and v* =. I -43 1 v* for the re- lation of the discharging velocities, v , from an orifice, and, Vty from a short tube. The height due to the re- 1 t' 2 sistance is therefore, ('.oi 4 2 ~~ l) o 1 for short pris- matic tubes, and (79742 ~~l) ~o - ' for orifices in thin plates. These are to each other as -508 to 054 x 1-431, or as 5-08 to -773, that is to .say, the loss of mechanical power arising from the re- sistance in passing through short tubes is 6 '5 7 times as great as when the water passes through thin plates or mouth-pieces, as in Fig. 4 ; and the discharging mechanical power in plates, is to that in tubes as 1*431 to 1, or as 1 : *698, the heads and quantities discharged being the same^ ol9V ^ j The whole loss of mechanical power in the passage is 5*4 per cent, for the plates, and about 51 per cent. for short tubes. If the loss compared with the whole head be sought, we get, when v is the discharging velocity, ^j for the theoretical velocity due to the mi& oilBflighq ho/Ia irgiJO'iJ) ouc8S./$q 9|[J head in short tubes, and its square Tgrp = TgTjo is as the whole head ; therefore, the whole head is to .9iHa oil,t gniod ssiiioolsv guigidBdoaibi^dJ bn^j the head due to the discharging velocity as ^ to v\ or as 1 to *663 ; and as -508 is the coefficient of re- sistance* for the discharging velocity, '508 x '663 zz '337 is the coefficient of resistance due to the :injJaiao'i i * Table, p. 171. ,jTl brm iTJ .qq ORIFICES, WEIRS, PIPES, AND RIVERS. 173 whole head ; this is equal to a loss of 34 per cent, nearly, or about one-third. In like manner, we find 974 2 x '054 = -0512 for the coefficient when the dis- charge takes place through thin plates, or 5| per cent, of the whole head. DIAPHRAGMS! '^> ^ When a diaphragm, o R, Fig. 27, is placed at the entrance of a short tube, we have shown, page 168, A 2 janiwollirt orfKIwijsIiiofBo r (^ c ~ 1 )^ A ui _ that a loss of head equal - -^ - takes place when v is the discharging velocity, whence the co- efficient of resistance is equal to ( l) ,* according .31/roAar-fiAirt ao't aD/tATaraaa ax A ,sfo;. ^ ^c to our definition. The coefficient of contraction c c , as we have before shown, page 170, should be taken equal to -590 in the application of formula (63) ; and, as it must also be taken equal to about *621 when the area of the tube i'ls very large compared with the area a of the orifice o R in the diaphragm, we may assume that when is equal to A 0, .!. -2, -3* -4, -5, -6, -7, -8, -9, and 1 O.aoU^jTLiilO;! 10 JiTQi . , , , , , , , successively, the coefficient c c must be taken equal to 621, -TO^HO? 48, '609, -606, -603, -600, -597, -593, and -590, in the same order. As the approaching section c ' * tor the sudden alteration in the velocity passing through a diaphragm, we must reject the hypothesis of D'Aubuisson, Traite d'Hydraulique, p. 238, and adopt that of Navier, taking the loss of head to correspond to the square of the difference and not to the difference of the squares of the velocities in and after passing the orifice. The coefficient of contraction must, however, be varied to suit the ratio of the channels, as it is in this and the following pages. 174 THE DISCHARGE OF WATER FROM may be considered exceedingly large, the value of the coefficient of discharge or velocity, as the tube o R s T is supposed full, in equation (61), becomes (66.) * and the coefficient of resistance ( 67 -) from which equations and the above values of c c , cor- responding to -, we have calculated the following A. values of the coefficients of discharge and resistance through the tube o K s T, Fig. 27. COEFFICIENTS OF CONTRACTION, DISCHARGE, AND RESISTANCE FOR DIAPHRAGMS. Ratio 1 1 1 Ratio 1 "S .2 1 a 6 ^ s j 3 I* a 3 fe I* ss *** A * 1 1 1 A * 1 1 1 o-o 621 000 infinite. 0-6 603 493 3-115 0-1 618 066 231- 0-7 600 587 1-907 0-2 615 139 50-8 0-8 597 675 1-198 0-3 612 219 . 19-8 0-9 593 753 762 0-4 609 307 9-6 1-0 590 821 483 0-5 606 399 5-3 - In this table c c is the coefficient of contraction, c d the coefficient of discharge, suited to the larger section of the pipe A, at s T ; and c r the coefficient of re- sistance. The discharge is found from equation (61), as c is here very large compared with A, to be 1 (67A.) D = A ORIFICES, WEIRS, PIPES, AND RIVERS. 175 A ^ The coefficient of resistance c r is here equal ( 1) , and the coefficient of discharge c 6 zz ^r-- -TI-* v* T c *) The tube must be so placed, that the water, after passing the diaphragm, shall fill it ; for instance, between two cisterns, when the height h must be measured between the water surfaces, or when the tube is sufficiently long to be filled ; in this case, however, the height must be determined from the dis- charging velocity , as a portion of the head is required to overcome the friction, which we shall have im- mediately to refer to more particularly. The table shows that the head due to the resistance is 5*3 times that due to the discharging velocity, when the area of the diaphragm is half the area of the tube ; that is, the whole head required is 6-3 times that due to the velocity, and that the coeffi- cient of discharge is reduced to -399. In order to find the coefficients suited to the smaller area of the orifice in the diaphragm o R, when it is to be used in calculations of the discharge, we have only to divide the numbers corresponding to into those of c d , op- A posite to them in the table. Thus, when - zz -8, we A have the coefficient of discharge suited to the area a, * For the loss sustained by contraction in the bore of a pipe by a diaphragm, see equations (123), (124), and (125). The actual value of c c in equation (67A) depends on the thickness of the diaphragm as well as on the relation of a and A. The form of the orifice a also affects the value of c c . 178 THE DISCHARGE OF WATER FROM 67 Fig-,28 od i equal -^- = -844, and so of other values of the ratio -. The cc efficients in the table, page 174, are A for the larger orifice A in the formula D = A c d \/2 g h. ^onuiaflKHOWi TUBES i SOBLIQUE AT THE JUNCTION. When a tube is at- fp tached obliquely, as in Fig. 28, we have found that if the number of degrees in the angle TO s, formed by the direction of the tube o s, with the perpendicular o T, be represented by <, then -814 -0016 will give the coefficient of discharge corresponding to the obliquely attached short tube in the Figure. This formula is, however, empirical, but it is simple, and agrees pretty closely with experimental results. As the coefficient [ . .UGG- . 1 '(. at as'fGf/Dsib lo irr^b of resistance is equal -, 1, equation (64), we have odi To -co isucma 9dKc3 Dan here ^r ^i , __ -6oi6 V> 2 ~" ^ ; ^ rom ^ ese equations we Have calculated the following table for heads measured to the middle of the outside orifice ; :-rm(f COEFFICIENTS OF DISCHARGE AND RESISTANCE FOR OBLIQUE JUNCTIONS. in degrees. Coefficient of discharge. Coefficient of resistance. in degrees. Coefficient of discharge. Coefficient of resistance. 814 508 35 758 740 5 806 539 40 750 778 10 15 20 "E 569 603 sr 635 (A 45 i B i ,-'742' - 734 81<6 [ 3,^87 25 . h7T.4 E '! ingJWth-i 60 718 940;;ili 30 766 704 ilV tidj gfO 710 984 ORIFICES, WEIRS, PIPES, AND RIVERS. 177 The coefficient of resistance for a tube at right angles to the side, is to the like coefficient when it makes an angle of 45 degrees as -508 to -816, or as 1 to 1-6 nearly ; and the loss of head is greater in the same proportion. If the short tube be more than three or four diameters in length, friction will have to be taken into account. The head h is measured to the outside orifice. FORMULA FOR FINDING THE TIME THE SURFACE OF WATER IN A CISTERN TAKES TO SINK A GIVEN DEPTH. DIS- CHARGE FROM ONE VESSEL OR CHAMBER INTO ANOTHER. LOCK CHAMBERS. In experiments for find- ing the value of the coeffi- cients of discharge, one of the best methods is to ob- serve the time the water discharged from the orifice takes to sink the surface in a prismatic cistern a given depth; the ratio of the observed to the theoretical time will then give the coefficient sought. A formula for finding the theoretical time is, therefore, of much practical value. In Fig. 29, the time of falling from s t to s T, in seconds, is 1 F 1 i *F Fi.29 _ _ ---i 1 . ' ' R ...^ in which a is the area of the orifice o R, and A the area of the prismatic vessel at st or ST; this formula is for measures in feet. For measures in inches, we have EXAMPLE VII. A cylindrical vessel 5 *74 inches in diameter has an orifice *2 inch in diameter at a depth 178 THE DISCHARGE OF WATER FROM 0/"16 inches below the surface, measured to the centre ; it is found that the water sinks 4 inches in 51 seconds ; what is the coefficient of discharge ? The theoretical time t is found from equation (69), equal 5-74*X-785 6> _ ^3^ JL ij y X *^j X io04: "OOO 1 T.fJKfJf? Q~f .Q = -556 ~ X '5359 =31-8 seconds; hence,-gpzz-624 is the coefficient sought. When the orifice o R and the horizontal section of the vessel are similar figures, A S T 2 - is equal ^ ; and therefore, for circular cisterns tt OR and orifices, it is unnecessary to introduce the mul- tiplier -7854. We have given above, formulae for -the time water in a prismatic vessel takes to fall a given depth, when dis- Fig. 29<*. charged from an orifice at the side or bottom. The time ' the surface s T, diagram 1, Fig. 29#, takes to rise to s t, when supplied through an orifice or tube o R, from an upper large chamber or canal, whose surface 2 A/* s 1 tf remains always at the same level, is "TTo^'* * The time of rising from s to s is exactly double the time it would take, if the pressure/ remained uniform, to fill the same depth below R. ORIFICES, WEIRS, PIPES, AND RIVERS. 179 and we thence get the time of rising from R to s for measures in feet and for measures in inches ( 69B -) '= in which A is the area of the horizontal section at s T ; a the sectional area of the communicating channel or orifice o R ; c d the coefficient of discharge suited to it, and ^ and/, as shown in the diagram. In order to find the time of filling the lower vessel to the level s T, supposing it at first empty, we have the contents of the portion below o R equal to A^ 2 > and the time of filling it equal to (69c.) 8-025 c d ah\ then the time of filling up to any level s T, for mea- sures in feet, is equal to the sum of (A) and (c) ; that is, 8-025 c d ah$ and for measures in inches 27-8 c d a "When s T coincides with * t A (2 A, + fl) T = 8-025 c,ahi> for measures in feet, and N3 180 THE DISCHARGE OF WATER FROM A (2 fr + ft,) 27-8 c,ah\ > for measures in inches. These equations are ex- actly suited to the case of a closed lock-chamber filled from an adjacent canal. When the upper level s' T' is also variable, as in Diagram 2, the time which the water in both vessels takes to come to the same uniform level s' t' s t, is 2AQ(A 1 +/ 1 -j)* = 2AO(/+/ 1 )*_ ' c d a(A+c)v/2^ c d tf(A+c)v/2#' in which h + jfj h = f -f f^ is the difference of levels at the beginning of the flow ; c the horizontal section of the upper chamber ; and the other quanti- ties as in Diagram 1. As c^ =: Af, we find Now, in order to find the time of falling a given depth d below the first level s' T', we have the head above s't's t equal to^ d in the upper vessel, and the depth below it in the lower vessel equal to ^ -- ; whence the difference of levels in the two A vessels at the end of the fall d, is The time of falling through d is, therefore, from equation (69 H), 2AC ORIFICES, WEIRS, PIPES, AND RIVERS. 181 in which \^g = 8*025 for measures in feet, and equal 27*8 for measures in inches. The whole time of filling to a level the lower empty vessel, is found by adding the time of filling the portion below R, de- termined in a manner similar to equations (68) and (69) to be to the time of filling above R, given in equation (H), when h is taken equal to zero. Equations (H), (i), and (K) are applicable to the case of the upper and lower chambers of a double lock, after making the necessary change in the diagrams. The above equations require further extensions when water flows into the upper vessel while also flowing from it into the lower ; such extensions are, however, of little practical value, and we therefore omit them. For sluices in flood-gates with square arrises, c d may be taken at about -545, but with rounded arrises, the coefficient will rise much higher. See SECTIONS III. and VI. 182 THE DISCHARGE OF WATER FROM SECTION VIII. FLOW OF WATER IN UNIFORM CHANNELS. MEAN VELOCITY. MEAN RADII AND HYDRAULIC MEAN DEPTHS. BORDER. TRAIN. HYDRAULIC INCLINATION. EFFECTS OF FRIC- TION. FORMULA FOR CALCULATING THE MEAN VELOCITY. APPLICATION OF THE FORMULA AND TABLES TO THE SOLUTIONS OF THREE USEFUL PROBLEMS. In rivers the velocity is a maximum along the central line of the surface, or, more correctly, over the deepest part of the channel ; and it decreases thence to the sides and bottom : but when backwater arises from any obstruction, either a submerged weir, Fig. 22, or a contracted channel, Fig. 23, the velocity in the channel approaching the obstruction is a maximum at the depth of the backwater below the surface, and it decreases thence to the surface, sides, and bottom. When water flows in a pipe of any length, the velocity at the centre is greatest, and it decreases thence to the sides or circumference of the pipe. If the pipe be supposed divided into two portions in the direction of its length, the lower por- tion or channel will be analogous to a small river or stream, in which the velocity is greatest at the central line of the surface, and the upper portion will be simply the lower reversed. A pipe flowing full may, therefore, be looked upon as a double stream, and we shall soon see that the formulae for the dis- charge from each kind are all but identical, though a pipe may discharge full at all inclinations, while the inclinations in rivers or streams, having uniform motion, never exceed a few feet per mile. ORIFICES, WEIRS, PIPES, AND RIVERS. 183 MEAN VELOCITY. It is found, by experiment, that the mean velocity is nearly independent of the depth or width of the channel, the central or maximum velocity being the same. From a number of experiments, Du Buat derived empirical formulae equivalent to v= V f=v^ v*+i, v b =(v* - I) 2 , and v=(v|+l) 2 ; in these equations v is the mean velocity, v the max- imum surface velocity, and v b the velocity at the sides, or bottom, expressed in French inches. Tables calculated from these formulae do not give correct results for measures in English inches, though they are those generally adopted. Disregarding the dif- ference in the measures, which are as 1 to 1*0678, it will be found that, in the generality of channels, the mean velocity is not an arithmetical mean between the velocity at the central surface line and that at the bottom, though nearly so between the mean bottom and mean surface velocities. Dr. Young,* modifying Du Buat's formula, assumes for English inches that v + v* zz v, and hence vzzv + i (v + 1)* This gives results very nearly the same as the other formula for v, but something less, particularly for small surface velocities. For instance, Du Buat's formula gives 5 inch for the mean velocity when the central surface velocity is 1 inch, whereas Dr. Young's makes it -38 inch. For large velocities both formulas agree very closely, disregarding the differ- ence between the measures, which is only seven per * Philosophical Transactions, 1808, p. 487. 184 THE DISCHARGE OF WATER FROM cent. They are best suited to very small channels or pipes, but unless at mean velocities of about 3 feet per second, they are wholly inapplicable to rivers. Prony found, from Du Buat's experiments, that /2-37187 + vx for measures in metres v (Q.I^QTO ! ) v > m which v is also the maximum surface velocity. This, re- duced for measures in English feet, becomes and for measures in English inches, 93*39 For medium velocities v = -81 v. The experiments from which these formulae were derived were made with small channels. We have calculated the values of v from that of v, equation (71), and given the results in columns 3, 6, and 9, in TABLE VII. Ximenes, Funk, and Briinning's experiments in larger channels give the mean velocity at the centre of the depth equal '914 v, when the central or maximum surface velocity is v ; but as the velocity also decreases in nearly the same ratio at the surface from the centre to the sides of the channel, we shall get the mean velocity in the whole section equal * Francis, Lowell Experiments/p. 150, finds this formula to give 1 5 per cent, less than the result found hy weir measurement from the formula D = 3 '33 (I ! n h) hi, the quantity discharged being about 250 cubic feet per second, and the velocity about 3 '2 feet. It appears, however, that Francis uses the mean surface velocity, and not the maximum surface velocity required by the formula : if the latter were used, the difference would be reduced to 6 per cent., or thereabouts, in equation (72). ^ , V*X^. fc^X^oL V^UStAjfc-^jj. JU 2. v v ^ "^<\\\, . 0. . * C . Q O ORIFICES, WEIRS, PIPES, AND RIVERS. 185 914 x '914 v z= '835 v ; and hence we have, for large channels, (72.) v = -835 v, in which equation v is the maximum velocity at the surface. We have also calculated the values of v from this formula, and given the results in columns 2, 5, and 8 of TABLE VII. This table will be found to vary considerably from those calculated from Du Buat's formula in French inches, hitherto generally used in this country, and much more applicable for all practical purposes. MEAN KADITJS. HYDRAULIC MEAN DEPTH. BORDER. COEFFICIENT OF FRICTION. If, in the diagrams 1 and 2, Fig. 30, exhibiting the sections of cylindrical and rectangular tubes filled with flowing water, the areas be divided respectively by the perimeters A c B D A and A B D c A, the quotients are termed " the mean radii" of the tubes, diagrams 1 and 2 ; and the perimeters in contact with the flowing water are termed "the borders." In the diagrams 3 and 4, the surface A B is not in contact with the channel, and the width of the bed and sides, taken together, A c D B, becomes " the border." " The mean radius " is equal to the area A B D c A divided by the length of the border A c D B. " The hydraulic mean depth" is the same as " the mean radius" this latter term being perhaps most applicable to pipes flowing full, as in diagrams 1 and 2 ; and the former to streams and rivers which 186 THE DISCHARGE OF WATER FROM have a surface line A B, diagrams 3 and 4. We shall, throughout the following equations, designate the value of the " mean radius," " hydraulic mean depth," . area A B D c A Jt or quotient, border BDCA ? b J the letter r > remarking here that for cylindrical pipes flowing full, or rivers with semicircular beds, it is always equal to half the radius, or one-fourth of the diameter. Du Buat was the first to observe that the head due to the resistance of friction for water flowing in a uniform channel increased directly as the length of the channel /, directly as the border, and inversely as the area of the cross-flowing section,! very nearly ; that is, as -. It also increases as the square of the velocity, nearly; therefore the head due to the re- v* I v z I sistance must be proportionate to - If c t X ^ zz h { , then c f is the coefficient for the head due to the resistance of friction, as h f is the head necessary to overcome the friction ; c is therefore termed " the co- efficient of friction" * M. Girard has conceived it necessary to introduce the coeffi- cient of correction 1-7 as a multiplier to the border for finding r, to allow for the increased resistance from aquatic plants ; so that, according to his reduction, area A* .... . , 1*7 border* See Eennie's First Keport on Hydraulics as a Branch of En- gineering ; Third Eeport of the British Association, p. 167 ; also, equation (85), p. 201. f Pitot had previously, in 1796, remarked that the diminution arising from friction in pipes is, cateris paribus, inversely as the diameters. ORIFICES, WEIRS, PIPES, AND RIVERS. 187 HYDRAULIC INCLINATION. TRAIN. If / be the length of a pipe or channel, and h { the height due to the resistance of friction of water flowing in it, then -/ is the hydraulic inclination. In i Fig. 31 the tubes A B, c D, of the same length /, and Fig.31 whose discharging extremities B and D are on the same horizontal plane B D, will have the same hy- draulic inclination and the same discharge, no matter what the actual inclinations or the depth of the en- trances at A and c may be, so they be of the same kind and bore ; and as the velocities in A B and c D are the same, the height h due to them must be the same when the circumstances of the orifices of entry A and c are alike. We have the whole head H h + h f (see pp. 166 and 167). The hydraulic inclination is not therefore the whole head H, divided by the length / of the pipe, as it is sometimes mistaken, but the height h f (found by subtracting the height h, due to the entrance at A or c, and the velocity in the pipe, from the whole height) divided by the length /. When the height h is very small compared with the whole height H, as it is in very long tubes with moderate heads, j may be substituted for j without error ; but for short pipes up to 100 feet in length 188 THE DISCHARGE OF WATER FROM the latter only should be used in applying Du Buat's and some other formulae ; otherwise the results will be too large, and only fit to be used approximately in order to determine the height h from the velocity of discharge thus found. When the horizontal pipe c D, Fig. 32, is equal in every way to the inclined pipe A B, and the head at A is that due to the velocity in c D, the discharge from the pipe A B will be equal to that from c D ; but a peculiar property belongs to the pipe A B in the position in which it is here placed ; for if we cut it short at any point e, or lengthen it to any extent, to E, the discharge will re- main the same and equal to that through the hori- zontal pipe c D. The velocity in A B at the angle of inclination ABC, when A c = h { , and A B zi c D, is therefore such that it remains unaffected by the length A E or A e, to which it may be extended or cut short ; and at this inclination the water in the pipe A B is said to be " in train." In like manner a river or stream is said to be "in train" when the inclination of its surface bears such a relation to the cross section that the mean velocity is neither decreased nor increased by the length of the chan- nel ; and we perceive from this that the acceleration caused by the inclination is exactly counterbalanced ORIFICES, WEIRS, PIPES, AND RIVERS. 189 by the resistances to the motion when the moving water in a pipe or river channel is in train. o As h = (1 + r ) o where c r is the coefficient of the height due to the resistance at the orifice of entry V 2 l A or c, and h { = c { * , we therefore get (73.) H = (l+c t ) + c f X r and hence we find the mean velocity of discharge as c\ zz j-y, equation (65). We have also C T -f- 1 (74B.) v z the values of the second member on the right hand side of this equation, or of fcM?feb$ 58S UiP r] are given, for different values of c , c d , and -, in the small table at p. 152, and below at p. 191. When h is small compared with A f , or, which comes to the same thing, 1 + c t small compared with c t x - f 190 THE DISCHARGE OF WATER FROM Iv* (75.) v = c t and TT If, in the last equation, we substitute s for y, equal the .sine of the angle of inclination A B c, we then have (77.) || . = The average value of c f for all pipes with straight channels, with velocities of about 1*5 foot per second, is -0069914, from which we find equation (77) be- comes, for measures in feet, (78.) v = 96 \/rs. As the mean value of the coefficient of resistance c y for the entrance into a tube is '508, and as 2# z= 64-403, and c f = -0069914, equation (74), for measures in feet, becomes f 64 -403 H U v= ~~T , or 1-508 + -0069914- -0234r+-OOOi085/J ~(234-hl-085/ (79.) * \\ 58c/+l-085/J * This, multiplied by the section, gives the discharge. For velocities between 2 and 2J feet per second, c t zz -0064403, and therefore ORIFICES, WEIRS, PIPES, AND RIVERS, Hr ^1 191 UI 50 0234 r + -0001 / in which d = 4 r i= diameter of a pipe. The following table is calculated from equation VALUES or - 3 + * X - C 7* N umber of diame- cers in the length of the pipe. Corresponding coeffici- ents of discharge. Number of diame- ters in the length of the pipe. Corresponding coeffici- ents of discharge. 2 diameters 986 814 715 900 diameter 239 236 233 5 957 791 698 950 " + 234 230 227 10 919 769 683 1000 228 225 222 15 886 749 669 noo 218 215 213 20 855 731 656 1200 209 207 205 25 828 713 643 1400 194 192 191 30 804 698 632 1600 182 180 179 35 781 683 620 1800 173 171 170 40 760 668 610 2000 163 162 161 45 741 655 600 2200 156 155 154 50 723 643 590 2400 149 149 148 100 595 548 514 2600 144 143 142 150 518 485 462 2800 139 138 137 200 464 440 422 3000 134 133 133 250 424 405 391 3200 130 129 129 300 392 378 366 3400 126 125 125 350 367 356 345 3600 122 121 121 400 346 336 329 3800 119 119 118 450 329 319 314 4000 116 116 115 500 314 307 300 4200 113 113 113 550 301 295 289 4400 111 111 111 600 289 283 278 4600 108 108 108 650 279 273 269 4800 106 106 106 700 269 265 261 5000 104 104 104 750 ;-;, 261 257 253 5200 102 102 102 800 253 249 246 5400 100 100 100 850 246 242 239 5600 098 098 098 192 THE DISCHARGE OF WATER FROM (74B) for a velocity of about 20 feet per second, when c f zr -004556, and for different orifices of entry, in which c d varies from -986 for a rounded orifice, to '715 when the pipe projects into the vessel. It gives directly the coefficient, which, multiplied by H, gives the velocity in the pipe, taking friction into account. The small table, SECTION VI., p. 152, gives the like coefficients of v/2#H in equation (74B), when c f = 00699 suited to a velocity of about 18 inches per second, and can be applied in like manner. The value of \/2#H is given, in inches, in column 2, TABLE II. For feet it is equal 8v/i nearly. DU BUAT'S FORMULA. The coefficient of friction c t is not, however, con- stant, as it varies with the velocity. That which we have just given answers for pipes when the velocity is 20 feet per second. For pipes and rivers it is found to increase as the velocity de- creases ; that is, the loss of head is proportionately greater for small than for large velocities. Du Buat found the loss of head to be also greater for small than large channels, and applied a correction accord- ingly in his formula. This, expressed in French inches, is (80 .) = - - , - -3 (H - -I), (^-hyp-iog^-M-e/ maintaining the preceding notation, in which s j. In this formula -1, in the numerator of the first term, is deducted as a correction due to the hydraulic ORIFICES, WEIRS, PIPES, AND RIVERS. 193'*ITY \s^U ff mean depth, as it was found that 297 (ri 0-1) agreed more exactly with experiment than 297 rf simply. The f \ \* second term hyp. log. (^ + I'^J, of the denominator is also deducted to compensate for the observed loss of head being greater for less velocities, and the last term -3 (rl _ -1) is a deduction for a general loss of velocity sustained from the unequal motions of the particles of water in the cross section as they move along the channel. These corrections are empirical ; they were, however, determined separately, and after being tested by experiment, applied, as above, to the radical formula v = 297 \/r~s. Du Buat's formula was published in his Principes d'Hydraulique, in 1786. It is, as we have seen, partly empirical, but deduced by an ingenious train of reasoning and with considerable penetration from about 125 experiments, made with pipes from the 19th part of an inch to 18 inches in diameter, laid horizontally, inclined at various inclinations, and vertical ; and also from experiments on open chan- nels with sectional areas from 19 to 40,000 square inches, and inclinations of from 1 in 112 to 1 in 36,000. The lengths of the pipes experimented with varied from 1 to 3, and from 3 to 3600 feet. In several experiments by which we have tested this formula, the resulting velocities found from it were from 1 to 5 per cent, too large for small pipes, and too small for straight rivers in nearly the same proportion. As the experiments from which it was derived were made with great care, those with pipes particularly so, this was to be expected. Expe- 194 THE DISCHARGE OF WATEK FROM riments with pipes of moderate or short lengths should have the circumstances of the orifice of entry from the reservoir duly noted ; for the close agree- ment of this formula with them must depend a great deal, in such pipes, on the coefficient due to the height h, which must be deducted from the whole head H before the hydraulic inclination, -j- zz s, can be obtained ; but for very long pipes and uniform channels this is not necessary. The experiments from which Du Buat's formula was constructed are given in full by the late Dr. Eobinson in his able article on "rivers" in the Encyclopaedia Britannica, pp. 268, 269, and 270, where the calculated and observed velocities are placed side by side in French inches per second. In all these experiments Du Buat carefully deducted the head due to the velocity and orifice of entry before finding the hydraulic inclination s, and those who attempt to calculate the velocity from the head and length of the channel only, without making this deduction, will find their calculated results very dif- ferent from those there given. If there were bends, curves, or contractions, deductions would have to be made for these in like manner before finding s. Under all the circumstances, and after comparing the results obtained from various other formulae, we have preferred calculating tables for the values of v from this formula reduced for measures in English inches, which is = OEIFICES, WEIKS, PIPES, AND RIVERS. 195 or more simply, , Q1 x 307 (r 1) (81.) v = --- - - -_. 3r *_.i. This gives the value of v a little larger than the original formula, but the difference is immaterial. For measures in English feet it becomes (82.) = t 88-51 (r*- -03) _ _ . 084 (fi _ . 03> (-)-hyp.log.(- + l-6)' The results of equation (81) are calculated for different values of s and r, and tabulated in TABLE VIII. , the first eight pages of which contain the velocities for values of r varying from Tsth inch to 6 inches ; or if pipes, diameters from | inch to 2 feet, and of various inclinations from horizontal to vertical. The last five pages contain the velocities for values of r from 6 inches to 12 feet, and with falls from 6 inches to 12 feet per mile. EXAMPLE VIII. A pipe, 1J inch diameter and 100 feet long, lias a constant head of 2 feet over the discharging extremity ; what is the velocity of dis- charge per second ? II 3. , .,100 1 The mean radius r=- = - inches, and zz50n:-, 48 2 S is the approximate hydraulic inclination. At page 2 of TABLE VIII., in the column under the mean radius o -, and opposite to the inclination 1 in 50, we find 8 f 30*69 inches for the velocity sought. This, however, is but approximative, as the head due to the velocity should be subtracted from the whole head of 2 feet, before finding the true hydraulic inclination. This o3 196 THE DISCHAKGE OF WATEE FEOM head depends on the coefficient of resistance at the entrance orifice, or the coefficient of discharge for a short tube. In all Du Buat's experiments this latter was taken at -8125, but it will depend on the nature of the junction, as, if the tube runs into the cistern, it will become as small as -715 ; and, if the junction be rounded into the form of the contracted vein, it will rise to -974, or 1 nearly. In this case, the co- efficient of discharge may be assumed -815,* from which, in TABLE II., we find the head due to a velo- city of 30-69 inches to be 1- = 1-87 inch nearly, which is the value of h ; and hence, H h h t = 24 - 1-87 = 22-13 inches ; and| = ^^ = 54 ' 2 =p the hydraulic inclination, more correctly. With this O i ' new inclination and the mean radius -, we find the 8 velocity by interpolating between the inclinations 1 in 50 and 1 in 60, given in the table to be 30-69 _ 1-34 ~ 29-35 inches per second. This operation may be repeated until v is found to any degree of accuracy according to the formula ; but it is, practically, unnecessary to do so. If we now wish to find the discharge per minute in cubic feet, we can easily do so from TABLE IX., in which, for an inch and a half pipe, we get Inches. Cubic Feet. For a velocity of 20-00 per second, 1-22718 per minute. 9-00 -55223 30 -01841 04 -00245 29-34 .1-80027 * See EXAMPLE 16, pp. 28, 30. ORIFICES, WEIRS, PIPES, AND RIVERS. 197 The discharge found experimentally by Mr. Provis, for a tube of the same length, bore, and head, was 1-745 cubic foot per minute. If we suppose the coefficient of discharge due to the orifice of entry and stop-cock in Mr. Provisos 208 experiments* with li inch lead pipes of 20, 40, 60, 80, and 100 feet lengths, to be '715, the results calculated by the tables will agree with the experi- mental results with very great accuracy, and it is very probable, from the circumstances described, that the ordinary coefficient -815 due to the entry was reduced by the circumstances of the stop-cock and fixing to about -715 ; but even with -815 for the coefficient, the difference between calculation and experiment is not much, the calculation being then in excess in every experiment, the average being about 5 per cent., and not so much in the example we have given. TABLE VIII. will give the velocity, and thence the discharge, immediately, for long pipes, and TABLE X. enables us to calculate the cubic feet discharged per minute, with great facility. For rivers, the mean velocity, and thence the discharge, is also found with quickness. See also TABLES XI., XII., and XIII., and the table at pp. 42 and 43. EXAMPLE IX. A watercourse is 7 feet wide at the bottom, the length of each sloping side is 6*8 feet, the width at the surface is 18 feet, the depth 4 feet, and the inclination of the surface 4 inches in a mile ; what is the quantity flowing down per minute ? * Transactions of the Institution of Civil Engineers, pp. 201, 210, vol. ii. 198 THE DISCHAEGE OF WATEK FROM /TO I I7\ v> _ 4 _ ?. =2-4272 feet=29-126 inches 7 + 2 x 6-8 ""20-6 zzr, is the hydraulic mean depth ; and as the fall is 4 inches per mile, we find at the llth page of TABLE VIII., the velocity v - 12-03 - -16 = 11-87 inches per second ; the discharge in cubic feet per minute is, therefore, 50 x X 60 = 2967-5. 12 If 94-17\/ r s = v, we have v = 94-17-Y/2 -427 + 15840 = 94 ' 17 X = =M7 feet = 14-04 inches. Watt, in a canal of the fall and dimensions here given, found the mean velocity about 13 inches per second. This corresponds to a fall of 5 inches in the mile, according to the formula. Du Buat's formula is less by 12 J per cent, or ^th ; the common formula too much by 5 per cent. In one of the original experiments with which the formula was tested on the canal of Jard, the mea- surements accorded very nearly with those in this example, viz. - = 15360, and r = 29*1 French s inches ; the observed velocity at the surface was 15*74, and the calculated mean velocity, from the formula, 11-61 French inches.* TABLE VII. will give 12-29 inches for the mean velocity, corre- * These measures reduced to inches, give r = 31-014, v = 12-374; and the surface velocity 16-775 inches; reduced for mean velocity 13-101 inches. ORIFICES, WEIES, PIPES, AND RIVERS. 199 spending to a superficial velocity of 15*74 inches. This shows that the formula also gives too small a value for v in this case, by about rrth of the result, it being about ; part in the other. The 8'3 probable error in the formula applied to straight clear rivers of about 2 feet 6 inches hydraulic mean depth is nearly Ath or 8 per cent, of the tabu- lated velocity, and this must be added for the more correct result ; the watercourse being supposed nearly straight and free from aquatic plants. Notwithstanding the differences above remarked on, we are of opinion that the results of this formula, which we have calculated and tabulated, may be more safely relied on as applied to general prac- tical purposes than most of those others which we shall proceed to lay before our readers. Eivers or watercourses are seldom straight or clear from weeds, and even if the sections, during any improvements, be made uniform, they will seldom continue so, as " the regimen" or adaptation of the velocity to the tenacity of the banks, must vary with the soil and bends of the channel, and can seldom continue per- manent for any length of time unless protected. From these causes a loss of velocity takes place, difficult, if not impossible, to estimate accurately, but which may be taken at from 10 to 15 per cent, of that in the clear unobstructed direct channel ; but be this as it may, it is safer to calculate the drainage or mechanical results obtainable from a given fall and river channel, from formula which give lesser, than 200 THE DISCHARGE OF WATER FROM from those which give larger velocities. This is a principle engineers cannot too much observe. We have before remarked, that for both pipes and rivers the coefficient of resistance increases as the velocity decreases. This is as much as to say, in the simple formula for the velocity, v zz m \/r s, that m must increase with v, and as some function of it. This is the case in TABLE VIII., throughout which the velocities increase faster than \/r, the \/s, or the v/rsT In all formulae with which we are acquainted but Du Buat's and Young's, the velocity found is con- stant when \/r s'or r X s is constant. In Du Buat's formula for r x s constant, v obtains maximum values between r zz f inch and r zz 1 inch ; the dif- ferences of the velocities for different values of r above 1 inch, r x s being constant, are not much. We may always find the maximum value, or nearly so, by assuming r zz J inch, and finding the corre- 4- 7* c spending inclination from the formula ~o~~j which is equal to it. For example, if r 12 inches, and s zz , the velocity is found equal 9*52 inches ; but when r s is constant, the inclination s corresponding 4 X 12" 1 to r zz j- inch is zz , from which we find 3 X 10560 660' from the table vzzlO-25 inches for the maximum velocity, making a difference of fully 7 per cent. When r zz -01 of an inch, or a pipe is Ath part of an inch in diameter, Du Buat's formula fails, but it gives correct results for pipes ith of an inch in diameter, and two of the experiments from which it OEIFICES, WEIES, PIPES, AND EIVERS. 201 was derived were made with pipes 12 inches long and only -rsth part of an inch in diameter. COULOMB having shown that the resistance opposed to a disc revolving in water increases as the function a v + b v* of the velocity v, we may assume that the height due to the resistance of friction in pipes and rivers is also of this form ; and that (83.) k t = (av + bv*) 1 -, and consequently^ (84.) r s av + bv*, and v= G-IRARD first gave values to the coefficients a and b. He assumed them equal, and each equal to -0003104 for measures in metres, and thence the velocity in canals, (85.) v = (3221-016 rs + -25^ -5;* which reduced for measures in English feet becomes (v = (10567-8 r s + 2-67)* 1-64, or (v = 103 \/Ts 1-64, nearly. The value of a = b = -0003104 was obtained by means of twelve experiments by Du Buat and Chezy. Of course the value is four times this in the original, as we use the mean radius in all the formulae instead of the diameter. This formula is only suited for very small velocities in canals, between locks, con- taining aquatic plants ; it is inapplicable to rivers and channels in which the velocity exceeds an inch per second. PRONY found from thirty experiments on canals, * See Brewster's Encyclopedia, Article Hydrodynamics, p. 259. 202 THE DISCHAEGE OF WATER FROM that a = -000044450 and b = 000309314,* for mea- sures in metres, from which we find (87.) v = (3232-96 r s + -005 16)* -0719 ; this reduced for measures in English feet is, rt> = (10607-02 r* + -0556)* -236 ;f or (v = 103 \/rs -24 nearly : the velocities did not exceed 3 feet per second in the experiments from which this was derived. For pipes, Prony found,J from fifty-one experi- ments made by Du Buat, Bossut, and Couplet, with pipes from 1 to 5 inches diameter, from 30 to 7,000 feet in length, and one pipe 19 inches diameter and nearly 4,000 feet long, that a = -00001733, and b zz -0003483, from which values (89.) v = (2871-09 rs + -0006192)* -0249, for measures in metres, and for measures in English feet, (v = (9419-75 rs + -00665)* '0816; or ( ' (v = 97 v/ '08 nearly. Prony also gives the following formula applicable to pipes and rivers. It is derived from fifty-one selected experiments with pipes, and thirty-one with open channels : (91.) v = (3041-47 rs + -0022065)* -0469734,g * Reeherches Physico-Mathe'matiques sur la ThSorie des Eaux Courantes. j- For canals containing aquatic plants, reeds, &c., we must sub- stitute jTy for r. See note, p. 186. J Keeherches Physico-Mathematiques sur la The"orie du Mouve- ment des Eaux Courantes, 1804. Keeherches Physico-Mathe'inatiques sur la Theorie des Eaux Oourantes. A reduction of this formula into English feet is given at page 6, Article Hydrodynamics, Encyclopedia Britannica ; at ORIFICES, WEIRS, PIPES, AND RIVERS. 203 TABLE of the fifty-one Experiments referred to in Equation (89), the value of g in the sixth being taken at 9-8088 metres. It will be perceived that Prony did not take into calculation, in framing his formula, the head due to the velocity in the pipe and to the orifice of entry. Number of selected expe- riments. Names of Experimenters. Heads measured to the lower ori- fice in metres. n! w Length of the pipes in metres. Values of 9 r * V in metres. Experimental values of the velocity v in metres. Calculated velocity from formula (89) in metres. 1 Du Buat 0041 0271 19-95 000314 0430 0427 2 Couplet 1511 1353 2280-37 000404 0544 0591 3 Couplet 3068 1353 2280-37 000523 0854 0921 4 Du Buat 0135 02707 19-95 000459 0980 0926 5 Couplet 4534 1333 2280-37 000590 1117 1263 6 Couplet 6105 1333 2280-37 000638 1301 1330 7 Couplet 6497 1333 2280-37 000670 1411 1433 8 Couplet 6767 1333 2280-37 000683 1441 1467 9 Du Buat 0189 0271 3-75 001426 2352 2895 10 Du Buat 1137 0271 3-75 '001138 2826 3088 11 Du Buat 1137 0271 375 001309 2888 3088 12 Bossut 1083 0271 16-24 001337 3308 3359 13 Bossut 3248 0361 58-47 001446 3400 3553 14 Du Buat 1605 0271 19-95 001482 3604 3713 15 Bossut 3248 0361 48-75 001549 3807 3915 16 Du Buat 2106 0271 19-95 001713 4091 4287 17 Bossut 3248 0361 38-98 001687 4366 4402 18 Du Buat 2425 0271 19-95 001830 4408 4618 19 Bossut 3248 0544 58-47 001672 4433 4416 20 Du Buat 2425 0271 1995 001793 4500 4618 21 Bossut 3248 0544 48-73 001795 4955 4860 22 Bossut 6497 0361 58-47 '001922 5115 5122 23 Bossut 3248 0361 29-23 001918 5128 5122 24 Du Buat 3335 0271 19-95 002050 5411 5450 25 Bossut 3248 0544 38-98 '001981 5605 5458 26 Du But 3709 0271 19-95 '002174 5676 5766 27 Bossut 6497 0361 48-73 002073 5693 5634 28 Du Buat 3952 0271 19-95 002223 5916 5961 29 Bossut 3248 0271 26-24 002201 6032 5990 30 Bossut 3248 0361 19-49 002333 6323 6327 31 Bossut 3248 0544 29-23 002300 6444 6344 32 Bossut 6497 0361 38-98 002267 6498 6323 33 Bossut 6497 0544 58-47 002214 6695 6344 34 Bossut 6497 0544 48-73 002392 7436 6972 35 Bossut 6497 0361 29'23 002588 74 7343 36 Du Bu^t 6416 0271 19-95 002750 7761 7660 37 Bossut 3248 0544 19-49 002812 7908 7823 38 Du Butt 1624 0271 3-75 003620 7943 8930 39 Bossut 6497 0544 38-98 002656 8363 7819 40 Bossut 3248 0361 9-74 003287 8976 9048 41 Bossut 65 0361 1949 003161 9332 9048 42 Bossut 65 0544 29-23 003062 9681 9071 43 Couplet 3-9274 4873 1169-42 003785 1-0600 1-0592 44 Bossut 3248 0544 9'74 004073 1'0915 1-1164 45 Bossut 6497 0544 19-49 003821 1-1640 1-1164 46 Bossut 6497 0361 9-74 004491 1-3138 1-2896 47 Du Buat 4873 0271 3-17 006470 1-5784 1-7043 48 Du Buat 5671 0271 3-75 006307 1-5919 1-6898 49 Bossut 6497 0544 9-74 005578 1-5945 1-5890 50 Du Buat 7219 0271 3-17 007838 1-9301 2-0798 51 Du Buat 9745 0271 3-17 008882 2-2994 2-4205 204 THE DISCHARGE OF WATER FROM for measures in metres, which, reduced for measures in English feet, is v = (9978-76 r* + -02375)* -15412; or k ' v = 100 vr s -15 nearly. EYTELWEIN, following the method of investigation pursued previously by Prony, found from a large number of experiments, a = -0000242651, and b = 000365543 in rivers, for measures in metres ; and, therefore, (93.) v = (2735-66 rs + -001102)* -0332.* This reduced for measures in English feet, is v = (897543 r s + -0118858)* '1089 ; or (94.) v = 94-5 v/rs -11 nearly = 1-3 v//> -11 = v/f-fyv -11 when f is the fall in feet per mile. He also shows,f that irths of a mean proportional between the fall in two English miles and the hydraulic mean depth, gives the mean velocity very nearly. This rule for measures in inches is equivalent to (95.) v = 324 >/ ; and for measures in feet (96.) t? = 93-4\/r*. For the velocity of water in pipes, he found,J from the fifty-one experiments of Du Buat, Bossut, and page 164, Third Report, British Association, by Rennie ; and at pages 427 and 533, Article Hydrodynamics, Brewster's Encyclo- pedia. This reduction, v = 0-1541 + (-02375 + 32806*6 rs)* is entirely incorrect ; and. being the same in each of those works, appears to have been copied one from the other. * Memoires de 1'Academie de Berlin, 1814 et 1815. See equation (110). f Handbuch der Mechanik und der Hydraulik, Berlin, 1801. J Memoires de I'Acad&nie des Sciences de Berlin, 1814 et 1815. ORIFICES, WEIRS, PIPES, AND RIVERS. 205 Couplet, that a = -0000223, and b = -0002803, from which we get for measures in metres, (97.) v = (3567-29 r s + -00157,* -0397 ; which reduced for measures in English feet becomes r v = (11703-95 rs + -01698)* -1303 ; or (98>) \v = 108 \/rs -13 nearly. Another formula given by Eytelwein for pipes, which includes the head due to the velocity for the orifice of entry, is in which H is the head, I the length, and d the dia- meter of the pipe, all expressed in English feet. This is a particular value of equation (74) suited to velo- cities of about 2J feet per second. It must be here mentioned, that much of the valuable information presented by Eytelwein is but a modification of what Du Buat had previously given, to whom only for much that is attributed to the former we are primarily indebted. In the foregoing as well as in the following equa- tions for the velocity, we have, unless otherwise stated, maintained one class of standards. It is evident if we change these standards in part, or in whole, that apparently different forms of the equa- tions will arise ; thus if for s, the hydraulic incli- ryyi nation, we substitute T^on? we shall have the fall m in feet per mile, in place of the inclination s; so that equation (94), for instance, would become v = (l-7nr+-012)* -11 = (l*7mr) i -11 nearly, in which v is the velocity in feet per second, m the 206 THE DISCHARGE OF WATER FROM fall in feet per mile, and r the "hydraulic mean depth " in feet. In like manner equation (98) would become v = (2-2 m r + -02)* -13 = (2 -2 m r)* -13. The first of these reductions, viz. : v = (1-7 m r + -0119)* -109, is given in a book of tables calculated for river channels for the Commissioners of Public Works, Ire- land, the original equation being Eytelwein's, and not D'Aubuisson's, who merely copied it, and is suited for velocities averaging about 1*3 feet per second. Mr. Hawksley gives for pipes the formula itJiMi l,+ 1-5 in which / is the length in yards, H the head in inches, d the diameter in inches, and v the velocity in yards per second. For uniform feet measures, for, v, d, and H, this becomes which is only Eytelwein's equation (99) slightly modified. Eytelwein's equation expressed in the measures used by Mr. Hawksley would be very nearly d { ^ )i which is far the simplest of the two ; both, however, are but particular cases of the general equation 74), and .only suited for velocities of about 2j feet per second. DR. THOMAS YOUNG* also derives his formula from * Philosophical Transactions for 1808. OBIFICES, WEIRS, PIPES, AND RIVERS. 207 the supposition, that the head due to the resistance of friction assumes the form of equation (83) ; calling the diameter of a pipe d, he takes and the whole height H = h { + j^, expressed in inches. He found from some experiments of his own, those collected by Du Buat, and some of G-erstner's, that (100.) a = -0000002 and then as zz -00171, we get 586 / al vM a I \2/ + -00341 dJ I '" 2 b I + -00341 3/--J 2 b I + -00341 d' When the length / of the pipe is very great compared with the head due to the orifice of entrance and velocity, -00171 v*, we have n end a 2 * a TT or by substituting for ^ its value s, equal the sine of the inclination, n (sd a 2 J a (104.) v= -- + 208 THE DISCHARGE OF WATER FROM The values of a and b are for measures in inches. For most rivers, in which d must be taken equal 4 r, he finds for French inch measures, v = v/20000^ ; this reduced for English inches is (105.) ^ = 292x7^; which again reduced for feet measures, becomes (106.) v = 84-3v/. These latter values, for rivers, are even smaller than those found from Du Buat's formula ; less than the observed velocities, and less than those found from any other formula, with the exception of Girard's. The values of the coefficients a and b vary in this formula with the value of d = r; they are expressed generally in equations (101) and (102), from which we have calculated the following table for different values of d and r. An examination of this table will show that a obtains a minimum value when d is between 10 and 11 inches ; and b when the diameter is between J and f of an inch. Now, it appears from equation (102), that v increases with <\ nearly, or, which is the same thing, as b decreases, there must, cceteris paribus^ be a maximum value of v for a H d given value of -j-, or r s, when d is between J and } inch ; but as -~-. has a minimum value when d is nearly 12 inches, the maximum value of v referred to will be found between values of d from I inch to 12 inches ; in fact, when d 10 inches nearly. We have already pointed out a similar peculiarity in Du ORIFICES, WEIRS, PIPES, AND RIVERS. 209 Buat's general theorem, at page 195. It will not be a a* necessary to take out the values* of -^-j and ^ to more than one place of decimals. The values of 77-7- are also given in the following table, and may be used in equation (104) for finding the discharge from long pipes. It is, however, neces- sary to remark, that this equation is sometimes mis- applied in finding the velocity from short pipes, and those of moderate lengths. It is necessary to use equation (102), which takes into consideration the head due to the velocity and orifice of entry for such pipes. For a pipe 11 inches in diameter, the expression for the velocity, equation (104), becomes for.jnch measures, and for feet measures, also substituting 4 r for d, ( very nearly. For a pipe -7 inch in diameter we should find in a like manner for feet measures, (106s.) v = 118(r*)i-5, which is only suitable for pipes with very high velo- cities. SIR JOHN LESLIE states,* that the mean velocity of a river in miles per hour, is Ifths of the mean proportional between the hydraulic mean depth and * Natural Philosophy, p. 428. 210 THE DISCHAKGE OF WATEE FKOM ft U5 > CO co co co 10 10 o us o t>- OOOOOOiOOO c?ot^oOOOC? OKIFICES, WEIKS, PIPES, AND EIVEKS. 211 the fall in two miles in feet. This rule is equivalent, for measures in feet, to (107.) v zz 100 v/777 and is applicable to rivers with velocities of about 2i feet per second. D'AuBuissoN, from an examination of the results obtained by Prony and Eytelwein, assumes* for measures in metres that a zz -0000189, and b zz 0003425 for pipes, substituting these in equation (84) and resolving the quadratic (108.) v zz (2919-71 rs + -00074)^ _ -027 ; which reduced for measures in English feet becomes (v zz (9579 rs + -00813)* - -0902, or \v = 98 v/_-l nearly. For rivers he assumes with Eytelwein,f a zz 000024123 and b zz -0003655, for measures in metres, and hence (110.) v zz (2735-98 rs + -0011)* -033 ; which for measures in English feet is v zz (8976-5 rs + -012)' _ -109, or v zz 94-5 N/^S '11 nearly. When the velocity exceeds two feet per second, he assumes, from the experiments of Couplet, a zz 0, and b zz -00035875 ; these values give (112.) v zz v/ 2787-46 rs, for measures in metres, and (113.) v zz 95-6 \/7s zz v/ 9145 rs for measures in English feet. Equations (110) and (111) are the same as (93) and (94), found from Eytel- * Traite d'Hydraulique, p. 224. t Traite d'Hydraulique, p. 133. See Equation (93). p 3 212 THE DISCHARGE OF WATER FROM wein's values of a and b, and it may be remarked that D'Aubuisson's equations for the velocity generally, are simply those of Prony and Eytelwein. The values which we have found to agree best with experiments on clear straight rivers are a z= 0000035, and b = -0001150 for measures in English feet, from which we find v = (8695 -6 r s + -00023)*- '0152, or which for an average velocity of 1J foot per second will give v =. 92'3\A*s nearly, and for large velo- cities vzz93'3v/rA- ; for smaller velocities than 1J foot per second, the coefficients of \/r s decrease pretty rapidly. This formula will be found to agree more accurately with observation and experiment than any other we know of this form. WEISBACH is perhaps the only writer who has modified the form of the equation r s a v + b v*. In Dr. Young's formula, a and j b vary with r, but 7 j 2 Weisbach assumes that A f (a + -i)~ x ~o~ > and finds from the fifty-one experiments of Couplet, Bossut, and Du Buat, before referred to, one experi- ment by Guemard, and eleven by himself, all with pipes varying from an inch to five and a half inches in diameter, and with velocities varying from 1J inch to 15 feet per second, that a -01439, and b z= 0094711 for measures in metres ; hence we have for the metrical standard This reduced for the mean radius r is ORIFICES, WEIRS, PIPES, AND RIVERS. 213 (116.) from which we find for measures in English feet and thence (118.) ,. = (, + 5!") *; and by substituting for 2^7, its value 64*403, (119.) ,. = (-00005585 + /. In equation (117), (-003597 + - -4~ -) = c t is the coefficient of the head due to friction. The equation does not admit of a direct solution, but the coeffi- cient should be first determined for different values of the velocity v and tabulated, after which the true value of v can be determined by finding an approx- imate value, and thence taking out the corresponding coefficient from the table, which does not vary to any considerable extent for small changes of velocity. In the following small table we have calculated the coefficients of friction, and also those of t; 2 , in equa- tion (119), for different values of the velocity v. 214 THE DISCHARGE OF WATER FROM TABLE OF THE COEFFICIENTS OF FRICTION IN PIPES. Velocity in feet. Cf Cf 64-4 644 Cf v/^ 4 Cf Velocity in feet. Cf Cf 64-4 644 c r ^/m 1 017159 0002664 3078-07 55-5 2-4 006365 0000988 10121-5 100-5 2 013186 0002047 4885-2 69-9 2-5 006309 0000979 10214-5 101-0 3 011427 0001774 5636-9 75-08 2-6 006257 0000972 10288-1 101-4 4 010378 0001611 6270-3 78-8 2-7 006207 0000964 10373-4 101-8 5 009662 0001500 6666-6 81-6 2-8 006160 0000956 10460-2 102-2 6 009133 0001418 7052-2 84-0 2-9 006115 0000949 10537-4 102-6 7 008723 0001354 7385-5 85-9 3- 006073 0000943 10604-4 102-9 8 008391 0001303 7674-6 87-6 3-5 005890 0000914 10940-9 104-6 9 008117 0001260 7936-5 89-1 4- 005741 0000891 11223-3 105-9 1-0 007886 0001224 8169-9 90-4 5- 005514 0000856 11682-2 108-0 1-1 007686 0001193 8382-2 91-5 6- 005348 0000830 12048-2 109-7 1-2 007512 0001166 8576-3 92-6 7- 005218 0000810 12345-6 111-1 1-25 007433 0001154 8665-5 93-1 8- 005113 0000794 12632-2 112-4 1*8 007358 0001142 8756-5 93-5 9- 005026 0000780 12820-5 113-3 1-4 007221 0001121 8920-6 94-4 10- 004953 0000769 13003-9 114-0 1-5 007098 0001102 9074-4 95-2 15- 004704 0000730 13698-6 117-0 1-6 006987 0001085 9216-5 96-0 16- 004669 0000725 13793-1 117-4 1-7 006886 0001069 9354-5 96-7 20- 004556 0000707 14144-2 118-9 1-75 006839 0001062 9416-2 97-03 25- 004455 0000691 14471-7 120-3 1-8 006794 0001054 9487-6 97-4 30- 004380 0000680 14705-9 121-2 1-9 006715 0001042 9596-9 97-9 35- 004322 0000671 14903-1 122-0 2- 006629 0001029 9718-2 98-5 40- 004275 0000664 15060-2 122-7 2-1 006556 0001018 9823-2 99-1 45- 004236 0000658 15197-5 123-3 2-2 006488 0001007 9930-5 99-6 50- 084203 0000653 15313-8 123-7 2-3 006424 0000997 10003- 100- 100- 004208 0000625 16000-0 126-4 If the value of the equation v = 644 here found, be substituted in c f rs. we shall have the value of v. According to this table the coefficient of fric- tion for a velocity of six inches is more than twice that for a velocity of twenty feet, and the velocity is less in the proportion of 81*6 to 118*9, or of 81*6 (rs)* ORIFICES, WEIRS, PIPES, AND RIVERS. 215 to 118*9 (rs)*. On comparing these coefficients and those for pipes in the preceding formulae, with those for rivers of the same hydraulic depth, we perceive that the loss from friction is greatest in the latter, as might have been anticipated ; but this evidently arises from lesser velocities. It has been already remarked that the coefficient of friction decreases as the velocity increases. The only general formula which properly meets this de- fect in the common formulae is Weisbach's, but it does not give the velocity v directly, as this quantity is involved in both sides of his equation. As for several hydraulic works it is necessary to convey water through pipes to work machines under high heads, and for which the common formula would give results considerably under the true ones, it appeared to me desirable to obtain some simple ex- pression for the velocity which might be easily remembered and applied, which would be equally correct with other formulae for medium velocities of from one to two and a half feet, and which at the same time would give practically correct results for lesser and greater velocities within the limits of experiment. By reducing the velocity found from experiment to the form v = m ^/^~ s for every case, and afterwards applying a correction of the form n \/^rs to meet the increasing value of m as v in- creased, I discovered that the expression (119 A) v= 140 (rs)* - 11 (r*)* gave results not differing more from experiments than these frequently do from each other. The following table exhibits the velocities compared with those 216 THE DISCHARGE OF WATER FROM obtained from the experiments made by Du Buat, Couplet, Watt, Mr. Provis, and Mr. Leslie, in the Minutes of the Institution of Civil Engineers for February 1855. The last experiment was furnished to me by Mr. Hodson of Lincoln. Numbers 34 and 35 were made by myself, and give the mean results of several experiments made with great care ; the coefficient of the orifice of entry was found to be -860.* The measures have been all reduced to English feet. The results found by the same experi- menters, at the same time, with the same apparatus, sometimes differ by three or four per cent., as may be seen by referring to Mr. Provis' experiments, (Trans- actions of the Institution of Civil Engineers, vol. n., p. 203,) and the difference in the experiments shown in the table are apparent. The difference in the velocities found from the experiments, do not exceed those inseparable from practical investigations, and they differ as much in themselves as from the formula, which for cylindrical pipes of diameter d may be thus expressed, v zz 70 (d 5)4 6-93 (d )*, or (119B.) 1 v = 70 (d *)4 - 7 (d s)* nearly. The expression fails when 70 (d sfi is equal to or less than 6-93 (d s$, but as this only happens when f\\ \8 r s =1^40) = '000000235, and for velocities below one inch per second, its practical value is not thereby affected. The expression of Du Buat fails with a * The coefficient for the orifice of entry was found by cutting off the pipe at two diameters from the cistern at the conclusion of the experiments, and finding the time of emptying. Vide p. 177. ORIFICES, WEIRS, PIPES, AND RIVERS. 217 TABLE showing the Experimental Results of observed Velocities in Water Channels, with the Author's general formula for Pipes and Rivers, viz. v = 140 (rsj*-Il (r s)*. c K Heads in feet (H). Lengths in feet (I). Values of r. Values of s. Values of r 8. Velocities 1 from experiment. I Velocities I from the 1 formula. 1 Velocities expressed in the form v = m \/ r s. Experimen- ters' Names. 1 08333 1086- 052083 000076 00000396 100 105 52-6 ^/T, Mr. Leslie 2 01332 65-37 022204 000116 00000434 140 113 54-0 Couwlet 8 14583 1086- 052083 000133 00000693 118 157 60-0 Mr. Leslie 4 49566 7482- 111000 000066 00000134 178 167 61-5 Couplet 6 20833 1086- 052083 000190 00000989 217 206 65-0 Mr. Leslie 6 45833 1086- 052083 000417 00002170 361 345 74-1 7 1-4876H 7482- 111000 000198 00002220 366 348 74-1 Couplet 8 g 1-448001086- 2-781251086- 052083 052083 001321 002538 0000688 0001322 715 1-085 711 1-050 85-7 91-3 Mr. Leslie 10 2-427200 000063 0001532 1-166 1-143 92-4 Watt" 11| -50000| 100- 031250 004741 0001482 1-023 1-122 92-2 Mr. Provis 12 2-78125 1086- 052083 004348 0002265 1-461 1-438 95-5 Mr. Leslie L3 4-76042 1086- 052083 006410 0003340 1-725 1-796 98-3 14 1-06580 127-9 044630 007748 0003458 1-839 1-840 98-9 Bossut 15 50000 40- 031250 010810 0003378 1-711 1-816 98-7 Mr. Frovis in 1-06580 95-92 044630 010050 0004485 2-111 2-124 100-3 Bossut 17 1-5 100- 031250 014156 0004422 2-005 2-103 100-5 Mr. Provis 18 9-9896 1086- 052083 009174 0004779 2-095 2-185 100-6 Mr. Leslie 1!) 8575 40- 031250 018042 0005638 2-380 2-414 101-7 Mr. Provis '20 2-1316 191-9 044630 010548 0004708 2-463 2-183 100-6 Bossut 21 2-1316 159-9 044630 012524 0005589 2-440 2-404 101-7 , 22 ft 052083 014286 0007440 2-800 2-823 103-5 , Mr. Leslie 2:) 2-1316 127-9 044630 015350 0006851 2-744 2-696 103-0 , Bossut 24 n n 044630 027921 0012465 3-819 3-760 106-5 , 25 n J} 052083 025000 0013021 3-783 3-852 106-7 , Mr. Leslie 36 3-27416 40- 031250 018040 002093 5-054 5-006 109-3 , Mr. Provis 27 2-3684 10-39155 022204 133689 0029679 6-322 6-048 111-0 , Du Buat 28 3-27416 20- 031250 111200 0034750 6-723 6-572 111-5 , Mr. Provis 2!) 3-4525 20- 031250 113900 0035594 7-086 6-668 111-9 , 30 7-135 62-8822 029605 098861 0029268 6-157 5-999 110-9 , Couplet 31 14-270 125-7644 029605 106151 0031426 6-151 6-239 111-3 :S2 21-405 188-6466 029605 108579 00321455 6-145 6-316 111-4 33 3-1974 10-39155 022204 176991 0039292 7-544 7-039 112-3 Du Buat 34 11-125 9-292 021250 713000 01515125 14-583 14-513 117-9 Mr. Neville 35 20-8 19-2 021250 814000 01729750 15-667 15-617 118-4 36 ISO- 100- 020833 1-400000 0291667 21-7 20-6 120-3^/71 Mr.Hodson tube of one twenty-fifth part of an inch in diameter, no matter what the head may be, as it then makes the velocity equal to nothing, although some of the experiments from which it was derived were made with tubes but the eighteenth part of an inch dia- 218 THE DISCHARGE OF WATER FROM meter. The following expression is free from this defect : (119o.) v = 60 (rsfi + 120 (rs)$, and will give results approximating very closely to those found from Du Buat's formula, and, therefore, with those experiments with which it most nearly coincides, but agreeing much more closely with Watt's and other experiments, on rivers. It gives higher results than the previous formula for velo- cities below six inches, but the results found by different experimenters differ very much in those. For higher velocities it appears to differ occasionally only about one-twentieth from observation, being in general less, as far as twenty feet per second, where it coincides very closely with Mr. Hodson's expe- riment. As the errors appear to be of an opposite kind generally, in the two last expressions, we may get by combining them (119D.) v = 100 (rs)* + 60 (r s)* 5 -5 (r*)*, an expression which, however, wants simplicity for ready practical application. When the length of the pipe does not exceed from 1000 to 2000 diameters, a correction is due to the velocity in it, and to the orifice of entry before finding the "hydraulic incli- nation" (s). The coefficient used in reducing the foregoing experiments for the orifice of entry was v 2 815, which gives 1*508 -^ for the height due to the joint effects of velocity and orifice. This must be deducted from the head (H) before dividing it by the length (/) to find the inclination (s) in our table. ORIFICES, WEIRS, PIPES, AND RIVERS. 219 The following table, calculated from the formula (119A), v = 140 (rs)? 11 (r s)?, gives the corre- sponding values of r s and v, so that when one is known the other is immediately found from inspec- tion. Thus, if r s = -03125, we find v 20-6 when rs= -029167 v 24-7 when r s = -041666 Difference 4-1 corresponds to -012499 03125 02917 Difference -00208 Whence -0125 : 4-1 : : -00208 : -7 nearly, and 20-6 4 7 z= 21-3 is the velocity sought; the same practically as found in EXAMPLE 26, p. 37. If allowance is to be made for the head due to the orifice of entry and velocity, this head can be de- termined from the velocity due to the value of r s in the table next less than the given value with sufficient accuracy. In this case, this velocity is 20-6 feet per second zz 247 inches nearly. If the orifice of entry be square, the coefficient is -815, and the head due to the velocity and this coefficient is, TABLE II., 10 feet nearly. If r be known separately, and also ,9, as well as the head H, and the length of the pipe /, we had at first H H 10 h j zz s, and, therefore, * ~ 7 "" s ' In EXAMPLE 26, p. 37, H = 150, and 1= 100 feet, therefore, the new value of j z= =r^ is 14 ; and as r must be equal '020833, rs = -02917 : the value 220 THE DISCHARGE OF WATER FROM TABLE for finding the Velocity in feet per second, from the product of the hydraulic mean depths and hydraulic inclinations, and the reverse calculated from the Authors formula v = 140 (rs) 4 11 (rs)*, in which r, s, and v, are feet measures. Values of r s. Velo- city V. Values of rs. Velo- city V. Values of r Velo- city V Values of rs Velo- city V. 00000296 083 0001302 1-04 000689 270 003559 6-67 00000332 091 0001322 1-05 000710 2-75 003599 6-71 00000395 104 0001420 1-09 000744 2-83 003630 6-74 00000427 111 0001482 1-12 000758 2-85 003788 6-90 00000543 133 0001532 1-14 000789 2-91 003929 7-04 00000592 142 0001578 1-16 000805 2'94 003946 7-05 00000690 158 0001610 1-17 000833 3-00 003977 7-08 00000734 167 0001657 1-19 000852 304 004104 7-20 00000947 198 0001736 1-21 000900 3-13 004167 7-27 00000989 206 0001776 1-24 000947 3'22 004356 7-44 00001184 231 0001815 1-26 001042 340 004546 762 00001263 241 0001894 1-30 001105 3-51 004630 7-69 00001420 261 0002052 1 35 001136 3-57 004735 7-78 00001578 280 0002131 1-38 001231 3-73 005556 8-49 00001677 292 0002265 1-43 001246 3'76 006944 9-61 00001894 316 0002367 1-47 001263 3'78 007576 10-0 00001973 325 0002552 1-50 001302 3-85 008333 10'5 00002170 345 0002604 1-55 001326 3-89 009259 11 1 00002367 365 0002652 1-57 001420 4-04 010417 118 00002565 385 0002778 1-61 001515 4'18 011905 12-7 00002841 411 0002841 1-63 001576 4-28 013889 13-8 00003255 448 0003030 1-69 001610 4-32 015151 14-5 00003354 457 0003157 1-73 001667 4-41 016667 15-3 00003551 473 0003220 1 75 001705 4'46 017297 156 00003748 489 0003314 1-79 001735 4'51 020833 17-1 00003946 505 0003378 1-80 001799 4'60 027778 20-2 OOC04143 521 0003409 1-81 001894 4-73 029167 20-6 00004340 536 0003551 1-85 001989 4-87 041666 24-7 00004632 558 0003630 1-89 002052 4-94 055556 28-8 00005130 594 0003706 1-90 002083 4-98 062500 30-6 00005327 608 0003788 1-92 002093 5-00 072916 33-2 00005524 622 0003946 1-98 002178 5-10 083333 35-6 00005919 648 0004022 1-10 002210 5-14 104167 40-0 00006314 674 0004103 2-02 002273 5-22 125 43-9 00006708 699 0004261 36 002375 5*35 145583 47-6 0000688 711 0004419 2-10 002462 5-46 166667 51-1 00007102 724 0004485 2-12 002533 553 208333 57-3 00007694 760 0004546 2-14 002652 56-8 229167 60-2 00008049 781 0004708 2-18 002683 5-72 250000 63-0 00008523 808 0004735 2-18 002841 5'90 270833 65-7 00008681 828 0004893 2-23 002968 6-05 312500 70-7 00009270 849 0005051 2-27 002999 6-08 333333 73-2 00009470 861 0005208 2-31 003030 6-11 354167 75-5 00010259 903 0005303 2-33 003143 6-23 375000 77-7 00010654 923 0005638 2-41 003157 6-25 395833 80-0 00011048 945 0006061 2-52 003214 6-31 416667 62-1 00011364 960 0006155 2-54 003220 6-32 437500 84-2 00011837 983 0006313 2-57 003314 6-42 458333 86-2 00012232 1-00 0006440 2-60 003409 6-51 479166 88-3 00012627 1-02 0006629 2-64 003475 6-58 500000 90-2 ORIFICES, WEIRS, PIPES, AND RIVERS. 221 corresponding to which, in our table, is 20*6, the velocity when allowance is made for the head due to the velocity and orifice of entry. *+ In general, by taking the value of v for the next less value of r s in the table, we shall find the velocity with sufficient accuracy, and also the value of r s from that of v by taking it as the next greater. If we had taken r s -0008523, the table would give v = 3-04 feet, the same practically as already found in EXAMPLE 27, p. 38. The value of r s, when known, determines the value of v. If r be assumed of any convenient dimen- sions, s is determined ; and, in like manner, any suitable value of s determines r ; thus : r s _ r s =s, and z= r. r s It is well to remark, here again, that for pipes the value of r is the fourth part of the diameter J, and that r zz j, and 4 r = d. In 1857, M. Darcy, inspecteur des ponts et chausees, published his Eecherches experimentales relatives au Mouvement de 1'Eau dans les Tuyaux,* the result of 198 experiments, in which the velocities varied from '03 to 5 or 6 metres per second, or from H inch to 16 or 19 feet, and with pipes varying from J inch to 20 inches diameter. The formula by which he presents the results is in metres, (a.) R j = x u 2 , in which R is the radius of the pipe, j the hydraulic * Morin's Hydraulique, deuxieme edition, Paris, p. 164. 222 THE DISCHARGE OF WATER FROM inclination, b-^ a variable coefficient dependent on the circumstances, and u the velocity per second. For wrought and cast iron pipes of the same state of bore, the value of b is expressed by M. Darcy, by the equation (ft.) Z, = -000507 + 0000 647 , ' R the agreement between which and experiment is shown in the following table. ;. Diameters in English inches. Diameters in metres. Values of ^ from experi- ments. Values of b^ by the formula. Kemarks. 5 0122 001'673 001568 1- 0266 000918 000993 1-5 0395 000785 000835 3-2 0819 000695 000665 5-4 1370 000553 000601 Well polished bore. 7.4 11-7 1880 2970 000584 000612 000576 000551 f Pipe already in use, \ but the bore cleaned. 19-7 5000 000509 000532 For iron coated with bitumen, the value of b in a pipe *196 metres in diameter was -0004334 ; for a newly cast pipe of -188 metres, b was -000584; and for a pipe -2432 metres in diameter, ^ was 001168 ; the relative proportions of b in these three instances, being as 1-1 to 1-5 and to 3 ; and, therefore, the velocities, or discharges, would be inversely as the square roots of these, or as 95 to -82 and to -58. By substituting our notation for that of M. Darcy, we shall have in metres, from equations (a) and (b), ORIFICES, WEIRS, PIPES, AND RIVERS, - 223 which for feet measures becomes (as 1 metre z= 3-281 feet) r " 3-281 x '0000016175! v 2 rs= {'0002535 + - ^- ~| x 3^281 : hence we get rs **= ^-00007726 + miQ ' 2 and, therefore, -00007726 For all half-inch pipes this becomes -00023278 for all inch pipes, t f* C \ TV v = 1-000155021 = for all two-inch pipes, for all four-inch pipes, c =1-0000967) = for all six-inch pipes, v = 1-000090221 = 105 ' 3 224 THE DISCHARGE OF WATER FROM for all nine-inch pipes, = 107-8 v for all twelve-inch pipes, *= 1- -00008374 for all eighteen-inch pipes, * =1- 00008158) for all twenty-four-inch pipes, rs ={: 0000805) = 111-5 and when r is large, as for very large pipes and channels, we get the velocity f v -00007726J = 113-8 vW There is evidently, on an examination of these results, a great error in the formula of M. Darcy. As long as the diameter of a long pipe continues constant, the velocity is always represented by a given fixed multiple of \/~rs, or of the square root of the product of the hydraulic inclination and hydraulic mean depth, no matter how small or great the velocity In the pipe may be. For an inch pipe this multiplier for feet measures is 80-3. Now with a lead pipe I have found, from several experiments, that for a velocity of about 15 feet per second, the multiplier to be 117 or 118 ; and for a velocity of about 22 feet per second, Mr. Hodson's experiment gives a multiplier of about 120. Taking the other ORIFICES, WEIRS, PIPES, AND RIVERS, 225 extreme for large pipes, the multiplier derived from M. Darcy' s formula is 113*8, no matter how small the velocity may be. Now we have experiments in abundance to prove that for velocities of about 12 or 13 inches per second, the multiplier cannot exceed 95. We, therefore, look upon these researches of M. Darcy as partial and defective, and his formula as a representation, at best, of a limited range of velocities, in which those at either side are omitted or not perceived. For small pipes, any obstruction arising from de- fective bore, decomposition, encrustation, or from diminished bore, affects the discharge much more considerably than the same obstructions in a large pipe. In order to compare correctly the effects of the state of the bore on the discharge, we must use pipes of exactly the same diameter, and determine the value of b : from experiments in which the velo- city is the same, otherwise the results, as deduced by M. Darcy and given by Morin, cannot be de- pended upon. COEFFICIENTS DUE TO THE ORIFICE OF ENTRY. PROBLEMS. Unless where otherwise expressed, the head due to the velocity and orifice of entry is , not considered in the preceding equations. In equation (74), where it is taken into calculation generally, Cf X - /1\ 2 in which 1 + c r is equal to ( ) , c r being the coefficient 226 THE DISCHARGE OF WATER FROM of resistance clue to the orifice of entry, and c v the coefficient of velocity or discharge from a short tube. If the tube project into the reservoir, and be of small thickness, c v will be equal -715 nearly, and therefore c r zz -956 ; if the tube be square at the junction, the mean value of c v will be -814, and therefore c r = -508; and if the junction be rounded in the form of the contracted vein, c v is equal to unity very nearly, and C T z= 0. For other forms of junction the coefficients of discharge and resistance will vary between these limits, and particular attention must be paid to their values in finding the discharge from shorter tubes and those of moderate lengths ; but in very long tubes 1 + c r becomes very small compared with c f x , and maybe neglected without practical error. These remarks are necessary to prevent the misapplication of the tables and formulae, as the height due to the velocity and orifice of entry is an important element in all calculations for short tubes. We have considered it unnecessary to give any formulae for finding the discharge itself, because, the mean velocity once determined, the calculation of the discharge from -the area of the section is one of simple mensuration ; and the introduction of this element into the three problems to which this por- tion of hydraulic engineering applies itself, renders the equations of solution complex, though easily derived ; and presents them with an appearance of difficulty and want of simplicity which excludes them, nearly altogether, from practical application. The three problems are as follows : ORIFICES, WEIRS, PIPES, AND RIVERS. 227 I. Given the fall, length, and diameter of a pipe or hydraulic mean depth of any channel, to find the discharge. Here all that is necessary is to find the mean velocity of discharge, which, multiplied by the area of the section (equal d 2 x *7854 in a cylindrical pipe), gives the discharge sought. TABLE VIII. gives the velocity at once for long channels, accord- ing to Du Buat, or it can be found from equation (119A) by calculation. TABLE IX. gives the dis- charge in cubic feet per minute for different diameters of pipes, and velocities in inches per second, when found from TABLE VIII. or formula (119A). See also TABLES XI. and XII. For a pipe 6 inches in diameter, the velocity per second is practically equal to the discharge in cubic feet per minute. See also the tables, pp. 42, 43, 252, and 253. II. Given the discharge and cross section of a channel, to find the fall or hydraulic inclination. If the cross section be circular, as in most pipes, the hydraulic mean depth is one-fourth of the dia- meter ; in other channels it is found by dividing the water and channel line of the section, wetted peri- meter, or border, into the area. The velocity is found by dividing the area into the discharge, and reducing it to inches per second ; then in TABLE VIII., under the hydraulic mean depth, find the velocity, corresponding to which the fall per mile will be found in the first column, and the hydraulic inclination in the second. This result can be cor- rected by trial and error to accord with formula (11 BA), and the table for the values of r s and v, p. 220, Q3 228 THE DISCHARGE OF WATER FROM calculated from it. See also the tables, pp. 42, 43, 252, and 253. III. Given the discharge, length, and fall, to find the diameter of a pipe, or hydraulic mean depth and dimensions of a channel. This is the most useful problem of the three. Assume any mean radius r a , and find the discharge D a by Problem I. We shall then have for cylindrical pipes rf : 7*5 : : D a : D : : 1 : ; and as r a , D, and D a are known, r% becomes also known, and thence r. TABLE XIII. will enable us to find r with great facility. Thus, if we had assumed r & =. 1 and found D a = 15, D being 33, we then have 1 : r*:: 1 : - :: 1 : 2*2, therefore r* = 22 ; 15 and thence by TABLE XIII., r = 1*37, the mean radius required, four times which is the diameter of the pipe. For other channels, the quantity thus found must be the hydraulic mean depth ; and all channels, however varied in the cross section, will have the same velocity of discharge, when the fall, length, and hydraulic mean depth are constant. If r a be as- sumed equal to 1J inch, the velocity found from TABLE VIII. will then be the discharge in cubic feet per minute nearly, and this "mean radius" can always be assumed for the first term of the pro- portion. See also the tables, pp. 42, 43, 252, and 253. In order to find the dimensions of any polygonal channel whatever, which will give a discharge equal to D, we may assume any channel similar to that OEIFICES, WEIES, PIPES, AND RIVERS. 229 proposed, one of whose known sides is s a , and find the corresponding discharge, D a , by Problem L, or from TABLES XI. and XII. ; then, if we call the like side of the required channel, s, we shall have sn:s a ( j, and thence the numerical value from TABLE XIII. The result, as before, can be corrected to accord with any of our formulae by the method of trial and error. As it frequently happens that deposits in and en- crustations on a pipe take place from time to time, which diminish the flowing section considerably, it is always prudent, when calculating the necessary diameter, to take the largest coefficient of friction, c f ,or to double its mean value, particularly for small pipes, when calculating the diameter from any of the for- mulae. Some engineers, as D'Aubuisson, increase the quantity of water by one-half to find the dia- meter ; but much must depend on the peculiar cir- cumstances of each case, as sometimes less may be sufficient, or more necessary. Tne discharge increases in similar figures, nearly as r% or as d? 9 that is, as the square root of the fifth power of the diameter, and the corresponding increase in the diameter for any given or allowed increase in the discharge can be easily found by means of TABLE XIII., as shown above. If we increase the dimensions by one-sixth, the discharge will be increased by one-half nearly, and by doubling them the discharge is increased in the proportion of 5 1 to 1. For shorter pipes, we have to take into considera- tion the head due to the velocity and orifice of entry. 230 THE DISCHARGE OF WATER FROM Taking the mean coefficient of velocity or discharge, we find from TABLE II. the head due to the velocity, if it be known ; this subtracted from the whole head, H, leaves the head, h iy due to the hydraulic inclina- tion, which is that we must make use of in the table. If the velocity be not given, we can find it approxi- mately ; the head found for this velocity, due to the orifice of entry, when deducted, as before, will give a close value of h t , from which the velocity may be determined with greater accuracy, and so on to any degree of approximation. In general, one approxi- mation to h t will be sufficient, unless the pipes be very short, in which case it is best to use equation (74). EXAMPLE VIII., p. 195, and the explanation of the use of the tables, SECTION I., may be usefully referred to. TABLES XI., XII., and XIII. enable us to solve with considerable facility all questions connected with discharge, dimensions of channel, and the ordi- nary surface inclinations of rivers. The discharge corresponding to any intermediate channels or falls to those given in TABLES XI. or XII., will be found with abundant accuracy, by inspection and simple interpolation ; and in the same manner the channels from the discharges. Rivers have seldom greater falls than those given in TABLE XII., but in such an event we have only to divide the fall by 4, then twice the corresponding discharge will be that re- quired. TABLE XIII. gives the comparative dis- charging powers of all similar channels, whether pipes or rivers, and the comparative dimensions from the discharges. We perceive from it, that an increase OEIFICES, WEIKS, PIPES, AND RIVERS. 231 of one-third in the dimensions doubles, and a de- crease of one-fourth reduces the discharge to one- half. By means of this table, we can determine by a simple proportion, the dimensions of any given form of channel when the discharge is known. See EXAMPLE 17, p. 30. See also the tables pp. 42, 43, 252 and 253. The mean widths in TABLES XI. and XII. are cal- culated for rectangular channels, and those having side slopes of 1J- to 1. Both these tables are, how- ever, practically, equally applicable to any side slopes from to 1 up to 2 to 1, or even higher, when the mean widths are taken and not those at top or bottom. A semihexagon of all trapezoidal channels of equal area has the greatest discharging power, and the semisquare and all rectangles exactly the same as channels of equal areas and depths with side slopes of H to 1. The maximum discharge is ob- tained between these for the semihexagon with side slopes, of nearly \ to 1, but for equal areas and depths the discharge decreases afterwards as the slope flattens. The question of "HOW MUCH?" is here, however, a very important one ; for, as we have already pointed out in equations (28) and (31), the differences for any practical purposes may be imma- terial. This is particularly so in the case of chan- nels with different side slopes, if, instead of the top or bottom, we make use of the mean width to calcu- late from. We then have only to subtract the ratio of the slope multiplied by the depth to find the bottom, and add it to find the top. If the mean width be 50 feet, the depth 5 feet, and the side slopes 232 THE DISCHARGE OF WATER FROM 2 to 1, we get 50 (2 x 5) z= 40 for the bottom, and 50 + (2 x 5) = 60 for the top width. Side slopes of 2 to 1 present a greater difference from the mean slope of H to 1, than any others in general practice when new cuts are to be made. A triangular channel having slopes of 2 to 1, and bot- tom equal to zero, differs more in its discharging power from the half square, equal to it in depth and area, than if the bottom in each was equally in- creased, yet even here it is easy to show that this maximum difference is only 5 per cent. If the bottom be increased so as to equal the depth, it is only 4J per cent. ; when equal to twice the depth, 3 -8 per cent. ; and when equal to four times the depth, to 2 per cent. ; while the differences in the dimensions taken in the same order are only 2-2, 1-8, 1-5, and 0-8 per cent. For greater bottoms in proportion to the depth the differences become of no comparative value. It therefore appears pretty evident, that TABLES XL and XII. will be found equally applicable to all side slopes from to\upto1to\,by taking the mean widths. When new cuts are to be made, we see no reason whatever in starting from bottom rather than mean widths, to calculate the other dimensions ; indeed, the necessary extra tables and calculations involved ought entirely to preclude us from doing so. Besides, the formulas for finding the discharge vary in themselves, and for different velocities the coefficient of friction also varies.* Added to which * The coefficient m in the formula v = m (r s)* in rivers for velocities from 3 inches to 3 feet per second, varies from about 72 to 103 ; yet, strange to say, most tables are calculated from ORIFICES, WEIRS, PIPES, AND RIVERS. 233 the inequalities in every river channel, caused by bends and unequal regimen, preclude altogether any regularity in the working slopes and bottom, though the mean width would continue pretty uniform under all circumstances. The quantities in TABLE XII. are calculated, from the velocities found from TABLE VIII., to correspond to a channel 70 feet wide and of different depths, the equivalents to which are given in TABLE XI. In order to apply these tables generally to all open channels, the latter are to be reduced to rectangular ones of the same depth and mean width, or the re- verse, as already pointed out. If the dimensions of the given channel be not within the limits of TABLE XI., divide the dimensions of the larger channels by 4, and multiply the corresponding discharge found in TABLE XII. by 32 ; for smaller channels, multiply the dimensions by 4, and divide by 32. In like manner, if the discharge be given and exceed any to be found in TABLE XIII., divide by 32, and multiply the dimensions of the suitable equivalent channel found in TABLE XI. by 4. If we wish to find equiva- lent channels of less widths than 10 feet for small discharges, multiply the discharge by 32, and divide the dimensions of the corresponding equivalent by 4, Many other multipliers and divisors as well as 4 and 32 may be found from TABLE XIII., such as 3 and one coefficient alone ; or, rather, from a formula equivalent to 94-17 (r s)$, which gives results suited only to a velocity of 16 inches. Dimensions of channels calculated by means of this formula are too small in one case, and too large in the other. In pipes the variation of the coefficients is shown in the small tables, pp. 214 and 217. 234 THE DISCHARGE OF WATER FROM 15-6, 6 and 88-2, 7 and 130, 9 and 243, 10 and 316, 12 and 499, &c. The differences indicated at pages 198 and 199, must be expected in the application of these rules, which will give, however, dimensions for new channels which can be depended on for doing duty. It will be seen from TABLE XIII. that a very small increase in the dimensions increases the dis- charging power very considerably. TABLE XII. also shows that a small increase in the depth alone adds very much to the discharge. If we express in this latter case a small increase in the depth, d, by - , 7i then it is easy to prove that the corresponding in- crease in the velocity, v, will be ; and that in the A 1 discharge D, o~, if the surface inclination continue unchanged ; but as it is always observable in rivers that the surface inclinations increase with floods, the differences in practice will be found greater than these expressions make it. As in a large river the surface inclination must be very small, four times the fall will add very little to the sectional area ; yet this increase of fall will double the discharge, and we thence perceive how tributaries can be absorbed into the main channel without any great increase to its depth. OKIFICES, WEIRS, PIPES, AND EIVERS. 235 SECTION IX. BEST FORMS OF THE CHANNEL. REGIMEN. VELOCITY. EQUALLY DISCHARGING CHANNELS. We have seen above, that the determination of the hydraulic mean depth does not necessarily determine the section of the channel. If the form be a circle, the diameter is four times the mean radius ; but, though this form be almost always adopted for pipes, the beds of rivers take almost every curvilineal and trapezoidal shape. Other things being the same, that form of a river channel, in which the area divided by the border is a maximum, is the best. This is a semicircle having the diameter for the sur- face line, and in the same manner, half the regular figures, an octagon, hex- agon, and square, in Fig. 33, are better forms for the channel, the areas and side slopes being constant, than any others of the same number of sides. Of all rectangular channels, Diagram 4, in which A B c D is half a square, is the best cross section ; and in Diagram 3, A c D B, half a hexagon, is the best trapezoidal form of cross section. When the width of the bottom, c D, Diagram 3, is given, and the slope Aa = n, then, in order that the discharge may be the greatest possible, we must have 236 THE DISCHARGE OF WATER FROM A ** -12 in which A is the given area of the channel. As, however, we have never known a river in which the slope of the natural banks continued uniform, even though made so for any improvements, we consider it almost unnecessary to give tables for different values of n. If, notwithstanding, we put $ for the inclina- tion of the slope A c, equal angle c A a, we shall find, 1 as cot. < = n, and vri 2 + 1 = - - , that the fore- sin. going equations become ( A sin. - C{ CH Cq rH lf| 333333333 2 " CO t* 00 CO ^O 00 O^ CO CO O^J CO O 00 b* - - (NcpcpcpcpqpTh E- (^1 C^l C^ O^ CO CO CO TH C 1 ^ O 3333333333 CO CD ^^^^ cs^o-^iotoooo CO CO CO CO CO CO CO CO T^-lAlCNICSIC^ICSIC^rH OOiccOr-ii>cqoQO 0.2 "~*o COt^i-tt-^Ht-OOO 0^3,- CO CO O O ^5 a 1 ' , |S||^S!>S? ll>t~t-tCOCOCOt- TH .S S a, w : ^g 86 i TS S K * i ^3333333g| OHNMHtrH HcoHNeto^OQ qg 13 J-i O 1 1 CL-fi'Z " OKIFICES, WEIES, PIPES, AND RIVERS. 239 When the sectional area is given, the above table shows that the semicircle is the best discharging channel, and the complete circle the worst; the latter is so, however, only compared with the open channels given in the table, it being the very best form for an enclosed channel flowing full. T/ie best form of channel is particularly suited for new cuts in flat, marsh, callow, and fen lands, in which it is also often advisable to cut them with a level bed, up from the discharging point, in order to increase the hy- draulic mean depth, and consequently the velocity and discharge. As the quantity of water coming down a river channel in a season varies very considerably, we have observed it in one case to vary from one to thirty, and occasionally in the same channel from one to seventy-five, the proportion of the water section to the channel itself must also vary, and those relations of the depth, sides, and width to each other, above referred to, cease to hold good and be the best under such circumstances. If the object be to construct a mill-race, temporary drain for un- watering a river, or other small channel, in which the depth remains nearly constant, channels of the form of a half hexagon, diagram 3, Fig. 33, will be, perhaps, the best, if the tenacity of the banks per- mit the slope ; but rivers, in which the quantity of water varies considerably, require wider channels in proportion to the depth ; and also, that the velocity be so proportioned to the tenacity of the soil, or as it is termed " the regimen" that the banks and bed Fig. 34 240 THE DISCHAKGE OF WATER FROM shall not vary from time to time to any injurious extent, and that any deposits made during their summer state, and during light freshes, shall be carried off periodically by floods. Another circum- stance, also, modifies the effects of the water on the banks. It is this, that at curves, and turns, the current acts wit,h greatest effect against the bank, concave to the di- rection in which it is moving; deep- ening the channel there ; undermin- ing also the bank, as at A, Fig. 34 ; and raising the bed towards the opposite side B. The reflexion of the current to the opposite bank from A acts also in a similar man- ner, lower down, upon it ; and this natural operation proceeds, until the number of turns, increased length of channel, and loss of head from reflexion and unequal depths, bring the currents into regimen with the bed and banks. At all bends it is, therefore, prudent to widen the channel on the convex side B, Fig. 34, in order to reduce the velocity of approach ; and if the bed be here also sunk below its natural inclination, as we see it in most rivers at bends, the velocity will be farther reduced, and the permanence of the bed better established. The circumstances to be considered in deciding on the dimensions and fall of a new river course, after the depth to which the surface of the water is to be brought has been decided on, are the following : ORIFICES, WEIRS, PIPES, AND RIVERS. 241 The mean velocity must not be too slow, or aquatic plants will grow, and deposits take place, reducing the sectional area until a new and smaller channel is formed within the first with just sufficient velocity to keep itself clear. This velocity should not in general be less than from ten to fourteen inches per second. The velocity in a canal or river is increased very considerably by cutting or re- moving reeds and aquatic plants growing on the sides or bottom.* The mean velocity must not be too quick, and should be so determined as to suit the tenacity and resistance of the channel, otherwise the bed and banks will change continually, unless artificially protected ; it should not exceed 25 feet per minute in soft alluvial deposits. 40 clayey beds. 60 sandy and silty beds. 120 gravelly. 180 strong gravelly shingle. * " M. Girard a fait observer, avec raison, que les plantes aqua- tiques, qui croissent toujours sur le fond et sur les berges des canaux, augmentent conside"rablement le perimetre mouille, et par suite la resistance ; il a rapelle* que Du Buat, ayant mesure la vitesse de 1'eau dans le canal du Jard, avant et apres la coupe des roseaux dont il e"tait garni, avait trouve un resultat bien moindre avant qu'apres. En consequence, il a presque double la pente donne"e par le calcul . . ." Traite d'Hydraulique, p. 135. When the fall does not exceed a few inches per mile, the velocity, as determined from the inclination, is very uncertain, and for this reason it is always prudent to increase the depths and sectional areas of channels in flat lands, as far as the regimen will permit. In such cases the section of the channel should approximate towards the best form. See p. 238. R 242 THE DISCHARGE OF WATER FROM 240 feet per minute in shingly. 300 ., shingly and rocky. 400 and upwards in rocky and shingly.* A velocity of 180 feet per minute will remove an- gular stones the size of an egg. Mr. Phillips, under the Metropolitan Commissioners of Sewers, states that 2| feet per second, or 150 feet per minute, is sufficient to prevent soil depositing in sewers. The fall per mile should decrease as the hydraulic mean depth increases, and both be so proportioned that floods may have sufficient power to carry off the deposits, if any, periodically. The proportion of the width to the depth of the channel should not be derived, for new cuts or river courses, from any formula, but taken from such portions of the old channel as approximate in depth and in the inclina- tion of the surface to that proposed. When the depth is nearly half the width, the formula shows, cceteris paribus, that the discharge will be a maxi- mum ; but as (altogether apart from the question * TABLE OF VELOCITIES OP SOME MOVING BODIES COMPARED WITH THOSE OF RIVERS. Objects in motion. Miles per hour. Feet per second. Objects in motion. Miles per hour. Feet per second. Current of slow rivers . . Currents of ordinary rivers, up to A ii 1 Railway trains, French . . German . Sound when atmosphere is at 32 Fahr 27 24 743 89* 53i 1 090 Currents of rapid rivers . 7 lof Ditto 60 Fahr 765 1,122 Man walking 3 4? 850 1 247 Horse trotting 7 10i TVtt 'h fh h t ' Swiftest race-horse . . . Moderate winds .... 60 7 36 88 101 52$ stands at 30 inches . . Common musket-ball . . Rifle-ball 917 850 1000 1,344 1,247 1/67 80 117t 1 091 1 600 Swift English steamboats navigating the channels . Swift American Kiver steamers 14 18 **t 20| 26^ Bullet discharged from air- gun, air being compress- ed into the hundredth 477 700 Fast sailing vessels . . . Railway trains, English . American . Belgian . 12 32 18 25 Hi 47 26* 36f A point on earth's surface at the equator moving round the axis .... Earth moving round sun . 1,040 68,182 1,525 100,000 ORIFICES, WEIRS, PIPES, AND RIVERS. 243 of expense) the quantity of water discharged daily, at different seasons, may vary from one to seventy, or more, and " the regimen" has to be maintained, the best proportion between the width and depth of a new cut should be obtained, as we have stated, from some selected portion of the old channel, whose general circumstances and surface inclination ap- proximate to those of the one proposed ; and the side slopes of the banks must be such as are best suited to the soil. The resistance of the banks to the current being in general less than that of the beds, which get covered with gravel, and the neces- sary provision required for floods, appears to be the principal reason why rivers are in general so very much wider than about twice the depth, the relation which gives the minimum of friction. The following table is given by Bennie, as an approximation, generally, to the actual state of rivers.* The surface inclinations, however, given in this table for the first and second classes, are very considerable for large rivers, and would give velocities which would effectually scour them. For a hydraulic mean depth of 12 feet, the velocity, with a fall of 10000' would be 2 feet 8 inches per second by Du Buat's formula ; and 3-3 feet per second by our formula. The description, therefore, can only apply to small channels. In fact, 4 inches to a mile, or , is a considerable inclination for a large river. * Eeport to the British Association 1834. R3 244 THE DISCHARGE OF WATER PHOM i* 2 2 a S tjj ||| *8J i?l 'S'o DISTINCTIVE ATTBEBUTE8 **f - 'o "3 6 | ** * 00 *H rfl '> 09 THE f "o > &o i.'|| o" a! 2 1 .& I-H VABIOUS KINDS OF BIVEBS. g | fl 'S'wo fi "3 * o-g B& 13 1*1 u - a "! a ill III t$l -||o Channels wherein the resist- ance from the bed, and other obstacles, equal the quantity of current acquired from the de- clivity ; so that the waters would 1st. 0" 12000 14 stagnate therein, were it not for the compression and impulsion of the upper and back waters . Artificial canals in the Dutch i and Austrian Netherlands . . J 2nd. 1 180 1000 8 Rivers in low flat countries, "] full of turns and windings, and of a very slow current, subject to frequent and lasting inunda- - 3rd. 1 120 10 5200 6 Rivers in most countries that \ are a mean between flat and hilly, which have good currents, but are subject to overflow ; also 4th. li 80 15 Wtro 4| the upper parts of rivers in flat Rivers in hilly countries with\ a strong current, and seldom subject to inundations ; also all 1 rivers near their sources have ( 5th. *i 55 21* T2"0(7 3f this declivity and velocity, and often much more J Rivers in mountainous coun- \ tries having a rapid current and ( straight course, and very rarely [ 6th. 3 40 30 2600 3 Rivers in their descent from\ among mountains down into the 1 plains below, in which plains [ 7th. 5 24 50 "2000 2J they run torrent- wise. . . . j Absolute torrents among i 8th. 8 15 80 x 2 mountains . . i 1 7 ORIFICES, WEIRS, PIPES, AND RIVERS. 245 The following information with reference to the sur- face inclinations of the Thames, is from Rennie's Report on Hydraulics,* as a branch of engineering science. Names of places. Length. Fall Fall in feet per mile. Ratio of inclinations. From Lechdale at St. John's Bridge to Oxford at FoUy Miles. Fur. 28 Feet. In. 47 1-68 TTTTS From Oxford to Abingdon 9 13 11 1-73 1 From Abingdon to Walling- ford Bridge .... 14 27 4 1-95 From Wallingford to Read- in s Bridsre 18 24 1 1-81 From Reading to Henley Bridge 9 19 3 2-14 TT A & 7 From Henley to Mario w Bridge 9 12 2 1-35 From Marlow to Maiden- head Bridge 8 15 1 1-86 From Maidenhead Bridge to Windsor Bridge .... From Windsor 'to Staines Bridge 7 8 13 6 15 8 1-93 1-96 2 1*8" 3 ^ ff,T3TF From Staines to Chertsey Bridge .... 4 6 6 6 1-44 From Chertsey to Tedding- ton-Lock . 13 6 19 8 1-45 s-T^rr From Teddington-Lock to London Bridge .... From London to Yanlet Creek 19 40 2 9 2 1 145 052 3.64 1 TnTi4'14 TO 1637 From Lechdale to Yanlet Creek 186 4 218 Deduct . . . 40 .- From Lechdale to London . 146 4 For enclosed channels, the circular form of sewer will have the largest scouring power, at a given hydraulic inclination. For the area of the sections being the same, the velocity in the circular channel will be a maximum. When the supply is intermit- tent, and the channel too large, the egg-shaped form * Report, for 1834, of the British Association. 246 THE DISCHARGE OF WATER FROM with the smaller end for the bottom, or the sides vertical with an inverted ridge-tile or V bottom for drains, will have a hydrostatic flushing power to remove soil and obstructions, which a cylindrical channel, only partly fall, does not possess ; because a given quantity of water rises higher against the same obstruction, or obstacle, to the flow in the pipe. It must be confessed, however, that for small drains and house-sewage, this gain is immaterial, and is at best but effected by a sacrifice of space, mate- rial, and friction in the upper part of drains, from 6 to 12 inches in diameter. Besides this, the mere hydrostatic pressure is only intermittent, and during an ordinary, or heavy, fall of rain, the hydrodynamic power is always more efficient in scouring properly- proportioned cylindrical drains ; and the workman- ship in the form and joints is less imperfect than for more compound forms, as those with egg-shaped and inverted tile bottoms. The moulds and joints of cylindrical stone- ware drains, exceeding 12 inches in diameter, are seldom, however, in large quantities perfect ; and the expense will exceed that of brick, stone, or other sufficient drains in most localities. As to the increased discharging power which it is asserted by some, stone-ware cylindrical drains pos- sess over other ordinary drains, no doubt it is true for small sizes, because the form, jointing, and sur- face are in general more smooth and circular ; and for sewage matter* the friction and adherence to the sides and bottom is less ; any advantage from these causes becomes, however, immaterial for the larger * Weisbach found the coefficient of resistance 1'75 times as great for small wooden as for metallic pipes. All permeable pipes ORIFICES, WEIRS, PIPES, AND RIVERS. 247 sizes, as these can be constructed of brick or stone abundantly perfect to any form, and sufficiently smooth for all practical purposes, for in the larger properly-proportioned sizes the same amount of sur- face roughness opposed to the sewage matter is, com- paratively, of no effect. The judicious inclination and form of the bottom, and properly curved junc- tions, are the principal points to be attended to. Smaller drains tile-bottomed, with brick or stone sides, and flat-covered, have one great advantage over circular pipes,* They can be opened up, for examination and repairs at any time with facility, and at the smallest expense ; but greater certainty must be attached to the working of small stone-ware drains than to equally-sized small brick or stone drains, and they will be found, in general, also cheaper. This, however will depend on the locality. It may be observed in numerous experiments, that water flowing from a pipe does not entirely fill the orifice of exit, when the velocities are not con- siderable, and yet the results are found to be but slightly affected if a little more than three-fourths of the circumference be full. It is easy to demonstrate that the full circle does not give the maximum dis- charging velocity as has been generally believed, but present greater resistance than impermeable ones ; hence the prin- cipal advantage derived from glazing. * Half-socket joints at bottom would remedy this imperfection in small pipes, and they could be better laid and cemented. A semicircular flange laid on at top would effectually protect the joint on the upper side. Latterly Doulton has cut off an upper segment from the pipe, which can be removed for cleaning. And it may be demonstrated, that when this is a segment of 78J degrees, the lower portion will discharge more than a full pipe at the same inclination. 248 THE DISCHARGE OF WATER FROM when filled to the height of the chord ac of arc a e c of 784 degrees, and where the velocity is 9i per cent, over that due to the full circle, for then ,1 area ad c the =- is a maxi- arc aae mum, and the length of the arc adc is equal to the tangent of the supple- mental arc a e c, as may be without difficulty demon- strated. The hydraulic mean depths of the circle and larger segment are to each other as -5 to -6, and their square roots, which are as the velocities or scouring powers, are as 1 to 1-095. The discharging powers are to each other as 1x3*1416 to 1-095 X 2*946, or as 1 to 1-026, which shews that the seg- ment adc has also a greater discharging power than the whole circle of nearly three per cent. These facts, which were first pointed out by the author, are not unimportant in matters connected with pipe-drains and sewerage. The effects of greater velocity and discharge here pointed out, are sometimes increased, in short pipes, from the fall between the surface a c, and the surface from which the head is measured, being greater than the fall to the top of the pipe at e, or from the inclination of the surface of the water in the pipe being greater than the inclination of the pipe itself. EQUALLY DISCHARGING CHANNELS. In order that different channels should have the same discharging power, the inclination of the sur- face being the same, the areas must be inversely as ORIFICES, WEIRS, PIPES, AND RIVERS. 249 the square roots of the hydraulic mean depths. The channel a dcv, Fig. 35, will have the same discharge as the channel A D c B if they be to each other as f ADCB ,i to , I A D 4- DC 4- C B J ( AD + DC + CB ad + dc + CB and hence the square root of the cube of the channel area, divided by the border, must be constant. With a fall of one or more feet to a mile, two channels, one 70 feet wide and 1 foot deep, and the other 20 feet wide and 2i feet deep, will have the same dis- charge. If we put w for the width and d for the depth of any rectangular channel, then w w y + 2 d) m ; we therefore have the cubic equation (122.) d--d = w 3 w 2 for finding the depth, d of any other rectangular channel whose width is w, of the same discharging power. We have calculated the depths d for different widths of channel from this equation, assuming a width of 70 feet and different depths to find m from. The results are given in TABLE XI., which will be found sufficiently accurate for all practical purposes, when the banks are sloped, by taking the mean width. This table is equally applicable to any measures whatever, to their multiples, and sub-multiples. If the hydraulic inclinations vary, then the \/rs must be inversely as the areas of the channels when \/rs x channel or the discharge is constant ; and if 250 THE DISCHARGE OF WATER FROM the area of the channel and discharge be each con- stant, r must vary inversely as s; and r s be also constant. For instance, a channel which has a fall of four feet per mile, and a hydraulic mean depth of one foot, will have the same discharge as another chan- nel of equal area, having a hydraulic mean depth of four feet, and a fall per mile of only one foot. If in TABLE XII. we take the same discharge from the columns for different inclinations, we shall get the mean rectangular dimensions corresponding to them in the first column, and thereby be enabled to select an equally discharging channel from TABLE XL, suited to an increase or decrease of the hydraulic inclina- tion.* We have, however, calculated for this edition the table at p. 252, of equally discharging river channels, with a primary channel having a mean width of 100, instead of 70, as in TABLE XI. ; and in the table at p. 253 we have given the discharges at different in- clinations from this new primary channel, to find those from its equivalents. The tables at pp. 42, 43, 253, and TABLE VIII., have been calculated from Du Buat's formula. For slow velocity of only a few inches per second, the dimensions should be increased by about one-sixth, and the discharges by about one- half. With reference to pipes, it is apparent that a given depth of roughness or contraction arising from any * Tables similar to numbers XL, XII., and XIII., but on a much more extended scale, have been printed and published by MB. WEALE, on a separate sheet for office use, and may be had from him. ORIFICES, WEIRS, PIPES, AND RIVERS. 251 cause will have a greater effect the smaller the dia- meter becomes. Now in practice, it is necessary to increase the diameter beyond what is found by calcu- lation. For small service pipes half-an-inch is the smallest diameter in general use. For mains and sub-mains the value of c t in equation (74B), or at p. 214, should at least be doubled, or the discharge taken at one and a half times its amount to find the dia- meter. By enlarging the diameter by one-seventh, one- half the amount will be added to the discharge, very nearly; and by increasing the diameter by one-third, the; discharge will be doubled. In a broad and practical sense, and considering the losses arising from depo- sitions,* pipes under two inches should have one- third or more added to their calculated dimensions, and larger pipes from one-third to one- seven th even after making allowance for junctions, bends, and con- tractions. For large conduits or channels the allow- ance need not be so large, if the maximum quantity to be conveyed be duly estimated. * Mr. Bateman lately in giving evidence says :^ " He wished to mention a circumstance which might he useful with regard to the spongillse found in the Dublin water pipes. At Manchester, before the introduction of soft water, the city was supplied with hard water, which favoured the growth of a small fresh- water mussel, which thickly line the reservoirs and pipes. There were myriads of them, and they lay in the pipes as thick as paving stones. These were caused by the large quantity of lime in the water. He was curious to see what would be the effect of passing water without lime. This was done ten or eleven years ago, and the result was that these mussels had entirely disappeared. There was no longer anything from which they could make their shells, and for years, on their discharge, the small pipes were found choked with them. If soft water were supplied to Dublin in place of the present hard water, which probably favoured the growth of spongillaD, they would probably disappear." TABLE of mean widtJis and depths of equally discharging trapezoidal River-channels, or Sewers, with side slopes up to 2 J to 1. Practically all river-channels may be reduced to rectangular sections of equal areas and depths to find the discharge. Primary Channel Mean rectangular dimensions of equally discharging water-channels or sewers, in any measures whatever, inches, feet, yards, fathoms, or their aliquot parts, or multiples. Primary Channel Mean width 100 Mean width 90 Mean width 80 Mean width 70 Mean width 60 Mean width 50 Mean width 40 Mean width 30 Mean width 20 Mean width 15 Mean width 10 Mean width 100 1 11 12 13 14 16 18 22 29 35 47 1 125 13 14 16 17 20 23 28 37 45 60 125 2 21 23 25 28 32 37 45 60 73 98 2 25 27 29 32 35 40 46 56 75 92 1-25 25 3 32 35 38 42 48 56 68 90 1-11 1-52 3 375 40 44 .48 53 60 70 85 1-13 1-40 1-94 375 4 43 46 51 56 64 74 91 1-21 1-50 2-08 4 5 54 58 64 71 80 93 14 1-53 1-90 2-67 5 6 64 70 76 85 96 1-12 37 1-85 2-31 3-28 6 625 67 73 79 88 1-00 1-16 43 1-93 2-42 3-44 625 7 75 81 89 99 1-12 1-31 61 2-17 2-73 3-92 7 75 80 87 95 06 1-20 1-41 73 2-34 2-95 425 75 8 86 93 1-02 13 1-29 1-50 85 251 3-17 4-59 8 875 94 02 1-12 24 1-40 1-64 2-02 2-76 3-50 5-10* 875 9 97 05 1-15 27 1-45 1-69 2-08 2-84 3-61 5-28 9 1-0 1-07 16 1-27 42 1-61 1-88 2-32 3-18 4-07 5-99 1-0 1-125 1-21 31 1-43 60 1-81 2-13 2-63 3-62 4-64 6-92 1-25 1-2 1-29 40 1-53 70 1-94 2-27 2-81 3-88 5-00 7-50 1-2 1-25 1-35 46 1-60 78 2-02 2-37 2-94 4-06 5-24 7-89 1-125 1-3 1-40 51 1-66 85 2-10 2-47 3-06 4-24 5-48 8-29 1-3 1-375 1-48 60 1-76 1-96 2-23 2-62 3-25 4-51 5-85 8-89 1-375 1-4 1-50 63 1-79 1-99 2-27 2-66 3-31 4-60 5-97 9-10 1-4 1-5 1-61 75 1-92 2-14 2-43 2-86 3-56 4-97 6-47 9-92 1-5 1-6 1-72 1-86 2-05 2-28 2-60 3-06 3-81 5-34 6-U8 10-78 16 1-625 1-75 1-89 2-08 2-32 2-64 3-11 3-87 5-43 7-11 11-00 1-625 1-7 1-83 1-98 2-17 2-43 2-76 3-26 4-06 5-72 7-50* 11-66 1-7 1-75 1-88 2-04 2-24 2-50 2-85 3-36 4-19 5-91 7-77 12-10 175 1-8 1"93 2-10 2-30 2-57 2-93 3-45 4-32 6-09 8-03 12-54 1-8 1-875 2-02 2-19 2-40 2-68 3-05 3-60 4-51 6-38 8-43 13-23 . 1-875 1-9 2-04 2-22 2-43 2-71 3-10 3-65 4-57 6-48 8-57 13-46 1-9 2-0 2-15 2-33 2-56 2-86 3-26 3-86 4-83 6-87 9-11 14-39 2-0 2-1 2-26 2-45 2-69 3-01 3-43 4-06 5-09 7-27 9-67 15-35 2-1 2-2 2-37 2-57 2-82 3-15 3-60 4-26 5-36 7-66 10-23 1632 2-2 2-3 2-47 2-69 2-95 3-30 3-77 4-4'J 5-62 8-07 10-80 17-31 2-3 2-4 2-58 2-80 3-08 3-44 3-94 4-67 5-89 8-48 11-38 18-33 2-4 2-5 2-69 2-92 3-21 3-59 4-11 4-87 6-16 8-79 11-97 19-35 2-5 2-6 2-80 3-04 3-34 3-74 4-28 5-08 6-42 9-31 12-57 20-40 2-6 2-7 2-91 3-16 3-47 3-88 4-55 5-28 6-69 9-73 13-17 21-46 2-7 2-8 3-01 3-27 3-60 4-03 4-62 5-49 6-97 10-16* 13-78 22-52 2-8 2-9 3-12 3-39 3-73 4-18 4-79 5-70 7-24 10-59 14-40 23-63 2-9 3-0 3-23 3-51 3-86 4-42 4-96 5-91 7-52 11-02 15-03 24-75 3-0 3-1 3-34 3-63 3-99 4-47 5-13 6-12 7-79 11-46 15-68 3-1 3-2 3-45 3-75 4-13 4-62 5-30 6-33 8-07 11-90 16-32 el 3-2 3-3 3-55 3-86 4-26 4-77 5-48 6-54 8-35 12-35 16-97 ,. 3-3 3-4 3-66 3-98 4-39 4-92 5-65 6-75 8-64 12-80 17-63 || 3-4 3-5 3-77 4-10 4-52 5-06 5-82 6-96 8-92 13-26 18-29 PL 3-5 3-6 3-88 4-22 4-65 5-21 6-00 7-18 9-21 13-71 18-96 in 3-6 3-7 3-99 4-34 4-78 5-36 6-17 7-39 9-49 14-18 1965 gs 3-7 3-8 4-09 4-46 4-91 5-51 6-35 7-60 9-78 14-65 20-34 ^ 3-8 3-9 4-20 4-58 5-05 5-66 6-52 7-82 10-07 15-12 *j "5-^ 3-9 4-0 4-31 4-69 5-18 5-81 6-70 8-04 10-36 15-59 ilif-si 4-0 4-1 4-42 4-81 5-31 5-96 6-87 8-25 10-66 16-07 "t* o3 H^'n fl> Ss^g-s^ 4-1 4-2 4-53 4-93 5-44 6-11 7-05 8-47 10-95 16-55 ^llt = l 4-2 4-3 4-64 5-05 5-57 6-26 7-23 8-69 11-25 17-04 2**lid 4-3 4-4 4-74 5-17 5-71 6-41 7-40 8-91 11-55 17-53 KfS-&8 4-4 4-5 4-85 5-29 5-84 6-56 7-58 9-13 11-85 18-02 IHi^s 4-5 4-6 4-96 5-47 5-97 6-72 7-76 935 12-15 18-52 !*uHi 4-6 4-7 5-07 5-53 6-10 6-87 7-94 9-57 12-45 19-02 ftSI&:J 4-7 4-8 5-18 5-64 6-24 7-02 8-12 9-79 12-75 19-53 li*tgS1 4-8 4-9 5-29 5-76 6-37 7-17 8-29 10-02 13-06 20-04 S|.sB*-8 4-9 5-0 5-40 5-88 6-50 7-32 8-47 10-24 13-37 ?iliii* 5-0 TABLE of the Discharges in cubic feet per minute from the primary Channel in the opposite page, taken in feet; and from the corresponding equivalent Channels, also taken in feet. Depths of the prln.a- 1 ry channel in oppo- 1 site table, having a 1 mean width of 100 ; 1 In feet. Discharges In cubic feet per minute. Interpolate for Intermediate falls or Inclinations: divide greater falls or Inclinations by 4, and donble the corresponding discharges. If the dimensions be in Inches, divide the discharges here given by 500 ; if in tenths, by 316 ; If in eighths, by ibl ; if in sixths, by 88; if in fifths, by 56; if in quarters, l>y 32; if in thirds, by 15-6; and if in halves, dy 5-66. Reverse the operation and multiply for like multiples. Depths of the prima- ry channel In oppo- site table, having a mean width of lOo; in feet. 4 inches per mile lin 15840. 1 6 inches per mile lin 10560. 9 inches per mile lin 7040. ;12inches per mile lin 5280. 15inchef per mile lin 4224. 18inche per mile lin 3520. 21 inches per mile, lin 3017-1. 24 inches per mile, lin 2640. 80 inches per mile, lin 2112. 36inche per mile lin 1760. 1 IK 140 176 207 235 260 284 305 345 383 1 125 157 198 250 294 332 373 402 433 490 543 125 2 325 409 515 606 6*6 760 828 b91 1,009 1,117 2 25 455 675 725 853 966 1,070 1,166 1,256 1,422 1,574 25 3 601 760 957 1,125 1,275 1,412 1,539 1,658 1,876 2,076 3 375 844 1,068 1,344 1,582 1,793 1,985 2,162 2.329 2,637 2,919 375 4 931 1,178 1,484 1,745 1,977 2,189 2,386 2,569 2,900 3,220 4 5 1,308 1,653 2,()81 2,447 2,775 3.071 3,347 3.6<6 4,0*3 4,513 5 6 1,721 2,178 2.743 3,227 3,657 4,047 4,410 4,752 5.401 5,956 6 625 1,830 2,316 2,917 3,431 3,887 4,303 4,690 5,053 5,795 6,332 625 7 2,177 2,750 3,463 4,0'2 4,614 5,109 5.568 5,999 6,936 7,516 7 75 2,414 3,029 3,844 4,520 5,123 5.674 6,180 6.660 7,567 8,342 75 8 2,660 3,363 4,236 4,982 5,646 6,253 6,811 7,340 8,309 9,194 8 875 3,044 3,850 4,846 5,703 6,463 7,157 7,770 8,401 9,513 10,527 875 9 3,175 4,017 5,060 5,951 6,743 7,467 8,082 8,765 9,926 10,984 9 1-0 3,731 4,711 5,933 6,973 7,903 8.750 9,513 10,273 11,634 12,877 1-0 1-125 4,441 5,614 7,071 8,313 9,421 10,430 11,369 12,216 13,867 15,347 1-125 1-2 4,88!) 6,186 7,791 9,163 10,381 11,494 12,521 13,494 15,280 16.914 1-2 1-25 5,207 6,582 8,291 9,752 11,048 12,232 13,336 14,361 16,261 18,000 1-25 1-3 5,529 6,981 8,793 10,357 11,718 12,974 14,146 15,234 17,246 19,091 1-3 1-375 6,004 7,591 9,561 11,245 12,734 14,107 15,386 16,576 18,752 20,756 1-375 1-4 6,167 7,797 9,821 11,544 13,087 14,491 15,794 17,031 19,262 21,318 1-4 1-5 6,844 8,653 10,^98 12,818 14,524 16,081 17,523 18,917 21,376 23,658 1-5 1-6 7,538 9,520 12,002 14,115 15,994 17,709 19,296 20,829 23,539 26,053 1-6 1-625 7,705 9,741 12,272 14.428 16,348 18,102 19,724 21,286 24,061 26,631 1-625 1-7 8.252 10,432 3,139 15,432 17,509 19,360 21,126 22,780 25,769 28,523 1-7 1-75 8,617 10,893 3,719 16,134 18,282 20,241 22,060 23,776 26,907 29,784 1-75 1-8 1-875 8,993 9,561 11,369 12,088 4,318 15,226 16,851 19,079 17,905 |20,287 21,124 22,463 23,024 24,476 24,821 26,372 28,081 29,860 31,085 33,052 1-8 1-875 1-9 9,741 12,316)15,515 18,245 120,672 22.890 24,946 26,872 30,426 33,682 1-9 2-0 10,515 13,297 16,753 19,702 22.320 24.718 26,935 29,019 32,852 36,358 2-0 2-1 11,307 14,300 18,020 21,192 23,991 26,561 29,074 31,213 35,334 39,106 2-1 2-2 12,110 15,314 19,297 22,689 25,708 28,467 31,024 33,424 37,838 41,878 2-2 2-3 12,935 16,357 ! 20,608 24,235 27,456 30,407 33,134 35,694 40,410 44,724 2'3 2-4 13,781 17,425 121,954 25,816 29,250 32,32 35,299 38,022 43,048 47,643 2-4 2-5 14,647 18,520 123,332 ! 27,436 31,087 34,425 37,516 40,407 45,750 50,634 2-5 2-6 15,538 19,645 '24,747 129,100 132,974 36,514 39,794 42,856 48,526 53,706 2-6 2-7 16,430 20,773 26,167 30,770 34,867 38,610 42,078 45,316 51.311 56,789 2-7 2-8 17,333 21,915 27,605 32,462 36,784 40,733 44.390 47,809 54,131 59,913 2-8 2-9 18,257 23,084 29,076 ;34,193 38,744 42,905 46,755 50,359 57,017 63,110 2-9 3-0 19,203 24,280 30,581 35,963 (40,750 45,127 49,175 52,968 59,968 66,379 3-0 3-1 20,167 25,498 32,120 37,767 (42.794 47,392 51,640 55,634 62,986 69,709 3-1 3-2 21,146 26,737 33,673 39,600 144,871 49,692 54,148 58,327 66,033 73,097 3-2 3-3 22,118 27,969 35,225 41,425 46,939 51,978 56,640 61,017 69,077 76,465 3-3 3-4 23,106 29,220 36,798 13,275 49,036 54,302 59,171 63,745 72,164 79,879 3-4 3-5 24,115 30,497 38,407 45,166 51,1*0 56,675 61,758 66,534 75.322 83,371 3-5 3-6 25,139|31,795 40,040 47,086 53,356 59,084 64,384 69,366 78,526 86,915 3-6 3-7 26,18233.116 41,702 49,041 55,572 61,532 67,058 72,249 81,789 90,524 3-7 3-8 27,23:334,446 43,379 51,013 57,807 64,009 6:>,753 75,158 85,078 94,162 3-8 3-9 28-287 35,777 45,060 52,989 50,046 66.489 72,455 78.061 88.371 97,810 3-9 4-0 29,356 37,128 46,766 54.9M 52,318 69,006 75,197 81,012 91,710 101,512 4*0 4-1 30,438 38,495 48,492 57,024 64,616 n,553 77,973 83, '.*99 95,093 105,259 4-1 4-2 31,538 39.H84 50,246 59,0*6 66.960 74.141 80,7^3 87,033 98,535 109,065 4-2 4-3 32,654 41,294 "2,027 161,180 6.327 76,769 83,655 90,116 102,025 112,93" 4-3 4-4 33,776 +2,712 53,816 63,283 71,709 79,406 86,529 93.209 105,531 116,811 4-4 4-5 34,908 44,138 55,613 !6r>,394 ' "4.100 32,052 89.413 96,318 L09.054 I2<),709 4-5 4-6 36,041 45,579 ;57,429 67,527 76,500 J84,725 92,327 99,460 Ll26i4 124,647 4-6 4-7 37,193 47,034 59,2(i2 6.*, 682 78,955 (87.426 95,271 102,632 116,209 ; 128,625 4-7 4-8 38,36:3 18,514 61,128 71,874 |81,438 |90,173 98,266 105,860 119,866 132,672 4-8 4-9 39,544 50,009 ^3,011 74.087 83,944 < )2,946 01,289 : L09.119 123,559 ] L36.758 4-9 5-0 40,725 51,507 54,895 7 6,298 |86,450 < >5,720 04,313 : 112,376 127,248 ] L40,841 5-0 254 THE DISCHARGE OF WATER FROM SECTION X. EFFECTS OF ENLARGEMENTS AND CONTRACTIONS. BACKWATER WEIR CASE, LONG AND SHORT WEIRS. When the flowing section in pipes or rivers expands or contracts suddenly, a loss of head always ensues ; this is probably expended in forming eddies at the sides, or in giving the water its new section. A side current, moving slowly upwards, may be frequently observed in the wide parts of rivers, when the channel is unequal, though the downward current, at the centre, be pretty rapid ; and though we may assume generally that the velocities are inversely as the sec- tions, when the channels are uniform, we cannot properly do so when they are not, and the motions so uncertain as those referred to. When a pipe is contracted by a diaphragm at Fig. 3 6 the orifice of entry, Fig. 27, we have seen (equation 60), that the loss of head is, (123.) h = When the diaphragm is placed in a uniform pipe, Fig. 36, then A zz c, and we get the loss of head (124.) h= - 27 > and the coefficient of resistance as in equation (67). The coefficient of discharge c d is ORIFICES, WEIRS, PIPES, AND RIVERS. 255 here equal to the coefficient of contraction c c , o * very nearly. Now we have shown in equation (45), and the remarks following it, that the value of the co* efficient of discharge, c d , varies according to the ratio of the sections, A * and in TABLE V. we have calcu- a lated the new coefficients for different values of the ratios, and different values of the primary coefficient c d . If we assume c d , when A is very large compared with a, to be '628, we then find by attending to the remarks at pp. 109 and 128, that the different values of c d corresponding to -807 x^, taken from TABLE V., a are those in columns Nos. 2 and 5 of the next small TABLE OF COEFFICIENTS FOR CONTRACTION, BY A DIAPHRAGM IN A PIPE, a A Pd C r a A CA c r 628 infinite 6 713 1-790 1 630 221-2 7 753 807 2 636 47-1 8 807 301 3 647 17-2 85 845 154 4 661 7-7 9 890 062 5 683 3-7 1 1-000 000 * The general value of c c , as given by Professor Eankine, is 618 c c = ~ ^-7, which is equal to unity when = A, as it should be ; and equal to -618, when a is very small, compared with A, as it also should be when the diaphragm is a thin plate, but not otherwise. If the thickness of the diaphragm be twice the dia- meter of the orifice a, the coefficient of discharge would be -815 ; and if the higher arris be rounded, this would be increased to 1, in which cases the expression would clearly fail ; the thickness of the diaphragm and the form of the aperture a must also be considered. 256 THE DISCHARGE OF WATER FROM table, the values of the coefficient of resistance, in columns 3 and 6, being calculated from equation (125) for the respective new values of the coefficient of discharge thus found. The table shows that when the aperture in a diaphragm is Aths of the section of the pipe, that 47 times the head due to the velocity is lost thereby. If the aperture in the diaphragm be rounded at the arrises, the loss will not be so great, as the primary coefficient c d will then be greater than that due to an orifice in a thin plate : see coefficients, p. 174 When there are a number of diaphragms in a tube, the loss of head for each must be found separately, and all added together for the total loss. If the diaphragms, however, approach each other, so that the water issuing from one of the orifices a, Fig. 36, shall pass into the next before it again takes the velocity due to the diameter of the pipe, the loss will not be so great as when the distance is sufficient to allow this change to take place. This view is fully borne out by the experiments of Eytelwein with tubes 1-03 inch in diameter, having apertures in the diaph- ragms of -51 inch in diameter. Venturi's twenty-fourth experiment, with tubes varying from -75 inch to -934 inch in diameter at the junction with the cistern, so as to take the form of the contracted vein, and expanding and contracting along the length from -75 to 2 inches and from 2 inches to *75 inch alternately, shows the great loss of head sustained by successive enlargements and con- tractions of a channel, even when the junction of the parts is gradual. Calling the coefficient for the short ORIFICES, WEIRS, PIPES, AND RIVERS. 257 tube, with a junction of nearly the form of the con- tracted vein, 1, then the following coefficients are derivable from the experiment : Short tube with rounded junction . . , . 1- One enlargement ......... "741 Three enlargements ........ -569 Five enlargements , -454 Simple tube with a rounded junction of the same length, 36 inches, as the tube with the five enlarged parts . . . . , , . ? 786 The head, in the experiment, was 32 J inches. Venturi states that no observable differences occurred in the times of discharge when the enlarged portions were lengthened from 3i to 6^ inches. See tables, pp. 152 and 191. With reference to this experiment, so often quoted, it is necessary to remark that the diameters of the enlarged portions were 2 inches each, while the lengths varied only from 3 to 6-J- inches, and consequently were at most only 3 times the diameter. Now with such a large ratio of the width to the length of the H enlarged portions, a A B b, Fig. 37, it is pretty clear that a good deal of the head is lost by the impact of the moving water on the shoulders A and B. That this is so is evident from the fact, stated by the experimenter, of the time of discharge remaining the same when a A, in five dif- ferent enlargements, was increased from 3i to 6i inches ; though this must have lengthened the whole 258 THE DISCHARGE OF WATER FROM tube from 36 to 50 inches,* thereby increasing the loss from friction proportionately, but which happened to be compensated for by the reduction in the resist- ances from impact at A and B, and in the eddies, by doubling the lengths from a to A. If, however, the length from a to A be very large compared with the diameter, and the junctions at a, A, B, and by be well grafted, less loss will arise from the enlargement than if the smaller diameter con- tinued all along uniform. The explanation is clear, as the resistance from friction is inversely as the square roots of the mean radii ; and the length being the same, the loss must be less with a large than a small diameter, These remarks, mutatis mutandis, apply equally to rivers as to pipes. We have already, pp. 140 and 147, pointed out the effects of submerged weirs and con- tracted river channels, and given formulae for calcu- lating them. BACKWATER FROM CONTRACTIONS IN RIVERS. A river may be contracted in width or depth, by jetties or by weirs ; and when the quantity to be dis- charged is known, we have given, in formulae (9), (55), and (57), equations from which the increase of head may be determined. The effect of a weir, jetty, or contracted channel of any kind, is to increase the depth of water above ; and this is sometimes neces- sary for navigation purposes, or to obtain a head for mill power. When a weir is to rise over the surface, we can easily find, from the discharge and length, the discharge per minute over each foot of length, with which, and the coefficient due to the ratio of the * The dimensions throughout this experiment are given as in the original, viz. in French inches. ORIFICES, WEIRS, PIPES, AND RIVERS. 259 sections, on and above the weir, found from TABLE V., we can find the head from TABLE VI. For submerged weirs and contracted widths of channel, the head can be best calculated, by approximation, from the equa- tions above referred to. The head once determined, the extent of the back- water is a question of some importance. If F c o D, Fig. 38, be the original surface of a river, and a A B p the raised surface by backwater from the weir at a, then the extent a F of this backwater, in a regular channel, will be from 1-5 to 1*9 times ac drawn parallel to the horizon to meet the original surface in c. This rule Fig-. 38 will be found useful for practical purposes ; but in order to determine more accurately the rise for a given length, B : B 2 or B! B, of the channel, it is neces- sary to commence at the weir and calculate the heights from A to B, B to B I? and from BJ to B 2 sepa- rately, the distance from A to B 2 being supposed divided into some convenient number of equal parts, so that the lengths A B, BB^ &c., may be considered free from curvature. Now, as the head A D is known, or may be calculated by some of the preceding for- mulae, the section of the channel at the head of the weir also becomes known, and thence the mean velo- city in it, by means of the discharge over the weir. Putting A for the area of the channel at A n, d for its depth A H, and v for the mean velocity ; also A X for s3 260 THE DISCHARGE OF WATER FROM the area of the channel at B i, d 1 for its depth, and v l for its mean velocity ; b m the mean border between the sections at AH and BI; r m the mean hydraulic V -j V-i depth ; ' the mean velocity ; AD A; B o zz ^ ; the sine of angle o D E zz s ; and the length A B zz D o nearly zz /; we get A x v = A X x v i and r m zz ' , -; ^ ^m but as, in passing from B to A, the velocity changes from v l to v, there is a loss of head equal -^ ? and if c t be the coefficient of friction, there is a loss of head from this cause equal c f X -- X ^*m hence the whole change of head in passing from B to A is equal to c { x X Q -^ . But this * change of head is equal to BE ADZZBO + OE AD zz h^ + Is h, whence we get (126.) AV A + A! , , or as v l = . and r m zz ( ', . we get, by a few re- A! ZO m ductions and change of signs, (127.) A- A! = *- * x ft m X X and therefore we get ORIFICES, WEIES, PIPES, AND RIVERS. 261 _A 2 -A* , 6 m x (A + A,) v ^ ~ Ct ' 2 A? v 2# from which we can calculate the length / corre- sponding to any assumed change of level between A and B. Then, by a simple proportion we can find the change of level for any smaller length. To find the change of level directly from a given length does not admit of a direct solution, for the value of h k in equation (127) involves A 1? which depends again on h h ly and further reduction leads to an equation of a higher order ; but the length corresponding to a given rise, k ly is found directly by equation (128). When the width of the channel, w, is constant, and the section equal to w x d nearly, the above equa- tions admit of a further reduction for A X = d l w and A = dw, by substituting these values in equation (127) it becomes, after a few reductions, (129.) h h l =dd 1 dxd l t or, as it may be further reduced, J^ s ~~ Ct X X (130.) A-A, zz d\ f *g Now, we may take in this equation for all practical purposes, approximately, b being the border of the section at 262 THE DISCHARGE OF WATER FROM AH; and also, , a l = -?, approximately; there- ! a fore we shall have f dw 2 a (131.) h-h 1 = - -3 ^r--X/5 and f 2 - (132.) /= b d j" 9 Now, as -T = -, 2# r 64-4, and the mean value of the coefficient of friction for small velocities c f -z 0078, we shall get 64 -4 ds- -0078 ^ 2 (133.) *, = A 64-4 rf- 2 t/ X /5 " / and 64-4^5- -0078 - r v* very nearly* Having by means of these equations found A B from B o or B E, and B o from A B, we can in the same manner proceed up the channel and calcu- late B! c, B 2 G!, &c>, until the points B, B!, Ba in the curve of the backwater shall have been determined, and until the last nearly coincides with the original surface of the river. When A x = 0, we shall have 64-4 tfs -0078- v 2 r ORIFICES, WEIRS, PIPES, AND RIVE ^ If we examine equation (134) it appears that w 64-4 d -=L 2 v 2 , /must be equal to zero; or. when d v 2 2 == gr-ij equal the height due to the velocity v. When / is infinite, 64 -4 d must exceed 2# 2 , and 64 equal to -0078 ^ 2 ; 64*4 7*5 > .Q078 This is the velocity clue to friction in a channel of the depth d, hydraulic mean depth r, and inclination 5 ; and, as in wide rivers r d nearly, v m 90'9 \/d~s, but when the numerator was zero we had from it v m \/32-2 6/ ; equating these values of v, we get 5 ='0039 n --nearly : see p. 139. Now, the larger the fraction s is, the larger will the velocity v become ; and the larger v becomes, the more nearly, in all practical cases, will the terms 64-4 d -2 v 2 and 64-4tfs - -0078 ^ 2 , in the numerator and denominator of equation (134), approach zero ; when 64-4 d 2 v 2 becomes zero first, / = ; when 64'4 ds -0078 - v 2 becomes zero first, / equals infinity ; and when they both become zero at the same time, l = h h l9 and 5 m , see p. 139 ; if 5 be larger than this fraction, the numerator in equation (134) will generally become zero before the denominator, or negative, in which cases / will also be zero, or negative ; and the backwater will take the 264 THE DISCHARGE OF WATEE FROM form F c 2 b 2 j b a 1 a, Fig. 38, with a hollow at C* Bidone first observed a hollow, as F c 2 b 2 , when the inclination s was -. When the inclination of a river 30 channel changes from greater to less, the velocity is obstructed, and a hollow similar to F c 2 b 2 sometimes occurs ; when the difference of velocity is consider- able, the upper water at b 2 falls backwards towards c 2 and F, and forms a bore, a splendid instance of which is the pororoca, on the Amazon, which takes place where the inclination of the surface changes from 6 inches to ^th of an inch per mile, and the velocity from about 22 feet to 4J feet per second. WEIR CASE, LONG AND SHORT WEIRS. When a channel is of very unequal widths, above a weir, we have found the following simple method of calculating the backwater sufficiently accurate, and the results to agree with observation. Having as- certained the surf ace fall due to friction in the channel at a uniform mean section, add to this fall the height which the whole quantity of water flowing down would rise on a weir having its crest on the same level as the lower weir, and of the same length as the width of the channel in the contracted pass. The sum will be the head of water at some distance above such pass very nearly. A weir was recently constructed on the river Blackwater, at the bounds of the counties Armagh and Tyrone, half a mile below certain mills, which, it was asserted, were injuriously affected by backwater thrown into the wheel-pits. The crest of the weir, 220 feet long, was 2 feet 6 inches below the ORIFICES, WEIRS* PIPES, AND RIVERS. 265 pit; the river channel between varied from 50 and 57 feet to 123 feet in width, from 1 foot to 14 feet deep ; and the fall of the surface, with 3 inches of water passing over the weir and the sluices down> was nearly 4 inches in the length of half a mile. Having seen the river in this state in summer, the writer had to calculate the backwater produced by different depths passing over the weir in autumn and winter, which in some cases of extraordinary floods were known to rise to 3 feet. The width of the channel about 60 feet above the weir averaged 120 feet; The width, 2050 feet above the weir and 550 feet below the mills, was narrowed by a slip in an adjacent canal bank, to 45 feet at the level of the top of the weir, the average width at this place as the water rose being 55 feet. The channel above and below the slip widened to 80 and 123 feet. Between the mills and the weir there were, therefore^ two passes ; one at the slip, averaging 55 feet wide ; another above the weir, about 120 feet wide. As- suming as above, that the water rises to the heights due to weirs 55 and 120 feet long, at these passes, we get, by an easy calculation, or by means of TABLE X., the heads in columns two and four of the follow- ing table, corresponding to the assumed ones on the weir, given in the first column. As the length of the river was short, and the hydraulic mean depth pretty large, the fall due to friction for 60 feet above the weir was very small, and therefore no allowance was made for it ; even the distance to the slip was comparatively short, being less than half a mile, and as the water approached it 266 THE DISCHARGE 0V WATER FROM TABLE OF CALCULATED AND OBSERVED HEIGHTS ABOVE M'KEAN's WEIR ON THE RIVER BLACKWATER. Heights at M'Kean's weir 220 feet long, in inches. Heights 60 feet above the Weir channel 120 feet wide. Heights 2050 fef>t above the weir channel 55 feet wide ; average. Calculated inches. Observed inches. Calculated inches. Observed inches. 1* 2J 2J 4J 5* 2 :. ^ii ?'!< 3 4* i . 7i 7 4 6 .. 10 9 5 7 * . 12* llj 6 9 9 15 !% 7 10* 10* 17* 18| 8 12 '" t?::v 20 20* 9 13* 12* 22* 20| 10 15 . . 24J 20 11 16* . . 27| 24 12 18 17 30 31 13 19* 18* 32| 33 15 22* 21 37| 40 18 27 25 45J 46 21 31* 29| 53 54 24 36 34 604 62 with considerable velocity, this was conceived, as the observations afterwards showed, to be a sufficient compensation for the loss of head below by friction. The observations were made by a separate party, over whom the writer had no control, and it is neces- sary to remark, that with the same head of water on the weir, they often differed more from each other than from the calculation. This, probably, arose from the different directions of the wind, and the water rising during one observation, and falling during another. ORIFICES, WEIRS, PIPES, AND RIVERS. 267 The true principle for determining the head at #, Fig. 39, apart from that due to friction, is that pointed out at pages 142 and 147 ; but when the passes are very near each other, or the depth d 2 , Fig. 23, is small, the effect of the discharge through d 2 is incon- siderable in reducing the head, as the contraction and loss of vis-viva are then large, and the head d^ be- comes, that due to a weir of the width of the con- tracted channel at A, nearly. The reduction in the extent of the backwater, by lowering the head on a longer weir, is found by taking the difference of the amplitudes due to the heads at g, Fig. 39, in both cases, as determined from equations (56), (128), etseq. This will seldom exceed a mile up the river, as the surface inclination is found to be considerably greater than that due to mere friction and velocity, and hence the general failure of drainage works designed on the assumption that the lowering of the head below, by means of long weirs, extends its effects all the way up a channel. We must nearly treble the length of a weir before the head passing over can be reduced by one-half, TABLE X., even supposing the circum- stances of approach to be the same : surely several engineering appliances for shorter weirs, during periods of flood, would be found more effective and far less expensive than this alternative, with its extra sinking and weir basin for drainage purposes. The advocates for the necessity of weirs longer than the width of the channel, for drainage purposes, must show that the reduction of the head and extent of backwater above ^, Fig. 39, is not small, and that the effects extend .the whole way up the channel, 268 THE DISCHAKCiE OF WATEK FBOM or at least as far as the district to be benefited. Practice has heretofore shown, that long weirs have failed (unless after the introduction of sluices or other appliances) in producing the expected arterial drainage results, notwithstanding the increased leak- age from increased length, which must accompany their construction. The deepening in the weir basin a b B E A is mostly of use in reducing the surface inclination between a b and A B by increasing the hydraulic mean depth ; but, thereby, the velocity of approach is lessened, and therefore the head at E increased. When the length of a weir basin a E exceeds that point where these two opposite effects balance each other, there will be a gain by the difference of the surface in- clinations in favor of the long weir : but unless a E exceeds half a mile, this difference cannot amount to more than 3 or 4 inches, unless the river be very small indeed ; and if the channel be sunk for the long weirs B A or b a ly it should also be sunk to at least the same depth and extent for the short weirs B e, b a, otherwise there is no fair comparison of their separate merits. The effect of the widening between a b and A B, the depth being the same, is also to reduce the OKIFICES, WEIES, PIPES, AND EIVEES. 269 surface inclination from a toE ; but, as before, unless a D be of considerable length, this gain will also be small. Now A B, at best, is but a weir the direct width of the new channel at A B, and if the length a E be considerable, we have an entirely new river channel with a direct weir at the lower end, and the saving of head effected arises entirely from the larger channel, with a direct transverse weir at the lower end. By referring to TABLE VIII., it will be found that for a hydraulic mean depth of 5 feet a fall of 74 inches per mile will give a velocity of 2 feet per second ; if we double the depth, a fall of 4 inches will give the same velocity ; and for a depth of only 2 feet 6 inches, a fall of 121 inches is necessary, This is a velocity much larger than we have ever observed in a weir basin, yet we easily perceive that the difference in the inclinations for a short distance, E a of a few hundred feet, must be small. If one section be double the other, the hydraulic mean depth remaining constant, the velocity must be one-half, and the fall per mile one-four-th, nearly. This would leave 7| 2 = 5| inches per mile, or 1 inch per 1000 feet nearly, as the gain with a hydrau- lic mean depth of 5 feet for a double water channel. For greater depths the gain would be less, and the contrary for lesser depths. Is the saving of head and amplitude of backwater we have calculated worth the increased cost of long weirs and the consequent necessity and expense of sinking and widening the channels for such long distances ? We think not ; indeed, the sinking in the basin immediately at the weir is absolutely injurious, 270 THE DISCHAEGE OF WATER FROM by destroying the velocity of approach, and increasing the contraction. The gradual approach of the bottom towards the crest, shown by the upper dotted line b E in the section, Fig. 39, and a sudden overfall, will be found more effective in reducing the head, unless so far as leakage takes place, than any depth of sinking for nearly 80 or 100 feet above long weirs. In most instances, the extra head will be only per- ceived by an increased surface inclination, which may extend for a mile or more up the channel, according to the sinking and widening. It is a general rule that, for shorter weirs, the co- efficients of discharge decrease ; this arises from the greater amount of lateral contraction, and is more marked in notches or Poncelet weirs, than for weirs extending from side to side of the channel ; but for weirs exceeding 10 feet in length the decrease in the coefficients from this cause is immaterial, unless the head passing over bear a large ratio to the length ; and we , even see from the coefficients, page 80, de- rived from Mr. Blackwell's experiments, that with 10 inches head passing over a 2-inch plank, the co- efficient for a length of 3 feet is *614 ; for a length of 6 feet *539 ; and for a length of 10 feet -534 ; show- ing a decrease as the weir lengthens, but which may, in the particular cases, be accounted for. We have before referred to other circumstances which modify the coefficients, yet we may assume generally, without any error of practical value, that the coefficients are the same for different weirs extending from side to side of a river. If, then, we put w and w l for the lengths of two such weirs, we shall have the relation ORIFICES, WEIRS, PIPES, AND RIVERS. 271 of the heads d and d l for the same quantity of water passing over, as in the following proportion : d : d l ; : w^ : w 3 ; and therefore (135.) #=(-)* X d, M0i' By means of this equation we have calculated TABLE X,, the ratio being given in columns 1, 3, w l l 5 and 7, and the value of ( IL Y, or the coefficient by \Wif which d is to be multiplied, to find d l in columns 2, 4, 6 and 8. It appears also, that if we take the heads passing over any weir in a river in an arithmetical progression, the heads then passing over any other weir in the same river must also be in arithmetical progres- sion, unless the quantity flowing down varies from erogation or supply, such as drawing off by millraces, &;c. If c d be the coefficient for a direct weir, *94 d will answer for an obliquity of 45, and -91 c d for an angle of 65. SECTION XL BENDS AND CURVES. BRANCH PIPES. DIFFERENT LOSSES OF HEAD. GENERAL EQUATION FOR FINDING THE VELOCITY. HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZOMETER. CATCHMENT BASINS. RAIN-FALL PER ANNUM. The resistance or loss of head due to bends and curves has now to be considered. If we fix a bent pipe, F B c D E G, Fig. 40, between two cisterns, so as 272 THE DISCHARGE OF WATER FROM to be capable of re- volving round in col- lars at F and G, we shall find the time the water takes to sink a given distance from/ to F in the upper cistern the same, whether the tube occupy the position shown in the figure or the hori- zontal position shown by the dotted line YbcdeG. I This shows that the resistances due to friction and to \ bends are independent of the pressure. If the tube were straight, the discharge would depend on the length, diameter, and difference of level between f and a, and may be determined from the mean velocity of discharge, found from TABLE VIII. or equation (79). Here, however, we h^ve to take into consideration the logs sustained at the bends and curves, and our illustration shows that it is unaffected by the pressure. / The experiments of Bossut, Du Buat, and others, show that the loss of head from bends and curves like that from friction increases as the square of the velocity ; but when the curves have large radii, and the bends are very obtuse, the loss is very small. With a head of nearly 3 feet, Venturi's twenty- third experiment, when reduced, gives for a short straight tube 15 inches long, and 1^ inch in diameter, having the junction of the form of the contracted vein, very nearly *873 for the coefficient of discharge. When of the same length and diameter, but bent as in Diagram I, Fig. 40, the coefficient is reduced to -785 ; and when bent at a right angle as at H, Fig. 40, the co- ORIFICES, WEIRS, PIPES, AND RIVERS. 273 efficient is further reduced to -560. In these respec- tive cases we have therefore* 1. 2. , _ v = -873 \/^g~h, and h = 1-312 x 3-; -2 vzz-785 , and k = 1-623 x _ 3. w = -560 v/2 # A, and A = 3-188 x j- ; showing that the loss of head in the tube H, Fig. 40, v 2 from the bend, is 1-876 x o~> or nearly double the theoretical head due to the velocity in the tube. The v 2 loss of head by the circular bend is only -3 11s, or not quite one-sixth of the other. Du Buat deduced, from about twenty-five experi- ments, that the head due to the resistance in any bent tube ABCDEFGH, diagram 1, Fig. 41, depends IT O on the number of deflections between the entrance at A and the departure at H ; that it increases at each * It is stated that the time necessary for the discharge of a given quantity of water through a straight pipe being 1, the time for an equal quantity through a curve of 90 would be 1-11, with a right angle 1-57; two right angles would increase the time to 2'464, and two curved junctions to only 1*23. Vide REPORT ON THE SUPPLY OF WATER TO THE METROPOLIS, p. 337, APPENDIX No. 3. 274 THE DISCHARGE OF WATER FROM reflection as the square of the sine of the deflected angle, A B R for instance, and as the square of the velocity; and that if <, < 1? < 2 , < 3 , &c., be the number of degrees in the angles of deflection at B, c, D, E, &c., then for measures in French inches the height A b , due to the resistances from curves, is v 2 (sin. 2 4> + sin. 2 ^ + sin. 2 4> 2 + sin. 2 4> 3 + &c.) *t>.) /**= 3000 which for measures in English inches becomes ^ 2 (sin. 2 < + sin. 2 ^ + sin. 2 < 2 + sin. 2 (137.) h*- 3197 and for measures in English feet, 58.) K- 266-4 or, as it may be more generally expressed for all measures, (139.) ^ = (sin/^+sin. 2 ^ + sin. 2 < 2 + sin. 2 4> 3 + &c.) X 2 , in which 8>27 zz ^grj = '00375 v 2 in feet. The angle of deflection, in the experiments from which equation (136) was derived, did not exceed 36. We have already shown the loss of head from the circular bend in diagram L Fig. 40, where the angle of deflection is nearly 45, to be *311 K = 00483 v\ but as the sin. 45 = -707 ; sin. 2 45 = -5 we get -00483 v* = -00966 v* x sin. 2 45, or more than two and a half times as much as Du Buat's formula would give; and if we compare it with Rennie's experiments,* with a pipe 15 feet long, J inch dia- meter, bent into fifteen curves, each 3 inches radius, * Philosophical Transactions for 1831, p. 438. ORIFICES, WEIRS, PIPES, AND RIVERS. 275 we should find the formula gives a loss of head not much more than one half of that which may be derived from the observed change, -419 to -370 cubic feet per minute in the discharge. See p. 278. Dr. Young* first perceived the necessity of taking into consideration the length of the curve and the radius of curvature. In the twenty-five experiments made by Du Buat, he rejected ten in framing his formula, and the remaining fifteen agreed with it very closely. Dr. Young finds (140.) ^OOOOM^x*. where is the number of degrees in the curve N P, diagram 2, Fig. 41, equal the angle N o P ; P = o N the radius of curvature of the axis ; h b the head due to the resistance of the curve, and v the velocity, all expressed in French inches. This formula reduced for measures in English inches is (141.) ^ = -000004*^ X*. P and for measures in English feet, (U2 .) , b Equation (140) agrees to ^ n of the whole with twenty of Du Buat's experiments, his own formula agreeing so closely with only fifteen of them. The resistance must evidently increase with the number of bends or curves ; but when they come close upon, and are grafted into each other, as in diagram 1, Fig. 41, and in the tube F B c D E G, Fig. 40, the motion in one bend or curve immediately affects those * Philosophical Transactions for 1808, pp. 173 175. T3 276 THE DISCHARGE OF WATER FROM in the adjacent bends or curves, and this law does not hold. Neither Du Buat nor Young took any notice of the relation that must exist between the resistance and the ratio of the radius of curvature to the radius of the pipe. Weisbach does, and combining Du Buat's experiments with some of his own, finds for circular tubes, (143.) Ab = Tf0 X {-131 + 1-847 (^J) x $*;. s 2^ and for quadrangular tubes, d_ 7 p- (144) A b=T f X ("124 4 3-104(^7) x ; in which < is equal the angle N o P = N i R, diagram 2, Fig. 41 ; d the mean diameter of the tube, and p the radius N o of the axis. When - exceeds -2, the value of -131 + 1-847 * exceeds -124 + 3-104 and the resistance due to the quadrangular tube ex- ceeds that due to the circular one. We have ar- ranged and calculated the following table of the numerical values of these two expressions for the more easy application of equations (143) and (144). This table will be found of considerable use in cal- culating the values of equations (143) and (144), as the second and fifth columns contain the values of 131 + 1-847 o~ 5 and the third and sixth columns the values of -124 + 3-104 (:;)> corresponding to ORIFICES, WEIRS, PIPES, AND RIVERS. 277 different values of ~- ; and it is carried to twice the & P extent of those given by Weisbach. TABLE OF THE VALUES OF THE EXPEESSIONS 131 + 1-847 (-\* and -124 + 3-104 ( - d 2p ' Circular tubes. Quadrangular tubes. d V Circular tubes. Quadrangular tubes. 1 131 124 6 440 643 15 133 128 65 540 811 2 1S8 135 7 661 1-015 25 145 148 75 806 1-258 3 158 170 8 977 1-545 35 178 203 85 1-177 1-881 4 206 250 9 1-408 2-271 45 244 314 95 1-674 2-718 5* 294 308 1-00 1-978 3-228 For bent tubes, diagrams 3, 4, and 5, Fig. 41, the loss of head is considerably greater than for rounded tubes. If, as before, we put the angle N i R = <, i R being at right angles to i o the line bisecting the angle or bend, we shall find, by decomposing the V* V 2 motion, that the head ^ becomes ^ x cos. 2 $ from the change of direction ; and that a loss of head (145.) A b = (1 - cos. 2 2 $) ^- = sin. 2 2 < f- must take place. When the angle is a right angle, v 2 as in diagram 4, cos. 2 <#> = 0, and k b z= ^- ; that ^9 is to say, the loss of head is exactly equal to the * The values corresponding to Q =-55 are -350 and *507 for circular and quadrangular tubes. 278 THE DISCHARGE OF WATER FROM theoretical head. When the angle or bend is acute, v 2 as in diagram 5, the loss of head is (1 + cos. 2 2 ~- 9 for then cos. 2 < becomes negative. Weisbach does not find the loss of head in a right angular bend v 2 greater than -984 - ; while Venturi's twenty-third experiment, made with extreme care, p. 273, shows the loss to be 1-876 s When the pipes are long, however, the value of o - is in general small, and this correction does not affect the final results in any material degree. Rennie's experiments,* with a pipe 15 feet long, J inch in diameter, and with 4 feet head, give the dis- charge per second Cubic feet. 1. Straight, see table, p. 152 . . -00699 2. Fifteen semicircular bends . . -00617 3. One bend, a right angle 8J inches from the end of the pipe . . -00556 4. Twenty-four right angles . . -00253 From these data we find consecutively, the theoreti- cal discharge being -021885 cubic feet per second, Q and the theoretical head H s , that 1. v = -319 \/2#H, and therefore H = 9-82 x j- ; 2. v = -282 \X2^H, H = 12-58 x^-; * Philosophical Transactions for 1831, p. 438. ORIFICES, WEIRS, PIPES, AND RIVERS. 279 ^2 3. v n -254 \/2#H,and therefore H = 15*50 x ^2 4. v = -116 v/2#H, HIZ 74-34 x g-- The loss of head, therefore, by the introduction of 15 semicircular bends, is 2*76 ; by the intro- 2 duction of one right angle, 5-68 n ; and by the 2 introduction of 24 right angles, 64-52 -, or about 12 times the loss due to one right angle. This shows that the resistance does not increase as the number of bends, as we before remarked, p. 256, when they are close to each other. The loss of head from one right angle, 5*68 o , is more than double the loss from 15 semicircular bends, or 2-76 ^~. The loss of head for a right angular bend, determined v 2 from Venturi's experiment, is 1-876 s-; formula v 2 (145) makes it ^- ; and Weisbach's empirical for- v 2 mula, (-9457 sin. < + 2-047 sin. 4 <) H-, makes it only v 2 984 s . The formulae now in use give, therefore, results considerably under the truth. It appears to us, that the velocity of the water moving directly towards the bend must be taken into consideration, and also the loss of mechanical ef- fect from contraction, and eddies at the bend, as 280 THE DISCHAEGE OF WATEE FEOM well as the loss arising from the mere change of direction. BRANCH PIPES. When a pipe is joined to another, the quantity of water flowing below the junction B, diagram 1, Fig. 42, must be equal to the sum of the quantities Fig*. 42 flowing in the upper branches in the case of supply ; and when the branch pipe draws off a portion of the water, as in diagram 2, the quantity flowing above the junction must be equal to the quantities flowing in the lower branches. Both cases differ only in the motion being from or to the branches, which, in pipes, are generally grafted at right angles to the main, for practical convenience, as shown at bb, and then carried on in any given direction. The loss of head arising from change of direction, equation (145), 2 is sin. 2 2 < n~, in which 2 < zz angle ABO; but as in general 2 <#> is a right angle for branches to mains, v 2 this source of loss becomes then simply s . I n addition to this, a loss of head is sustained at the junction, from a certain amount of force required to unite or separate the water in the new channel. In the case of drawing off, diagram 2, this loss was estimated by D'Aubuisson, from experiments by Genieys, to be about twice the theoretical head due ORIFICES, WEIRS, PIPES, AND RIVERS. 281 2 v 2 to the velocity in the branch, or o , so that the whole loss of head arising from the junction is v 2 2v 2 3 v 2 o + ~ - =: ^ , or three times the theoretical head 2# n 1g 2g> due to the velocity. In the case of supply, the loss is probably nearly the same. The actual loss is, however, very uncertain; but, as was before ob- served when discussing the loss of head occasioned v 2 by bends, two or three times ^ is in general so comparatively small, that its omission does not ma- terially affect the final results. A loss also arises from contraction, &c. See pp. 175, 176. The calculations for mains and branches become often very troublesome, but they may always be simplified by rejecting at first any minor corrections for contraction at orifice of entry, bends, junctions, or curves. If, in diagram 2, Fig. 42, we put h for the head at B, or height of the surface of the reservoir over it ; h & for the fall from B to A ; A d for the fall from B to D ; / equal the length of pipe from B to the reservoir ; / a equal the length B A ; 4 equal the length B D ; r equal the mean radius of the pipe B c ; r a the mean radius of the pipe B A ; r d the mean radius of B D ; v the mean velocity in B c ; v a the velocity in B A ; and v d the velocity in B D, we then find, by means of equation (73), the fall from the reservoir to A equal to (146.) A+A> the fall from the reservoir to D equal to 282 THE DISCHARGE OF WATER FROM (147.) k + ht and, as the quantity of water passing from c to B is equal to the sum of the quantities passing from B to A and from B to D, (148.) vr 2 = v a rl + v d rl By means of these three equations we can find any three of the quantities A, A a , A d , r, r a , r d , b, b a , b A , the others being given. Equations (146) and (147) may be simplified by neglecting c r , the coefficient due to the orifice of entry from the reservoir, and 1, the coefficient of velocity. They will then become (148A.) h + A. = c t x - X + X and (149.) A + A d = c f The mean value of c { for a velocity of 4 feet per second is -005741. and of ^, -0000891. The values *9 for any other velocities may be had from the table of coefficients of friction given at p. 214. When /, h, and r are given, the velocity v can be had from the equation, v = ( x -y) , or more immediately from TABLE VIII. GENERAL EQUATION FOR MEAN VELOCITY. We are now enabled to give a general equation for finding the whole head H, and the mean velocity v, in any channel ; and to extend equations (73) and (74) so as to comprehend the corrections due to bends, ORIFICES, WEIKS, PIPES, AND EIVERS. 283 curves, &c. Designating, as before, the height due to the resistance at the orifice of entry by h r , and the corresponding coefficient by c r ; h { the head due to friction, and c ( the coefficient of friction ; 7i b the head due to hends, and c b the coefficient of hends ; h c the head due to curves, and c c the coefficient of curves ; h e the head due to erogation, and c e the coefficient of erogation ; /i x the head due to other resistances, and c x their mean coefficient ; then we get (150.) n = h t + h f + h l) + h c +h e + ^ + ~; that is to say, by substituting for h r , h f , &c., their values as previously found, v* I v 2 v 2 V 2 V 2 V 2 X + Ce X + Cx X or, more briefly, 7 (151.) H = from which we find (152.) v= c t x - It is to be observed here, that for very long uniform channels, the value of the mean velocity will be found in general equal to I ^ rH | a s the \ Cf I ) other resistances and the head due to the velocity are all trifling compared with the friction, and may be rejected without error ; but, as we before stated, it is advisable in practice, when determining the diameter of pipes, p. 229, to increase the value of c 284 THE DISCHARGE OF WATER FROM table, p. 214, or to increase the diameter found from the formula by one-sixth, which will increase the discharging power by one half. (See TABLE XIII.) In equations (74) and (151), the coefficient of fric- tion c { depends on the velocity v, and its value can be found from an approximate value of that velocity from the small table, p. 214. If, however, we use both powers of the velocity, as in equation (83), we shall get, when H is the whole head, and h the head from the surface to the orifice of entry I v 2 (av + bv 2 )- + (I + ;)*- + h=n, a quadratic equation from which we find f (H-fe)2<7r / gal 1(1 + c T )r + 2gbl (1 + c r )r for a more general value of the velocity than that given in equation (74). If now we put c s = c r + c b + c c + c e + c x in equation (151) we shall find gal Zgbl for a more general expression of equation (152), when the simple power of the velocity, as in equation (83), is taken into consideration. For measures in English feet, we may take a -0000223 and b = -0000854, which correspond to those of Eytelwein, in equation (97). The value of a is the same in English as in French measures, but the value of b in equation (83), for measures in metres, must be divided by 3-2809 to find its corresponding value for measures in v 2 English feet. In considering the head ^- c n due to contraction at the orifice of entry as not implicitly comprised in the primary values of a and b, equation ORIFICES, WEIRS, PIPES, AND RIVERS. 285 (83), Eytelwein is certainly more correct than D'Au- buisson, Traite d'Hydraulique, pp. 223 et 224, as this head varies with the nature of the junction, and should be considered in connection with the head due to the velocity, or separately. It can never be correctly considered as a portion of the head due to friction. In all Du Buat's experiments, this head was considered as a portion of that due to the velo- v 2 city, and the whole head, (1 + c r ) 9 ? deducted to find the head due to friction and thence the hydraulic inclination. VALUES OF a AND I FOR MEASURES IN ENGLISH FEET. Equation (88.) (90.) (94.) (98.; (109.) (HI.) (114.) Mean values for all straight channels, pipes, or rivers These mean values of a and b give the equation r s = -0001040 v 2 + -0000221 v, from which we find 9615 r s =v 2 + -21 v, and thence (153.) v =(9615r* + -Oil)* -105 = 98 v/~ 1, very nearly, suited to velocities of about 2 feet, p. 217. a. b. ) 0000445 0000944 ) 0000173 0001061 ) 0000243 0001114 ) 0000223 0000854 ) 0000189 0001044 ) 0000241 0001114 ) 0000035 0001150 0000221 0001040 286 THE DISCHARGE OF WATER FROM HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZOMETER. When water is at rest in any vessel or channel, the pressure on a unit of surface is proportionate to the head at its centre,* measured to the surface, and is expressed in Ibs. for measures in feet, by 624 H s, in which H is the head, and s the surface exposed to the pressure, both in feet measures. This is the hydro- static pressure. In the pipe A B c D F E, Fig. 43, the pressure at the points B, c, D, F, and E, on the sides of the tube will be respectively as the heads B b, c c, D d, F/J and E e, if all motion in the tube be prevented by stopping the discharging orifice at E. In this case the pressure is a maximum and hydrostatic ; but if the discharging orifice at E be partially or entirely open, a portion of each pressure at B, c, D, F, &c., is ab- sorbed in overcoming the different resistances of friction, bends, &c., between it and the orifice of entry at A, and also by the velocity in the tube, and the difference is the hydraulic pressure. * This is only correct when the surface is small in depth com- pared with the head. If H he the depth of a rectangular surface in feet, and also the head of water measured to the lower hori- zontal edge, then the pressure in Ibs. is expressed by 31 H 2 ; and the centre of pressure is at f rds- of the depth. OEIFICES, WEIES, PIPES, AND EIVEES. 287 Bernoulli first showed that the head due to the pressure at any point, in any tube, is equal to the effective head at that point, minus the head due to the velocity. When the resistances in a tube vanish, the effective head becomes the hydrostatic head, and by representing the former byA ef we shall have, adopting the notation in equation (150), and consequently the head due to the hydraulic pres- sure equal v * X If small tubes be inserted, as shown in Fig. 43, at the points B, c, D, and F, the heights B b 1 , c c 1 , D d 1 , F/ 1 , which the water rises to, will be represented by the corresponding values of A p in the preceding equation; and the difference between the heights c c 1 , F/ 1 , at c and F, for instance, added to the fall from c to F will, evidently, express the head due to all the re- sistances between c and F. When H = E e, and the orifice at E is open, we have, from equation (150), , and therefore Ap=0, that is, the pressure at the discharging orifice is nothing. The vertical tubes at B, c, D, F, when properly graduated, are termed piezometers or pressure gauges; they not only show the actual pressure at the points where placed, but also the difference between any two ; D d 1 B b 1 , for instance, added to the difference of head between D and B, or D d 2 will give D d 1 sb l + D d 2 for the head or pressure due to the resistances 288 THE DISCHARGE OF WATER FROM between B and D. This instrument affords, perhaps, the very best means of determining the loss of head due to bends, curves, diaphragms, &c. The loss of head due to friction, bend, diaphragms, &c., between K and L, Fig. 43, is equal to K k L / + K v. If M be the same distance from L as K is, L / M m will be the height due to the friction (L and M being on the same level) ; therefore K k L / + KV L / + M m m K k + K v + M m 2 L / is the head due to the diaphragm and bend both together. If the diaphragm be absent, we get the head due to the bend, and if the bend be absent, the head due to the diaphragm in like manner. When the discharging orifice, as at E, is quite open, we have seen that the pressure there is zero; but when, as at G, it is only partly open, this is no longer the case, and the hydraulic pressure increases from zero to hydrostatic pressure, as the orifice decreases from the full section to one indefinitely small com- pared with it. A piezometer, placed a short distance inside G, will give this pressure ; and the difference between it and the whole head will be the head due to the resistances and velocity in the pipe : from which, and also the length and diameter, the discharge may be calculated as before shown. Again, by means of the head M m 1 , and that due to the velocity of ap- proach, we can also find the discharge through the diaphragm G ; see equation (45) and the remarks fol- lowing it. This result must be equal to the other, and we may in this way test the formulae anew or correct them by the practical results. The velocity of discharge of the tube A c D E, may ORIFICES, WEIRS, PIPES, AND RIVERS. 289 be calculated by means of any piezometric height c c 1 ; for by putting the whole fall from c 1 to E equal (2flrrH c lH to H C I, we get, disregarding bends, v = \ - \ , j n 1 Me 1 J which I C \G E. This is evident from equation (152), as we have supposed that no part of the head is ab- sorbed in generating velocity, or in overcoming the resistance of bends. If the bend at D were taken into consideration, then v = ' -fX^r + SECTION XII. RAIN-FALL CATCHMENT BASINS. DISCHARGE INTO CHANNELS. DISCHARGE FROM SEWERS. LOSS FROM EVAPORATION, ETC. A catchment basin is a district which drains itself into a river and its tributaries. It is bounded gene- rally by the summits of the neighbouring hills, ridges, or high lands forming the water-shed boundary ; and may vary in extent from a few square miles to many thousands ; that of the Shannon is 4,544 square miles. The average quantity of water which dis- charges itself into a river will, cceteris paribus, depend on the extent of its catchment basin, and the whole quantity of rain discharged on the area of the catch- ment basin, including lakes and rivers. The quantity of rain which falls annually varies with the district and the year ; and it also varies at different parts of the same district. The average quantity in Ireland may be taken at about 34 inches 290 TABLE of some Catchment Basins in Ireland. Names of Drainage-districts, or Rivers. Counties or Towns. Area of Catchment in acres. Area of Catchment in square miles. Avonmore . . Wicklow and Wexford 128 000 200- Wicklow . . 179,840 281- Ballinasloe . Mayo 70000 110- Barrow, Nore, and Suir . . . Waterford 2,176,000 3400- Blackwater and Boyne .... Meath, &c 695,040 1086- Blackwater .... Waterford Youghal 780,160 1219- Blackwater Armagh 336,640 526- Blackwater Meath and Kildare 50,000 78-1 Bandon River Cork 145,920 228- Bann, Upper and Lower, and the Main 810,240 1266- Boyne Meath, Westmeath Kildare and King's 304,139 478-2 Brusna (Ferbane) King's . . . 389,120 608- Wexford 26,752 41-8 Ballinamore and Bally connel Cavan, Fermanagh, Leitrim, and Roscommon 101,455 158-5 Breeogue Sligo . . 180,408 282- Cork 23,500 36-7 Cappagh 34,856 54-4 Sliffo 90744 141-8 Camoge Limerick . . . 61,184 95-6 Galway, Mayo, and Roscom- mon . . . 96,161 150-2 Dodder Dublin 35,200 55- Deel Meath and Westmeath 64,000 100- Dee Louth and Meath 78,000 121-9 Erne Belturbet Enniskillen 1,014,400 1585- Fovle Londonderry 944,640 1476- Fergus Clare and Galway 134,400 210- Fane Louth 87,400 136-6 Glyde Louth, Meath Monaghan and Cavan 176,813 276-3 Inny . . Meath Westmeath Long- ford, and Cavan 231,116 361-1 Kilbeggan . . Westmeath and King's 88,030 137-5 lifi'ey and Tolka Dublin, &c 328,320 513-0 Lee ... Cork 470,400 735- Lough Gara and Mantua . . Loughs Oughter and Gowna and River Erne Lough Neagh Roscommon, Mayo, and Sligo Cavan, Leitrim, and Longford Londonderry Antrim Down 128,000 260,480 200- 407- and Armagh 1,411,320 2205-2 Lough Mask and River Robe Mayo and Galway 225,000 351-5 Loughs Corrib, Mask, and Carra . Galwav and Mayo 780,000 1218-7 Lonford Longford . 72320 113-0 Moy . Mayo, Ballina 661,120 1033- r ? Main . , Antrim 37,600 90- 54,000 84-4 Ma^hera . Down 19000 29-7 Nobber Meath 40,000 62-3 Quoile Down 57000 89-1 Rinn and Black River .... Strokestown Leitrim and Longford .... Roscommon 74,000 70000 115-6 109-4 Different Counties, Towns of Athlone, Limerick 2,908 160 4544- Slaney . . Wexford 521 600 815- ORIFICES, WEIRS, PIPES, AND RIVERS. 291 deep, that which falls in Dublin being 27 inches, and that in Cork 41 inches nearly. The average yearly fall in Dublin for seven years, ending with 1849, was 26*407 inches ; and the maximum fall in any month took place in April 1846, being 5-082 inches. "The average fall in inches per month for seven years, ending with 1849, was as follows : October, 3 -060 ; August, 2-936 ; January, 2-544 ; April, 2-503 ; No- vember, 2-300 ; July, 2-116 ; June, 2-005 ; Decem- ber, 1*938; September, 1-860 ; May, 1-814; March, 1-739 ; February, 1-534."* A gauge at London- derry, 1795 to 1801, gives 31 inches average ; one at Belfast, from 1836 to 1841, gives 35 inches; at Mountjoy, Phoenix Park, 182 feet above low water, 1839 and 1840, there is an average of 33 inches ; and at the College of Surgeons, 52 feet over low water, the average is 30 inches for the same two years. Sir Robert Kane assumes that 36 inches is the average fall in Ireland, and that out of that depth 12 inches, or one-third, passes on to the sea, two- thirds being evaporated and taken up by plants. The quantity varies a good deal with the altitude of the district. In parts of Westmoreland it rises sometimes to 140 inches ; in London, an average of 20 years' observations, gives a fall of nearly 25 inches. Forty years' observation at Greenwich, Kent, at 155 feet above the level of the sea, gives the following results : * Proceedings of the Koyal Irish Academy, vol. v., p. 18. TJ3 292 THE DISCHAEGE OF WATER FROM Description of fall. Winter. Spring. Summer Entire Years. Mean annual fall . . . inches. 7-86 inches. 7-25 inches. 10-47 inches, f 25-48 Maximum fall; being a mean of five of the wettest years during forty years 11-05 10-86 14-96 { 25-58 ( 34-00 Minimum fall ; being a mean of five of the driest years during forty years . . 5-22 4-05 6-80 { 36-87 ( 18-40 (16-07 In this table Winter comprises November, De- cember, January, and February ; Spring, the next four months ; and Summer, the months of July, August, September, and October. The last column contains means of two classes of years : the first figures showing the ordinary years from January to December, and the second, under the first, years from November to October.* We see here that the mean maximum is fully double of the mean minimum, and about one-and-a-half times the mean annual fall, and therefore the necessity for calculating from the minimum fall for all water works in which it is an element, and from the maximum for sewerage works where it is not intended to pass off a portion on the surface or through other available channels. In the district surrounding the Bann reservoirs in the County Down, the average fall has been so high as 72 inches. In Keswick, the average fall is said to be 67| inches, and in Upminster, Essex, only 19| inches. Indeed, it is requisite to obtain the fall from observation for any particular district, when it * See Mr. James Simpson in the Metropolitan Main Drainage Report, 1857, p. 115. ORIFICES, WEIRS, PIPES, AND RIVERS. 293 is necessary to apply the results to scientific pur- poses ; and not the mean average fall alone, but also the maximums and minimums in a series of years and months in each year. Mr. Symons gives (see Builder for I860, p. 230) the following heavy falls of rain during 1859 : Wandsworth, June 12th, 217 inches in two hours ; Manchester, August 7th ? 1*849 inches in twenty-four hours ; Southampton, September 26th, 2-05 inches in two-and-a-quarter hours ; Truro, October 25th, during the day, 2-4 inches. The mean falls in the South Western Counties were 39-1 inches ; in the South Eastern Counties, 30-2 inches ; in the West Midland Counties, 28 inches ; in the Eastern Coun- ties, 2 5 *4 inches ; in the North Midland Counties, 24 inches ; in the North Western Counties, 39 inches ; in the Northern Counties, 55 inches ; and the average of all England, 31-857 inches. As an instance of extraordinary rain-fall, in connexion with the sewage question, it is stated that 4 inches of rain fell in one hour in the Holborn and Finsbury sewers' district, on the 1st of August, 1846 ; at Highgate, 3-5 to 3 -3 inches ; and at Greenwich, 0-95 inches.* In the upland districts about Manchester, Mr. Homershamf gives the result of observations at Fairfield, Bolton, Rocksdale, Marple, Comlis re- servoir, Belmont, Chapel-en-le-Frith, and Whiteholme reservoir, for four years. These give a maximum fall of 61-4 inches at Belmont Sharpies in 1847, and * Metropolitan Main Drainage Keport, p. 16. f Report on the Supply of Water to Manchester. WEALE. 294 THE DISCHARGE OF WATER FROM a minimum of 24-8 at Whiteholme reservoir in 1844. The general average for the four years being 42-49 inches. April is the driest month, and October, or about it, the wettest month, and the average fall during the year varies sometimes as two to one. The proportion between the quantity which falls, and that which passes from a catchment basin into its river, also varies very considerably. When the sides of a catchment basin are steep, and the water passes off rapidly into the adjacent river or tri- butaries, there is less loss by evaporation and perco- lation than when they are nearly flat. The soil, subsoil, and stratification, have also considerable effect on the proportion. Reservoirs being generally constructed adjacent to steep side falls, give a much larger proportion of the quantity fallen than can be obtained from rivers in flatter districts ; besides, the quantity of rain which falls on the high summits, near reservoirs, almost always considerably exceeds the average fall. As 640 acres is equal to one square mile, and one acre is equal to 43,560 square feet, a fall of one inch of rain is equal to 3,630 cubic feet per acre, and to 3,630 x 640 = 2,323,200 cubic feet per square mile : the proportion of this fall, for each acre, or square mile of the catchment basin, which enters the river, must depend entirely on the district and local circumstances, the full or maximum quantity being retained on lakes. A stream de- livering 53 cubic feet per minute supplies an equiva- lent to 12 inches of rain-fall collected per square mile per annum. ORIFICES, WEIRS, PIPES, AND RIVERS. 295 It is too often taken for granted that the discharge from a catchment basin takes place, into the con- veying channels, in nearly the same time that a given quantity of rain falls. Perhaps the largest registry on record in Great Britain is a fall of four inches in an hour. The maximum fall in any hour of any year seldom exceeds half of this amount, and then perhaps only once in several years. The quan- tity which falls will not be discharged into the channels in the same time. The quantity discharged, QUANTITY PER ACRE FOR A GIVEN DEPTH OF FALL. Fall in inches. Cubic feet per acre. Fall in inches. Cubic feet per acre. Fall in inches. Cubic feet per acre. Fall in inches. Cubic feet per acre. 2 7260 i 1815 1 454 JL_ 181 If 6352 -i 1361 | 403 TO- 121 ii 5445 907 1 363 1 91 2 4 TO 4: U 4537 i 726 A 302 50 73 i 3630 i 605 yV 259 "g"0" 61 I 2723 i T 519 TV 227 TV 52 and time, will depend a good deal on the season and district. The arterial channel receives the supply at different places and from different distances, and the water in passing into and from it does not encounter the same amount of resistance as if it all passed first into the upper end. Less sectional area is therefore necessary than if the whole discharge had to pass through the whole length of the channel and during the time of fall. The relation of the quantity of rain -fall to the portion which flows into the main channel, as well as the time which it takes to arrive at it, and the places of arrival, must be known 296 THE DISCHARGE OF WATER FROM before the proper size of a new channel can be determined, particularly sewers in urban districts. A pipe sufficient to discharge the water from 200 acres need not be 20 times the discharging power of one exactly suited to 10 acres of the same district, for the discharge from the outlying 190 acres will not arrive at the main in the same time as that from the adjacent 10 acres. The following table of rain-fall, at Athlone, central in Ireland, was furnished to the Royal Irish Academy by General Sir H. D. Jones, and is printed in the Proceedings.* The average for four years, gives 29 inches, and the effect on the Upper and Lower Sills of the Lock as affecting the rise and fall of the Shannon, affords valuable data, although not analysed. The rise and fall on the sills is the sum of the monthly risings and fallings for each year, and must be divided by 12 to get the average monthly rise and fall. In 1845 the greatest rise was in January, 2 feet 9 inches at the upper sill, and 3 feet 1 1| inches at the lower sill. In 1846 the greatest rise was 2 feet 5 inches in October, at the upper sill ; and 5 feet 6J inches on the lower sill, in August. Upper SiU. Lower Sill. Maximum rise in Maximum rise in one month. one month. 1845 .... 2 ft. 9 in. January . . 3 ft. !! in. January. 1846 . . . . 2 ft. 5 in. October . . 5 ft. 6 in. January. 1847 .... 3ft. 1 in. November. .4ft. 6 in. May. 1848 .... 3 ft. 3 in. February . . 4ft. 11 in. February. The sum of the risings and fallings for each month, taken as a mean of four years, is nearly the same * Vol. iv. ORIFICES, WEIRS, PIPES, AND RIVERS. 297 ^ 05 OO b- CO jS 00 O CO CO 1 0? 0? S ^ rH b- Oi b- TH f 1 PC o *- .; rH t~- C5 00 1 CO TH CO TH s rH H* 5 i qjuog HS TH o c? co ^ OZ rH O? rH b- b- rH fc 55 10 rH b- o g 11 5-S | 1 S! | Oi r^T TH "S 00 TH H b- rH CM OJ N * RIVER S 06,4 if! lit ! * gj HM HM 2 00 O 00 O rH rH i5i in 1 1 2 rH 00 O? O .f5 rH ,H rH rH rH CO g rH s" 8 ! -TCI-ML 6 ^H 0? oo TH ^J rH rH rH s5 i mBH qjJAl J O5 O rH CO Jo"^H. M b- TH CO , rH C~ O5 CO 1 b- b- g iq^TK i -ons aao ; S OO O Ot rH j} rH TH 00 rH | rH rH 6 rH rH " IS 1 S OS b- O CO ,S cs co TH 2 cb o? cb cb M O? CO Ol CO 1 rH | O* f! i g b- O O b- OS O CO b- rl rH rH rH rH O5 00 1 2 i ft JO* b- O CO 00 00 CO TH TH -- rH rH rH rH f CCi o O op TH rH 1 O CO b- 00 00 00 00 00 Amount for \ 1 four years, j P 298 THE DISCHARGE OF WATER FROM at either sill. The general average of the rise and fall for the upper sill, is about 1 foot 3i inches each way, and 1 foot 101 inches at the lower sill. These would give 2 feet 7 inches for the average difference of level in the Shannon above, and 3 feet 9 1 inches for that in the Shannon below. In Lough Allen catchment of 146 square miles, the maximum rise was sometimes 6 inches in 24 hours, calculated at 568 inch of depth of rain, over the catchment area. Above Killaloe, the catchment is 3,611 square miles, and the floods about once a year rose 6 inches in 24 hours, or *296 inch in depth of rain over the catchment. Once, in 1840, it is reported to have risen 12 inches, or *6 inch of rain over the catchment in one day. MAXIMUM DISCHARGES OF THE SHANNON AND ERNE, AND A TRIBUTARY OF THE LATTER, THE WOODFORD RIVER. BIVERS IN IRELAND. Extent of catchment, statute acres. Square miles. Maximum discharge per minute in cubic feet. Cubic feet per minute from each acre. Cubic feet per minute from each square mile Shannon, at Killaloe, measured previous to the commence- ment of Shannon Works, 3,000,000 4687'5 1,000,000 0-33 211 Lower Erne, measured during the very high floods of Jan. 1851, at BeUeek 974,000 1521-9 657,511 0-67 429- Upper Erne, measured during the very high floods of Jan. 1851, at Belturhet 309,000 482-8 257,771 0-83 531 Woodford River, Counties of Lei trim and Cavan, measured during the very high floods of Jan. 1851, at Ballyconnell 90,000 140-6 101,035 1-12 717 Yellow River, or upper portion of the Woodford River, mea- sured during the very high floods of Jan. 1851, Co. Leitrim 5,000 7-8 52,125 10-43 6675- ORIFICES, WEIRS, PIPES, AND RIVERS. 299 These results show how difficult it is to draw any inference from discharge and area of catchment alone, as the discharge, per minute per acre, must vary with the contour and elevation of the district in the same course ; and with the climate also, in different countries. We have observed ourselves the maximum discharges to vary up to 6 cubic feet per minute per acre, the lesser maximums being due to broad flat districts, and the greater maximums to higher and steeper districts, near the sources. In the Proceedings of the Institution of Civil Engi- neers, Ireland, vol. iv., from which we have collected and arranged some of the foregoing information, it is stated, p. 96, that the ratio of the discharge to the rain-fall, on a catchment on the Glyde, of 79,433 acres, for three months, ending March 13th, 1851, was 1*49 to 1 up to January 13th ; 1*39 to 1 up to February 13th ; and 3-86 to 1 up to March 13th, making a general average of 1*59 to 1 ; the whole rain-fall for the three months being only 5 '89 inches, while the discharge was 9-35 inches ! We fancy there is a mistake here. The whole catchment of the Glyde is 176,813 acres, and there is no data to show the discharge previous to or after the rain-fall from which to calculate the difference due to it per se for the three months; nor is the place or method of gauging stated. The supply from springs and the actual discharge before and after rain-fall must be correctly gauged before the proportion passing into the main channel in a given time, can be pro- perly estimated ; the results just stated clearly con- tradict themselves. The following anomalous results 300 THE DISCHARGE OF WATER FROM from p. 47 of the same work are also worthy of note. In five different districts the discharge is gauged, or estimated, greater than the fall, as shown in the following table. It is not stated, however, if District. River. Catchment in Acres. DECEMBER 1850. JANUARY 1851. Total fall of Bain by gauge in inches. Total depth of discharge off catch- ment in inches. Total fall of Rain by Castlebar gauge in inches. Total depth of discharge off catch- ment in inches. Saleen . . . Saleen 2,625 20,640 33,500 70,000 3,200 32,000 . 3-55 , J4-00 6-26 5-46 5-46 6527 5-705 . 6-33 . 9-20 $55 8-18 7-39 Ijannagh . . . Castlebar . . Manulla . . . Robe Balla Mask and Robe . DaUa . . . . I Dalla Owenmore . I the depths passed off, estimated over the catchments, include the flow before the commencement of the rain. If so the results are so far useless ; and if they do not include it, there must be an error somewhere. Indeed, in the Eobe we have evidence that not more than 58 per cent, passed from the catchment to the river, from Mr. Betagh's paper, the results of which are arranged below. Also, in July 1850, it is shown that in the Lannagh district only -53 inch in depth passed off the catchment from a fall of 1'83 inches, or about one-third of the depth. The method of determining this was un- objectionable. Where such discrepancies as above exhibited exist, it is important that the method of gauging, and the whole calculation, should be shown, in order that other engineers should be able to judge ORIFICES, WEIRS, PIPES, AND RIVERS. 301 of their accuracy ; otherwise the results should be rejected, no matter under whose authority they may be published. The following information has been collected and arranged by us from a paper by Mr. Betagh, in the Proceedings of the Institution of Civil Engineers, Ireland, vol. iv. In January 1851, 3*41 inches of rain fell in seven days, producing the maximum dis- charge of 85,836 cubic feet; while in December 1852, 3-17 inches, also falling in seven days, pro- duced 115,656 for the maximum. At the beginning of the first fall there was flowing 2 6, 640 feet, leaving the effects of the seven days' rain 85,836 - 26,640 zz 5 9,1 96 cubic feet, while in the second year the quantity flowing at first was 75,360 cubic feet, leaving the effects of the seven days' rain-fall equal to 115,656 - 75,360 zz 40,296 cubic feet. The effect of the previous state of the weather on the catch- ment must always modify, to a considerable extent, the discharge from a given rain-fall, and this has more to do with the results than the effect of arterial drainage itself, unless so far as one is a result of the other. Taking the mean of 1851 and 1852, it ap- pears that the evaporation in the Ballinrobe catch- ment was to the rain-fall as 41 '6 to 98*7, or about 42 per cent. This is certainly, from the nature of the catchment, less than the average through Ireland, which cannot be less than 60 per cent. In high, steep districts, fully three-fourths or 75 per cent, of the rain-fall can be collected, and at times, when the catchment is saturated, nearly the whole ; even in some few limited cases, when springs or hidden 302 THE DISCHARGE OF WATER FROM TABLES showing in detail, for the years 1851 and 1852, the Monthly Fall of Rain and the corresponding Discharge of the River Robe, at Ballinrobe, County Mayo ; the catchment basin being 70,000 acres, or 110 square miles ; the lower end 100 feet, the upper end 336 feet ; and the average height of the surface about 180 feet above the level of the sea. The average fall of the river, not including the rapids, is from one to two feet per mile ; the catchment is about 20 miles long, about one- tenth of the area bog or low marsh, and nine-tenths clayey and gravelly. The river is about 33 miles long. OBSERVATIONS IN 1851. MONTHS. Eain-fall each Month in inches. Discharge each Month of rain-fall in inches. Discharge in cubic feet per mi- nute, from a catchment of 70,000 acres, for each month. Discharge in cubic feet per minute, per acre, for each month. Maximum minimum. Average. Maximum. Minium m . Average. January 9.2 7.4 85,836 20,133 43,373 1-158 287 620 February . 6-8 4.7 72,448 18,420 30,410 1-034 263 434 March . . 4.4 3-6 49,137 10,860 20,945 702 155 300 April . . 3-4 2-5 24,200 5,760 14,355 345 082 205 May . . . 1-0 8 5,820 4,125 5,001 083 059 071 June . . . 3-8 8 7,040 1,114 4,230 100 016 060 July. . . 3-8 5 4,920 1,500 2,558 070 021 036 August . . 2-4 0-9 17,055 1,240 4,866 243 017 069 September . 1-9 0-5 4,746 1,200 2,854 067 017 040 October . . 5-0 1-6 23,980 6,940 12,588 342 099 179 November . 1-3 1-2 12,852 6,000 7,827 183 085 111 December . Total 2-6 2-5 44,715 6,210 14,373 638 088 205 45-6 27- 352,749 83,502 163,380 4-965 1-189 2-33 supplies are re-tapped, a larger discharge may take place than that due to the catchment and rain-fall ; but these do not affect the general question. " The future population of the suburbs of London is calculated at 30,000 inhabitants per square mile. According to the following data, some of the densest portions of our large towns have a population of 220 persons to an acre. The population on the ORIFICES, WEIRS, PIPES, AND RIVERS. 303 RIVER ROBE OBSERVATIONS IN 1852. Continued from last page. MONTHS. Eain-f all each Month in inches. Discharge each Month of rain-fall in inches. Discharge in cubic feet per mi- nute, from a catchment of 70,000 acres, for each month. Discharge in cubic feet per minute, per acre, for each month. Maximum. Minimum . Average. Maximum. Minimum. Average. January. . 7-5 54 41,600 12,852 28,730 594 183 410 February . 4-8 4-3 56,400 8,190 25,296 805 117 361 March . . 1-0 0-7 9,600 2,737 6,702 137 039 095 April . ' *. 1-1 0-5 3,931 1,468 2,477 056 020 035 May. .' ; . 1-9 0-4 3,931 1,050 1,861 056 015 026 June . . . 6-6 1-2 22,764 1,400 6,547 325 020 093 July. . .. 2-5 1-0 15,439 3,172 6,057 220 045 087 August . . 4-5 0-6 3,856 2,236 3,070 055 032 043 September . 1-8 0-5 3,427 2,642 2,874 048 037 041 October . . 3-9 1-0 32,040 1,114 5,932 457 016 084 November . 5-5 5-2 45,360 17,000 30,742 648 242 439 December . Total 12-0 9-5 115,656 23,232 54,846 1-657 331 783 53-1 30-1 354,004 77,093 175,134 5-058 1-097 2-497 north side of the Thames is about 75 persons per acre, and on the south side 28 persons per acre. Taking the average density of population in our twenty-one principal towns, there appear to be 5045 inhabitants to the square mile ; but, from the fol- lowing table, extracted from Dr. Duncan's report on Liverpool, it will be seen that if we select five of our most populous cities, the average in these is much greater, while in others, it is equally certain that the crowding is far less than the general stand- ard to which we have referred : 304 THE DISCHARGE OF WATER FROM Inhabitants to a Square Mile. Towns. Total Area. Builded Area. Leeds 20,892 . . 87,256 London .... 27,423 . . 50,000 Birmingham . . 33,669 . . 40,000 Manchester . . . 83,224 . . 100,000 Liverpool . . . 100,899 . . 138,224 Dr. Duncan, however, states that there is a district in Liverpool containing 12,000 inhabitants crowded together on a surface of only 105,000 square yards, which gives a ratio of 460,000 inhabitants to the geographical square mile. In the East and West London Unions, Mr. Farr has estimated that there are nearly 243,000 inhabitants to a geographical square mile ; but, great as this overcrowding is, the maximum density of Liverpool exceeds that of the metropolis by nearly double."* The amount of sewage from each person is cal- culated about FIVE CUBIC FEET PER PERSON, including the supply from manufactories, breweries, distilleries, &c. SEVEN FEET PER HEAD has been recommended as data to calculate from by Captain Galton, Messrs. Simpson and Blackwell, in their Eeport on the Main Drainage, and it has been found that about half of the estimated quantity of sewage would pass off in six or eight hours. In calculating the size of sewers, however, the rain-fall must be provided for, in addition to the sewage matter from houses and public establish- ments. Mr. Bazalgette calculated this for the Lon- don sewerage at ith of an inch fall in 24 hours in the urban districts, and ith of an inch for the suburban districts. Captain Galton and the Messrs. * Illustrated News, September 8th, 1855. ORIFICES, WEIRS, PIPES, AND RIVERS. 305 Simpson and Blackwell assumed fths of an inch fall during eight hours' maximum flow. This would be ]452 feet per acre. Assuming the highest data, we shall have to provide sewers to discharge in eight hours 1,452 cubic feet of rain water per acre, 3J cubic feet of sewage nearly per person. Assuming a population of 80 persons per acre, then these figures would become in eight hours, or 1,452 cubic feet for ram, ' OOA i,- f 4. f \ at)OUt 3 i cubic f eet 280 cubic feet for sewage. per minute, per acre, which shows that the sewage is not more than -Hh of the rain water ; and that, in calculations for the size of sewers, the surface water is the most im- portant element to be considered. If we had as- sumed a larger fall of rain, the difference between sewage and rain would be greater. On the 20th June, 1857, the day after heavy rain, the referees on the Metropolitan Drainage question found the Nor- folk-street sewer to discharge 3 feet ; the Essex- street sewer 5| feet; the Northumberland-street sewer 3J feet ; and the Savoy-street sewer 20J feet per minute per acre ; but the last result has been controverted. It appears that the daily amount of sewage varies from 4*8 cubic feet per head in the more thickly inhabited portions of London, occupied by a larger portion of the poorer classes, to 8 cubic feet per head in the western districts, where the value of water is more appreciated, and the cost less a matter of consideration; and the average of the whole 306 THE DISCHARGE OF WATER FROM metropolitan districts appears to be 5-8 cubic feet per head per diem. If the day be divided into three periods of eight hours each, the amount of the maximum flow is between nine A.M. and five P.M. and 49 per cent. of the whole, whilst only 18 per cent, flows during the eight hours of minimum flow, which occur be- tween eleven P.M. and seven A.M.* The advantage of storm flows in flushing is shown by the heavy rain which occurred on the 20th of June, causing a flow in the Savoy-street sewer .which was equivalent to 20 times the ordinary flow at the time. This was six times the maximum flow, and although the sewer had been scoured, to a considerable extent, by a heavy fall of rain on the previous night, the sample con- tained more than double the amount of total impurity contained in specimens of ordinary sewage. In a town district, such as that drained by the Savoy and Northumberland-street sewers, the quan- tity running off into sewers, within six hours after the fall, varies from 10 to 60 per cent, of the quan- tity fallen. Of the rain during the storm of the 20th June, 1857, nearly one inch-and-a-quarter in an hour, 65 per cent., ran off within 15 hours of the fall, viz. : 46 per cent, in 45 minutes after the rain ceased, 14 in the next 6| hours, 5 in the next 7i hours. In a suburban locality, such as the Counter Creek sewer drain, the quantity reaching the sewers would * Metropolitan Main Drainage Keport, pp. 15, 17. OEIFICES, WEIKS, PIPES, AND KIVEES. 307 vary from to 30 or 40 per cent, in 24 hours after the rain.* In the Holborn and Finsbury divisions Mr. Roe calculated that an 18-inch cylindrical pipe, laid at an inclination of 1 in 80, is sufficient for 20 acres of house-sewage, while a 5-inch pipe, laid at an inclina- tion of 1 in 20, is necessary for 1 acre, and a 3-inch pipe, laid also at I in 20, for j acre. A pipe 30" in diameter, laid with an inclination of 1 in 200, would discharge 1700 cubic feet per minute, and per- fectly drain 200 acres of urban land covered with houses to the extent of 4000 or upwards, and each house having a water supply of 150 gallons per diem. In each of these cases, however, the dis- charge must depend on the head and length of the pipe as well as the inclination at which it is laid. Assuming the inclination of those pipes to corre- spond with the hydraulic inclination, we have calcu- lated their discharging powers with water to be respectively 807, 72, 20, and 1700 cubic feet per minute, the areas to be drained being 20, 1, J, and 200 acres. In all calculations of this kind it is necessary , for accuracy, to ascertain not only the max- imum rain-fall per hour, but also the proportions dis- charged per hour, according to the season and district, into the main channel, as well as the junctions or places of arrival. In urban districts, 1500, 2100, and sometimes 3600 cubic feet per hour per acre, have to be discharged after extraordinary rain-falls. These may be taken as maximum results. The * Metropolitan Main Drainage Keport, pp. 75, 76. x3 308 THE DISCHARGE OF WATER FROM gaugings of the Westminster sewers in summer give 53 cubic feet per hour for the urban, and 17 feet for the suburban, according to Mr. Hawkins. In urban districts, however, a much larger quantity of water is conveyed more rapidly, cceteris paribus, to the mains, than in suburban districts and catch- ment basins generally, in which the maximum dis- charge per acre per hour, even in the steeper and higher districts, seldom exceeds 700 cubic feet, and varies from about 20 cubic feet for the larger and flatter districts upwards. This arises from the impervious nature of the surfaces it falls upon in towns, and the lesser waste in passing to the drains, as well as a large portion of the supply being often artificial. From 70 to 90 cubic feet* per acre per hour, is generally taken for the maximum discharge from the average number of catchment basins ; this is nearly equal to a supply of one-fiftieth part of an inch in depth from the whole area. Thorough-drainage in- creases the supply and discharge. Every catchment basin has, however, its own peculiar data, and a knowledge of these is necessary before we can draw any correct conclusions for new waterworks in connec- tion with it. It may be remarked, however, that any conclusions drawn from experiments on the supply of tributaries, particularly in high districts, are wholly inapplicable to the main channel into which they flow. The flow into tributaries and mountain streams, or rivers, is always more rapid than into * Some interesting observations on rain-fall and flood dis- charges are given in the Transactions of the Institution of Civil Engineers, Ireland, for 1851, pp. 19-33, and pp. 44-52. OBIFICES, WEIES, PIPES, AND RIVERS. 309 main channels and rivers in flat districts, and the supply from springs often forms a large portion of the water flowing in them. TABLE showing Summer Discharges of some English Rivers, as collected from various authorities, re-arranged, showing to some extent the effect of Springs in supplying Channels in different places. NAMES OF RIVEKS. 1 Valley. Hill. Catchment in 1 square miles. 1 B 11 Discharge per 1 square mile in I cubic feet per 1 minute. Representing inches of rain-fall I per annum. Total average 1 rain-fall in inches 1 per annum. Gade, at Hunton Bridge, chalk . . 150 to 500 69-5 2,500 36-2 8-19 Lea, at Lea Bridge, chalk. (Rennie, April 1796) . . . 30 to 600 570-0 8,880 15-58 3-53 - Loddon, (Feb. 1850,) green sand . . . 110 to 700 221-8 3,000 13-53 3-01 25-4 Medway, driest sea- sons,(Rennie!787,) clay 481-5 2,209 4-59 1-04 Mimram, atPanshan- ger, chalk . . . 200 to 500 29-2 1,500 51-4 11-58 26-6 Medway, ordinary summer run, (Ren- nie, 1787,) clay . . 481-5 2,520 5-23 2-19 Nene, at Peterbo- rough, oolites, Ox- ford clay, and lias . 10 to 600 620-0 5,000 8-45 1-88 23-1 Plym, at Sheepstor, granite .... 800 to 1,500 7-6 500 71-4 15-10 45-0 Severn, at Stone- bench, silurian . . 400 to 2,600 3,900 33,111 8-49 1-98 Thames, at Staines, chalk, green sand, Oxford clay, oolites, &c 40 to 700 3,086 40,000 12-98 2-93 24-5 Verulam, at Bushey Hall, chalk . . . 150 to 500 120-8 1,800 14-9 3-37 Wandle, below Car- shalton, chalk .;-.. 70 to 350 41-0 1,800 43-9 9-93 24-0 Trent, at its mouth, oolites and Oxford 100 to 600 3,921 310 THE DISCHARGE OF WATER FROM The above information has- been obtained from Mr. Beardmore and Mr. Hughes' books, and from Rennie's reports. The effect of the geology and fissures in the chalk and mountain limestone forma- tions on the springs of a catchment basin, and on maintaining the summer discharge, should be carefully noted as one of the elements entering into catchment basin statistics. Indeed, the maximum and minimum discharges from catchments are of as much impor- tance to the engineer as the averages, and, for many purposes, more important. There were abundant opportunities of acquiring this information for all our Irish rivers, but we are not aware if these were turned to account. The effects of evaporation are very variable ; some- times 58 or 60 per cent, of the annual fall is carried off in this way from ordinary flat tillage soils, and other estimates are much higher ; much, however, depends on the soil, subsoil, inclination, stratification, and season. The evaporation from water surfaces exceeds the annual fall in these countries by about one-third ; and that from flat, marsh, and callow lands exceeds the evaporation from ordinary tillage, porous, and high lands. When the flat lands along the banks of rivers extend considerably on both sides, an extra fall is necessary into the main channel, along the normal drains, otherwise such lands must suffer from excessive evaporation as well as floods. Evaporation also varies with the climate, and in this country we may assume that one-third of the whole rain-fall passes on to the sea. In a paper in the Journal of the Royal Agricultural OKIF1CES, WEIRS, PIPES, AND RIVERS. 311 Society of England, vol. v, part 1, 1844, Mr. Josiah Parkes shows, that 42 1 per cent, of the whole annual rain of England filters through the soil, and 57| per cent, evaporated, being the mean results of eight years' observations, from 1836 to 1843, both included. The mean evaporation and filtration for each month during this period is shown and arranged by us in the following table : MONTHS. Total falling. Evaporated. Remaining. Deposited in Tons and Cubic feet per acre. Inches. Inches. per cent. Inches. per cent. Cubic feet Tons. January .... 1-847 540 29-3 1-307 70-7 4,744 132 February . . . 1-971 424 21-6 1-547 78-4 5,616 156 March .... April 1-617 1-456 1-856 2-213 540 1-150 1-748 2-174 33-4 79-0 94-2 98-3 1-077 0-306 0-108 0-039 66-6 21-0 5-8 1-7 3,910 1,111 392 142 109 39 11 4 ]\lav . June July 2-287 2-245 982 0-024 1-8 87 2-4 August .... 2-427 2-391 98-6 0-036 1-4 131 3-6 September . . . 2-639 2-270 80-1 0-369 13-9 1,339 37 October .... 2-823 1-423 50-5 1-400 49-5 5,082 141 November . . . 3-837 0-579 15-1 3-258 84-9 11,826 328 December . . . 1-641 0-164 00-0 1-805 ico-o 6,552 182 Yearly averages . 26-614 15-320 57-6 11-294 42-4 40,932 1145 The maximum quantity, 32-10 inches, fell in 184.1, and the minimum in 1837, 21-10 inches. The maxi- mum quantity which fell in January was 3-95 inches, and the minimum -31 inch; in February 2*85 and 1-02 inches; in March 3-65 and 0*34 inches; in April, 2-57 and -34 inches; in May 5-00 and -70 312 THE DISCHARGE OF WATER FROM inches ; in June 3-31 and 1*33 inches ; in July 4-36 and 1-30 inches; in August 3-65 and 0-95 inches; in September 4-50 and 0*63 inches ; in October 4-82 and 1*41 inches ; in November 5*77 and 2*05 inches; and in December 3-02 and *40 inches. The greatest quantities fall in September, October, and November; and the least in February, March, and April. The general mean fall for England is said to be 3 H inches, and near London 25 inches. The amount of rain varies, not only at different places and different elevations, but also at different elevations in the same place. The following table shows the amount of rain collected in each month in 1855 at Greenwich observatory, at different elevations: MONTH IN 1855. Osier's anemo- meter gauge, inches. On the roof of the library. Cylinder partly sunk in the ground. January 0-2 1*0 1 5 February 0'2 1-4 1-0 0*5 1'3 2-0 AprU... 0-1 0-1 0-1 May ., 0-5 1-5 1-8 0'5 0*7 0'9 July 3*1 4'8 5.3 August 0-6 0*8 1-4 September 0'8 1-1 2-0 2'6 4*5 6'2 0'5 ri 1*5 0*4 0-9 1-1 Totals 10-0 19-2 23-8 The cylinder gauge was placed 155 feet above the OKIFICES, WEIES, PIPES, AND EIVERS. 313 level of the sea; the gauge on the roof of the library 22 feet over the cylinder gauge, and Osier's anemometer gauge 28 feet higher than the gauge on the roof of the library. In the valleys in the lake districts, Westmoreland and Cumberland, the annual fall varies occasionally from 50 to 100 inches, and the maximum fall is said to obtain at about 2000 feet above the level of the sea on high catchments. At Ballinrobe, a gauge placed on the church tower, 60 feet above the ground, indicated 42 per cent, less rain than one on the ground ; and another experi- ment with a change of gauges, gave 68 per cent, less at the greater elevation. At Kinfauns Castle, Scotland, a gauge 600 feet high on a hill, gave 41 J inches, while one at the base, 580 feet lower, gave only 25 \ inches. In Keswick, the fall is 65! inches, and in Carlisle only 30 inches. At Kendall the fall is 60 inches ; at Manchester 33 inches ; at Lancaster 45 inches ; at Liverpool 34 inches. From the 23rd of February to the 6th of June, 1 860, the rain at Dublin was 8 inches. At the Leefin Moun- tain, which is 2000 feet high, the rain was 13-1 inches. From the 23rd of February to the 9th of July, the rain at Dublin was 10-674 inches ; and at the same time, on the Leefin Mountains (over Ballysmutten), 181 inches ; that is, an increase of nearly 80 per cent, in that time. From the 23rd February to the 21st August, inclusive, the rain-fall at Dublin was 17 inches ; at Blessington 21 inches ; at Ballysmutten, on the site of a proposed reservoir, 27 inches. This showed an increase over Dublin of 10 inches. It would appear that from 50 to nearly 80 per cent, more rain fell at Ballysmutten than at Dublin. 314 THE DISCHARGE OF WATER FROM Experiments were made at York in 1832, 1833, and 1834, for the British Association, with three gauges, the first placed on a large grass plot in the grounds of the Yorkshire Museum ; the second at a higher elevation, 43 feet 8 inches, on the roof of the Mu- seum; and the third on a pole 9 feet above the battlements of the great tower of the Minster, at an elevation over the gauge on the ground of 212 feet 10i inches. The quantities received were as follows : Average depth for Depth for three years. one year. First gauge . . 64-430 inches . . 91-477 inches Second gauge. . 52-169 \.-.. 17-389 Third gauge . . 38-972 . . 12-991 Professor Phillips gives the following formula for calculating the difference between the ratios of rain falling on the ground and at any height h in the same place 1 the temperature of the season, and c a co- efficient dependent upon it ; then the difference d is > - ' d==ch Tw' The mean height at which rain begins to be formed by this formula is 1,747 feet over the ground; and at 356 feet high, the depth which falls is one-half of what falls on the ground.* A discussion of the mean temperature in connexion with the fall of rain, has been made at Greenwich for the years 1852, 1853, and 1854 ; and at Oxford for the years 1855, 1856, and 1857. The result shows an average of 160-3 rainy days at Greenwich for each year, and 146-6 at Oxford. The difference of the mean temperatures of the day of rain and the day before is less than that of the day of rain and the day after. * Vide Civil Engineer and Architect's Journal for 1860, p. 167. ORIFICES, WEIRS, PIPES, AND RIVERS. 315 Mean tempera- Mean tempera- Mean tempera- ture, day ture, day of ture, day before rain. rain. after rain. Greenwich observations 49-25 . 49-27 . 48-98 Oxford do. 49-50 . 49-63 -V 49-44 Dividing the winds into two groups, northerly and southerly, the Oxford observations give the di- rection for 218'5 days' fair weather. The wind was northerly for 131-5 days, and southerly for 87 days. For the remaining 146-5 rainy days, the wind was northerly for 64-5 days, and southerly for 82 days. SECTION XIII. WATER SUPPLY FOR TOWNS. STRENGTH OF PIPES. SEWER- AGE ESTIMATES AND COST. THOROUGH-DRAINAGE. ARTERIAL DRAINAGE. SUPPLY. QUALITY. The supply of water to towns has become latterly a subject of considerable importance. Three points have to be considered, first, a sufficient supply at high pressure, when it can be obtained within a reasonable expenditure ; secondly, the quality ; and, thirdly, the cost. The advantages in towns of high pressure are now apparent to all in overcoming fire ; fronts of houses and pavements may also be cleaned, and streets watered if the supply be abundant. The highest apartments ean be supplied, and even mecha- nical power can be obtained for many purposes, as grinding coffee, at a reasonable cost. Mr. Glynn says,* " In many parts of London water is supplied at 4< for 1000 gallons, at a pressure of 150 feet : a gallon of water weighs 10 Ibs., so that 1000 * Power of Water. WEALE. 316 THE DISCHARGE OF WATER FROM gallons of water falling 150 feet, are equal to 1,500,000 Ibs. falling one foot ; and if 1500 gal- lons of water be used in one hour, they are equal to 37,500 Ibs. falling one foot in one minute, or somewhat more than a horse's power, which is 33,000; therefore, it may be assumed, that the cost of a horse's power for an hour in such cases, is only 6<" The number of gallons of water required for the supply of each person, including all collateral uses, has been differently estimated, and varies in almost every town, and even in the same city London, for instance, when supplied by different companies and under different systems. 44 gallons per head, per diem, were supplied by the several companies of London in 1853, while evidence has been given to show that the actual average consumption for all purposes did not exceed 10 gallons per head, per diem ; the remainder having been wasted under an imperfect system of distribution. It is asserted that when the supply is 25 gallons per head, per diem, that 5 gallons of it are used for purposes requiring nitration, 10 gallons for purposes not requiring fil- tration, and 10 gallons wasted, or two-fifths of the supply. As there must be a considerable, loss under even the best system of supply, we may assume, with the Board of Health, that a minimum supply of 75 gallons per house, per diem, or 15 gallons per person, per diem, is necessary. The following is an abstract of the average num- ber of gallons of water furnished per diem, by different water companies in London, during the year 1853, to each house, including manufactories and public establishments as houses : ORIFICES, WEIRS, PIPES, AND RIVERS. 317 New River Company Gallons. Per House. Per Person. 193 38-3-5 East London Water Works . . , . 187 37-4-5 West Middlesex Water Works 204 40-5-5 Grand Junction Water Works f319 (336 f 63-4-5 1 67-1-5 Southwark and Vauxhall Companies' Houses 175 35 Ditto average houses, manufactories, public 209 227 41-4-5 45-2-5 Chelsea Water Works . . . . ..- Hampstead Water Works .... 111 22-1-5 Kent Water Works Mean Values . 270 55 2233 446-3-5 223-3-10 44-3-5 These quantities have been calculated from the par- liamentary returns made in 1854; and if there be any truth in the calculations and returns of the quantities actually consumed per person said to be 10 gallons we get the proportion, as 10 is to 34 so is the quantity consumed to the quantity wasted. But, even assuming the quantity consumed to be 20 gal- lons per head, what an immense loss is here exhibited from want of a suitable system of distribution. For large towns it is safe to provide for many purposes, besides mere personal or house wants ; and it is safer, where it can be done without much cost, to provide for a supply of 40 gallons to each inha- bitant, even if this quantity shall not be used or 318 THE DISCHARGE OF WATER FROM raised. For high pressure, the supply required will generally vary from 15 to 42 gallons, or from 3 to 7 cubic feet to each inhabitant, or an average of about 30 gallons, including the supply to stables, offices, manufactories, and breweries. The quality of water for drinking, washing, or cooking, is also an important element in selecting a source of supply. Hardness is measured by the number of grains of chalk or carbonate of lime to a gallon of water, each called a degree. The average hardness of spring water is about 26, that is, 26 grains of carbonate of lime to one gallon of water. Eivers and brooks have an average hardness of 13, and water derived from surface drainage 5 ; hence the great advantage of the latter kinds of water in washing. The average hardness of the London pipe waters is from 10 to 16. The following report and analyses furnished to me, in 1855, by Professor Sullivan, of the Museum of Irish Industry, Dublin, will show what is generally required on this head : " On the annexed page you will find the numerical results of my analyses of the four samples of water which you left with me for examination. From the table you will perceive that the water of the Mattock Eiver appears to be the purest, so far as the nature and the amount of the foreign substances held dis- solved in it is concerned. The water of the Boyne comes next in quality to that of the Mattock River, the pump water being in every sense the worst, so far as amount of ingredients can be taken as a test of the quality of a water ; in this respect, indeed, it resembles the water of the deep wells of London and elsewhere. ORIFICES, WEIRS, PIPES, AND RIVERS. 319 " As the ordinary mode in which the quality of a water, for drinking and for culinary and like pur- poses, is judged of is, by the comparative amount of organic matter, the total amount of dissolved matter, and its hardness, according to the 'soap test,' I shall give in the following table the numbers repre- senting each of these qualities : TABLE showing the number of grains of Organic Matter, and the number of grains of Solid Matter, in an imperial gallon of Water from Number of Grains of Organic Matter, per Imperial Gal. Number of Grains of Solid Matter, per Imperial Gal. Degree of Hardness according to the Soap Test. No. 1. Tullyescar . . . 8-975 grs. 31,175 15 8-10ths. 2. Eiver Mattock . 2- (about) 15,360 9 l-10th. 3. Kiver Boyne . . 3-250 22,700 14 9-10ths. 4. Burn's Pump . . 7-100 76,850 34 4-10ths. " In order to render this table more instructive, it may be well to subjoin a few of the results obtained from the analyses of the waters of other localities. TABLE showing the number of grains of Solid Matter contained in one gal- lon of the following Water : Thames, at Greenwich . *' i*-> . V*? 27-9 grains. London .,.*,, . .. . : . .... , . 5 ., .28-0 Westminster ...._;"..;* ..= * f . 24-4 Twickenham . . . 22-4 Teddington .... 17-4 New River (London) . . . . 19-2 Lea ... ;'- 23-7 Trafalgar Square Fountain, Deep Well . 68-9 Well in St. Giles', Holborn . . . 105-0 Artesian Well at Grenelle (Paris) . T"; 9-86 " The following are some of the results obtained from an examination of the waters in the neighbour- 320 THE DISCHARGE OF WATER FROM hood of Dublin, or which have been proposed as a source of supply :* Locality from whence Water was obtained. Total Number of Grains per Imperial Gallon. Total Number of Grains of Organic Matter. Degree of Hard- ness according to the Soap Test. Royal Canal (12th Lock) . . 21-0 2-80 degs. 14-0 Grand Canal (7th Lock) . . 16-300 2-30 10 3-4ths. River Liffey, at Kippure . . 3-525 1-90 2-10ths. at Phoulaphouca 5-125 1-50 2-10ths. Lough Dan, Co. Wicklow . . 2*800 1-225 8-10ths. River Dodder, at City Weir . Lough Owel , 8-350 10-225 1-625 1-550 1 8-10ths. 6 7-10ths. " The quality of a water for drinking purposes depends in a great degree upon the condition in which the organic matter is found, much more than upon its quantity. This is, however, a question outside of the domain of chemistry, and can only be solved by the aid of the microscope. I may, however, venture to remark that the organic matter contained in the water of the Boyne and the Mat- * While these pages were passing through the press, Dr. Apjohn gave the following analyses : * Total matter Organic dissolved, matter. Hardness. Grand Canal mean of seven analyses . 20-78 -95 15-9 Royal Canal mean of five analyses . 20-76 1-64 14-1 Liffey mean of eleven analyses . . 8*62 1-77 6*1 Analysis of the deposition on pipes from the Portobello basin : Water . . . "..;.' ... ' . 2-20 Organic Matter '; . ' . ; . , . 9-71 Sand . . . . 10-20 Per Oxide of Iron and Alumina . 3-50 Carbonate of Lime . .... . 74-20 Carbonate of Magnesia . '..' . -19 100 ORIFICES, WEIRS, PIPES, AND RIVERS. 321 tock River is of vegetable origin, and would not, so far as I believe, be injurious to health. " As a general rule, I believe that the water of clear flowing rivers, even though it may contain a large amount of solid matter, and even of organic matter, will be found wholesomer than well water, especially in towns. u For certain manufacturing purposes, and for culinary purposes, too large an amount of lime is injurious, but I believe that a certain quantity pre- sent in water, is not only not injurious, but in my opinion is of the greatest utility, and renders the waters wholesome. I think the rage for extracting pure water containing only one grain of solid matter to the gallon, or thereabouts, for supplying towns, is carried too far. Such water is, no doubt, the best on a hill side ; but, I question whether it is equally well adapted for resting in basins, tanks, pipes, &c., with that containing some lime. The Eiver Dodder and Lough Owel waters appear to me the best adapted for city and town supplies. The River Mattock contains rather more than either, but it is decidedly better than the water of either of the canals from which our Dublin supply is drawn. " Drogheda is rather badly situated for a supply of very soft water, as almost the whole drainage basin of the Boyne is either situated upon limestone, or the feeders of that river rise through the cal- careous drift gravel which covers so much of the country. The water of the Boyne appears to be an excellent water for most purposes, and perhaps the difference between it and the Mattock River 322 THE DISCHARGE OF WATER FROM Tabular Results of the Special Analyses of Four Samples of Water from the neighbourhood of Drogheda. Nature of dissolved matter. No. 1. Tullyescar. No. 2. Mattock River. No. 8. Boyne River. No. 4. Burns's pump water. Observations. Carbonate of lime . . . Carbonate of magnesia . 9-350 0-429 7-302 0-510 11-648 0-888 21-475^ 0-585J (^Inclusive of a very small quantity of -j phosphate of lime and iron not sepa- t^rated from the lime. Sulphate of lime . . . 9-043 2-514 4-459 4-568 Chloride of magnesium . 0-743 1-258 1-685 8-445 Chloride of calcium . . .. 9-524 Chloride of soldium . . .. 0-991 .. Magnesia existing as cre- nate, &c., in the water . 0*464 .. .. .. Lime do. do. . . 0-548 Silica do. do. . . 0-627 0-322 2-212 Potash and eoda existing in water, as nitrates, crenates, and other or- ganic salts ..... 1-644 2-785 0-448 22-393 Organic matter . . . . Total number of grains per Imperial gallon . . 8-975 8-250 7-100 31-175 15-360 22-700 76-850 may in part be accounted for by its being taken near the banks, or more probably, perhaps, because it was above and close to where some small stream entered. " The quantity of solid matter in it, however, was not more than I would expect considering the nature of the locality. I did not draw attention in my Report to a point of some importance namely, the proportion of lime and magnesia existing as car- bonates, and as sulphates, and chlorides. The whole ORIFICES, WEIRS, PIPES, AND RIVERS. 323 of the lime and magnesia existing as carbonates, and as sulphates, and chlorides, is precipitated by boil- ing, the water being thus proportionably rendered less hard ; lime and magnesia existing as sulphates or chlorides, on the other hand, are not precipitated. This difference is of great consequence in culinary operations, as where boiled water is used, the car- bonates of lime and magnesia are not injurious, and if no sulphates or chlorides be present, the water may be soft after boiling. The same observation applies to water applied to washing clothes when boiled. And lastly, sulphate of lime forms one of the worst elements of fur or deposits upon steam boilers." The saving in soap effected by a reduction of 10 degrees in hardness, is found to be over 50 per cent. Some of the metropolitan waters analyzed by Dr. Eobert Dundas Thomson, F.R.S., were found, in May 1860, much more impure than others, the sam- ples of which had been taken at the beginning of the month, before the impurities conveyed by the rains had contaminated them. The supply afforded by large and small rivers, as in London, in this table, contrasts most unfavourably with that afforded by the drainage of mountain ridges, as at Glasgow and Manchester. The specimens of water from the two latter cities were taken by the instructions of Mr. Bateman, F.R.S., the engineer, from the main pipes during the month. It should be the object of the London Companies to avoid pumping the water in its most impure state, and to store it when in the condition of the greatest purity. Y3 324 THE DISCHARGE OP WATER FROM Total Impurity per gallon. Organic Impurity per gallon. Grs., or . o-o Grs., or . o-o Loch Katrine water, new supply to Glasgow 3-16 0-96 Manchester water supply .... 4-32 0-64 THAMES COMPANIES : Chelsea . 17-84 17-08 1-48 1-64 Grand Junction 20-72 2-00 West Middlesex 20-08 20-80 2-08 2'40 OTHER COMPANIES": New Kiver 18-52 23-64 1-56 3-20 21-68 2-96 The table is read thus : Loch Katrine water con- tains in the gallon 3-16 degrees or grains of foreign matter in solution, of which *96 degrees or grains are of vegetable or animal origin. Professor Apjohn gives the following analyses of waters furnished to the city of Dublin in 1860. It shows how necessary it is to distinguish the time of taking specimens for analysis, and the previous state of the weather as affecting the foreign matters in the water. The specimens were collected on the 5th and 19th of May, 1860. The quantity operated upon in each instance was an imperial gallon, or 277-273 cubic inches : ORIFICES, WEIRS, PIPES, AND RIVERS. 325 Carbonate of lime . . . Carbonate of magnesia . . j Sulphate of lime and chlo- rides of sodium and mag- CITY WATER COURSE, DODDER. 5th May. 1 4-056 19th May. 7-308^1 2-269 nesium Silex 0-166 Organic matter 1*811 8-302 0-700 2-171 0-526 Specific gravity of specimen (5th May) 1-00011. Specific gravity of specimen (19th May) 1-00014. 11-086 PORTOBELLO BASIN. 7-687 Carbonate of lime Carbonate of magnesia Sulphate of lime and chlo- ) rides of sodium and mag- j- 4-058 nesium j Silex 0-073 Organic matter 3-308 15-126 11-660- 0-764 3-751 0-194 2-289 18-658 Specific gravity of specimen (5th May) 1-00023. Specific gravity of specimen (19th May) 1-00031. It will be observed that the quantities of saline and other ingredients found in specimens of same water collected at the two separate periods above mentioned are materially different; those obtained at the later date (May 19) containing the larger portion of foreign matters. The extent of this variation is very considerable, and it appears to Dr. Apjohn to have been the consequence of a very con- siderable fall of rain, which took place in the in- terval between the periods at which the specimens were taken up for analysis. When the means of the preceding analyses are taken, we obtain the following results : City Water Course. Portobello Basin. Mean amount of saline matter . 8-598 14-094 organic matter . 1-456 2*798 326 THE DISCHARGE OF WATER FROM SOURCES AND GATHERING GROUNDS. The sources from which a water supply for towns may be derived are lakes, rivers, and streams, springs, wells, and gathering grounds. Of the latter it may be said that, however ably put forward under the auspices of the Board of Health, it is far safer to resort to good river waters than trust to what has been termed, with some satirical truth, "new fangled schemes of pot-piped gathering grounds." Springs and wells afford, at best, but a partial sup- ply unless for villages or manufactories; and we must almost always trust to lakes, rivers, or streams, with sometimes reservoirs, for stowage, for a suffi- cient supply for large towns. The Grot on aqueduct, conveying water with an average of three degrees of hardness, to New York, is perhaps the noblest work for water supply of modern times. The length of the aqueduct is about 44 miles, with a channel in- clination of about 15 inches per mile. The receiving reservoir is about two miles higher up the channel than the distributing reservoir, which latter is 115 feet over the level of the sea, and commands the highest buildings of the city. In the driest weather the supply is equal to 28,000,000 gallons.* The cost of the work, including the purchase of land and water rights, was 8,575,000 dollars, or 8 per lineal foot nearly. The cost of distributing pipes was 1,800,000 dollars. Latterly we have had the Loch Katrine and Glasgow aqueduct, also a noble work, * Schrainke's Croton Aqueduct, New York. ORIFICES, WEIRS, PIPES, AND RIVERS. 327 constructed after this model by Mr. Bateman, not- withstanding the previous supply of that city, or a portion of it, the G-orbals, from gathering grounds at a high level. It is, however, sometimes necessary to make use of such grounds, particularly when TABLE showing the Quantities of Gathering Ground and Reservoir Room to supply a given population with 15, 30, and 40 gallons of water per head per diem. The reservoir room is calculated to hold 12 inches in depth of rain-fall per mile as a guide for lesser depths. For 4 inches the results are to be divided by 3; and for 6 inches by 2. ttd *4 SB S.g-1 1 *|^* S f*J JH si it *i 11 a li Ifftl |P| CJ 1a!l Si 1 Reservoir room per square mile in millions of cu- bic feet of water. Contents of Reservoir in mil- 1 lions of cubic 1 feet. Ashton 59 40-0 15-5 39 21-0 12 Albany Works, U.S 29 1-1 32 Ballinrobe, Ireland 11-0 49-3 28-5 58 Belmont (moorland, mean of four years) 2-81 54-5 39-6 72 26-8 75 Bolton 80 25-6 20 Bute (low country) 45-4 23-9 53 Bateman's Evidence on the drainage area of Long- dendale : First half of 1845, very dry . . 21-2 13-5 64 .. .. Second half of 1845 38-6 27-25 71 First half of 1846 22-5 17-5 78 Oct., Nov., and Dec., 1846 10-2 8-67 85 Bann Reservoir (moorland .. 72- 48-0 66 Drainage areas on south side of Longridge Fell, near Preston, May 1852, i :: 54-' 15-5 18-0 oo.ft 29 33 40 to April 1853 / A/Si U 4bO r DilworthReservoir of Pres ton Works, Lancashire 092 .. .. 54-0 5 6-00 37-0 22-3 60 7-66 46 Grreenock 7-88 60-0 41-0 68 38* 300 Homersham's estimate of 24,000 cubic feet of Re- servoir to each acre oi drainage 1 15-36 15-36 Longdendale 23-8 12'3 292 Proposed Reservoir for Wolverhampton Works 22- .. 7 16 Rivington Pike 16-25 55-5 24'25 44 29-6 481 Sheffield 1-42 36"5 52 Turton and Entwistle . 3-18 46-2 41-0 89 31-43 100 330 THE DISCHARGE OF WATER FROM COST. With reference to cost, the following tables, ar- ranged by us from various sources, will afford in- formation from works executed. The actual cost of all works for house service varies very much in different towns, and with the quantities supplied, from a general average of Id. per house per week, to 2d., and from an annual rate of 9d. in the pound to Is. 6<, and higher. The cost of raising and supplying 1000 gallons from a height of 135 feet in Nottingham, is said to be 3d., and the charge for house service to vary from 5s. to 60s. annually. In Eugby, the average cost per house is 19s. per year, 4&d. per week, or an annual charge of 3s. 3d. per year, or ^d. per week per head of the population, and for a bare supply of 13 gallons. In Croydon, for a supply of only 14 gallons per head, the cost of works varied from lid. to 2%d. per house per week. The parliamentary returns, showing the number of houses supplied, and cost of supply, by different water companies of London, in 1834, give the following results : COMPANIES. Number of Houses. Daily average Supply in Gallons. Height of Supply over Thames. Amount of charge per Company. 73,212 241 145 s. d. 166 Chelsea . 13891 168 135 1 13 3 West Middlesex 16000 185 155 2 16 10 Grand. Junction 11,140 350 152 286 46,421 120 107 129 South London 12,046 100 80 15 Lambeth 16,682 124 185 17 7,100 156 60 113 ORIFICES, WEIRS, PIPES, AND RIVERS. 331 Cost of house apparatus for private supply from street mains, as averaged by the Board of Health, for first-rate houses, is 3 13s. Id. ; second-rate houses, 2 18s. 6d. ; third-rate, 2 3s. 3d. ; fourth- rate and cottages, 17s. 5d. ; average cost for houses and cottages, 2 8s. Id. The actual cost of private works to take water from mains for the supply of cottages is shown in the following table : Work executed in NAME OF PLACE. Mean Expense of Private Works for each Cottage. Annual Value of each Cottage. Jan. 1852 Mar.1852 1852 Aug.1852 Mean valu< Rugby, mean of 6 Cottages s. d. 1 12 11 200 1 18 li 2 11 10 s. d. 5 10 400 326 10 Barnard Castle 11 Tottenham .... 6 .... 3s for each Cottage 209 5 13 1J The water rate charged by the Local Board at Tottenham, is given as follows : In the Special District Rate Assessment. Water Bate per week. Water Rate per annum. Above And not exceeding On Premises assessed. s. d. 8. d. 10 8. d. 8. d. 026 10 15 n 039 15 20 H 050 n 20 25 n 063 tt 25 30 n 080 n 30 40 H 11 40 50 14 and 3s. for every additional rate of 10. 332 THE DISCHARGE OF WATER FROM p ^ O h H * I papuadxg ^ -idsouo pred ^ ^ rH CO O CO CO O O t- i> CSJ O CO O* rH CO O .-H CO 0* '^T 1 : : ; O O O ' ' '. i I t I CO saiqB, S 03 ' : O O O O t> O O O i-H rH rH CO Oi r^ CO O CO co CO O ^ 898UOH 1 rH CO 00* ssBio qs ^ CO 00 rH I 1 sasnoH o * : 0? CO o* rH o rH rH O o rH rH 89SUOH i>4 ^^ s : S o rH O 2 o? rH rH CO rH i r sasnoH BSBIQ pug ^ <* * "^ CO 00 CO CM o o rH CO O CO rH rH o* rH 53% o : 0? rH CO g "* O : : : : s sraooa 91 Suture^uoo 898HOH - : S rH ; /. O ii : : : : 00 ; SUTUIB^UOO B98UOH ^ oj 00 H? CO O CO CO !!:=:: ^ <* CO sraoog OT Smure^uoo sWnoH r>* O O ^00 CO ^ rH CO o o T-H 9 8 ' I CO 00 i O O -^ c? rH ^ rH rH O a rH CO O O 0? O rH rH 1-1 rH rH Sutura;uoo B98UOH r>4 O O . o O ^ rH rH O ; O O O O 0* O rH rH rH TH rH O 00 1-1 sraooa e CO rH ' ' 00 l-H t- O co Smuro^uoo ns o -s CO fc. CO ' ' rH O m 1 ' j VauxhaU M d o ; lal Charge COMPAQ New River . . East London Southwark & 1 1 ill J I rl 6 M W Average Annt ON | rH (M CO 2 O CO t~ 00 OS ORIFICES, WEIRS, PIPES, AND RIVERS. 333 PUBLIC WORKS OF WATER SUPPLY, PRESTON. Yards. Cost of Pipes. s. d. 44 of 2-in. iron pipes, including valves, fire-plugs, outlet-pipes, and all appurtenances, at Is. 7^. ... 398 1,496 of 3-in. ditto, at 3s. 40 CO *t>COCOCO i 73* *O If Si I H rt 78 rH ST. THOMAS'S, EXETEE. Pipes laid 8 to 7 feet deep. 73 CO CD CO 00 CO ~ O O 00 rH <* *CO " O 00 10 CO * Tfl CO O rH nil 73' t- 00 ^ co co co d 73 10 00 O O rH CO drHrHrHOO* iW 73* .0 co co * oj CO CO d rH 73 <* IO O rH 00 OJ =5 ' t~ IQ CO 0? rH rH |3 .s. j!h 73* O O O t-H t- co ' CO CO d d f 73 O {O 73 00 d rH t> IO co O -^ d d eco co d J> -i os o rfCO ' rH CO CO <*< CO CO 1 1 73 d rH rH CO O d 00 CO aj rHO5tO^ COdd rH if 73'CO CO d ^ CO d 73 COOJCOCOO OO5OS iW 73O O O CO d CO O aSCO OOCOCOdrHO rH ^ * "oOtU5COd rHOO sadid |o ja^eureid gddrHrHiH -tivo inches deep, thirty inches wide at top, twenty-four inches wide at bottom ; the materials used in them shall be . " The side walls in them shall be twelve inches in height, six inches thick, and well at bottom. They shall be covered with . SUBMAINS. " These shall be cut forty-two inches deep, eighteen inches wide &t top, fourteen inches wide at bottom. They shall be car- ried along the low side of the fields, or portions of land to be drained, at a distance from the fences of thirteen feet, and through natural hollows, where necessary. No submain to be allowed to run beyond the length of one hundred and fifty yards, without discharging itself into a covered or open main drain. / x o f* o'S * .SP.fH H P* H * 2 I H ri 1 33 88 15-85 so- 13 10 10 9 275- 4358 1411 324 340 775 2 30 86 15-0 so- 12 10 9 6 264-7 3970 1266 320 350 740 3 27 82 13-7 28- 11 2 8 6 243- 3329 1044 315 340 750 4 24 78 12-3 27-7 9 10 7 5 235- 2890 901-4 312 355 753 5 21 76 11-4 25-9 8 10 6 5 214- 2439 735-7 302 345 732 6 18 70 9-95 23-5 6 10 5 5 199- 1970 561-8 285 336 802 7 15 65 8-54 23-4 5 2 4 4 178-5 1524 442-5 290 360 830 8 12 60 7-29 22- 3 10 3 5 161- 1173 328 280 377 910 9 9 52 5-47 19- 2 12 2 8 134- 733 213-7 290 365 910 10 6 42 3-55 16- 1 12 1 10 114- 404-7 117-0 282 380 930 11 24 84 14-2 30-75 13 10 10 14 342- 4890 1505 307 366 790 12 21 81 13-5 29- 11 10 9 6 297- 4009 1223 301 362 805 13 18 72 10-5 26- 9 10 8 7 285- 2993 975 325 360 875 14 15 69 9-6 25- 7 10 6 14 277- 2659 774 292 362 900 15 12 63 8-0 25- 5 10 4 14 234- 1872 549 294 397 870 16 9 56 6-37 23- 4 3 13 201- 1280 390 305 410 950 17 6 46 4-25 21- 2 8 2 4 167-5 712 212 298 455 900 18 15 72 10-5 29- 11 10 9 6 357- 3748 1210 323 402 805 1 19 12 66 8-75 26-75 8 10 7 6 330- 2887 878 305 405 810 20 9 58 6-8 24-5 5 8 5 255- 1734 541 301 422 910 21 6 48 4-7 23-5 3 2 3 228- 1064 317 299 490 960 22 12 68 9-3 27- 9 2 8 6 359- 3338 1006 302 397 917. 1 23 9 58 6-8 26-25 6 2 5 13 332- 2257 686 904 452 950 24 6 48 4-7 24-5 3 12 3 8 262- 1231 385 313 510 935 25 9 60 7-29 27-3 6 12 6 6 355- 2588 785 303 455 945 1 26 6 50 5-03 24-6 4 6 4 1 307- 1544 450 292 490 930 I 27 6 50 5-03 26- 4 15 4 9 360- 1811 534 295 520 925 1 ORIFICES, WEIRS, PIPES, AND RIVERS. 381 Smeaton derived the following " maxims " from the foregoing experiments. Their truth, independent of any experiment, will be apparent : /. That the virtual or effective head being the same, the effect will be nearly as the quantity of water expended. II. That the eoopense of water being the same, the effect will be nearly as the height of the virtual or effective head. III. That the quantity of water expended being the same, the effect is nearly as the square of the velocity. IV. The aperture being the same, the effect will be nearly as the cube of the velocity of the water. FOR TURBINES OR HORIZONTAL WHEELS, a USeM effect of two-thirds or '67 may be assumed, or 49,500 foot-pounds in a minute for a horse-power, and the efficiency varies from -5 to -8, or less.* Pon- celet's turbine gives an efficiency of *5 to *6. Float- ing wheels -38, impact wheels from -16 to *4, and Barker's mill from -16 to -35. We believe that the efficiency of the turbine has been too often over- estimated, and that the great advantage of this wheel, as a medium of power, is derived from its capability of employment for all falls, whether large or small, without any considerable loss of effect. In Ireland, Mr. Gardner, of Armagh, was amongst the first, if not the first, to apply this wheel to prac- tical purposes ; and Professor Thomson has, in his * In our first edition we gave an efficiency of -821, on the authority of a paper by Dr. Robinson, Armagh, in the Proceed- ings of the Royal Irish Academy, vol. iv., p. 914. On again glancing over this paper, we believe there are mistakes, which vitiate the results there given; first, in the formula for calcu- lating the discharge over the weir, and next, in the formula for finding the effect of the brake. Francis gives an efficiency of -88, p. 3, his book. 382 THE DISCHARGE OF WATER FROM vortex wheels, produced, we believe, the highest effi- ciencies which have yet been obtained in practice. In the experiments on the Ballysillan wheel, higher efficiencies would probably have been attained with a supply pipe of larger diameter. It will be seen from the remarks, at pp. 171 and 172, and the tables, at pp. 152 and 191, that quite apart from bends, &c., a loss of mechanical power always results from the passage through orifices and pipes ; and that it is necessary to take this loss into account, before the head acting on the wheel can be accurately used to determine its effective power. The table, next page, contains the experiments on the Ballysillan turbine. The following remarks on the vortex turbine, read at the meeting of the British Association at Belfast, in 1852, are also by Professor Thomson : "Numberless are the varieties, both of principle and of construction, in the mechanisms by which motive power may be obtained from falls of water. The chief modes of action of the water are, however, reducible to three, as follows : First, the water may act directly by its weight on a part of the mechanism which descends while loaded with water, and ascends while free from load. The most prominent example of the application of this mode is afforded by the ordinary bucket water wheel. Secondly, the water may act by fluid pressure, and drive before it some yielding part of a vessel by which it is confined. This is the mode in which the water acts in the water pressure engine, analogous to the ordinary high-pres- sure steam-engine. Thirdly, the water, having been brought to its place of action subject to the pressure ORIFICES, WEIRS, PIPES, AND RIVERS. 383 the ppo nts time o the xperi at atin f th Remarks ments, VOM a Aq :mo UOO cococbcocbcb cocbcbcbcbcoco luacujjadxg Sujjnp uoijniOAa^ jo aaqoim spunod ui y pioo uo }n3j CO O t-- CO C* CO O rH ^H O rH l>OCOCOrtHr-IOO in COCO7OCO CO'CO CO TtfrJtCOCCCOOO T-IC?COTJv 1 J gallons. This ram worked under a head of 37 feet, discharg- ing in use 31i gallons each minute, and raising 3-85 gallons a height of 195 feet. The largest ram employed by Eytelwein in his ex- periments had the following dimensions Length of the body pipe or injection pipe 43 feet 9 inches. Diameter of ditto . . . .0 feet 2 inches. Contents of air-chamber . . ; 1-94 gallons. Area of tail or escape valve . -^ ,' T ' . 3-74 square inches ; and his experiments led to the following practical formula by D'Aubuisson OKIFICES, WEIKS, PIPES, AND EIVEKS. 397 in which D is the water used per minute in gallons, d the quantity raised in gallons, h the head used, and h' the lift of the quantity d. By a slight re- duction we get dh' = 1-42 D (h - -28 v/AAO for the effect produced, which is reduced nearly one- sixth for practical application, giving the formula dA'=l-2 D (A - -2 v/A"A') for the work done. EXPERIMENTAL RESULTS. HYDEAULIC RAM. umber of strokes minute. Height in feet of Ratio of leights. Gallons of water per minute. ** Ratio D d~ Fall h Elevation h' h 1 h Expended D Raised d Experi- ments. Formula. Ft. Iu. Ft. In. 66 icy o" 26' 4" 2-63 10-65 3-39 97 2-92 04 10 2 32 4 3-18 13-97 383 873 92 3-67 00 9 11 38 8 3-9 12-01 2-622 85 87 4-58 52 8 32 4 4- 8-16 1-687 847 85 4-72 45 8 9 38 8 4-4 10-85 2-09 845 84 5-2 42 7 5 38 8 5-21 9-92 1-5 787 78 6-62 36 6 38 8 6-5 8-89 1-05 754 71 8-62 26 4 6 32 4 7-2 5-23 495 672 67 10-7 31 5 38 7 7-7 8-05 704 667 65 11-54 23 4 1 38 8 9-47 11-11 649 548 56 17-2 17 3 32 2 10-7 10-8 479 473 51 22-6 15 3 3 38 8 11-9 12-34 363 352 45 33-8 14 2 6 38 8 15-5 11-95 22 284 32 54-6 10 \, MIJ 38 8 19-3 9-81 088 181 18 106-6 39 THE DISCHARGE OF WATER FROM Eytelwein recommends, that the length of the body pipe should not be less than three-fourths of the height to which the water is to be raised ; its diameter in inches equal -58 V/D"; the diameter of the rising pipe -3 \/:D ; and the contents of the air- chamber equal to that of the rising pipe. The following table gives the result of experiments made by Montgolfier and his son : TABLE OF EXPEEIMENTAL KESULTS HYDEAULIC HAM. Height. Water per Minute. dh Dh Mean Batio dlif Dh Fall h Elevation h Expended D Delivered d Ft. In. Ft. In. Gallons. Gallons. 8' 6" 52' 8" 15 1-37 57 , . 37 2 195 31 3-85 653 34 9 111 11 18-5 3-74 651 65 3 3 14 11 437 59-18 629 .. 22 10 196 10 2-86 0-22 671 '" Latterly, the Messrs. Easton and Amos have patented improvements in this machine, and have raised water to a height of 330 feet. The injection pipe is laid by them at an inclination of about one in four for high falls, and varies down to one in eighteen for smaller falls. The quantities raised in their practice vary up to six gallons per minute. WATER PRESSURE ENGINES give a useful effect varying up to 70 per cent, for the best constructed. An immense amount of mechanical skill and inven- tion has been brought to bear on their construction, ORIFICES, WEIRS, PIPES, AND RIVERS. 399 and in Weisbach's book* a useful effect of 83 per cent, has been calculated ; this is, however, a result seldom obtained in practice, where two-thirds, or 66 per cent., is nearer to the general efficiency. Jordan got a maximum efficiency of '66 from one of the Clausthal engines, making four strokes per minute, and -71 making three strokes per minute. These re- sults were for the combined engine and pumps, from which it was calculated that the efficiency of the engine alone, was in the first case -83, and in the second -85. It would be a great mistake to calculate on such high efficiencies. CORN MILLS will grind about a bushel of corn per horse-power per hour, but much depends on the state of the stones and of the grain. The value of the work done in an hour being once known, the value of the standard horse-power can be determined accordingly. * Vol. ii., p. 342. 400 THE DISCHARGE OF WATER FROM TABLE I. Coefficients of Discharge from Square and differently proportioned Rectangular Lateral Orifices in thin Vertical Plates, arranged from Poncelet and Lesbros. , Square orifice Rectangular Rectangular 8" X 8". orifice 8" X 4". orifice 8" X 2", J" p " *i ( Ratio of the sides Ratio of the sides Ratio of the sides . _-- S Itol. 2tol. 4 to 1. |!|i iff |J Jfj Li Sjj 1 | |l ^ 1 | ill 1 1 IM! i ill 'rt i ~ J "o fi If w a 11 w s o-ooo 619 667 713 0-197 025 597 630 668 0-394 050 595 618 607 642 0-591 075 594 593 615 612 639 0-787 100 572 594 596 614 615 638 1-181 150 578 593 600 613 -620 637 1-575 200 582 593* 603 612 623 636 1969 250 585 593 605 612* 625 636 2-362 300 587 594 607 613 627 635 2-756 350 588 594 609 613 628 635 3-150 400 589 594 610 613 629 635 3-545 450 591 595 610 614 629 634 3-937 500 592 595 611 614 630 634 4-724 600 593 596 612 614 630 633 5-512 700 595 597 613 614 630 632 6-299 800 596 597 614 615 631* 631 7-087 900 597 598 615 615 630 631 7-874 1-000 598 599 615 615 630 630 9-843 1-250 599 600 616 616 630 630 11-811 1-500 600 601 616 - 616 629 629 15-748 2-000 602 602 617 617 628 629 19-685 2-500 603 603 617* 617* 628 628 23-622 3-000 604 604 617 617 627 627 27-560 3-500 604 604 616 616 627 627 31-497 4-000 605 605 616 616 627 627 35-434 4-500 605* 605* 615 615 526 626 39-371 5-000 605 605 615 615 626 626 43-307 5-500 604 604 614 614 625 625 47-245 6-000 604 604 614 614 624 624 51-182 6-500 603 603 613 613 622 622 55-119 7-000 603 603 612 612 621 621 59-056 7-500 602 602 611 611 620 620 62-993 8-000 602 602 611 611 618 618 66-930 8-500 602 602 610 610 617 617 70-867 9-000 601 601 609 609 615 615 74-805 9-500 601 601 608 608 614 614 78-742 10-000 601 601 607 607 613 614 118-112 15-000 601 601 603 603 606 606 See pages 71, 72, and 73. ORIFICES, WEIRS, PIPES, AND RIVERS. 401 TABLE J. -Coefficients of Discharge from Square and differently proportioned Rectangular Lateral Orifices in thin Vertical Plates t arranged from Poncelet and Lesbros. Rectangular Rectangular Rectangular o ,0 orifice 8" Xl-18". orifice 8" X 0-8". orifice 8"X 0-4". 3 j *o Ratio of the sides Ratio of the sides Ratio of the sides r-j 03 60 7 to 1 nearly. 10 to 1. 20 to 1. J^ J S-I W . si If |l 3s 1 sl Has a 1 1 Jll ll'!l li 1 %'% 1 1 Sell IF -I] W Heads 1 back fro orifk ~m~a o -H ^ *o g *>J 5 *! 6 '6^ 2-78 2-71 2-66 2-39 2-27 2-22 0^ 3-48 3-38 3-32 2-99 2-83 278 0^-g. 6-95 6-77 6-64 5-98 5-66 5-56 Oi 9-829 957 9-40 8-45 8-01 7-86 -JL 12038 11-72 11-51 1035 9-81 9-63 Oi 13-900 13-54 1329 11 95 11-33 o o-j? 15541 15-14 14-86 13-36 12-67 12-43 Of 17024 16-58 16-27 1464 1387 13-62 S of 18-388 19-658 17-91 19 15 17-58 1879 15-81 16-91 14-99 16-02 14-71 1573 T 9 20850 20-31 19-93 1793 1699 16-68 Of 21-978 21-41 21-01 18-90 17-91 17-58 o ott 23-051 22-45 22-04 19-82 18-79 1844 o of 24-076 23-45 23-02 20-70 19-62 1926 o olf 25-059 24-41 24-00 21-55 20-4-2 20-05 Oi 26-005 25-33 24-86 22-36 21-19 2080 0^-f 26-917 26-22 25-73 23-15 21-94 21-53 1 27-800 27-08 26-58 23-91 22-66 22-24 1 29-486 28'72 28-19 25-36 24-03 23-59 o u 31-081 30-27 29-71 26-73 2533 24-87 l| 32-598 3175 31-16 28-03 26-57 26-08 1| 34-048 33-19 32-58 29-30 27-75 2726 l| 35-438 3452 33-88 30-48 28-88 28-35 1| 36-776 3582 35 16 31-63 29-97 29-4-2 13 38-067 3708 3639 32-74 31 02 30-45 2 8 39-315 38-29 37-59 33-81 32-04 31-45 2i 40525 39-47 38-74 34-85 33-03 3242 2| 41-700 40*62 3987 35-86 33-99 33-36 2s 42-843 4173 40-96 36-84 34-92 3427 2| 43-956 42-81 42-02 37-80 35-82 3516 2| 45-041 43-87 43-06 38-74 36-71 3603 2f 46-101 44-90 44-07 39-65 3757 36-88 2 47-137 45-90 45-06 40-54 38-42 37-71 3 48-151 46-90 46-03 41-41 39-24 38-52 3$. 49-144 47-87 4698 42-26 40-05 39-32 3| 50-117 4881 47-91 43-10 40-85 40-09 3f 51072 49-74 4882 4392 4162 4086 3| 52-009 50-66 49-72 44-73 4239 41 61 3| 52-930 5155 5060 45-52 43 14 42-34 3f 53-834 52-43 51-47 46-30 43-88 43-07 3| 54725 53-30 52-3-2 47-06 44-60 4378 ORIFICES, WEIRS, PIPES, AND RIVERS. 403 TABLE II. For finding the Velocities from the Altitudes, and the Altitudes from the Velocities. Altitudes feet T g- inch to feet 3| inches. Coefficients of velocity, and the corresponding velocities of 1 discharge in inches per second. 1 Hi t*g *sS "o^ *ag **.$ ,M ^ ^* ffi r * fc ^P re " f/3 -* so r* ^ o> 10 ** 2 ill S3 a QO .2 fS li?l J oo "a g "S 5 S 9 to "S c5 "3 J ?||'.a ^ 85 S -1 a 05 .2 f> cq jS gal 1 e ^ Q -'ij -i| 1 1 ii| -i| 6// 4 55-600 54-15 53-15 47-82 45-31 44-48 4ft 56-462 54-99 53-98 48-56 46-02 45-17 4j 57-311 55-82 54-79 49-29 46-71 45-85 4f 58-148 56-64 55-59 50-01 47-39 46-52 4J 58-973 57-44 56-38 50-72 48-06 47-18 4f 59-786 58-23 57-16 51-42 48-73 47-83 4f 60-589 59-01 57-92 52-11 49-38 48-47 4|- 61-368 59-77 58-67 52-78 50-02 49-09 5 62-163 60-55 69*43 53-46 50-66 49-73 5J 62-935 61-30 60-17 54-12 51-29 50-35 51 63-698 62-04 60-90 54-78 51-91 50-96 5| 64-452 62-78 61-62 55-43 52-53 51-56 5 65-197 63-50 62-33 56-07 53-14 52-16 5f 65-933 64-22 63-03 56-70 53-74 52-75 5f 66-662 64-93 63-73 57-33 54-33 53-33 5 67-383 65-63 64-42 57-95 54-92 53-91 6 68-096 66-33 65-10 58-56 55-50 54-48 61 69-500 67-69 66-44 59-77 56-64 55-60 6| 70-876 69-03 67-76 60-95 57-24 56-70 6f 72-227 70-35 69-05 62-11 58-86 57-78 7 73-552 71-64 70-32 63-25 59-95 58-84 7i 74-854 72-91 71-56 64-37 61-01 59-88 7$ 76-133 74-15 72-78 65-47 62-05 60-91 7| 77-392 75-38 73-99 66-56 63-07 61-91 0> 8 78-630 76-59 75-17 67-62 64-08 62-90 8J 79-849 77-77 76-34 68-67 65-08 63-88 8 81-050 78-94 77-48 69-70 66-06 64-84 8| 82-234 80-10 78-62 70-72 67-02 65-79 9 83-40 81-23 79-73 71-72 67-97 66-72 9 84-550 82-35 80-83 72-71 68-91 67-64 9J 85*685 83-46 81-92 73-69 69-83 68-55 Of 86-805 84-55 82-99 74-65 70-75 69-44 10 87-911 85-63 84-04 75-60 71-65 70-33 10i 89-004 86-69 85-09 76-54 72-54 71-20 10 90-082 87-74 86-12 77-47 73-42 72-07 lOf 91-148 88-79 87-14 78-39 74-29 72-92 11 92-202 89-80 88-15 79-29 75-14 73-76 111 93-244 90-82 89-14 80-19 75-99 74-59 Hi 94-274 91-82 90-13 81-08 76-83 75-42 11| 95-294 92-82 91.10 81-95 77-66 76-23 1 96-302 93-80 92-06 82-82 78-49 77-04 ORIFICES, WEIRS, PIPES, AND RIVERS. 405 TABLE II. For finding the Velocities from the Altitudes, and the Altitudes from the Velocities. Altitudes feet 4 inches to 1 foot. Coefficients of velocity, and the corresponding velocities of ti discharge in inches per second. ' | S~ w .g **| H* ** *ftg J J 1 S 13 Hi |I| Hj ||j I1J jl -1*1 ij OS J SfjJ rfi| **j 1 38-92 37-03 34-92 34-31 33-69 32-47 6 I 39-52 37-60 35-46 34-84 34-22 32-97 4* 40-12 38-17 35-99 35-36 34-73 33-47 4J- 40-70 38-73 36-52 35-88 35-24 33-96 4f 41-28 39-28 37-03 36-39 35-74 34-44 4 41-85 39-82 37-55 36-89 36-23 34-92 4f 42-41 40-35 38-05 37-38 36-72 35-38 4| 42-96 40-87 38-54 37-86 37-19 35-84 4| 43-51 41-40 39-04 38-35 37-67 36-30 5 44-05 41-91 39-52 38-83 38-14 36-75 6k 44-59 42-42 40-00 39-30 38-60 37-20 5i 45-12 42-92 40-48 39-77 39-06 37-64 51 45-64 43-42 40-94 40-23 39-51 38-07 5| 46-15 43-91 41-41 40-68 39-96 38-51 5f 46-66 44-40 41-86 41-13 40-40 38-93 5| 47-17 44-88 42-32 41-58 40-83 39-35 5 47-67 45-35 42-76 42-02 41-27 39-77 6 48-65 46-29 43-65 42-88 42-12 40-59 6J 49-61 47-20 44-51 43-73 42-95 41-39 6 50-56 48-10 45-36 44-56 43-77 42-18 6| 51-49 48-99 46-19 45-38 44-57 42-95 7 52-40 49-85 47-01 46-18 45-36 43-71 7J ^3-29 50-70 47-81 46-97 46-14 44-46 7f 54-17 51-54 48-60 47-75 46-90 45-20 7f 55-04 52-37 49-38 48-51 47.65 45-92 8 55-89 53-18 50-15 49-27 48-39 46-63 8J 56-74 53-98 50-90 50-01 49-12 47-33 8| 57-56 54-77 51-64 50-74 49-83 48-02 8f 58-38 55-54 52-38 51-46 50-54 48-71 9 59-19 56-31 53-10 52-17 51-24 49-38 9| 59-98 57-07 53-81 52-87 51-93 50-04 9| 60-76 57-81 54-51 53-56 52-60 50-69 9| 61-54 58-55 55-22 54-24 53-27 51-34 10 62-30 59-28 55-89 54-92 53-94 51-98 10 63-06 60-00 56-57 55-58 54-59 52-61 10J 63-80 60-70 57-24 56-24 55-24 53-23 10f 64-54 61-41 57-90 56-89 55-87 53-85 11 65-27 62-10 58-56 57-53 56-51 54-45 Hi 65-99 62-79 59-70 58-17 57-13 55-06 HJ 66-71 63-47 59-84 58-80 57-75 55-65 llf 67-41 64-14 60-48 59-42 58-36 56-24 1 406 THE DISCHABGE OF WATEK FEOM TABLE II. For finding the Velocities from the Altitudes, and the Altitudes from the Velocities. Altitudes 1 foot O inch to 5 feet 3 inches. *2 Coefficients of velocity, and the corresponding velocities of 1 discharge in inches per second. i| *8 *" "i 8 r^ Is * -U |.s? !~;i ,5 ^ .3 "3 - H a " 5 * *? g i H 9 a k | -3 ^ ^i ^ i > S3 - 8 -3 > * 1 52 * ei & J rfij 4>| >! rfV| 0} 98-288 95-73 93-96 84-53 80-10 78-63 1 100-234 97-63 95-82 86-20 81-69 80-19 1 102-144 99-49 97-65 87-84 83-25 81-71 2 104-018 101-31 99-44 89-46 8477 83-21 2} 105-859 103-11 101-20 91-04 86-28 8469 3 107-669 10487 102-93 9-260 87-75 86-14 3} 109 449 10660 104-63 94-13 89-20 87-56 4 111-200 108-31 106-31 95-63 90-63 88-96 4} 112-924 10999 107 96 97 11 9203 90-34 5 114-6-22 111-42 109-58 9858 93-42 91-70 5} 116-296 113-27 111-18 100-01 94-78 93-04 6 117-945 114-78 11276 10143 96-13 94-36 7 1-21177 118-03 115-85 10421 98-76 96-94 8 124 325 121 -09 11886 106-92 101-33 99-46 9 127-896 124-08 121-79 109-56 103-83 101-92 10 130-394 127-00 12466 112 14 106-27 10431 11 183-324 129-86 .127-46 114-66 108-66 106-66 2 136-192 132-65 130-20 117-12 111-00 108-95 2 H 140-383 13673 13421 120-73 11441 112-31 2 3 144-453 140-70 138-10 1-24-23 117-73 115-56 2 4} 148-411 144-55 141-88 127-64 120-96 118-73 2 6 152-267 148-31 14557 130-95 124-10 12181 2 7} 156027 151-97 149-16 134-18 127-16 124-82 2 9 159-699 155-55 152-67 137-34 130-15 127-76 2 10} 163-288 159-04 156-10 140-43 133-80 130-63 3 166-800 162-46 159-46 143-45 13594 133-44 3 1} 170-240 165-81 162-75 14641 138-75 136-19 3 3 173-611 169-10 165-97 149*31 141-49 13889 3 4} 176-918 172-32 169-13 152-15 144-19 141-53 3 6 180-165 175-48 172-24 154-94 146-83 144-13 3 7} 183-354 178-59 175-29 157-68 149-43 146-68 3 9 186-488 181-64 178-28 1' 0'38 151-99 149-19 3 10} 189-571 184-64 181-23 163-03 154-50 151-66 4 192604 187-60 184-13 165-64 156-97 154-08 4 2 196576 191-46 187-93 169-06 16021 157-26 4 4 200-469 195-26 191-65 172-40 16338 160-37 4 6 204287 198-98 195-30 17569 16I5-49 163-43 4 8 208-036 202-63 198-88 178-91 169-55 166-43 4 10 211-718 206-21 202-40 182-08 172-55 16937 5 215338 209-74 205-86 185-19 17550 172-27 5 3 220-656 214-92 210-95 189-76 179-83 176-52 OEIFICES, WEIES, PIPES, AND EIVEES. 407 TABLE II. For finding the Velocities from the Altitudes, and tlie Altitudes from the Velocities. Altitudes 1 foot OJ inch to 5 feet 3 inches. Coefficients of velocity, and the corresponding velocities of discharge in inches per second. 9 s *j o^i **g oa ** S M 1*1 s o to .^ 1 is ! 2 ^ QJ CO " |S1 H ? s fj > 7 i 3 * > 5 -3 s| s| II fi *J1 ~- ^ 3 ^< ^ I *!* |j 5iS Jij P S | 10 ?> J CJ Coefficient of velocity, and the corresponding velocities of 1 S*" discharge in inches per second. 2| **ij **f **S "* CO "3^5; 00 **| JI l^'rt "I Is! 1 -*> 3s J9 j I > * 1 '* Ill u I III i 1 . W jj Jijt ci s> 8 *M II 1 . 6 3-50 036 662 681 038 679 700 3-25 042 663 686 044 681 705 3-0 049 666 692 052 684 711 275 059 669 699 062 687 720 2-50 073 673 709 077 692 731 2-25 091 679 723 096 698 745 2-0 118 687 742 125 707 766 1-95 125 689 747 132 709 771 1-90 133 692 752 140 712 -777 1-85 141 694 758 149 715 783 1-80 150 697 764 159 718 790 1-75 160 700 771 170 721 797 1-70 172 704 779 182 725 805 1-65 184 707 786 195 729 814 1-60 198 711 795 210 733 823 1-55 213 716 805 227 738 833 50 231 721 816 '246 744 846 45 251 727 828 268 751 859 40 275 734 842 293 758 874 35 302 742 858 322 764 888 30 333 751 876 356 776 911 1-25 371 761 896 398 788 934 1-20 415 773 920 446 802 961 1-15 469 788 949 506 818 992 1-10 537 806 983 580 838 1-030 1-05 621 828 T024 675 863 1-076 i-oo 732 855 1-074 800 894 1-133 See the auxiliary table, p. 136. ORIFICES, WEIES, PIPES, AND RIVERS. 423 TABLE V. Coefficients of Discharge for different Ratios of the Channel to the Orifice. Coefficients for heads in still water N /~ 7 5~ = -7071 and 1. Coefficient '7071 for heads in still Coefficient 1-000 for heads in still water. water. Ratio of the Ratio of Coefficients Coefficient Ratio of Coefficients Coefficients channel the height for orifices for weirs : the height for orifices for weirs : to the due to the the heads the heads due to the the heads the heads orifice. velocity of measured measured velocity of measured measured approach to the the full approach to the the full to the head centres. depth. to the, head centres. depth. 30- 001 707 708 001 1-001 1-002 20- 001 708 708 003 1-001 1-004 15- 001 708 709 005 1-002 1-006 lo- 005 709 712 010 1-005 1-014 o- 006 709 713 013 1-006 1-017 s' 008 710 714 016 1-008 1-021 7- 010 711 717 021 1-010 1-028 6- 014 712 721 029 1-014 1-038 5-5 017 713 723 034 1017 1-045 5-0 020 714 727 041 1-021 1-055 4-5 025 716 731 052 1-026 1-067 4-0 032 718 737 067 1-033 1-084 3-75 037 720 742 077 1-038 1-096 3-50 043 722 747 089 1-044 rno 3-25 050 724 753 105 1-051 1-127 300 059 728 760 125 1-061 1-149 2-75 071 732 770 152 1-073 1-178 2-50 087 737 783 190 1-091 1-216 2-25 110 745 801 246 1-116 1-269 200 143 756 826 333 1-155 1-347 T95 151 759 832 356 1-165 1-367 1-90 161 762 839 383 1-176 T389 T85 171 765 846 412 1-188 1-413 1-80 182 769 854 446 1-203 1-441 1-75 195 773 863 484 1-218 1-471 1-70 209 778 873 529 1-237 1-505 1-65 225 783 883 579 1-257 1-543 1-60 243 788 895 641 1-281 1-589 1-55 263 795 908 711 1-308 1-638 1-50 286 802 923 800 1-342 1-699 1-45 312 810 939 903 1-379 1-767 1-40 342 819 958 1-042 1-429 1-854 1-35 378 830 980 1 216 1-489 1-958 1-30 421 842 1'003 1-449 1-565 2-088 T25 471 857 1-033 1-778 1-667 2-259 1-20 532 875 1-066 2-273 T810 2-499 1-15 608 897 1-107 3-100 2-025 2-844 1-10 704 923 1-155 4-762 2-400 3-440 1-05 830 957 1-216 9756 3-280 4-803 1-00 '1-000 1-000 1-293 infinite. infinite. infinite. See the auxiliary table, p. 136, also p. 138, 424 THE DISCHARGE OF WATER FROM TABLE VI. The Discharge over Weirs or Notches of one foot in length, in Cubic feet per minute. Depths I inch to 10 inches. Coefficients -667 to -617. GREATER COEFFICIENTS. The Formulce at the heads of the Columns give the Value of the Discharge, D, in Cubic feet per minute, when I, the length of the Weir, is taken feet, and the head, h, in inches. For I \fh* we may substitute I h retaining tJie same standards.. Heads in inches. Theoretical discharge, D = 7-72 l\/h3. Coefficient 667, D = 5-15 l\fh*. Coefficient 650. D = 5-02 J\A3. Coefficient 639. D = 4-93 l\/hf. Coefficient 628. D = 4-85 Z\A 3 - Coefficient 617. D = 4-76 l\/h*. 25 965 644 627 617 606 596 5 2730 1-821 1-775 1-744 1-714 1-684 75 5-016 3-345 3-260 3-205 3-150 3-095 1- 7-722 5-151 5-019 4-934 4-849 4-764 1-25 10-792 7-198 7-015 6-896 6*777 6-659 1-5 14186 9-462 9221 9-065 8-909 8-753 1-75 17-877 11-924 11-620 11-423 11-227 11-030 2- 21-842 14-569 14-197 13957 13-717 13-477 2-25 26-062 17-383 16-940 16-654 16-367 16-080 2-5 30-524 20-360 19-841 19-505 19-169 18-833 2-75 35-215 23-489 22-890 22-503 22'115 21-728 3- 40-125 26-763 26-081 25-640 25-199 24-757 3-25 45-244 30-178 29-408 28-911 28-413 27-915 3-5 50563 33-726 32-866 32310 31-754 31-197 3-75 56-077 37-403 36-450 35-833 35-216 34-599 4' 61-777 41-205 40-155 39-476 38-796 38-116 4-25 67-658 45-128 43978 43-233 42-489 41-745 4-5 73-714 49-167 47-914 47-103 46*292 45-482 4-75 79-942 53-321 51-962 51-083 50-203 49324 5' 86-335 57-585 56-118 55-168 54-218 63269 5-25 92-891 61-958 60-379 59-357 58-335 57-314 5-5 99-604 66-436 64-743 63-647 62-551 61-456 5-75 106-472 71-017 69-207 68-036 66-864 65-693 6- 113-491 75698 73-769 72521 71-272 70024 625 120 657 80-478 78-427 77-100 75-772 74-445 6-5 127-969 85355 83-180 81-772 80-365 78-957 6-75 135-422 90-326 88-024 86-535 85-045 83-555 7* 143-015 95-391 9-2-960 91-387 89-813 88-240 7-25 150-744 100-546 97'983 96-325 94-667 93-009 7-5 158-608 105-792 103-095 101-350 99 606 97-861 7-75 166-604 111-125 108-292 106-460 1046-^7 102-795 8' 174-731 116-546 113-575 111-653 109731 107-809 8-25 182-984 122-05] 118-940 116927 114-914 112-901 8'5 191-365 127-640 124-387 122-282 120-177 118-072 875 199-869 133-313 129915 127716 1-25-518 123-319 9- 208-496 139-067 135-52-2 133-229 130935 128-642 9-25 217243 144-901 141-207 138-818 136-428 134-039 9'5 226-111 150-816 146972 144-485 141-997 139-510 9-75 235-093 156-807 152-810 150225 147-639 145-053 10- 244-193 162-877 158-725 156-039 153-353 150-666 See pp. Ill to 127. ORIFICES, WEIRS, PIPES, AND RIVERS. 425 TABLE VI. The Discharge over Weirs or Notches of one foot in length, in Cubic feet per minute. Depths 10-25 inches to 32 inches. Coefficients -667 to -617. GEEATER COEFFICIENTS. The Formula at the heads of the Columns give the Value of the Discharge D, in Cubic feet per minute, when I, the length of the Weir, is taken in feet, and the head, h, in inches. For I \fh? we may substitute I h \/Ji, retaining the same standards. Heads in inches. Theoretical discharge D- 7-72 I \A 3 - Coefficient 667. D = 5-15 l\/lfl. Coefficient 650. D = 5-02 I \/W. Coefficient 639. D = 4-93 I retaining the same standards. Heads in inches. Coefficient 606. D = ^eSZ-v/k 3 - Coefficient 595. D = 4-59 l\/W. Coefficient 584. D = 4-51 1 \/l#. Coefficient 562. D = 4-34 1 -vA. Coefficient 540. D = 4-17 l\/h*. Coefficient 518. D = 4 I */h\ 10-25 153-565 150-777 147-990 142-415 136-840 131-265 10-5 159-217 156-327 153-437 147-657 141-876 136-096 1075 164-937 161-943 158-949 152961 146-974 140-986 11- 170-724 167625 164-526 158-328 152-130 145-933 11-25 176-577 173372 170 167 163-756 157-346 150-936 11-5 182-496 179-183 175-870 169-245 162-620 155-995 11-75 188-479 185-059 181-636 174-794 167-9,52 161-109 12- 194-526 190-995 187-464 180-402 173-340 166-278 12-5 206-810 203-056 199 302 191-794 184-286 176-778 13- 219-342 215-360 211-379 203-415 195-453 187-490 13-5 232-117 227903 223690 215263 206-837 198-410 14- 245-131 240-682 236-232 227333 218-434 209-535 145 258-379 253-689 248-999 239619 230-239 220-859 15- 271-858 266-924 261 989 252-119 242250 232-380 15-5 285-564 280-381 275-197 264.830 254-463 244-096 16- 299-492 294*056 288-6-20 277-747 266875 256001 16-5 313-640 307*947 302-253 290-868 279-481 268-095 17- 328-004 322-050 316-096 304-189 29-2-281 280-373 17-5 342-581 336-362 330-144 317-707 305-270 292-833 18- 357-367 350-880 344-394 381-4-0 318-446 305-472 18-5 372-352 365 594 358-835 345-317 331-799 318-241 19 387-557 380'o22 373-487 359-418 345-348 331-278 19-5 402-956 395-642 388-3-27 373-699 359-070 344-441 20- 418*553 410-959 403-358 388-163 372-968 357-773 20'5 434-343 426-458 418-574 402-806 387-038 371-270 21- 450-334 442-159 433-985 417-636 401-288 384-939 21'5 466-513 458-045 449-577 432-641 415-704 398-768 22- 482-880 474-115 465-350 447-819 430-289 41-2759 22-5 499-436 490-370 481 -304 463-173 445-042 426-910 23- 517-176 506-806 497-437 478-698 459*959 441-219 23'5 533-098 523-421 513-745 494-391 475-038 455 685 24- 550-203 540-215 530-228 510254 490280 470-305 25- 584943 574-326 563*708 542-472 521-237 500-001 26- 620-391 609-130 597 869 575-346 552'824 530-301 27- 656-525 644-608 632-691 608-857 585-023 561-188 28- 693334 680-749 668-164 642-993 61 7-828 592-652 29' 730-806 717-540 734-275 677-744 651-213 624-682 30- 768-932 754-974 711-017 713-102 6^5-187 657-272 31- 807-697 793-036 778-374 749-052 739730 690-407 32- 847-092 831-716 8A6-340 785-587 7b4-835 724-082 See pp. Ill to 127. OKIFICES, WEIES, PIPES, AND EIVEES. 429 TABLE VI. The. Discharge over Weirs or Notches of one foot in length, in Cubic feet per minute. Depths 33 inches to 72 inches. Coefficients -606 to '518. LESSER COEFFICIENTS. The Formula at the heads of the Columns give the Value of the Discharge, D, in Cubic feet per minute, whence I, the length of the Weir, is taken in feet, and the head, h, in inches. For l^/ h 3 we may substitute I h^/ h, retaining the same standards. TT__ J_ Coefficient Coefficient Coefficient Coefficient Coefficient Coefficient Heads 606. 595. 584. 562. 540. 518. in D= D = D = D = D = D = inches. 4-88 I\A*' 4-59 I \/W. 4-51 1 \/h*. 4-34 14/h*. 4-17/ * S * -d M " ri . "S "3, "3 1 *O >F * ^ .M "-" fl '111 'i ','> '^"3 Ml B g 3S > a >" p, ill '3 C 2o m d QQ 11$ P. ?ll Sis, 1 "is & G S lls w a ss '5 * J gal ||I Pi 00 Sf &>J3 III Pi- |&| III 3*1 m ".9 l-i 333 g P 3*3 s ^l S P i 84 75 41 34-24 33-37 81 67-64 68-86 2 1-67 1-51 42 35-07 34-23 82 68-47 69-77 3 2-51 2-27 43 35-91 35-09 83 69-31 7068 4 3-34 3-04 44 36-74 35-95 84 70-14 71-59 5 4-18 3-81 45 37-58 36-82 85 70-98 72-50 6 5-01 4-58 46 38-41 37-69 86 71-81 73-42 7 5-85 5-36 47 3925 38-56 87 72-65 74-33 8 6-68 6-14 48 40-08 39-43 88 73-48 75-24 9 7-52 6-92 49 40-92 40-30 89 74-32 76-16 10 8-35 7-71 50 41-75 41-17 90 75-15 77-08 11 9-19 8-50 51 42-59 42-05 91 75-99 77-99 12 10-02 9-29 52 43-42 42-92 92 76-82 78-91 13 10-86 10-09 53 44-26 43-80 93 77-66 79-83 14 11-69 10-88 54 45-09 44-68 94 78-49 80-75 15 12-53 11-69 55 45-93 45-56 95 79-33 81-67 16 13-36 12-49 56 46-76 46-44 96 80-16 82-59 17 14-20 13-30 57 47-60 47*32 97 81-00 83-51 18 15-03 14-11 58 48-43 48'2t 98 81-83 84-43 19 15-87 14-92 59 49-27 49-09 99 82-67 85-36 20 16-70 15-73 60 50-10 49-98 100 83-50 86-28 21 17-54 16-55 61 50-94 50-87 101 84-34 87-20 22 18-37 17-37 62 51-77 51'76 102 85-17 88-13 23 19-21 18-19 63 52-61 52-65 103 86-01 89-06 24 20-04 19-02 64 53-44 53*54 104 86-84 89-98 25 20-88 19-85 65 54-28 54-43 105 87-68 90-91 26 21-71 20-68 66 55-11 55-33 106 88-51 91-84 27 22-55 21-51 67 55-95 56-22 107 89-35 92-77 28 23-38 22-34 68 56-78 57-12 108 90-18 93-69 29 24-22 23-18 69 57-62 58-02 109 91-02 94-62 30 25-05 24-02 70 58-45 58-91 110 91-85 9555 31 25-89 24-86 71 59-29 59-81 111 92-69 96-49 32 26-72 25-70 72 60-12 60-71 112 93-52 97-42 33 27-56 26-54 73 60-96 61-61 113 94-36 98-35 34 28-39 27-39 74 61-79 62-52 114 95-19 99-28 35 29-23 28-24 75 62-63 63-42 115 96-03 100-21 36 30-06 29-09 76 63-46 64-32 116 96-86 101-15 37 30-90 29-94 77 64-30 65-23 117 97-70 102-08 38 31-73 30-79 78 65-13 66-13 118 98-53 103-02 39 32-57 31-65 79 65-97 67-04 119 99-37 103-95 40 33-40 32-51 80 66-80 67-95 120 100-20 104-89 ORIFICES, WEIRS, PIPES, AND RIVERS. 431 TABLE VIII, For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. Diameters of pipes \ inch to 2 inches. Falls per mile 1 inch to 12 feet. Falls per mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations one in ^inch. J inch. J inch. inch. J inch. F. I. 1 63360 14 24 38 49 57 2 31680 22 37 59 76 90 3 21120 28 48 75 97 1-15 4 15840 34 57 89 1-15 1-36 5 12672 38 65 1-02 1-30 1-55 6 10560 42 72 1-13 45 1-72 7 9051 46 78 1-24 58 1-88 8 7920 50 85 1-33 71 202 9 7040 53 '90 1-43 83 2'16 10 6336 57 96 1-51 94 2-30 11 5760 60 I'Ol 1-60 96 2'42 1 5280 63 1-06 1-68 2'15 2'54 1 3 4224 71 1'20 1-90 2-43 2'88 1 6 3520 79 1'33 2'10 2'69 3-19 1 9 3017 87 1-45 2-29 2'94 3'48 2 2640 93 1'56 2-47 3-16 375 2 3 [Interpolated. 99 T67 2'63 3-37 399 2 6 2112 1-05 1-77 279 3-58 4-24 2 9 Interpolated. 1 11 1-87 294 3'77 4-47 3 1760 1-16 1-96 3'09 3'96 4-69 3 3 [nterpolated. 1-21 2-05 3-23 4-14 4-91 3 6 1508 1-26 2-14 3'37 4-32 5-12 3 9 [nterpolated. 1-31 2-22 3'50 4-48 5'31 4 1320 1-36 2-30 3-63 4-65 5-51 4 6 Interpolated. 45 2-45 3'87 4-96 5'88 5 1056 54 2'6t 4-11 5-27 6-24 5 6 Interpolated. 62 2-75 4-33 5'55 6'58 6 880 71 2-89 4-55 5'83 6-91 6 6 Interpolated. 78 3-02 4'76 6'10 7-22 7 754 1-86 3-15 4-97 6-36 7'54 7 6 Interpolated. 193 3-27 5-16 6'61 7'83 8 660 2-01 3-39 5'35 6-86 8-12 8 6 Interpolated. 2-07 3-51 5-53 7-09 8'40 9 587 2-14 3-62 5-72 7-32 8-68 9 6 Interpolated. 2-20 3'74 5-89 7-55 8'94 10 528 2-28 3-85 6-07 I'll 9-21 10 6 Interpolated 2-33 3-95 6-24 7'99 9-47 11 480 2-40 4-06 6-40 8'20 972 11 6 Interpolated. 2-46 4*16 657 8-41 9-97 12 440 2-52 4-27 6-73 8-62 10-21 See p. 195. 432 THE DISCHAEGE OF WATEE FEOM TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. Diameters of pipes J inch to 2 inches. Falls per mile 13 feet to 5280 feet. Falls per mile in feet, and the hydraulic inclination. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations one in T V inch. J- inch. | inch. inch. \ inch. 13-2 400 2-66 4-50 7-10 9-10 10-78 13-6 Interpolated. 2-71 4-59 7-24 9-27 1098 14-1 375 2-76 4-67 7-37 9-44 11-18 14-6 Interpolated. 2-82 4-76 7-52 9-63 1141 15-1 350 2-87 4-85 7-66 9-82 11-63 15-6 Interpolated. 2-94 4-96 7'83 10-03 11-88 16'2 325 3-00 5-07 7-99 10-24 12-13 17'6 300 3-14 5-30 8'37 10-72 12-70 19'2 275 3-30 5-58 8-80 11-27 13-35 21-1 250 3-48 5-89 9-39 11-90 14-10 23-5 225 3-70 6-26 9-87 12-65 14-99 26-4 200 3'96 670 10-57 13*54 16-04 30-2 175 4-28 7-24 11-42 14-63 17-33 35'2 150 4-68 7-92 12-49 1600 18-96 37-7 140 4-88 8-24 13-00 16-66 19-74 42-2 125 5'21 8-81 13-90 1780 2r09 48- 110 5-62 9-50 14-98 19-19 22-74 52'8 100 5-94 10-05 15-85 20-30 24-06 587 90 6-33 10-69 16-87 21-61 25-60 66' 80 6-78 11-47 18-10 23-17 27'46 75'4 70 7'35 12-42 19-59 25-09 29*73 88' 60 8'05 13-61 21'48 27-51 32-60 105'6 50 8-99 15-19 23-96 30-69 36-37 117-3 45 9'57 16-18 25-53 32-70 38-75 132-0 40 10'28 17-37 27'41 35-11 41-60 150-8 35 iri4 18-84 29-71 38-06 45-10 176- 30 12-23 20-68 32'62 41-78 49-51 212-2 25 13-66 23-09 36-43 4667 55-30 264- 20 15-64 26-44 41-71 53-43 63-30 352' 15 18-61 31-46 4963 63-57 7533 528- 10 23-73 40-11 63-28 81-06 96-05 586-7 9 25-26 42-70 67-37 86-29 102-25 660' 8 27-08 4578 72-22 92-51 109-61 754-3 7 29-29 49-51 78-10 100-04 118-54 880-0 6 32-05 54-15 85-43 109-43 129-66 1056- 5 35-08 60-15 94-89 121-54 144-02 1320- 4 40-40 68-29 107-73 137-99 163-51 1760- 3 47-48 80-25 126-61 162-17 192-16 2640- 2 59-47 100-53 158-59 203-14 240-70 5280- 1 88-13 148-97 235-02 301-04 356-70 See p. 195. ORIFICES, WEIRS, PIPES, AND RIVERS. 433 TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. Diameters of pipes 2J inches to 5 inches. Falls per mile 1 inch to 12 feet.. Falls pei 1 mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations 4 inch. if inch. inch. 1 inch. 1 J in. in- one in O terpolated. F. I. 1 63360 65 73 79 85 96 2 31680 1-02 1-13 1-23 1-33 1-49 3 21120 1-30 1-45 1-58 1-70 1-91 4 15840 1-54 1-71 1-87 2-01 2-26 5 12672 1-76 1-95 2-13 2-29 2-58 6 10560 1-95 2-17 2-36 2-55 2-86 7 9051 2-13 2-37 2-58 2-78 313 8 7920 2-30 2-55 2-78 3-00 3-37 9 7040 2-46 2-73 2-98 3-21 361 10 6336 2-61 2-90 3-16 3-40 3'83 11 5760 2-76 3-06 3-33 3-59 4-04 1 5280 2-89 3-21 3-50 3-77 4'24 1 3 4224 3-28 3-64 3-97 4-27 4-81 1 6 3520 363 4-03 4-39 4-73 5'32 1 9 3017 3-96 4'39 4-79 5-16 5'80 2 2640 4-26 4-73 5-16 5-55 6-25 2 3 Interpolated. 4-55 5-04 5-50 5-92 6-66 2 6 2112 4-83 5'35 5-84 6-29 7-07 2 9 Interpolated. 5-09 5-64 6-15 6-12 7-46 3 1760 5-34 5'92 6-46 6-96 7-83 3 3 Interpolated. 5-58 6-19 6-75 7-27 8-18 3 6 1508 5-82 6-46 7-04 7-59 8-53 3 9 Interpolated. 6-05 6-71 7-31 7-88 8-86 4 1320 6-27 6-95 7-58 8-17 9-19 4 6 Interpolated. 6-69 7-42 8-09 8-71 9-80 5 1056 7-10 7-88 8-59 9-25 10-41 5 6 Interpolated. 7-48 8-30 9-05 9-76 10-97 6 880 7-86 8-72 9-51 10-25 11-53 6 6 Interpolated. 8-22 9-12 9-94 10-71 12-05 7 754 8-57 9-51 10-37 11-17 12-57 7 6 Interpolated. 8-92 9-89 10-78 11-62 13-06 8 660 9-24 10-25 11-18 12-04 13-54 8 6 Interpolated. 9-55 10-60 11-56 12-45 14-01 9 587 9-87 10-95 11-94 12-86 14--17 9 6 Interpolated. 10-18 11-28 12-31 13-26 14-91 10 528 10-48 11-62 1267 13-65 15-36 10 6 Interpolated. 10-77 11-95 13-03 14-03 15-78 11 480 11-00 12-27 1338 14-41 16-21 11 6 Interpolated. 1134 12-58 13-72 14-82 16-64 12 440 11-62 12-89 14-05 15-22 17-07 434 THE DISCHARGE OF WATER FROM TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. Diameters of pipes 2J inches to 5 inches. Falls per mile 13 feet to 5280 feet. Falls per mile in feet, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second; Falls. Inclinations one in g inch. | inch. I inch. 1 inch. 1 in. in- terpolated. F. 13-2 400 12-26 13-60 14-83 15-98 17-98 13-6 Interpolated. 12-49 13-86 15-11 16-28 18-31 14-1 375 12-72 14-11 15-39 16-58 1865 14-6 Interpolated. 12-98 14-39 15-70 16-91 19-02 15-1 350 13-23 14-68 16-00 17-24 19-40 15'6 Interpolated. 13-52 14-99 16-35 17-62 19-81 16-2 325 1380 1531 16-79 17-99 20-23 17-6 300 14-45 16-02 17-48 18-83 21-18 19'2 275 15-19 16-85 1837 19-79 22-26 21'1 250 16-04 17-80 1940 20-91 23-52 23-5 225 1705 18-91 20-62 22-21 24-99 26-4 200 18-25 20-24 22-07 23-78 26-75 30-2 175 19-71 21-87 23-85 25-69 28-90 35-2 150 21-57 23-92 26-09 28-11 31-62 37-7 140 22-45 24-91 27-16 29-26 32-92 42-2 125 23-99 26-62 29-03 31-27 35-18 48- 110 25-87 28-69 31-29 33-71 37-92 62'8 100 27-36 30-35 33-10 35-66 40-11 587 90 29-12 32-31 35-23 37-96 42-69 66' 80 31-23 34-64 37-78 40-70 45-79 75'4 70 33-82 37-51 40-91 44-07 49-58 88-0 60 37-08 41-13 44-86 48-33 54-36 105'6 50 41-37 45-78 50-04 5391 60-65 117-3 45 44-08 4889 53-32 57-44 64-62 132- 40 47-32 52-49 57-25 61-67 69-37 150'8 35 51-30 56-90 62-06 66-86 75-20 176- 30 56-32 62-47 68-13 73-40 82-56 211-2 25 62-90 69-77 76-09 81-97 92-21 264- 20 72-01 79-87 87-11 93-84 105-56 352- 15 85-68 95-05 103-66 111-67 125-61 528- 10 109-26 121-19 132-17 142-39 160-17 586-7 9 116-31 129-01 140-70 151-58 170-50 660- 8 124-68 138-30 150-83 162-49 182-78 754-3 7 134-84 149-57 163-12 17573 197-67 880- 6 147-69 163-60 . 178-42 192-22 21622 1056- 5 163-82 181-71 198-17 213-50 240-15 1320- 4 185-99 206-31 225-00 242-39 27266 1760- 3 218-58 242-46 264-42 284-86 320-43 2640- 2 273-79 303-70 331 22 356-82 401-37 5280- 1 405-74 450-07 490-84 528-79 594-82 ORIFICES, WEIRS, PIPES, AND RIVERS. 435 TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. OPEN DEAINS AND PIPES, Diameters of pipes 6 indies to 13 inches. Falls per mile 1 inch to 12 feet. Falls per mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities iu inches per second. Falls. Inclinations, IT? inch. If- in. in- 2 inches. 2 J inches 3 inches. one in terpolated. F. I. 1 63360 MW 1-15 1-24 1-40 1-55 2 31680 1-66 1-80 1-94 2-19 2-41 3 21120 2-12 2-30 2-48 2-80 3-08 4 15840 2-52 2-73 2-94 334 3-65 5 12672 2-86 3-11 3-35 3-77 4-16 6 10560 3-18 3-45 3-72 4-19 4-62 7 9051 3-47 3-77 4-06 4-58 5-04 8 7920 3-75 4-06 4-38 4-94 5-44 9 7040 4-01 4-34 4-68 5-28 5-81 10 6336 4-25 4-61 4-97 5-60 617 11 5760 4-49 4-86 5-24 5-91 6-51 1 5280 4-71 5-11 551 6-21 6-84 1 3 4224 5-34 5-79 6-24 7-03 775 1 6 3520 5-91 6-41 6-91 7-79 8-58 1 9 3017 6-44 6-99 7-53 8-49 9-35 2 2640 6-94 7-53 8-11 9-14 10-07 2 3 [nterpolated. 7-40 8-03 8'65 9-74 10-74 2 6 2112 7-86 8-52 9-18 10-35 11-40 2 9 Interpolated. 8-28 8-98 9-67 10-90 12-01 3 1760 8-70 9-43 10-16 11-45 12-62 3 3 [nterpolated. 9-09 9-85 10-62 11-97 13-19 3 6 1508 9-48 10-28 11-08 12'48 13-76 3 9 [nterpolated. 9-84 10-67 11-50 12-96 14-29 4 1320 10-21 11-07 11-93 13-44 14-81 4 6 interpolated. 10-89 11-80 12-72 14-34 15-80 5 1056 11-56 12-54 13-51 15-23 16-78 5 6 Interpolated. 12-18 13-21 14-24 1604 17-68 6 880 1280 13-88 14-96 16-86 18-58 6 6 Interpolated. 13-38 14-51 15-64 17-62 19-42 7 754 13-96 15-14 16-32 18-39 20-26 7 6 Interpolated. 14-51 15-73 16-95 19-10 21-05 8 660 15-05 16-32 17-58 19-82 21-84 8 6 Interpolated. 15-56 16-87 18-18 20-49 2-2-58 9 587 16-07 17-4*3 18-78 21-17 23-32 9 6 Interpolated. 16-57 17-97 1936 21-82 24-04 10 528 17-06 18-50 1994 22-47 24-76 10 6 Interpolated, 17-54 19-01 20-49 23-09 25-45 11 480 18-01 1953 21-04 23-72 26-13 11 6 Interpolated. 18-47 20-02 21-57 24-32 26-79 12 440 18-92 20-51 22-11 24-91 27-45 PF2 436 THE DISCHARGE OF WATER FROM TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers, For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. Diameters of pipes 6 inches to 14 inches. Falls per mile 13 feet to 5%8Qfeet. Falls per mile in feet, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Tails. Inclinations, one in 1 J inch. 2 inches. 2 J inches. 3 inches. i^- inches. 13-2 400 19-97 23-34 26-30 28-98 31-44 13-6 20-34 23-77 26-79 29-52 32-03 14-1 375 20-72 24-21 27-28 30-06 32-62 14-6 21-13 24-69 27-83 30-67 33-27 15-1 350 21-55 25-18 28-38 31-27 33-93 15-6 22-01 25-72 28-99 31-94 34-66 16-2 325 22-48 26-27 29-60 32-62 35-39 17-6 300 23-53 27-50 30-99 34-15 37-05 19-2 275 24-74 28-90 32-57 35-89 38-94 21-1 250 26-13 30-53 34-41 37-91 41-14 23-5 225 27-76 32-44 36-56 40-28 43-71 26-4 200 29-72 34-72 39-13 43-12 46-79 30-2 175 32-11 37-52 42-28 46-59 5055 35-2 350 35-13 41-04 46-26 50-97 55-30 37-7 140 36-57 42-73 48-16 53-07 57-58 42-2 125 39-08 45-66 51-46 56-71 61-53 48* 110 42-13 49-23 55-48 61-13 6633 52'8 100 44-57 52-07 58-69 64-67 70-17 587 90 47-43 55-42 62-46 68-83 74-68 66* 80 50-87 59-44 66-99 73-81 80-09 75-4 70 55-08 64-36 72-50 79-92 86-72 88- 60 60-39 70-57 79-53 87-63 95-09 105*6 50 67-38 78-73 88-73 97-77 106-08 117-3 45 71-79 83-88 94-54 104-17 113-03 132' 40 77-07 90-06 101-50 118-84 121-35 150-8 35 83-55 97-63 110-03 121-24 131-55 176- 30 91-72 107-18 120-79 133 10 144-41 211-2 25 102-44 11970 134-90 148-65 161-29 264- 20 117-28 137-03 154-44 170-18 184-65 352- 15 139-56 163-06 183-78 202-50 219-72 528- 10 177-95 207-92 234-33 258-21 280-16 586-7 9 189-43 221-34 249-45 274-87 298-24 660- 8 203-07 237-28 267-42 294-67 31972 754-3 7 219-61 256-61 289-20 318-67 345-77 880- 6 240-22 281-36 316-33 348-57 378-20 1056- 5 266-81 311-75 351-35 387-15 420-07 1320- 4 302-92 353-95 398-91 43955 47693 1760- 3 356-00 415-96 468-80 516-57 560-49 2640- 2 44593 521-04 587-22 647-06 702-08 5280- 1 660-84 772-16 87023 958-91 1040-44 ORIFICES, WEIRS, PIPES, AND RIVERS. 437 TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. Diameters of pipes 14 inches to 22 inches. Falls per mile 1 inch to 12 feet. Falls per mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations, one in 3 \ inches. 4 inches. 4J inches. 5 inches. 5 \ inches. F. I. 1 63360 1-68 1-80 1-91 2-02 2-13 2 31680 2-61 2-81 2-98 3-15 3-32 3 21120 3-34 3-59 3-82 4-03 4-24 4 15840 3-96 4-25 4-52 4-78 5-02 5 12672 4-51 4-84 5-15 5-44 5-72 6 10560 5-01 5-37 5-72 604 6-35 7 9051 5-47 5-87 6-24 6-60 6-94 8 7920 5-90 6-33 6-74 7-12 7-48 9 7040 631 6-77 7-20 7-61 8-00 10 6336 6-70 7-18 7-64 8-08 8-49 11 5760 7-06 7-58 8-06 8-52 8-96 1 5280 7-42 7-96 8-47 895 9-41 1 3 4224 8-41 9-02 9-60 10-14 10-66 1 6 3520 9-31 9-99 10-63 11-23 11-80 1 9 3017 10-15 10-89 11-58 12-24 12-86 2 2640 1093 11-73 12-47 13-18 13-86 2 3 Interpolated. 11-65 12-50 13-30 1405 14-77 2 6 2112 12-37 13-28 14-12 14-93 15-69 2 9 Interpolated. 13-03 13-68 14-88 15-72 16-53 3 1760 13-69 14-69 15-63 16-52 17-36 3 3 Interpolated. 14-31 15-35 16-33 17-26 18-14 3 6 1508 14-92 16-01 17-03 18-00 18-92 3 9 Interpolated. 15-50 16-63 17-69 18-70 19-65 4 1320 16-07 17-25 18-35 1939 20-38 4 6 Interpolated. 17-14 18-39 19-56 20-68 21-73 5 1056 18-21 19-53 20-78 2196 23-08 5 6 Interpolated. 19-18 20-58 21-90 23-14 24-32 6 880 20-16 21-63 23-01 24-32 25-56 6 6 Interpolated. 21-07 22-61 24-05 25-42 26-72 7 754 21-98 23-59 25-09 26-52 27-87 7 6 Interpolated. 22-84 24-50 26-07 27-55 28-96 8 660 23-69 25-42 27-04 28-58 30-04 8 6 Interpolated. 24-50 26-29 27-97 29-55 31-06 9 587 25-31 27-54 28-89 30-53 32-09 9 6 Interpolated. 26-09 27-99 29-78 31-47 3308 10 528 26-87 28-83 30-67 32-41 34-06 10 6 Interpolated. 27-61 29-62 31-52 3331 35-01 11 480 28-35 30-42 32-37 34-20 35-95 11 6 Interpolated. 29-07 31-19 33-18 35-07 36'86 12 440 29-79 31-96 34-00 35-93 37-77 438 THE DISCHARGE OF WATER FROM TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. For a full cylindrical pipe, divide the diameter by 4 to find the hydraulic mean depth. Diameters of pipes 16 inches to 2 feet. Falls per mile 13 feet to SSSOfeet. Falls per mile in feet and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations, one in. 4: inches. 4:3% inches. 5 inches. 5i-r inches. 6 inches. 13-2 400 33-74 35-89 37-93 3987 41-72 13-6 Interpolated. 34-37 36-56 38-64 40-61 42-50 14-1 375 35-00 3723 39-35 41-36 43-28 14-6 Interpolated. 35-70 37-98 40-14 42-19 44-15 15'1 350 36-40 38-73 40-92 43-02 45-01 15-6 Interpolated. 37-19 39-56 41-81 43-94 45-99 10-2 325 37-97 40-40 42-69 44-87 46-96 176 300 39-75 4229 44-69 46-97 49-16 19-2 275 41-78 44-45 46-97 49-38 51-67 21-1 250 4414 46-95 49-62 52-16 54-58 23-5 225 46'90 49-90 52-72 55-42 58-00 26-4 200 50-20 53-41 56-44 59-32 62-08 30-2 175 54-24 57-71 60-98 64-10 67-07 35-2 150 59-34 63-13 66-71 7012 73-37 37-7 140 61-78 65-72 69-45 73-00 76-39 42-2 125 66-02 70-23 74-22 78'Gl 81-64 48- 110 71-17 75-72 80-01 84-10 8800 52-8 100 75-29 80-09 84-64 sa-97 93-10 58-7 90 80 13 85-25 90-08 94*69 99-09 66- 80 85-93 91-42 96-61 101-54 106-26 75-4 70 93-04 98-98 104-60 109-95 115-05 88- 60 102-02 108-54 114-70 120-56 126-16 105-6 50 113-82 121-09 l-27'96 13450 140-74 117-3 45 121'27 129-01 136-34 143-30 14996 132- 40 130-20 138-51 146-38 153-86 161-00 150-8 35 141 14 150-16 158-68 166-79 17453 176- 30 154-95 164-84 174*20 183-10 191-61 211-2 25 173-05 184-10 194-56 204-50 214-UO 264- 20 198-12 210-77 222-73 234-11 244-98 352- 15 235-75 25080 265-04 278-58 291-52 528- 10 300-60 319-80 337-95 355-22 371-71 586-7 9 320-00 34043 359-76 378-14 395-70 660- 8 343-04 35965 385-67 40537 424-20 754-3 7 370-99 394-68 417-08 438-39 458-76 880- 6 405-79 431-70 456-21 47952 501-79 1056- 5 450-71 479-49 506-71 532-60 557-34 1320- 4 511-72 54439 575-30 604-69 63278 1760- 3 601-38 639-78 676-10 71064 743-65 2640- 2 753-29 80139 846 '89 890-16 931-50 5280- 1 1116-35 1187-62 1255-04 1319-17 1380 44 ORIFICES, WEIRS, PIPES, AND RIVERS. 439 TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes t Drains, Streams, and Rivers. The hydraulic mean depth is found for all channels, by dividing the wetted perimeter into the area. Hydraulic mean depths 6 inches to 10 inches. Falls per mile 1 inch to 12 feet. Falls per mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations, one in 6 inches 7 inches. 8 inches 9 inches 10 inches. F. I. 1 63360 2-23 2-41 2-58 2-75 2-90 2 31680 3'47 3-76 4-03 4-28 4-52 3 21120 4-43 4-80 5-15 5-47 5'78 4 15840 5-26 5-69 6-10 6-49 6-85 5 12672 5-98 6-48 6-95 7-39 7-80 6 10560 665 7-20 7-72 8-20 8-66 7 9051 7-26 7-86 8-43 8-96 9-46 8 7920 7-83 8-48 909 9-67 10-21 9 7040 837 9-07 9-72 10-33 10-91 10 6336 8-88 9-63 10-32 10-97 11-58 11 5760 9-37 10-16 10-89 11-57 12-22 1 5280 9-84 1067 11-43 12-15 12-83 1 3 4224 11-16 1209 12'95 13-77 14-54 1 6 3520 12-35 13-38 14'34 15-25 16-10 1 9 3017 13-46 14-58 15'63 16-61 17-54 2 2640 14-50 1571 16'84 17-90 18-90 2 3 Interpolated. 15-45 16-75 18'24 19-08 20-15 2 6 2112 16-42 17-79 19'64 20-26 21-40 2 9 Interpolated. 17-29 18-74 20*37 2134 22-54 3 1760 18-17 19-69 21-10 22-42 23-fi8 3 3 Interpolated. 18-99 20-57 22-05 23-43 24-75 3 6 . 1508 19-80 21-46 23'00 24-44 2581 3 9 Interpolated. 20-56 22-28 23'88 2o'38 26-80 4 1320 21-33 2311 24-77 26-32 27-80 4 6 interpolated. 22-74 2464 26-41 28-07 29-64 5 1056 24-16 26-17 28-05 2981 31-48 5 6 interpolated. 25-45 27-58 29-56 31-42 33-17 6 880 26-75 28-98 31-06 33-02 34-86 6 6 nterpolated. 27-96 3029 32'47 34-51 36-44 7 754 29-17 31-60 3387 36-00 3802 7 6 Interpolated. 3030 32-83 35-19 37-40 39-50 8 660 31-43 34-06 3650 38-80 40-97 8 6 Interpolated. 32-51 35-22 37-75 4012 42-37 9 587 33-58 3639 38'99 41-45 43-77 9 6 nterpolated. 34-61 3750 40'20 4272 45-11 10 528 35-65 38-63 41'40 4400 46-46 10 6 'nterpolated. 36-63 39-69 42'54 45-22 47-75 11 480 37'62 40-76 43 '69 4644 49-03 11 6 nterpolated. 38-57 41-79 4479 47-61 50-27 12 440 39-52 42-82 45-90 48-78 51-51 440 THE DISCHARGE OF WATER FROM TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. The hydraulic mean depth is found for all channels by dividing the wetted perimeter into the area. Hydraulic mean depths 11 inches to 91 inches. Falls per mile 1 inch to 12 feet. Falls per mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations, one in 11 inches. 12 inches. 15 inches. 18 inches. 21 inches. F. I. 1 63360 3-05 3-19 3-57 3-92 425 2 31680 4-75 4-97 5-57 6-12 6-62 3 21120 6-07 6-35 7-12 7-82 8-46 4 15840 7-19 7-53 8-44 9-27 10-03 6 12672 8-19 8-57 9-61 10-55 11-42 6 10560 9 10 9-52 10-67 11-72 12-68 7 9051 9-94 10-39 11-66 12-80 13-85 8 7920 10-72 11-21 12-57 13-81 1494 9 7041 11-46 11-99 13'44 14-76 15-97 10 6336 12-16 12-72 14-27 15-66 16-95 11 5760 12-83 13-42 15-05 16-53 17*88 1 5280 13-48 14-09 15-81 17-36 18-78 1 3 4224 15-27 15-97 17-91 19-67 21-28 1 6 3520 1691 17-68 19-83 21-78 23-56 1 9 3017 18-23 19-27 21-62 23-73 25-68 2 2640 19-85 20-76 23-28 25-63 27-66 2 3 Interpolated. 21-16 22'13 24'82 27-29 2949 2 6 2112 22 -4-8 23-51 26'36 28-95 31-32 2 9 Interpolated. 23-68 24-76 2777 30-49 3299 3 1760 24-88 26-02 29*18 32-04 34-67 3 3 Interpolated. 25-99 27-18 30-47 33-48 36-22 3 6 1508 27-11 28-35 31-77 34-92 37-78 3 9 Interpolated. 28-15 29'45 33-01 36-26 39-23 4 1320 29-20 30-54 34-25 37-60 40-69 4 6 Interpolated. 31-13 32-56 36-52 40-10 43-39 5 1056 33-07 34-59 38-79 42-59 46*09 6 6 Interpolated. 34-85 36-44 40-87 44-88 48-56 6 880 36'62 38-30 42-95 47-16 51-03 6 6 Interpolated. 38-28 40-03 44'90 49-30 5334 7 754 3993 41-76 46"84 51-43 55-65 7 6 Interpolated. 41-48 43-39 46"66 53-43 57-81 8 660 43-04 45-01 50'48 55-42 59-97 8 6 Interpolated. 44-50 46-54 52'20 57-32 62-02 9 587 45-97 48-08 53 "92 59-21 64-06 9 6 Interpolated. 47-39 49-56 55'58 61-03 66-04 10 528 48-80 51-04 57'24 62-85 68-01 10 6 Interpolated. 50-15 52-45 58'83 64-59 69-89 11 480 51-51 53-87 60'41 66-33 71-78 11 6 Interpolated. 52'81 55-23 61'94 6&-01 73-59 12 440 54-11 56-59 63'47 69-68 75-40 OEIFICES, WEIRS, PIPES, AND RIVERS. 441 TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. The hydraulic mean depth is found for all channels by dividing the wetted perimeter into the area. Hydraulic mean depths 24 inches to 4 feet. Falls per mile 1 inch to 12 feet. Falls per mile in feet and inches, and the hydraulic inclinations, " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations, one in 24 inches. 30 inches. 36 inches. 42 inches. 48 inches. F. I. 1 63360 4-54 5-09 5-59 6-04 6-47 2 31680 7-09 7-94 8-71 9-42 10-08 3 21120 9-06 10-15 11-14 12-04 12-89 4 15840 10-73 12-03 13-20 14-27 15-27 5 12672 1-2-22 13-69 15-03 16-25 17-39 6 10560 13-57 15-21 16-69 18-05 19-31 7 9051 14-83 16-61 1823 19-71 21-09 8 79-20 15-99 17-92 19-66 21-27 22-76 9 7041 17-10 19-16 21-02 22-73 24-33 10 6336 18-15 20-33 22-31 2413 25-82 11 5760 19-15 21-45 23-54 25-46 27-24 1 5280 20-11 22-53 24-72 2673 28-61 1 3 4224 22-78 25-53 28-01 3029 32-42 1 6 3520 25-23 28-27 31-02 33-54 35-90 1 9 3017 27-49 3081 3380 36-55 39-12 2 2640 29-62 33-18 36-41 3938 42-14 2 3 Interpolated. 31-57 35-38 38-82 41-98 44-92 2 6 2112 33-53 37-57 41-22 44-58 47-71 2 9 Interpolated. 3532 39-58 43-43 46-96 50-26 3 1760 37-11 41-58 45-63 49-34 5281 3 3 InterDolated. 38-78 43'45 47-68 51-56 55-18 3 6 1508 40-45 45-32 49-73 53*78 57-55 3 9 Interpolated. 4 13-20 42-00 4356 47-07 48-81 51-64 53-56 5585 57-92 5977 61-98 4 6 Interpolated. 46-45 5205 57-11 61 76 66-09 5 1056 4934 5528 60-66 65-60 70-20 5 6 Interpolated. 51-99 58-25 63-91 69-12 73-97 6 880 54-63 61-22 67-17 72-64 7774* 6 6 Interpolated. 57-11 6399 70-21 75-93* 81-25 7 754 59-58 66-76 73-25 7921 84-77 7 6 Interpolated. 6l-9 69-35 76-09* 87-29 88-06 8 660 64-21 71-94 78-94 85-37 91-35 8 6 Interpolated. 66-40 74-40 81-63 88-26 94-47 9 587 68-59 76-85* 84-32 91-19 97-59 9 6 Interpolated. 70-60 79-22 86-92 94'00 100-59 10 528 72-81 81-58 89-52 96-81 103-60 10 6 Interpolated. 74-83 83-84 91-99 99-49 106-47 11 480 76-84* 86-10 94-47 102-17 109-33 11 6 Interpolated. 78-78 88-28 96-86 104-75 112 10 12 440 80-72 90-45 99-25 107-33 114-86 442 THE DISCHAEGE OP WATER FROM TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. The hydraulic mean depth is found for all channels hy dividing the wetted perimeter into the area. Hydraulic mean depths 4 feet 6 inches to 7 feet. Falls per milel inch to 12 feet. Falls per mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or " mean radii," and velocities in inches per second. Falls. Inclinations, one in 54 inches. 50 inches. 66 inches. 72 inches. 34 inches. F. I. 1 63360 6-86 7-24 7-60 7-94 8-58 2 31680 10-70 11-29 11-85 12-38 13-39 3 21120 13-68 14-62 15-14 15-83 17-11 4 15840 16-21 17-10 17-95 18-76 20-28 5 12672 18-46 19-47 20-43 21-35 23-13 6 10560 2050 21-63 22-70 23-72 25-64 7 9051 22-39 23-62 24-79 25-90 28-00 8 7920 24-16 25-48 26-74 27-95 30-21 9 7041 25-83 2724 28-59 29-88 32-30 10 6336 27-41 28-91 30-34 31-71 34-28 11 5760 28-92 30-51 32-01 3346 36-17 1 5280 30-37 32-03 33-62 35-13 37-98 1 3 4224 34-41 36-30 3-iO 39-81 43-04 1 6 3520 38-10 40-19 42-18 44-08 47-65 1 9 3017 41-5-2 43-80 45-97 48-04 51-93 2 2640 44-73 47-18 49-52 51-75 55-94 2 3 interpolated. 47-69 50-30 2-79 55-17 59-64 2 6 2112 50-65 53-42 56-07 58-59 63-34 2 9 interpolated. 53-35 56-28 59-06 61-72 66-72 3 1760 56-06 59-13 62-05 64-85 70-10 3 3 interpolated. 58-57 61-79 64-84 67-76 73-25 3 6 1508 61-09 64-44 67-63 7067 76-40* 3 9 [interpolated. 63-44 6692 70-^3 73-39 79-35 4 1320 6580 69-41 72-84 76-11* 8229 4 6 [interpolated. 70-16 7401 77-67* 81-16 87-74 5 1056 74-52 78-61* 82-50 86-21 93-20 5 6 Interpolated. 78-52* 82-83 86-92 90-84 98-20 6 880 82-5-2 87-05 91-35 95-46 103-20 6 6 Interpolated. 86-25 90-98 95-58 99-78 107-87 7 754 89-99 9492 99-62 104-10 11254 7 6 Interpolated. 93-48 98-61 103-48 108-14 116-91 8 660 96-98 102-30 107-35 112-19 121-28 8 6 Interpolated 100-29 105-79 111-02 116-01 125-42 9 587 103-59 109-27 114-68 119-84 129-56 9 6 Interpolated 10678 112-64 118-21 123-53 133-55 10 528 109-97 116-01 121-74 127-22 137-54 10 6 Interpolated 113-02 119-22 12511 130-74 141-34 11 480 116-06 12-2-43 128-48 134-27 145-15 11 6 Interpolated 119-00 125-52 131-73 137-66 148-82 12 440 121-93 128-61 134-97 141-05 152-49 ORIFICES, WEIKS, PIPES, AND EIVEES. 443 TABLE VIII. For finding the Mean Velocities of Water flowing in Pipes, Drains, Streams, and Rivers. The hydraulic mean depth is found for all channels by dividing the wetted perimeter into the area. Hydraulic mean depths 8 feet to 12 feet. Falls per mile 1 inch to 19 feet. Falls per mile in feet and inches, and the hydraulic inclinations. " Hydraulic mean depths," or "mean radii," and velocities in inches per second. Falls. Inclinations, one in 96 inches. 108 inches. 120 inches. 132 inches. 14.4 inches. F. I. 1 63360 9-18 9-75 10-28 10-79 11-27 2 31680 14-32 15-20 16-03 16-82 17-57 3 21120 18-30 19-43 20-49 21-50 22-46 4 15840 2169 23-02 24-28 25-47 26-62 5 12672 24-70 26-21 27-64 29-00 30-31 6 10560 27-43 29-11 30-70 32-21 33-66 7 9051 29-96 31-80 33-53 35-18 36-76 8 7920 32-32 34-30 36-18 37-96 39-66 9 7041 34-55 36-67 38-67 40-58 42-40 10 6336 36-67 38-92 41-04 43-07 45-00 11 5760 38-69 41-06 43-31 45-44 47-48 1 5280 40-63 43-12 45-48 47-72 4986 1 3 4224 46-04 48-87 51-54 54-07 56-50 1 6 3520 50-98 54-11 57-06 59-87 62-56 1 9 3017 55-60 58-96 62-18 65-25 68-17 2 2640 59-85 63-52 66-98 70-28 73'44* 2 3 Interpolated. 63-80 67-72 71-41 74-93* 78-29 2 6 2112 67-76 71-91. 75-84* 79-58 83-15 2 9 Interpolated. 71-38 75-75* 79-89 83-83 87-59 3 1760 75-00* 7959 83-94 88-08 92-03 3 3 Interpolated. 7837 83-17 87-71 92-03 06-16 3 6 1508 81-74 86-75 91-48 95-99 100-30 3 9 Interpolated. 84-88 90-09 9501 9969 104-16 4 1320 88-03 93-43 98-53 103-38 108-02 4 6 Interpolated. 93-87 99-62 105-06 110-24 115-18 5 1056 99-70 105-82 111-59 117-09 122-34 o 6 Interpolated. 105-06 111-49 11758 123-38 128-91 6 880 110-41 117-17 123-57 129-66 135-48 6 6 Interpolated. 115-40 122-47 129 16 135-53 141-61 7 . 754 12D -40 127-76 134-75 141-39 147-73 7 6 Interpolated. 125-07 132-74 139-99 146-88 153-47 8 660 129-75 137-70 145-22 152-38 15921 8 6 Interpolated. 134-18 142-40 15U-18 157-57 164-64 9 587 138-00 147-10 155-13 162-77 170-07 9 6 Interpolated. 14287 151-63 159-91 167-78 175-31 10 528 147-14 156-16 164-68 172-80 180-55 10 6 Interpolated. 151-21 160-48 169-24 177-58 18555 11 480 155-29 164-80 173-80 182-36 190-54 11 6 Interpolated. 15921 168-97 178-19 186-97 195-38 12 440 163-13 173-13 182-59 191-58 200-17 444 THE DISCHARGE OF WATER FEOM TABLE IX. For finding the Discharge in Cubic Feet per Minute, when the Diameter of a Pipe, or Orifice, and the Velocity of Discharge are known ; and vice versa. sss $ .,3 e 5.s Discharge in cubic feet per minute, for different velocities. Velocity of 100 inches per second. Velocity of 200 inches per second. Velocity of 300 inches per second. Velocity of 400 inches per second. Velocity of 500 inches per second. i 170442 3409 5113 6818 8522 1 68177 1-3635 2-0453 2-7271 3-4089 1 1-53398 3-0679 4-6019 6-1359 7-6699 2-727077 5-4541 8-1812 10-9083 13-6354 it 4-26106 8-5221 12-7832 17-0442 21-3053 1} 6-13593 12-2718 18-4080 24-5437 30-6797 if 8-35167 16-7033 25-0550 33-4067 41-7584 2 10-90831 21-1817 32-7249 43-6332 54-5415 2J 13-80583 27-6117 41-4175 55-2233 69-0291 &t 17-04423 34-0885 51-1327 68-1769 85-2212 2f 2062352 41 2470 61-8706 824941 103-1176 3 24-54369 49-0874 73-6311 98-1748 121-7185 3i 28-80475 57-6095 864143 115-2190 144-0238 3 33-40669 66-8134 100-2201 133-6268 167-0335 3| 38-34952 76-6990 115-0486 153 3981 191-7476 4 43-63323 87-2665 130-8997 174-5329 218-1662 4J 49-25783 98-5157 147 7735 197-0313 2462892 if 5522331 110-4466 165-6699 2208932 276-1166 4| 61*52968 123-0594 184-5890 246-1187 307-6484 5 68-17692 136-3539 204-53U8 2727077 340-8846 6* 75-16506 150-3301 225-4952 300 6603 375-8253 6* 82-49408 1649882 247-4822 329-9763 412-4704 6f 90-16399 180-3280 270-49-20 360-6560 450-8200 6 98-17478 196-3495 294-5243 392-6991 490-8739 6* 106-5-2645 213-0529 319-5794 426-1058 532-6323 6J 115-2190 230-4380 345-6570 460-8760 576-0950 6| 124-25245 248-5049 3727574 497-0098 621-2623 133-6268 267-2536 400-8804 534-5072 668-1340 7* 143-34199 286-6840 430-0260 573 3680 716-7100 7* 153-39809 306-7962 460-1943 613-5924 766-9905 1 163-79507 327-5901 491-3852 655-1803 818-9753 8 174-53293 349-0659 523-5988 698-1317 872-6647 & 197-03132 394-0626 591-0940 788-1253 985-1566 9 22089325 441-7865 6626798 883-5730 1104-4663 9* 246-11871 492-2374 738-3561 984-4748 1230-5936 10 272-70771 545-4154 818-1231 1090 8308 1363-5386 IQi 300-66025 601-3205 901-9808 1202-6410 1503-3013 11 329-97633 6599527 989-9290 13199053 1649-8817 MJ 360-65595 721-3119 1081-9679 1442-6238 1803-2798 12 392-6991 785-3982 11780973 1570-7964 1963-4955 OEIFICES, WEIES, PIPES, AND EIVEES. 445 TABLE IX. For finding the Discharge in Cubic Feet, per Minute, when the Diameter of a Pipe, or Orifice, and the Velocity of dis- charge are known ; and vice versa. Discharge in cubic feet per minute, for different velocities. 5s 1-S1 sXs Velocity of 600 inches per second. Velocity of 700 inches per second. Velocity of 800 inches per second. Velocity of 900 inches per second. Velocity of 1000 inches per second. 1-0227 1-1931 1-3635 1-5340 1-7044 i 4-0906 4-7724 5-4542 6-1359 6-8177 i 9-2039 10-7379 12-2718 13-8058 15-3398 1 16-3625 19-0895 21-8166 24-5437 27-2708 25-5664 29-8274 34-0885 38-3495 42-6106 H 36-8155 429515 49-0874 55-2234 61-3593 *! 50-1100 58-4617 66-8134 75-1650 83-5167 if 65-4499 76-3582 87-2665 98 1748 109-0831 2 828350 96-6408 110-4466 124-2525 138-0583 *! 102-2654 119-3096 136-3538 153-3981 170-4423 2 1237411 144-3646 164-9882 185-6117 206-2352 2f 147-2621 171-8059 196-3496 220-8933 245*4369 3 172-8285 201-6333 230-4380 259-2428 288-0475 31 200-4401 233-8468 267-2535 300-6602 334-0669 3$ 230-0971 268-4467 306-7962 345-1457 383-4952 8| 261-7994 305-4326 349-0659 392-6991 436-3323 4 295-5470 344-8048 394-0626 443-3205 492-5783 J 331-3399 386-5632 441-7865 497-0098 552-2331 4 369-1781 430-7077 492 2374 553-7671 615-2968 4f 409-0615 477-2384 545-4154 6135923 681-7692 5 450-9904 526-1554 601-3205 676-4855 751-6506 fij 494-9645 577-4586 659-9526 742-4467 824-9408 ty 540-9839 631-1479 721-3119 811-4759 901-6399 fif 589-0486 687-2235 7853982 883-5730 981*7478 6 639-1587 745-6852 852-2116 958-7381 1065*2645 *i 691-3141 806-5330 921-7520 10369710 1152-1900 6* 745-5147 869-7672 994-0196 11182721 1242-5245 6f 801-7608 935-3876 1069-0144 1202-6412 1336-2680 7 860-0519 1003-3939 1146-7359 1290-0779 1433-4199 7i 920-3885 1073-7866 1227-1847 1380-5828 1533-9809 7 982-7704 1146-5655 1310 3605 1474 1556 1637-9507 7f 1047-1976 1221-7305 1396-2634 1570-7964 1745-3293 8 1182-1879 1379-2192 1576-2506 1773-2819 1970-3132 8 1325-3595 1546-2528 1767-1460 1988-0393 2208-9325 9 1476-7123 1722-8310 1968-9497 2215-0684 2461-1871 $$ 1636-2463 1908-9540 2181-6617 2454-3694 2727-0771 10 1803-9615 2104-6218 2405-2820 2705-9423 3006 6025 10,} 1979-8580 2309-8343 2639-8106 2969-7870 3299-7633 11 2163-9357 2524-5917 2885 2476 3245-9936 3606-5595 Hi 2356-1946 2748-8937 3141-5928 3534-2919 3926-9910 12 446 THE DISCHARGE OF "WATER FROM TABLE X. For finding the depths of Weirs of different lengths, the quantity discharged over each being supposed constant. See pages 270 and 371. Ratios of lengths. Coeffi. cients. Ratios of lengths. Coeffi- cients. Ratios of lengths. Coeffi- cients. Ratios of lengths. Coeffi- cients. 01 0464 405 5474 605 7153 805 8654 02 0737 410 5519 610 7193 810 8689 03 0965 415 5564 615 7232 815 8725 04 1170 420 5608 620 7271 820 8761 05 1357 425 5653 625 7310 825 8796 06 1533 430 5697 630 7349 830 .8832 07 1699 435 5741 635 7388 835 8867 08 1857 440 5785 640 7427 840 8903 09 2008 445 5829 645 7465 845 8938 10 2154 450 5872 650 7504 850 8973 11 2296 455 5916 655 7542 855 9008 12 2433 460 5959 660 7580 860 9043 13 2566 465 6002 665 7619 865 9078 14 2696 470 6045 670 7657 870 9113 15 2823 475 6088 675 7695 875 9148 16 2947 480 6130 680 7733 880 9183 17 3069 485 6173 685 7771 885 9218 18 3188 490 6215 690 7808 890 9253 19 3305 495 6258 795 7846 895 9287 20 3420 500 6300 700 7884 900 9322 21 3533 505 6342 705 7921 905 9356 22 3644 510 6383 710 7959 910 9391 23 3754 5J5 6425 715 7996 915 9425 24 3862 520 6466 720 8033 920 9459 25 3969 525 6508 725 8070 925 9494 26 4074 530 6549 730 8107 930 9528 27 4177 535 6590 735 8144 935 9562 2.8 4280 540 6631 740 8181 940 9596 29 4381 545 -6672 745 8218 945 9630 30 4481 550 6713 750 8255 950 9664 31 4580 555 6754 755 8291 955 8698 32 4678 500 6794 760 8328 960 9732 33 4775 565 6834 765 8365 965 9762 34 4871 570 6875 770 8401 970 9799 35 4966 575 6915 775 8437 975 9833 36 5061 580 6955 780 8474 980 9866 37 5154 585 6995 785 8510 985 9900 38 5246 590 7035 790 9546 990 9933 39 5338 595 7074 795 8582 9D5 9967 40 5429 600 7114 SCO 8618 1-000 1-0000 ORIFICES, TVEIRS, PIPES, AND RIVERS. 447 TABLE XL Mean relative Dimensions of equally Discharging Trape- zoidal Channels, with Side Slopes varying from to 1, up to % to 1. Half sum of the top and bottom is the mean width. The ratio of the slope, multiplied by the depth, subtracted from the mean width, will give the bottom ; and if added, will give the top. TABLE XII. gives the discharge in cubic feet per minute from the primary channel, 70 wide, and the corresponding depths taken in feet. For lesser or greater channels and discharges, see Eules, pp. 237, 229, 232, 249, and 252. The mean widths are given in the top horizontal line, and the corresponding depths in the other horizontal lines. They may be taken in inches, feet, yards, fathoms, or any other measures whatever. 70 60 50 40 35 30 25 20 15 10 125 13 15 17 20 23 26 29 35 -48 25 27 30 35 40 45 52 58 71 98 375 41 46 54 60 67 76 88 1-09 1-51 5 55 62 73 80 89 1-02 1-19 1-48 2-04 625 68 78 91 1-00 1-12 1-29 150 1-88 2-62 75 82 94 1-10 1-20 1-35 1-56 1-82 2-28 3-22 875 96 MO 1-29 1-41 1-58 1-83 2-14 2-69 3-86 1- 1-10 1-26 1-48 1-62 1-81 2-10 2-46 3-11 4-50 1-125 1-24 1-42 1-67 1-83 2-04 2-37 2-79 3-54 5-19* 1-25 1-39 1'58 1-86 2-04 2-28 2-65 3-12 3-98 5'89 1-375 1-53 1-74 2-05 2-25 2-51 2-92 3-46 4-43 6-60 1-5 1'67 1-90 2-24 2-46 2-75 3-20 3-80 4-88 7-31 1-625 1-81 2-06 2-43 2-67 2-99 3-47 4-15 5-34 8-08 Kfi 1-95 2-22 2-62 2-88 3-23 3-75 4-50 5-80 8-86 1-875 2-09 2-38 2-81 3-09 3-47 4-03 4-86 6-29 9-68 2- 2-23 2-54 3-00 3-31 3-72 4-32 5-22 6-78 10-50 2-125 2-37 2-70 3-19 3-52 3-96 4-61 5-58 7-29 11-37 2-25 2-51 2-86 3-38 3-73 4-21 4-91 5-95 7-81* 12-25 2-375 2-65 3-02 357 3-94 4-45 5-20 6-31 8-32 13-12 2-5 2-79 3-18 3-76 4-16 4-70 5-50 6-68 8-84 14-00 2-625 2-93 3-34 3-95 4-38 4-95 5-79 7-06 9-38 14-92 2-75 3-07 3-51 4-15 4-60 5-21 6-09 7-45 9-93 15-84 2-875 3-21 3-67 4-34 4-82 5-46 6'39 7-83 10-48 16-76 3- 3-35 3-84 4-54 5-04 5-72 6-69 8-22 11-03 17-68 3-125 3-49 4-00 4-73 5'26 5-97 7-00 8-62 11-60 18-68 3-25 3-63 4-17 4-93 5-49 6-23 7-31 9-02 12-17 19-68 3-375 3-77 4-33 5-13 5-72 6-49 7-62 9-42 12-74 20-68 3-5 3<91 4-50 5.33 5-95 6-75 7-93 9-82 1332 21-68 3-625 4-05 4-66 5-53 6-17 7-01 S-25 1023* 13-92 2276 3-75 4-19 4-82 5-73 6-40 7-28 8-57 1065 14-53 23-84 3-875 4-33 4-98 5-93 6-62 7-54 8-89 ii-or. 15-14 24-92 4- 4-48 5-14 6-13 6-85 7-81 9-21 11-48 15-75 26-00 4-25 4-76 5-46 6-54 7-30 8-35 9-85 12-33 16-98 28-18 4-5 5-05 5-79 6-95 7-75 8-90 10-50 1319 18-22 30-36 4-75 533 6-12 7-35 8-20 9-45 11-14 14-07 19-50 32-68 5- 5-62 6-45 7*75 8-66 10-00 11-79 14-96 20-80 35-00 5-25 5-90 6'78 8-16 9-14 10-55 12-51* 15-86 22-13 37-40 5-5 6-18 7-12 8-57 9-62 11-10 13-24 16-77 23-47 39-81 5-75 6-46 7-46 8-98 10-11 11-66 13-94 17-71 24-86 42-33 6- 6 f 75 7-80 9-40 10-60 12-22 14-65 18-65 26-25 44-86 448 THE DISCHARGE OF WATER FROM TABLE XII. Discharges from the Primary Channel in the first column of Table XI. If the dimensions of the primary channel be in inches, divide the discharges in this table by 500 ; if in yards, multiply by 15'6 ; if in quarters, multiply by 32 ; and if in fathoms, by 88-2, &c. : see pp. 233, 234. The final figures in the dis- charges may be rejected when they do not exceed one-half per cent., or 0-5 in 100. See pages 226 to 234. Depths of a channel whose mean width is 70: in feet. Falls, inclinations, and discharges in cubic feet per minute. Interpolate for intermediate falls ; divide greater falls by 4, and double the corresponding discharges. linch per mile, 2 inches per mile, 1 in 31680. 3 inches per mile, 1 in 21120. 6 inches per mile, 1 in 10560. 9 inches per mile, 1 in 7040. 12 inches per mile, 1 in 5280. 15 inches per mile. 1 in 4224. 125 47 72 93 139 175 205 233 25 136 210 268 403 506 596 675 375 249 389 498 746 940 1105 1252 50 387 603 770 1155 1454 1709 1935 625 541 849 1078 1617 2036 2395 2714 75 714 1112 1420 2128 2681 3153 3573 875 900 1401 1791 2685 3382 3978 4507 1- 1100 1714 2190 3283 4134 4862 5507 1-125 1310 2042 2614 3909 4927 5792 6577 1-25 1534 2384 3058 4581 5766 6780 7690 1-375 1767 2757 3521 5279 6661 7823 8863 1-50 2013 3142 4006 6016 7588 8915 10099 1-625 2268 3540 4525 6781 8541 10044 11381 1-75 2534 3950 5053 7570 9537 11210 12703 1-875 2812 4384 5599 8386 10570 12429 14083 2- 3090 4821 6161 9230 11628 13675 15513 2-125 3377 5273 6738 10092 12718 14956 16943 2-25 3674 5736 7331 10981 13833 16281 18435 2-375 3977 6210 7937 11889 14981 17645 19960 2-50 4293 6699 8563 12829 16161 19045 21534 2-625 4616 7203 9204 13800 17380 20434 23135 2-75 4947 7716 9865 14782 18624 21886 24800 2-875 5280 8233 10525 15773 19887 23360 26473 3- 5621 8762 11204 16788 21165 24833 28176 3-125 5972 9310 11900 17b'30 22454 26410 29925 3-25 6329 9862 12614 18897 23780 27994 31714 3-375 6689 10420 13320 19963 25145 29570 33507 3-50 7049 10995 14048 21052 26509 31262 35329 3-625 7418 11574 14785 22153 27906 32860 37186 3-75 7794 12163 15526 23284 29321 34479 39080 3-875 8178 12753 16283 24416 30756 36170 41013 4- 8566 13354 17070 25592 32225 37898 42954 4-25 9355 14582 18643 27936 35191 41368 46916 4-50 10173 15849 20267 30366 38254 44982 50973 4-75 11001 17140 21908 32818 41356 48630 55102 5- 11833 18454 23595 35355 44546 52378 59346 5-25 12696 19802 25362 37939 47795 56209 63688 5-50 13576 21172 27248 40564 51097 60079 68U97 5-75 14478 22580 29160 43253 54478 64058 72591 6- 15393 23995 31122 45969 57897 68082 77154 OftlFICES, WEIRS, PIPES, AND RIVERS. 449 TABLE XII. Discharges from the Primary Channel in the first column of Table XI. If the dimensions of the primary channel be in inches, divide the discharges in this table by 500 ; if in yards, multiply by 15*6, if in quarters, multiply by 32, and if in fathoms, by 88'2 etc. : see pp. 233 and 234. The final figures in the discharges may be rejected when they do not exceed one-half per cent., or 0*5 in 100. See pages 226 to 234. Falls, inclinations, and discharges in cubic feet per minute. Interpolate for intermediate falls; divide greater falls by 4, and double the corresponding discharges. Depths of a channel whose mean width is 0: in feet. 18 inches per mile, 1 in 3520. 21 inches per mile, 1 in 3017. 24 inches per mile, 1 in 2640. 27 inches per mile, 1 in 2347, 30 inches per mile, 1 in 2112. 33 inches per mile, 1 in 1920. 36 inches per mile, 1 in 1760. 258 281 303 323 343 362 380 125 748 815 877 936 993 1049 1100 25 1387 1511 1627 1736 1843 1952 2037 375 2145 2336 2515 2684 2852 3023 3155 50 3004 3274 3527 3753 4021 4207 4414 625 3957 4311 4645 4966 5287 5553 5817 75 4991 5422 5859 6274 6650 6992 7342 875 6097 6622 7159 7631 8107 8540 8974 1- 7266 7920 8531 9124 9660 10200 10693 1-125 8514 9284 9995 10658 11318 11923 12520 1-25 9816 10697 11539 12307 13045 13741 14479 1-375 11182 12185 13152 14007 14862 15656 16448 1-50 12601 13730 14821 15786 16750 17657 18552 1-625 14069 15331 16525 17616 18700 19698 20696 1-75 15593 16997 18306 19517 20728 21840 22944 1-875 17157 18697 20141 21469 22803 24017 25242 2- 18766 20446 22030 23480 24938 26269 27601 2-125 20410 22247 23965 25547 27129 28578 30027 2-25 22104 24087 25947 27662 29395 30934 32512 2-375 23848 25988 27992 29841 31701 33381 35096 2-50 25669 27953 30100 32069 34086 35910 37725 2-625 27479 29933 32247 34384 36512 38471 40415 2-75 29318 31947 34408 36697 38958 41055 43135 2-875 31206 34002 36624 39050 41464 43680 45896 3- 33141 36112 38897 41482 44048 46398 48747 3-125 35126 38266 41223 43954 46672 49174 51664 3-25 37109 40438 43556 46438 49330 51951 54586 3-375 39140 42631 45925 48963 51993 54775 57550 3-50 41184 44872 48343 51537 54728 57659 60580 3-625 43273 47158 50807 54162 57514 60585 63656 3-75 45407 49468 53300 56840 60341 63560 66784 3-875 47551 51818 55832 59514 63200 66576 69951 4- 51911 56586 60973 64974 69013 72694 76383 4-25 56448 61508 66176 70623 75017 79017 82994 4-50 61014 66500 71625 76408 81097 85426 89767 4-75 65713 71628 77140 82250 87351 92015 96653 5- 70509 76863 82779 88200 93731 98729 103745 5-25 75383 82159 88434 94344 100200 105550 110905 5-50 80379 87590 94348 100616 106823 112540 118254 5-75 85407 93093 100275 106911 113505 119616 125664 6- 450 THE DISCHARGE OF WATER FROM TABLE XIII. The Square Roots of the fifth powers of numbers for finding the Diameter of a Pipe, or dimensions of a Channel from the Discharge, or the Reverse; showing the relative Discharging Powers of pipes of different Diameters, and of any similar Channels whatever, closed or open. See pages 81, 230, 233, etc. If d be the diameter of a pipe, in feet, and D the discharge in cubic feet per minute, then for long straight pipes we shall ha,ve for velocities of nearly 3 feet per second, D = 2400 (d 5 s) J , and d = *044( \ ; or if D be the discharge per second, D = 40 (d 5 s)*, /D 2 \i 228 I - ) . See pages 190 to 224, and pages 42 and 43. V s ' and d Relative dimen- sions or Relative discharg- Relative dimen- sions or Relative discharg- Relative dimen- sions or Relative discharg- Relative dimen- sions or Relative discharg- diameters of pipes. ing powers. diameters of pipes. ing powers. diameters of pipes. ing powers. diameters of pipes. ing powers. 25 031 10-5 357-2 30-5 5138- 61- 29062- 5 177 11- 401-3 31- 5351- 02- 30268- 75 485 11-5 448-5 31-5 5569- 63- 315U3- 1- 1- 12- 498-8 32- 5793- 64- 32768- 1-25 1-747 12-5 552-4 32-5 0022- 65- 34063- 1-5 2-756 13- 609-3 33- (>256- 66- 35388- 1-75 4-051 13-5 669-6 33-5 6496- 67- 36744- 2- 5-657 14- 733-4 34- 6741- 68- 38131- 2-25 7-594 14-5 800-6 34-5 6991- 69- 39548- 2-5 9-882 15- 871-4 35- 7247- 70- 40996- 2-75 12-541 15-5 9459 35-5 7509- 71- 42476- 3- 15-588 16- 1024- 30- 7776- 72- 43988- 3-25 19-042 16-5 1105-9 36-5 8049- 73- 45531- 3-5 22-918 17- 1191-6 37- 8:327- 74- 47106- 3-75 27-232 17-5 1281-1 37-5 8611- 75- 48714- 4- 32- 18- 1374-6 38- 8901- 76- 50354- 4-25 37-24 18-5 1472-1 38-5 9197- 77- 52027- 4-5 42-96 19- 1573-6 39- 9498- 78- 53732- 4-75 49-17 19-5 1679-1 39-5 9806- 79- 55471- 5- 55-90 20- 1788-9 40- 10119- 80- 57243- 5-25 63-15 20-5 1902-8 41- 10764- 81- 59049- 5-5 70-94 21- 2020-9 42- 11432- 82- 60888- 5-75 79-28 21-5 2143-4 43- 12 125- 83- 62762- 6- 88-18 22- 2270-2 44- 12842- 84- 64669- 6-25 97-66 22-5 2401-4 45- 13584- 85- 66611- 6-5 107-72 23- 2537- 46- 14351- 86- 68588- 6-75 118-38 23-5 2677-1 47- 15144- 87- 70599- 7* 129-64 24- 2821-8 48- 15963- 88- 72645- 7-25 141-53 24-5 2971-1 49- 16807- 89- 74727- 7-5 154-05 25- 3125- 50- 17678- 90- 76843- 7-75 167-21 25-5 3283-6 51- 18575- 91- 78996- 8- 181-02 26- 3446-9 52- 19499- 92- 81184- 8-25 195-50 26-5 3615-1 53- 20450- 93- 83408- 8-5 210-64 27- 3788- 54- 21428- 94- 85668- 8-75 220-48 27-5 3965-8 55- 22434- 95- 87965- 9- 243- 28- 4148-5 56- 23468- 96- 90298- 9-25 260-23 28-5 4336-2 57- 24529- 97- 92668- 9-5 278-17 29- 4528-9 58- 25020- 98- 95075- 9-75 296-83 29-5 4726-7 59- 2(5738- 99- 97519- 10- 316-23 30- 4929-5 60- 27886- 100- 100000- ORIFICES, WEIRS, PIPES, AND RIVERS. 451 TABLE XIV. Weights and Measures, English and French, with their relative values. MEASUEES OF LENGTH. 12 inches 1 foot. 7-92 inches 1 link. 3 feet 1 yard. 5 \ yards = 16^ feet 1 pole or perch. 100 links = 22 yards 1 chain = 4 perches. 40 perches = 220 yards 1 furlong. 8 furlongs =17 60 yards 1 mile. 6 feet 1 fathom. 120 fathoms 1 cable's length. 1 Nautical mile 6082-7 feet. 69-12 miles 1 Geographical deg. 3 miles 1 league. The Irish perch is 21 feet, or seven yards. Three inches make a palm ; 4 inches a hand ; 5 feet a pace. In cloth measure 2 \ inches = 1 nail ; 4 nails = 1 quarter ; 4 quarters 1 yard. 11 Irish miles = 14 English. MEASUEES OP SUEFACE. 144 square inches 1 square foot. 62-7264 1 square link. 9 square feet 1 square yard. 30 \ square yards = 272 J square feet ... 1 square perch. 10,000 square links =4,356 ... 1 square chain. 10 square chains = 160 square perches... 1 acre. 1 rood = 210 square yards ... 40 perches. 4 roods =4,840 ... 1 acre. 640 acres =3,097,600 ... 1 square mile. The Irish perch is 49 square yards, or 441 square feet ; 1 Irish acre=la. 2r. 19-17p. statute; and 1 statute acre=0a. 2r. 18-77p. Irish. The Irish acre is to the English acre as 196 is to 121. 100 square feet, is a square of roofing, slating, or flooring. The Cunningham acre is = la. Ir. 6-61p. English; and 1 English acre is = Oa 3r. 3'904p. Cunningham measure. 452 THE DISCHARGE OF WATER FROM CUBIC MEASURES, AND MEASURES OF CAPACITY AND WEIGHT. 1728 cubic inches 1 cubic foot, 27 cubic feet 1 cubic yard. 16Jx lx 1 =24-75 cubic feet ... 1 perch of masonry 16Jxl6|x lf= 306 cubic feet ... 1 rod of brickwork, 21 x IJx 1 = 30 J cubic feet ... 1 Irish perch of masonry The standard gallon, imperial measure, contains 10 Ibs. avoirdupois, of distilled water at 62 Fahrenheit, the barometer standing at 30 inches. 6-232 gallons 1 cubic foot. 8*665 cubic inches ... ... ... 1 gill. 4 gills 34-659 cubic inches 1 pint. 2 pints 69*318 cubic inches... ... 1 quart. 2 quarts 138-637 cubic inches ... 1 pottle. 2 pottles 277-274 cubic inches ... 1 gallon. 2 gallons 554*548 cubic inches ... 1 peck. 4 pecks 2218- 191 cubic inches ... 1 bushel. The old Irish gallon contained 217-6 cubic inches, nearly, and 1 Irish gallon is therefore = -7850 imperial gallon. The Irish barrel of lime still measures 40 Irish gallons, or 31*321 imperial gallons, or 4 bushels, very nearly. It is measured by a cylindrical measure 12 inches high, and about 21 J inches in diameter, con- taining half the Irish barrel. In the old English liquid measures for ale and beer, 36 gallons = 1 barrel = 36 gallons, 3J quarts, imperial measure, nearly. For old dry measures, 82 bushels = 1 chaldron = 31 bushels, 1 pint, imperial measure, nearly. And 36 bushels of coal = 1 chaldron of coal = 34 bushels 3 pecks, and 1 gallon, imperial measure. The Irish barrel of wheat is 20 stone ; barley 16 stone ; oats 14 stone. TROY WEIGHT, 24 grains 1 pennyweight. 20 pennyweights ... ... ... 1 ounce 12 ounces ... 1 pound. One pound Troy = 22-816 cubic inches of distilled water, baro- meter 30 inches ; thermometer 62, ORIFICES, WEffiS, PIPES, AND RIVERS. 453 APOTHECARY'S WEIGHT. 90 Troy grains 1 scruple. 3 scruples 1 drachm. 8 drachms ... ... ... ... 1 ounce. 12 ounces .< 1 pound. The ounce weighs 480 grains, and the pound 5760 grains, both in Troy and Apothecary's weight. AVOIRDUPOIS OR COMMERCIAL WEIGHT. One pound Avoirdupois = 27*7274 cubic inches, when the barometer stands at 30 inches, and Fahrenheit's thermometer at 62. 16 drachms = 437-5 Troy grains 1 ounce 16 ounces = 7,000 Troy grains 1 pound 14 pounds =98,000 Troy grains 1 stone 8 stone = 112 pounds ... ... 1 cwt. 20 cwt. = 2,240 pounds 1 ton One pound Troy= -82286 pounds Avoirdupois, and one pound Avoirdupois, is equal to 1-2153 pounds Troy. One ton of water contains about 36 cubic feet, equal to 224 imperial gallons, nearly. Ten pounds of distilled water is equal to one gallon, the Barometer and Thermometer being as above stated. FRENCH MEASURES AND WEIGHTS COMPARED WITH ENGLISH. MEASURES OF LENGTH. 1 metre .. .. 3-2808992 feet 1 decimetre . . . . -3280899 1 centimetre . . . . -0328090 1 millimetre . . . . -0032809 1 Jdlometre(orlOOOmetres)-621383mile 1 foot English . . 1 inch 1 yard 1 perch, 5 yds. 1 mile 3047945 metre 0253995 9143835 5-0291092 1-60932 kilometre 1000 metres = 100 decametres = 10 hectometres = 1 kilometre =3280-849 feet. The metre is the 10,000,000th part of a quadrental arc of the meridian or 39-3708 inches English. 454 THE DISCHARGE OF WATER FROM MEASURES OF SURFACE 1 centiare (one square ) 1n 7ftlq metre) . . j U 1 declare.. .. 107-6430 1 are 1076-430 f . * 119-6033 sq. yds. 11-9603 1-1960 1 are 1 declare ( 1 centiare or sq. \ metre. 17 100 ares = 10 declares = 1 hectare = 2-471143 English acres, and hectares are nearly equal to 42 English acres. The old Paris foot is equal 1-06578 English feet ; the French inch = 1-06578 English inches ; the French line -08882 of an English inch ; thetoise is equal to 6 French feet = 76-736 English inches = 6-39468 feet. The perches is 18 French feet ; and the perch royal 22 French feet. The French square foot or inch = 1-13581 English square feet or inches, and the cubic foot or inch = 1-21061 English. MEASURES OF SOLIDITY AND CAPACITY. 1 millistere .. -0353166 cubic ft. T ^.^.^ 0610279 1 centistere .., -353166 1 decistere .. 3-53166 1 centilitre 610279 1 stere (one cubic > Q n.oi cc metre J 3 1 decilitre 1 litre . . 6-10279 .. 61-0279 1 decastere 353-166 1 decalitre ..610-279 1 hectostere 3531-66 1 hectolitre 6102-79 1 kilostere 35316-6 1 kilolitre 61027-9 The stere and kilolitre are each a cubic m^tre, and the litre is a cubic deci- mStre ; 50 litres are nearly 11 English gallons, and 1 hectolitre 2-751207 English bushels. MEASURES OF WEIGHT. 0648 gramme = 1 grain, and 7000 grains = 1 Ib. Avoirdupois. 1 milligramme 1 centigramme 1 decigramme 1 gramme 015432 grains 15432 1-5432 15-432 1 gramme . . 1 decagramme 1 hectogramme 1 killogramme 15-432 grains 02205 lb.avoir. 2205 2-2046 1-01605 tonnes = 1 ton ; and 1 tonne = -984206 ton. A gramme is the weight of a cubic centimetre of water and its maxim, density in vacuo, 1 kilogramme = 2-6795 Ibs. Troy=2-2046 Ibs. Avoirdupois. 1 metrical quintal 220-46 Ibs. Avoirdupois, and 10 quintals is equal to the weight of a cubic metre of water. ^VN. *OL/XJC-* JU V *V VXX WjiUkxx- * ~ rv> - R v . . - ^w- ^-^*; x , ORIFICES, WEIRS, PIPES, AND RIVERS, 455 TABLE XV. Shewing the Weight, Specific Gravity, strength and elasticity of various materials employed by the Physicist and Engineer. When used bi/ the Engineer only about one-fourth of the ultimate strengths here given should be calculated from. MATERIALS. Moduli of Rupture. Moduli of Elasticity. Crushing forces per sq. inch, inlbs. Tenacities persq. in. in Ibs. Weights of a cubic foot in Ibs. specific gravi- ties. Acacia, English Growth . . Ash 11,200 12,000 1,150,000 1,600,000 9,000 16,000 17,000 44-3 48-0 71 77 8,900,000 10,300 18,000 525-0 8-40 Beech 9,300 1,350,000 8,500 16,000 48-0 77 Brick Red 800 280 135-5 2-20 112-5 1-80 Do. Pale Red 550 300 130-3 2-08 Cedar, American, Fresh . . Do. do. Seasoned 490,000 5,600 4,900 11,400 19,000 56-8 47-0 538-0 0-91 0-75 8-61 Do. Sheet 30,000 549-0 8-80 Do. Wire-drawn 9,900 1,670,000 60,000 12,400 560-0 43-6 8-88 0-70 Do. Memel .... 10,400 1 530 000 37-0 0-60 Do. Norway Spruce Elm Seasoned 6100 700000 10,300 17,600 13500 21-2 36-8 0-34 0-59 Fir, New England 6 600 2 190000 10,000 34-5 0-55 Do. Riga 7,600 1,100,000 6,100 12,000 47-0 0-75 Glass 8,000,000 33,000 2,400 153-3 2-45 Iron, Wrought, English . . 57,000 57,000 481-2 487-0 7-70 7-80 Do. rolled in Sheets and Rivetted Cast Iron Carron, cold blast Do. Hot blast 38*500 37,500 17,270,000 16,080,000 106,000 108,000 31,000 16,700 13,500 487 441 440-0 7-8 7-07 7-04 Do. Buffery . 37 500 14 000 000 90,000 17,500 441-0 7-06 Larch, green 5,000 900,000 3,200 10,200 36-6 0-52 6,900 1,050,000 5,500 8,900 35-0 0-56 Lead, cast English Do. milled sheet Marble, white Italian .... Do. black Gal way .... Mortar, old, good Oak, English 1,100 2,700 10,000 720,000 2,520,000 1,450,000 250 6,600 1,800 3,300 "so 17,300 717-4 712-9 165-0 168-4 107-1 58-3 11-44 11-40 2-64 2-70 1-75 0-93 Do. Canadian 10500 2 150 000 6 500 10200 54-5 0-87 8700 1 190 000 12 700 47-4 76 Do. African . . . 13600 2 280 000 60-7 0-97 Do. Adriatic 8,300 970000 62-0 0-99 9,800 1,230,000 7,800 41-2 0'66 Do. red 8900 1 840 000 5300 41-2 0-66 Silver, Standard Slate, Welsh 11,800 15,800,000 40,900 12,800 644-5 180-5 10-31 2-89 Do. Westmoreland Do. Valentia 5200 1,290,000 174-4 180-0 2-70 2-88 Steel, soft 120.000 486-2 7-80 Do, razor tempered .... Stone, granite average .... Do. Rochdale 5,500 2400 29,000,000 8,000 150,000 490-0 168-0 161-0 7-84 2-70 2-58 Teak, dry 14800 2 400 000 12101 15000 41-1 0-66 Tin, cast 4 600 000 5,300 455-7 7-30 LONDON : PRINTED BY GEORGE PHIPPS, 18 & 14, TOTHILL STREET, WESTMINSTER. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 5O CENTS ON THE FOURTH DAY AND TO $I.OO ON THE SEVENTH DAY OVERDUE. YC