LIBRARY OF THE IMVKRSITY OF CALIFORNIA. IMIYSICS DEI'AimiKXT. r,ii : T 01 MISS ROSE WHITING. September, 1896. Accession J ELEMENTS OF PRACTICAL HYDRAULICS, FOR THE USE OF STUDENTS IN ENGINEERING AND ARCHITECTURE. PART I. WITH NUMEROUS WOODCUTS. BY SAMUEL DOWNING, LL.D., \\ PROFESSOR OF CIVIL ENGINEERING IN THE UNIVERSITY OF DUBLIN ; HON. MEMBER OF THE INSTITUTE OF MECHANICAL ENGINEERS ; ASSOCIATE INSTITUTION OF CIVIL ENGINEERS. LONDON: LONGMANS, GREEN, AND CO. 1875- DUBLIN f PRINTED AT THE UNIVERSITY PRESS, BY M. H. GILL. INDEX. INTRODUCTION. Definitions, units of weight and capacity, &c., \ ' ' . . _'-... CHAPTER I. DISCHARGE THROUGH AN ORIFICE. Velocity of water flowing from an orifice, j Law of Torricelli Experimental Proofs, J Velocity when other pressures than the atmosphere exist. Examples, Discharge, contraction of fluid vein, causes of, Effects of Form of and numerical dimensions, Discharge through a thin plate, coefficient, Contraction suppressed on one or more sides. For- mula of Bedone, Orifices in plates not being true planes, External and internal tubes attached. Limits of the value of the coefficient, . ' . Practical rules for discharge per minute and per second, ANTECEDENT VELOCITY. Effects on discharge ; formula for, CYLINDRICAL ADJUTAGES, discharge from, Illustration, mean coefficient, formula for, . . . Coefficients of discharge and velocity compared, CONICAL CONVERGING ADJUTAGES. Discharge, Coefficients for ; best angle of convergence, . CONICAL DIVERGING ADJUTAGES. Coefficient; discharge from ; best angle of divergence, Discharge with very small charges ; true formula ; error of approximate do. ; tabular statement of difference, &c., ...... True measurement of the charge or head, True coefficient of discharge. Example, WEIRS, WASTE-BOARDS, OR OVERFALLS. Formula for discharge by ; coefficients, Velocity of approach, effects of; formula including, Experimental tests of formula, .... Ratio of discharges to length of overfall, &c., Art. Tage. -5 1-6 6-10 7-16 11, 12 13-15 16-18- 19-24 19 20-22 24 3 25, 26 27 32 32 28 33 29 34 3 35 32-34 35.36 38-41 3 6 , 37 38,39 39 40-46 42,43 48 44-48 49 50-52 55 56, 57 53-55 56 57 61 62-66 58-61 69 82 11 Index. Overfalls with channels attached, . Drowned weirs, formula for, Gauging discharge by overfalls, &c., Experiments by Blackwell and Francis, Table giving coefficients by different experiment ers, &c., . Formula for weirs in Beardmore's Tables, CHAPTER II. FLOW OF WATER UNDER VARIABLE HEAD. General principles of velocity and discharge, Volume discharged ; proposition, . . . Time of complete and partial discharge, &c., Mean hydraulic charge, Basin receiving constant supply while discharging, Analogous cases of discharge by weirs, Basins not being prismatic, . Discharge from one reservoir into another, . , EXAMPLES AND PRACTICAL APPLICATIONS. SLUICES. Practical Rules, .... Various arithmetical examples, .... Measures of water on Italian irrigation canals, Apparatus for constant discharge, by Thorn, do. do. at Kilmarnock, by Gale, do. do. Canal of Isabella II., in Spain, do. do. Marseilles Canal, in France, do. do. Henares Canal, by Bateman, Pitot's tube for measuring velocity in rivers, ' . Ramsbottom's apparatus for filling tenders, . Floating Britannia tubes ; pressure on pontoons, The " catarast " in Cornish pumping -engines, Clepsydra or water-clock, WEIRS. Arithmetical examples, .... Self-acting separation of turbid from clear water, Lowell experiments, FLOW OF WATER UNDER A VARIABLE HEAD. Arithmetical examples, CHAPTER III. FLOW THROUGH PlPES, CHANNELS, AND RlVERS. General principles, . . . Laws of friction of solids and fluids ; formulae ; con- stant, hydraulic mean depth, .... Action of water on bottom and sides of channels, Best form of channel, Mean velocity by inspection of channel, Art. 69-72 73-79 80, 8i 82 83-86 87 88-90 92 93 94 95 97 98-101 102106 107 1 08 109 IIO 111-113 114 115 116 Page. 84 84,85 85-90 90-96 99 100, 101 102 105 105-108 108 108-110 110 no 111114 115-118 132 H3 M3 J 53 157 '59 161 163 168 169 171 172 179 185 187 194 196-216 218 218-229 229-232 INTRODUCTION. r I ^HE science of Hydraulics has for its object the A knowledge of the phenomena of fluids in motion, and of the laws which regulate the production of these phenomena. Applied as an art, its object is to render this know ledge available in the designs of the civil engineer, as in the determination of the dimensions of pipes for convey- ing water, gas, or air, and also in works for the collect- ing, conveying, and distributing the necessary supply of water, for mill-power, or for the summit-levels of canals; or for the supply of cities; and, generally, of all such works as depend for their suitable con- struction and proportions upon the result of calcu- lations requiring a knowledge of the pressure and motion of fluids. 2. Fluids are defined to be bodies whose particles, by reason of their extreme mobility, yield to every the least force ; they have, however, a certain degree of adherence B 4 Introduction. to the Ib. ; giving about 36 cubic feet to a ton, or 6 tons to a cubic fathom. By a like approximation we have 6.25 imperial gal- lons to the cubic foot. These numbers give rise to many convenient practical rules, which are given in the "Prac- tical Examples," for Chap. I. The Imperial Bushel, which is the dry measure of capacity, is equal to eight gallons, or 1.29 cubic ft. Throughout this work, the only units made use of are the foot and the cubic foot. We have in English works on Hydraulics a great variety of units : for volume, the gal- lon, the cubic foot, the ton, the cubic yard, and the hogs- head ; for length, the fathom, the yard, foot, and inch, which, coupled with the absence of decimal subdivisions in our weights and measures, is always perplexing to the reader. As soon as the student has become familiar with the value of the inches in a foot expressed decimally, it is hoped that this arrangement will be found useful. Of the eleven decimal fractions for the inches in a foot, five are well known, namely, those for , j, J, f , f , and the rest may be readily remembered. It will be observed, also, that the eighth of an inch is very nearly o.oi ft., and every other eighth has, in the place of hundredths, a corresponding figure, thus | = o.02o8 ft., = 0.0312 ft., 1 = 0.0416 ft., 1=0.0521 ft, | = o.o625 ft., I =0.0729 ft. Introduction. Table showing the Decimal Values of the Inch. Inches. Fractions of a Foot . Inches. Fractions of a Foot. I TT, 0.0833 7 i\, 0.5833 2 , 0.1666 8 f, 0.6666 3 , 0.2500 9 T, 0.7500 4 i, 0.3333 10 *, 0.8333 5 i* T , 0.4166 1 1 ti, 0.9166 6 ^, 0.5000 12 f|, I.OOOO The measure of the force of gravity is the velocity acquired in one second by a body falling freely from a state of rest, and is equal to 32.1948 feet per second, and always denoted by the letter g. 5. So many French works on Hydraulics, of great value, have been composed, that a notice of their weights and measures may here be useful. The Metre, adopted in France in 1798, as the unit of lineal measures, is supposed to be equal to the one ten millionth part of the quadrant of a Meridian of the earth ; the accuracy of this is not, however, essential to the value of the system ; expressed in English mea- sures it is equal to 39.37079 inches, or 3.280899 ft. ; which, in practice, may be taken, approximately, as 39.37 inches, and 3.281 feet. It is multiplied, decimally, into the Decameter, the Hectometer, and Kilometer, and is subdivided, decimally, into the Decimeter, the Centi- meter, and the Millimeter ; the Greek word being affixed for multiplication, and the Latin for division by ten. The unit of weight is the Gramme, which is equal to 6 Introduction. the weight of a cube of distilled water, at a tempera- ture of 44 centigrade, above zero (supposed to be its maximum density), and in vacua; the side of the cube being one centimeter in length. As the decimeter is equal to ten times the centimeter, its cube will be 1000 times the cubic centimeter; the Kilogram therefore (1000 grammes) is the weight of a cubic decimeter, or liter, of distilled water at the above temperature. It is equal to 2.20485 Ibs avoirdupois, hence 1000 kilos are nearly one ton or 36 cubic feet of water. The measures of length, area, capacity, and weight, are in this sytem mutually connected ; it is not so in the English weights and measures ; the side of a cube con- taining one gallon cannot be expressed by any whole number of inches, or any other lineal measure, as the foot, &c. ; it is a little greater than 6.5 inches. Hence the long columns of specific gravities, which are not needed in the metric system, as the weight of any body expressed in Kilograms, whose volume is stated in cubic meters, is also its specific gravity, or ratio of its weight to the weight of an equal bulk of water. ELEMENTS OF HYDRAULICS. CHAPTER I. ON THE FLOW, THROUGH AN ORIFICE, OF WATER CONTAINED IN A VESSEL. r T~ A HE vessel from whence water issues through an A orifice may be, first, maintained at a constant height of surface ; or, secondly, it may receive no sup- ply, and, consequently, be gradually exhausted ; or, thirdly, the orifice, instead of discharging freely into the air, may do so into another reservoir, under more or less resisting counter-pressure; and hence three divisions of this part of the subject. The second division also in- cludes the cases in which the level of the surface gradu- ally rises or falls, from the supply being greater or less than the discharge through the orifice. 7. The opening through which the water flows may be placed either in the bottom or in one of the sides of the experimental tank, most generally the latter, in which case the surface of the water in the basin should be above the upper edge of the orifice : this orifice is opened either in a thin plate, that is to say, in a plate whose thickness does not exceed half the diameter of the orifice, if circular, or smallest dimension, if rectangular ; or else it is furnished with an adjutage, or short tube, 8 Flow of Water General Principles. sometimes cylindrical, sometimes conical, converging towards an external point, less often diverging. An orifice placed in a very thick plate would evidently be equivalent to one of the same diameter if placed in a thin plate, with an adjutage attached. We may also have the surface of the fluid below the upper edge of the orifice ; that part of the border or cir- cumference is then without influence on the discharge, and very frequently it is not applied ; the opening, un- limited on its upper part, is then called an overfall or weir. The laws of the flow of water in this second case offer some peculiarities, and form the subject of a sepa- rate investigation. When the surface reaches to a very small height only above the opening, we also have special circumstances : this case is intermediate be- tween the two others first mentioned. Before entering upon them it is necessary to state briefly the general principles of the flow of water, and the modifications which the " contraction" of the fluid vein suffers in passing through the various orifices to be noticed. The vertical distance of the surface of the fluid above the centre of gravity of the orifice is called the charge of the water upon the orifice, or the head under which the flow takes place. This point is not the true depth at which the mean velocity is found, but may, in most cases, without any sensible error be taken to re- present it ; the exact determination of it will be found in a future page. 8. Velocity of Water flowing from an Orifice. Let a vessel X, maintained constantly full of water up to the level AB, have upon the horizontal faces CD and EF the open orifices M and N; the fluid will issue in vertical jets, which will rise almost to the level of the water AK ; they would rise fully up to it but for the resistance of the air. Now, by the first principles of Dynamics, in order Flow of Water General Principles. 9 that a body impelled in a vertical direction should reach to any height, it is necessary that at the point of depar- ture it should have had a velocity equal to that which it would have acquired in falling freely from that height ; consequently, the particles of the fluid must have had a Fig. i. velocity nearly equal to that due to the charge that is, to the height of the surface of the water above the ori- fices, the only supposition in the application of the principle to the flow of water being that the particles of the fluid are perfectly independent of each other after they leave the orifice. So also, if upon a vertical face BR an orifice be placed, the centre of which is at O, we shall see further on that, from the respective values of the lines OP and PQ, the fluid must have issued from O with a velocity due to the height OB. It would issue with a velocity due to BR, if the orifice had been opened in the bottom RT of the vessel ; and the velocity is the same in O, in d, and O 2 , the directions being different, but the charge the same. This truth holds good for different orifices, whatever may be the ratio of the area of the orifice to the horizontal section of the water in the vessel, provided that the level of the water is kept at the same constant height, and 10 Flow of Water General Principles. tranquil ; which last, however, cannot be attained, if the orifice be too large in proportion to that surface the water of the supply, in that case, producing disturbing movements in the reservoir. A second method of determining the velocity is by measuring the ordinates of the curve of the path of a jet of water issuing from an orifice in the vertical side of a cistern. To have a clear notion of this method, it is necessary to state the following principles : When a body is projected in any direction AY, with a certain uniform velocity, the combined action of this velocity with the force of gravity causes it to describe a curved path, AMB. By measuring A the absissa x and the corresponding ordinate y, we can compute the height from which a body must fall vertically by the action of gravity to acquire that velocity, and, lastly, comparing the height so computed with the actual height of the surface of the water above the centre of the orifice, they are found to be very nearly equal, and thus we have another proof that water issues from an orifice with a velocity equal to that it would acquire in falling from a height equal to the " head" or charge. We do not for this purpose require any of the pro- perties of the curve of the jet. If the velocity, and con- sequently the resistance of the air, be not very great, the curve is a parabola. The demonstration of this will appear from the computation of the quantity more im- mediately sought for, which results in the equation of that curve, the parameter being equal to four times the height due to the velocity of projection. Let v be the velocity with which the body is sent forth in the direction of AY, and t the time spent in Fig. 2. Flow of Water General Principles. 1 1 reaching the point N; then, since the velocity in the direction AN is uniform, AN = v x t ; on the other hand, if the body had been solely under the action of the accele- rating force of gravity, it would have descended from A to a point P, during that same interval /, such that we should have AP = J^/ 2 . If we complete the parallelogram APMN, the point M will have been reached under the joint action of these movements in the same time / in which the point P was attained under the sole accelerat- ing force ; and it will have, therefore, traversed the arc of the curve, whose abscissa will be AP, and ordinate MP, parallel to the axis AY. Let x = AP andjy = we have therefore from the laws of gravity and from the uniform velocity in the direction AY (*)>.;" y=vt. From [b] we have by division, /=-, and squaring / 2 = ; substituting this value of /* in (a\ we have = x 9 "V 2V* or 2V* X f and putting h for the height due to the velocity v, and 09 remembering that = h y we have , equal to that due to the charge/' Thus the velocity acquired by a body falling freely by the force of gravitation from the height H, is equal to that of the fluid as it issues from the orifice with that height for the charge ; that is called after Toricelli, its discoverer, the Torricellian theorem, in which H is the "charge," measured from the surface down to the centre of gravity of the orifice, and T 4 Floio of Water General Principles. g y the dynamical measure of the force of gravitation, being the rate, or number of feet per second, with which a body falling freely is moving at the end of the first second. 9. The following Table exhibits the results of expe- riments by Castel, D'Aubuisson, Bossut, Poncelet, &c., also proving that the velocity of issue is proportional to the square root of the charge. It will be observed that the charges vary from i to 200 and more, and the sections of the orifices from i to 500, and yet in all cases the velocities have followed the ratio of the square roots of the charges, minute dis- crepancies, sometimes giving too great a number, and sometimes too small, being inseparable from experiments of this nature. The actual object of measurement in the experiments was the quantity discharged in a given time ; but it is evident that, with the same orifice, the discharge is exactly proportional to the velocity with which the fluid issues, and, therefore, that column in the Table which expresses the gauged discharges, reference being made to some one discharge as a unit, also expresses the velocities. Thus, for example, take the ist and 4th lines with "square orifice/' The discharge of the former into a cubical vessel was 229 cubic feet in 40 seconds, and of the latter 258.5 cubic feet in 25 seconds; reducing both to the quantity for one second, we have 5.725 and 10.34 cubic feet respectively, dividing each by 5.725 their ratio is i to i. 806. The square roots of the charges are 1.1454 and 2.0652, dividing both by the first their ratio is i to 1.803, as * n the two last columns of the Table. Flow of Water General Principles. 1 5 TABLE showing that the Velocities are proportional to the square roots of the Charges. Diameter Charge above Serie sof of the Orifice. the Orifice. Square Roots of the Charges. Discharges or Velocities. Feet. Feet. 0.0328 0.085 .000 I.OOO 0.098 .074 1.064 0.131 .241 1.244 0.164 .386 1-393 0.196 .519 1.524 0.088 4.265 .000 I.OOO 9.580 .500 1.497 12.500 7 J 3 1.707 0.265 7.677 .000 I.OOO 12.500 35 1.301 22.179 .738 1.692 0-531 6.922 .000 I.OOO 12.008 .316 '315 Square Orifice 0.656 by 1.312 2.296 3.281 .000 323 1.581 I.OOO 1-330 1.590 0.656 4.265 1.803 1. 806 5-249 2.OOO 2.000 10. The general principle, that the velocities are as the square roots of the charges, as also the theorem of Torricelli ( 8) for cases in which it is applicable, ex- tends to every kind of fluids, to mercury, oils, alcohols, so that the velocity with which each of them issues from an orifice is independent of its particular nature, and of its density, it depends solely on the charge. Experiment demonstrates this, and very simple reasoning suffices to show its truth. Take the case of mercury : the particles situated immediately in front of the orifice, and in which it is necessary to create a certain velocity, are, it is true, fourteen times more dense than those of water, and they consequently oppose to motion a resistance fourteen 1 6 Flow of Water General Principles. times greater than it would do ; but the mass also which presses upon these particles, and produces the velocity of exit, the charge being the same, is greater in the same proportion, and therefore gives a motive force fourteen times greater. Thus a compensation exists, and the velocity impressed remains the same ; and, in like man- ner, it may be proved for a fluid lighter than water. ii. The proposition that has now been laid down with respect to the velocity of water issuing through an orifice is equally true in cases when the discharge takes place in vacuo, the velocity is always the same, with the same head, whatever be the pressure upon the free surface of the water in the vessel, provided the jet of water at its exit from the orifice be subject to an equal exterior pressure. But if the pressures on these two surfaces be not equal, the velocity will be very different from that due to H. If in the first place the pressure per square inch against the orifice at A be greater than that upon the free surface of the water BC (in the woodcut, Fig. 4), then the excess of the former above that on the free surface must be less than that of a column of the fluid whose height is the vertical distance of the orifice A, below the surface BC, for if they were equal, it is evident there could be no discharge. Let us, then, take an horizontal plane DE, below the plane BC, at such distance from it that the weight of a column of the water contained between the two planes, and whose base is the unit of surface, may be equal to the excess of pressure at A, of which we are speaking. The pressure, then, which exists upon any point in the plane DE will be equal to that upon any point in BC, plus the supposed excess of pressure against the exterior of the orifice ; and, therefore, the pressure upon the plane DE, will be the same as that against the Flow of Water General Principles. orifice at A. The liquid below the plane DE is then in the same condition as if that contained between BC and and DE were removed, and the free surface and exterior of the orifice were under equal pressures ; and thus the formula will represent the velocity ; h\ denoting the depth of the plane DE below BC. The water in the vessel being supposed to have but a slight degree of motion, on account of the relatively small area of the orifice to the surface BC, which is understood to subsist ; and therefore we may assume the pressures to be transmitted as if the water was in equilibrium. 12. If, secondly, the ex- terior pressure on A were less than that upon the surface BC, we may con- ceive the excess of pres- sure on BC to be produced by a liquid of the same specific gravity as that in the vessel, applied above BC and terminating in a Fig. free surface D' E', situated at such height that the ver- tical distance represents, as before, the column of the liquid, having for its base one square inch, or other unit of surface, whose pressure is equal to the excess of the pressure on BC above that against the orifice A. The flow, then, will take place with the same velocity as if the free surface of the liquid, instead of being in the plane BC, and supporting this excess of pressure, were at D' E', and supported the same pressure as the orifice at A ; the formula will therefore be in which h* is equal to the vertical distance of D'E', c i8 Flow of Water General Principles. above BC. We see thus that a diminution or augmen- tation of the pressure upon the free surface of the liquid in the vessel, without any change in that against the orifice at A, causes a corresponding diminution or aug- mentation in the velocity of the issuing fluid, and, on the contrary, that a diminution or augmentation of the pressure against the orifice, without any change in that upon the free surface, causes a corresponding augmen- tation or diminution in this velocity. The self-acting contrivance (of James Watt) for sup- plying the feed-water to low-pressure boilers comes under the first case. The pressure being supposed olbs. per inch above the atmo- sphere, it is required to place the cistern of the supply so high, that on the opening of the valve a y by the float b descending below the proper level, the water may enter against the pressure of the steam. Now, as the cubic foot of water weighs 62.5 Ibs., a column i foot high and i Fig. 3- 62.5 square inch base weighs - - = 0.434 Iks., an d, therefore, 144 the height of the column of water to balance any given pressure expressed in pounds per square inch is found by dividing that number by 0.434 i n this case, 5 -f- 0.434 = 1 1.52 feet : this gives exact equilibrium ; the additional head, in order that it may enter with due rapidity (from 2 to 4 feet per second generally), will depend upon the rate of evaporation of the boiler and the area of the supply pipe. It is evident that this mode of supply is not convenient in high-pressure boilers ; for suppose the pressure to be 50 Ibs. per inch, then the height to pro- Flow of Water General Principles. 1 9 duce equilibrium will be 115.2 feet. The pressure in a hydraulic press is frequently 3 tons per square inch, equal to (3 x 2240=) 6720 Ibs., and 67 20 -=-0.434 =15484 feet. If, instead of a free surface in the cistern, we had supposed a solid piston or plunger to press on the enclosed water, the head should in like manner be cal- culated, by turning the pressure per square inch on the piston into vertical feet of water. The condenser of a low-pressure steam-engine offers an example of the second case ; for, let us suppose a va- cuum of 25 inches of mercury to be maintained, and that the head of water in the cistern supplying the jet of cold water which effects the condensation, were 2 feet above the point at which it enters this partial vacuum, then the actual head producing the flow is 2 -f 28.25 = 3- 2 5 feet, for, pure mercury being 13.56 times heavier than water, we have the height of a column of water which would balance that of 25 inches of mercury equal to 25 x J 3-56 = 339 inches, or 28.25 feet. 13. Having thus established the law of the velocity of a fluid issuing from an orifice, let us proceed to apply it to the determination of its discharge, which is defined to be the volume of the fluid which escapes in the unit of time, that is, one second. If the mean velocity of all the particles was that due to the " charge" H, then this velocity, which is called the theoretic velocity, would be \/2^H ; and if at the same time the particles issued from all points of the orifice in parallel threads, it is evident that the volume of water flowing out in a second would be equal to the volume of a prism which would have the orifice for its base, and that velocity for its length ; and, calling ,5* the area or section of the orifice, the volume of water, or of the prism, would be This is the theoretic discharge. C 2 2O Flow of Water General Principles. 14. But the actual discharge is always less than this. In order to have an exact idea of the phenomena, let us consider the fluid vein a short distance after its issue from the orifice, and let us suppose it cut by a plane per- pendicular to its direction. It is manifest that the discharge will be equal to the product of the section by the mean velocity of all the several threads at the moment they intersect the plane of the section. If this section was equal to that of the orifice, and if this velo- city was that due to the charge, then the actual discharge would also be equal to the theoretic discharge. But whether from the section of the vein being considerably less than that of the orifice, as in the flow through orifices in a thin plate, or from the velocity being con- siderably less than that due to the charge, as in cylin- drical adjutages ; or, again, from a diminution in both the section and the velocity, as in certain conical adju- tages, it always results, that the actual discharge is in every case less than the theoretic, and, in order to reduce this last to the former, it is necessary to multiply it by some fraction. Let m represent this fraction, and Q the actual discharge in one second, we shall then have And representing the volume of water flowing off in the time T seconds by Q' we shall have Whether the diminution in the discharge arises from a diminution of the section, or of the velocity, it is always a consequence of the contraction which the fluid vein suffers in passing through the orifice, and thus the multiplier m, or " coefficient for the reduction of the theoretic to the actual discharge," is generally called the " coefficient of contraction" and is taken to represent the Flow of Water General Principles. 2 1 aggregate effect of all circumstances tending to diminish the discharge. Its accurate determination is of the greatest importance ; upon the degree of exactness with which it is ascertained depends that of the results we obtain when we would apply to practice formulae upon the flow of water. We shall now proceed to give the results of experiments on the value of the symbol m, making some preliminary statements upon the cause of the " contraction/' and the nature of its effects ; and also upon the form of the fluid vein the orifice being cir- cular its relative dimensions, and the effect of the form upon the discharge. 15. Cause of the Contraction. If we take a glass vessel in the side of which is an orifice through which the water flows, and render visible the movement of the molecules of the water in the vessel by disseminating through it a substance of equal specific gravity, and very minute, or by producing within the water some light chemical precipitation, such as occurs when we let fall a few drops of nitrate of silver in water slightly saline, we then see at a small distance from the orifice, as, for instance, about an inch, when its diameter is three-eighths of an inch the fluid molecules converge from all parts towards the orifice, describing curved lines, and, finally, as if approaching a centre of attraction, issue forth with a ra- pidly increasing motion. The convergence of the directions that they had within the vessel at the moment of their arrival at the orifice still continues for a short distance after they have passed out, so that we can plainly see the fluid vein gradually diminish, and become contracted up to the place where the particles, from the effect of their mutual action, and of the motions impressed upon them, take directions, either parallel to each other, or in some other lines. The vein thus forms a species of truncated pyra- 22 Flow of Water Contraction of the Fluid Vein, mid or cone, whose larger base is the orifice, and smaller the section of the fluid at its place of greatest contrac- tion, a section which is often called the "section of contraction." This figure, and all the phenomena of contraction, are thus a consequence of the convergence of the several threads of water when they arrive at the orifice. 1 6. Effects of the Contraction. When the orifice is in a thin plate, the contraction is completely external to the reservoir ; it is thus clearly visible, can be, and, in fact, it has been, measured, as we shall mention directly. When the orifice is circular, the fluid vein, after having reached the minimum section, continues of the same transverse area, and is thus cylindrical in form, having a velocity very nearly equal to that due to the charge. The discharge will, therefore, be the product of this section by the velocity, so that the effect of the contrac- tion is limited to the reduction of the value of the section which enters into the expression for the value of the discharge. The flow will take place as if the actual orifice had been replaced by another whose diameter was equal to the "section of contraction," but in which sup- posed orifice no true contraction took place. 17. Form and Dimensions of the Con- \ traded Vein of the Fluid. Let us next N examine the form that the contraction gives to the fluid vein issuing from an orifice, in the simple case of a circular orifice in a thin plate, truly plane. Everything being symmetrical around c the different points of the orifice, the direction as well as the velocity of the molecules, the contracted vein ought also to be of a symmetrical form, and, B consequently, a solid of revolution a / conoidal figure. It is actually so ac- Fig. 6. cording to the observations that have been made, and Flow of Water Contraction of the Fluid Vein. 23 which the figure ABab represents. Beyond ab the con- traction ceases, and the vein continues sensibly cylindri- cal for a certain length, until the resistance of the air and other causes entirely destroy this form. The earlier measurements that have been made give to the three principal dimensions AB, ab, and CD, the ratio of the numbers i.oo, 0.79, and 0.39. The length of the contracted vein would thus be about half the diameter of the smaller section, and 0.39 of the larger, that is, of the orifice. 1 8. Michelotti, from a mean of more recent experi- ments on a large scale, has adopted i.oo, 0.787, 0.498 : these D' Aubuisson follows. The ratio of the diameters AB and ab being thus i to 0.787, that of the sections is i to (o.787 2 =) 0.619, that, namely, of the squares of the former numbers ; thus, ifsbe the "contracted section," and S that of the orifice, we shall have s = 0.619 S, and, consequently, the discharge in one second will be , or 0.619 so that the value of w,-or the " coefficient of contraction," as determined by actual measurement, is, at the mean, equal to 0.619, being a little less than that which results from experiments on the gauged discharge. If the velocity at the passage of the "section of con- traction" was exactly that due to the charge, and that the flow took place through an adjutage of the exact form of the contracted vein, and that in the expression for the discharge the area, s, of the outer orifice of this adjutage, taken at the extremity, were introduced, then the calcu- lated would be equal to the actual discharge, and the 24 Flow of Water through Orifices in a Thin Plate. coefficient of the reduction of the one to the other would be equal to unity ; and Michelotti, in one of his experi- ments in which he employed a cycloidal adjutage, has reached 0.984. It is very probable he would have ac- tually reached i, if this form had more accurately been adapted to that of the fluid vein, and if the resistance of the air had not somewhat retarded the motion. 1 9. Flow of Water through an Orifice in a Thin Plate. We come now to the more direct determination of the coefficient for reducing the theoretical to the actual dis- charge. For this purpose it is necessary to gauge with care the volume of water discharged in a given time under a constant charge, from which we deduce the flow in one second, or the actual discharge ; and, dividing this by the theoretic discharge for the same head and same orifice, the quotient is the coefficient required. Thus in a cistern with a head of 4.012 feet above the centre of an orifice the diameter of which is 3.185 inches we have a theoretic discharge of 0.8903 cubic feet per second, obtained thus : with the above head we have a velocity of 16.07 f eet P er second, that is \/2 x 32.2 x 4.012. And the area of the orifice is equal to 3.i85 2 x 0.7854 = 7.97 square inches, and 12-L - 0.0554, its value in square feet. 144 This, multiplied into the velocity of issue, gives the vo- lume of the prism or cylinder equal to that of the water discharged ; that is ( 13), 0.0554 x 16.07 = 0.8903 cubic feet per second. But having found by experiment that in i J minutes the actual discharge was 49.68 cubic feet, reducing this to Flow of Water through Orifices in a Thin Plate. 25 its value for one second by dividing by 90, we obtain - = 0.552 cubic feet as the discharge in one second ; 90 hence, dividing the actual by the theoretic discharge, we find for the coefficient ' =0.620. 0.8903 Many hydraulicians have for a long time been en- gaged in its determination. The following Table, from D'Aubuisson, gives the principal results obtained by experiments up to the present time, and which, having been made under favourable circumstances, are generally received. They include circular, square, and rectangular orifices : TABLES of the Results of Experiments undertaken to deter- mine the " Coefficient of Contraction." CIRCULAR ORIFICES. Observers. Diameters. Charges. Coefficients. Feet. Feet. Mariotte, 0.0223 5.8712 0.692 Do. 0.0223 25.9120 0.692 Castel, . . 0.0328 2.1320 0.673 Do. 0.0328 1. 0168 0.654 Do. 0.0492 0.4526 0.632 Do. 0.0492 0.9840 0.617 Eytelwein, . 0.0856 2-37H 0.618 Bossut, . . 0.0889 4.2640 0.619 Michelotti, . 0.0889 7-3H4 0.618 Castel, . . 0.0984 0.5510 0.629 Venturi, . . 0-1345 2.8864 0.622 Bossut, . . 0.1771 12.4968 0.618 Michelotti, . 0.1771 7.2160 | 0.607 Do. 0.2657 7-347 2 0.613 Do. 0.2657 12.4968 0.612 Do. 0.2657 22.1728 0.597 ? Do. 0-53H 6.9208 | 0.619 Do. Q.53'4 12.0048 0.619 26 Flow of Water through Orifices in a Thin Plate. SQUARE ORIFICES. Observers. Side of Square. Charges. Coefficients. Feet. Feet. Castel, . . 0.0032 0.1640 0-655 Bossut, . . 0.0885 12.5000 0.616 Michelotti, . 0.0885 12.5000 0.607 Do. 0.0885 22.4078 0.606 Bossut, . . 0.1771 12.5000 0.618 Michelotti, . 0.1771 7-347 2 0.603 Do. 0.1771 12.5624 0.603 Do. 0.1771 22.2384 0.602 Do. 0.2689 7-3489 0.616 Do. 0.2656 12.5624 0.619 Do. 0.2656 22.3700 0.616 RECTANGULAR ORIFICES. Rectangle. Charges Coefficients Height. Base. Feet. Feet. Feet. 0.0301 0.0606 1.0824 0.620 0.0301 0.1213 1.0824 0.620 0.0301 0.2423 1.0824 0.621 0.0301 0.4847 1.0824 0.626 20. The experiments of Michelotti were carried on about three miles from Turin, at an hydraulic establish- ment constructed for experimental purposes, consisting of a building 26 feet high, supplied with water from the River Dora by a canal of derivation. The internal di- mension was a square of 3 feet 2 J inches ; on one of the sides was arranged a series of adjutages at the different depths deemed expedient, and upon the surface of the ground were arranged the different receptacles for the gauging of the actual discharges. It may be remarked upon the part of the Table given Flow of Water through Orifices in a Thin Plate. 27 by Michelotti that the coefficients obtained from the large orifices are higher than the others, and this con- trary to the rule that would be deduced from the experi- ments in general. The older writers supposed the deficiency in the dis- charge to arise from a diminution in the velocity, and not from the vend contractd. Thus Hutton writes : " The particles entering the orifice in all directions impede one another's motion : from whence it appears that the real velocity is less than that of a single particle only, urged with the same pressure of the superincumbent column of the fluid. And experiments on the discharge show that the velocity must be diminished rather more than a fourth, or such as to make it equal to that of a body fall- ing through half the ' charge' above the orifice." 21. In order to place the subject of the variation in the value of the coefficient, under different circumstances of area and charge, in a clear point of view, the following Table of MM. Poncelet and Lesbros' experiments at Metz, in the Province of Lorraine, in 1826 and 1827, is given. In these experiments the orifices were rectangu- lar, and all of the same breadth namely, o m . 20 = 0.656 feet; the heights were successively 0.656, 0.328, 0.164, 0.098, 0.065, an d 0.0328 feet. The charges extended from 0.33 feet to 5.58 feet. With the several orifices they re- peated the experiments, taking each of them with 8 or 10 charges, from the smallest, to the highest that the apparatus admitted, calculating the corresponding coeffi- cients. They then took the charges for the abscissae, and these coefficients for the ordinates of a curve con structed for each orifice, and by its aid they determined the ordinates, that is, the coefficients intermediate to those directly determined by experiment ; and thus gave a very extended Table, from which the following is taken : 28 Flow of Water through Orifices in a Thin Plate. TABLE showing the Results of Experiments to determine the Variation in the Value of the Coefficient of Contraction. Charge on Centre of Orifice. HEIGHT OF THE ORIFICES. Difference of maxi- mum and minimum coefficients. Feet. 0.656 Feet. 0.328 Feet, o. 164 Feet. 0.098 Feet. 0.065 Feet. 0.032 Feet. 0.032 0.709* 0.065 o. 660 O-693 0.098 0.638 0.660* o. 691 o. 131 0.612 0.640 0659 0.685 o. 164 0.617 0.640 0.659 0.682 0. 196 0.590 0.622 0-640* 0.658 0.678 o 262 0.600 o. 626 0.639 0.657 0.671 o 328 0.605 O-628 0.638 - 6 55 0.667 -393 0.572 0.609 0.630 0.637 0.654 0.664 0.092 0.492 0.585 0.611 o. 631 0.635 0.653 0.660 0.075 o. 656 0.592 O 613 0.634* 0.634 0.650 0.655 0.663 0.984 0.598 0.616 0.631 0.632 0.645 0.650 0.052 1.312 O 6OO 0.617* 0.631 0.631 0.642 0.647 0.047 1.640 o. 602 0.617 0.631 0.630 0.640 0.643 0.041 2 296 o. 604 0.616 o. 629 0.629 0.637 0.638 0.034 3.28l o. 605* 0.615 0.627 0.627 0.632 0.627 0.027 4.264 0.604 0.613 o. 623 0.623 o. 625 0.621 0.021 5.248 0.602 0.611 9.619 0.619 0.618 0.616 O.O17 6.562 0.601 0.607 o. 613 0.613 0.613 0.613 0.012 9^43 0.601 0.603 0.606 0.607 0.608 0.609 0.008 The woodcut illustrates this method of interpolation. From the point O, the several charges are laid off on the Fig. 7- line ON, as OX, OX t , &c., and the corresponding coeffi- cients XY, XiYj, &c. ; and the curve being traced through Y, YI, Y 2 , &c., we can obtain the coefficient proper to any Flow of Water through Orifices in a Thin Plate. 29 charge as Ox, by drawing the perpendicular line xy ter- minating in the curve. 22. All the numbers contained in this Table are the several values of the coefficient m in the formula Q= mSv / 2^H. But those in each column above the darker type are not the true coefficients for the reduction of the theoretic to the actual value, as will be shown hereafter. Casting the eye over each column, we may see that the coefficients increase as the charges are greater, but up to a certain point only, although the charge still in- creases : an asterisk in each column indicates the maxi- mum value. It may also be observed, that the coeffi- cients become more nearly equal in each column as the charges increase, the bottom line of figures, in which the charge was 3 = 9.84 feet, being almost identical in each column. 23. This Table, although constructed from experiments on rectangular orifices, can yet be extended to those of all other forms, the height of the rectangle, as given in the Table, corresponding to the smaller dimension of the orifice made use of ; for it is admitted that the dis- charge is altogether independent of the figure of the orifice when the area is constant, provided only that this figure has no re-entrant angles. 24. Although these experiments are on a considerable scale, yet there are some cases in practice in which the discharge is twenty or thirty times greater. Such are the sluices in lock-gates on canals of navigation. It is a matter of importance to determine directly the coeffi- cient of discharge for them, and to be able in practice to assign, with some confidence, the coefficient to employ in any particular case, when a direct experiment may not be possible. 30 Flow of Water through Orifices in a Thin Plate. TABLE showing the Value of the " Coefficient of Con- traction" in large Sluices: Canal Laquedoc. SLUICE. Width 4.25 ft. Charge upon the Centre. Discharge in one sec. Coefficient. Area. Height. Square Feet. Feet. Feet. Cubic Feet. 7.7442 .804 H-550 H5-3056 0.613 6.9928 .640 6.628 92.6438 0.641 6.9928 .640 6.245 88.2288 0.629 6.4664 .508 12.874 138.6302 0.641 6.7237 574 13.582 128.7759 0.647 6.7237 574 6.392 83-955I 0.616 6.7 2 37 1-574 6.215 79-8580 0.594 6.7172 i-574 6.478 85.2266 0.621 Mean Coefficient 0.625 The mean coefficient, 0.625, * s rather greater than that found by Poncelet (21), which is readily explained, as the flow of water did not take place in a thin plate, the contrac- tion being suppressed on some parts of the boundary. The wood-work which surrounded the sluice-way was 0.8856 feet thick, and on the sill was even 1.771 ft. Thus, when the sluice was raised but a small height, the contraction nearly ceased on four sides, and the coefficient was con- siderably increased. For example, when the sluice was raised only 0.393 ft., it gave a coefficient of 0.803 ; when raised 1.51 ft., it was 0.641. 25. Particular Cases in which the Contraction is sup- pressed on one or more Sides of the Orifice. In all the dif- ferent cases treated of hitherto, it has been assumed that the fluid arrived at the orifice from all parts equally, but frequently this is not so. For example, Fig. 8, when a rectangular orifice is at the bottom of a vertical plate, and its inferior edge is on the level of the bottom Flow of Water through Orifices in a Thin Plate. 3 i of the vessel or reservoir, the contraction is then destroyed on that side, the particles of water being compelled to take a direction parallel to the side of the vessel ; and, consequently, the discharge is increased. The question arises, therefore, how much will the discharge be aug- mented by the suppression of the contraction for a certain length of the periphery of the orifice ? The following Table gives the result of experiments instituted with the view of determining this point. The orifice was rectangular, 0.177 feet in base, and 0.089 feet in height. The plates, which were attached, some- times on one side, sometimes on two or three of the sides, were 0.22 feet long ; that is, they advanced this much into the reservoir. The flow was produced by charges from 6.56 feet to 22.56 feet in height : TABLE showing the Increase of the " Coefficient of Con^ traction " by its Suppression on Part of the Sides. Portion of Orifice without Contraction. Coefficient. Ratio of Increase. o .608 .000 i * o .620 .020 I o .637 .049 1 o .659 .085 t o .680 .119 1 o .692 139 26. In this Table the last column has for its unit the discharge when the orifice is perfectly free : the numbers, therefore, indicate the increase in the coefficients, and, 32 Flow of Water through Orifices in a Thin Plate. consequently, in the discharges. The formula deduced / yL by M. Bidone, the experimenter, is i + 0.152 -, in which n represents the length of the part of the perimeter in which the contraction is suppressed, and p the perimeter of the orifice. The greatest error of this formula being but the ^-gth part, it may be used for the value of the discharge when, in the case of rectangular orifices, there is no contraction on part of the boundary, and the actual discharge then is mS*/ . 1+0.152-) 27. Orifices in Plates not being true Planes. It has been hitherto always supposed that the sides or plates in which the orifices were placed were true planes ; they may, however, be of very differ- ently formed surfaces. In order to have a clear idea of the effect which any such alteration pro- duces upon the flow, it is neces- sary to recall to mind that if the threads of the fluid vein did ar- rive at the orifice mutually par- allel, the actual discharge would be equal to the theoretic, and that it is less than this only by reason of the oblique directions in which Fig. 9. they converge, from which necessarily results a de- struction of part of the acquired motion at the point of contact with the orifice. If, therefore, we imagine around the orifice a spherical surface of a radius equal to that of the sphere of action of the orifice, and this surface terminated by the sides of the vessel, then it must be intersected on every point, and in directions nearly per- pendicular, by the threads of the issuing fluid, as in the woodcut, Fig. 9 ; and the larger the portion of the complete Flow of Water through Orifices in a Thin Plate, 33 sphere this surface may be, and the more oblique, or even opposite, to one another, the threads of the fluid arrive at it, then the more the motion is destroyed at the en- trance of the orifice, and the less the discharge is found to be. When the sides are developed in one plane, then the supposed surface is a hemisphere, and the coefficient of this particular case is given above. But if they are disposed in the form of a funnel, or, if simply concave, towards the interior of the vessel, then the surface of this sphere is of less extent, and the discharge more considerable not, however, fol- lowing exactly the inverse pro- portion of the spherical surface. If, on the other hand, the side is convex, the discharge is dimin- Fig. 10. ished, and it will be less still in the case represented, Fig. 1 1 . Lastly, it will be at its minimum value if the supposed surface should become an entire sphere ; and this would happen if it was possible to carry an orifice into the midst of a mass of the fluid enclosed in the vessel. 28. Borda has succeeded in realizing this case almost completely. He has introduced into a vessel, as shown in Fig. 1 1, a tube of tin 0.443 f eet long and 0.105 feet in diameter, and under a charge of 0.82 feet he has caused the flow to take place so that the effluent water did not touch the tube at all. The actual discharge has been only 0.5 1 5 of the theoretic, and from various circumstances Borda was led to think Fi s- i r that he might have reduced it to 0.50. D 34 Flow of Water through Orifices in a Thin Plate. The woodcut shows the manner in which the fluid bends around the exterior edge, and enters the tube without touching the internal sides, the thickness being about T ! 2 of an inch, or 0.0069 f eet > an d the edges cut truly square : thus all that part of the sides within the exterior periphery is, as far as the discharge is concerned, as if totally removed ; and it is this external diameter that should be introduced in all calculations relative to internal adjutages. By taking it, M. Bidone has found, from two experiments in which the effluent fluid did not touch the sides, that the coefficient was nearly 0.50, that is, the area of the section of contraction was half the area of the orifice taken at the external circumference. Having subsequently surrounded the orifice of the entry of the tube with a border or rim, and having thus re- duced it to the condition of being in a plate perfectly plane, although in the centre of the fluid mass, he found the coefficient rise to 0.626. The same result might be obtained by employing a simple tube, but of a thick material. 29. Thus 0.50 and i.oo will express the limits of the coefficients of contraction, the limits to which they may approach very nearly, but which they can never actually attain. For orifices in a plate truly plane it does not descend below 0.60, or rise much above 0.70 ; and in ordinary practice it ranges between 0.60 and 0.64. As a mean term, 0*62 is generally taken : so that Q = mS ^2^H = o.62*5V7pl = 4.Q6.SVH ; from whence we have, as an approximate rule for the discharge in cubic feet per second Q = 5 x Area x \/H, and per minute, 300 x Area \/H ; Flow of Water having Velocity antecedently. 3 5 and if the orifice be circular of a diameter d y the area is expressed by d* x 0.7854 = S, and or approximately, Q = 4 x d 1 x -v/H per second, and 240 x d z x \/H per minute, the diameter being expressed in feet. For greater exactness in the coeffi- cient, recourse should, however, be had to the Table, page 28, 21, that one being chosen which has the near- est identity to the particular case. 30. Effects on the Discharge when the Fluid has Velocity antecedently. If the water contained in the reservoir, in- stead of being in a state of repose, was moving in the direction of the orifice, as when the vessel, having but a relatively small section, has a supply of water brought into it, and flowing directly up to the plate or side in which the orifice is opened, then the particles of the fluid would issue, not only in virtue of the pressure ex- erted by the fluid mass which is above it, but with the additional velocity that they had when they entered into the sphere of action of the orifice; we must, therefore, add to the actual charge measuring the pressure a new term, which will be the height due to this supposed ve- locity of arrival. Thus, if u represent this velocity, we 2 2 shall have (since is the height producing the velocity c> u) the expression ; Let u be equal to 4 ft. per second, then taking ^g as approximately equal to 64, we have = - = - = 0.25 ft. to be added to H. ig 64 4 D 2 36 Flow of Water through Cylindrical Adjutages. 31. Flow of Water with Cylindrical Adjutages. The addition of a tube to an orifice in a thin plate gives a discharge larger than that through an orifice in a thin plate ; but in order that it should produce this effect it is necessary that the water entirely fill the area of the ex- ternal mouth of the tube, and this is generally the case when the length of the tube is two or three times greater than its diameter : if it be less than this, the fluid vein, which is contracted at the entrance, does not always en- large so as to fill the interior of the tube ; the flow in that case takes place as if in a thin plate, and this is always the case when the length of the tube is less than the length of the contracted vein, which, as we have seen, is but half the diameter of the orifice, or even less. 32. The woodcut attached, which is a vertical sec- tion through the centre of the orifice and axis of the adju- tage, serves to illustrate the action which takes place ; the fluid threads arrive at the orifice converging, and there- fore the fluid will contract at the entrance. Experiments prove that this contraction is identical with that of the thin plate ; its position will, how- ever, be internal with respect to the mouth of the tube at- tached. Beyond the section of Fi s- I2 ' contraction, however, the attraction of the sides of the tube occasion a dilatation of the fluid vein ; the threads follow these sides, and issue parallel to each other and to the axis of the tube. The part which is darkened in the woodcut shows the space in which a partial vacuum is formed around the vend contractd: that such is the case is proved by this simple experiment ; a glass tube is in- serted, air tight, in the side of the adjutage, the other end Flow of Water through Cylindrical Adjutages. 3 7 being placed under the free surface of the water con- tained in a vessel at a lower level ; when the discharge takes place the water is seen to rise in the glass tube, to about f ths of the charge, affording a measure of the degree of vacuum formed in the adjutage. 33. TABLE showing the Increase in the " Coefficient of Contraction" by the Cylindrical Adjutage. Adjutage. Observers. Diameter. Length. Charge. Coefficient. Feet. Feet. Feet. Castel, . 0.0508 0.1312 0.656 0.827 Do. 0.0508 0.1312 i-574 0.829 Do. 0.0508 0.1312 3-247 0.829 Do. 0.0508 0.1312 6.560 0.829 Do. 0.0508 0.1312 9-938 0.830 Bossut, . 0.0754 0.1771 2.132 0.788 Do. 0.0754 0.1771 4.067 0.787 Eytelwein, 0.0852 0.2558 2.361 0.821 Bossut, . 0.0885 0.1344 12.628 0.804 Do. 0.0885 0.1771 12.693 0.804 Do. 0.0885 0-3542 12.857 0.804 Venturi, . 0.1344 0.4198 2.886 0.822 Michelotti, 0.2656 0.7084 7- I 5o 0.815 Square. Do. 0.2656 0.7084 12.464 0.803 Do. 0.2656 0.7084 22.008 0.803 Mean Coefficient, 0.817 34. The mean of these coefficients gives 0.817, its value is generally taken as 0.82, so that we have the fol- lowing formulae : Q = 0.82 S\~^R = 6.56 S i/H ; and if d be the diameter of a circular orifice 38 Flow of Water through Cylindrical Adjutages. 35. In the case when the jet issues with the tube full, in threads parallel to the axis of the orifice, and when, consequently, the section is equal to that of the orifice, the diminution of the discharge can only occur from a diminution of the velocity ; and the ratio of the actual to the theoretic discharge is the same as that of the actual to the theoretic velocity. TABLE showing the Identity of the "Coefficients of Discharge'' and Velocity, with the Cylindrical Adjutage. Observers. Coefficient of the Velocity. Discharge. Venturi, Castel, . Castel, . 0.824 0.832 0.832 0.822 0.827 0.829 Mean, 0.829 0.826 The three quantities measured were the "charge" on the centre of the tube, the velocity computed by measur- ing ordinate and abscissa, as in 8, and the volume discharged. The velocity due to the charge, compared with that so computed, gives the second column, and the product of the area of the tube into the velocity due to the charge, compared with the discharge, gives the third ; that is V(= V 2gH.) : computed Velocity, : : i : 0.824, and Sx V : Discharge : : I : 0.822. We must, therefore, conclude that the velocity of a jet of water at the extremity of a cylindrical adjutage is equal to 0.82 of that due to the charge, and that the head due to that velocity is but 0.67 of the actual head of the reservoir ; that is (o.82) 2 , because the heads or charges are as the squares of the velocities. Flow of Water through Conical Adjutages. 39 36. As to the cause of this increase of the coefficient from 0.62 to 0.82, D'Aubuisson ascribes it to the attrac- tion of the sides of the tube and the divergence of the fluid threads : after they have come in contact with the sides they are forcibly retained by some such attraction as that which causes the rise of fluids in capillary tubes : by this same force the outer threads draw after them the inner, and so all the vein issues with a full tube, and passes with an increased velocity through the contracted section. The immediate cause is the contact ; and every circumstance which favours that tends to produce an augmentation of the coefficient. 37. Flow\of Water through Conical Converging Adju- tages. Conical adjutages, properly so called, that is, those which are slightly converging to a point exterior to the reservoir, augment the discharge still more than the pre- ceding. They give jets of great regularity, and throw the water to a greater distance or height, and are hence frequently used in practice : the effects vary with the angle of convergence of the sides. Two distinct contractions of the fluid vein take place with this adjutage one internally, or at the entrance of the adjutage, which diminishes the velocity due to the charge ; the other at the exterior ; in consequence of which the true section of the fluid vein is slightly less than the area of the external mouth of the adjutage. If, therefore, we put S for the section of the external orifice, Z^for the velocity due to the charge, the actual discharge will be expressed by nS x n'V '= nn'SV y the two coefficients n and n' must be found by experiment, n being the ratio of the section of the fluid at its least diameter to that of the orifice, or the coefficient of the exterior contraction, and n' that of the actual velocity to the theoretic, or the coefficient of the velocity ', and nn' 9 their product, is the ratio of the actual discharge to the 40 Flow of Water through Conical Adjutages. theoretic, or the coefficient o>i the discharge. The knowledge of these two last is of practical importance in the case of jets of water, as in fountains and fire-engines. 38. In order to determine the coefficients above mentioned, and especially to ascertain the angle of con- vergence that gives the maximum discharge, experi- ments were undertaken with a number of adjutages Fig. 14. successively, in all of which the diameter of the orifice of final issue cd, in the wood engraving, and the length of the adjutage ab, remained constant ; but in each ex- periment the diameter of entrance, and consequently the angle of convergence, were altered. The flow of the water was produced under different charges with each of these varied adjutages. At every experiment the discharge was determined by actual gauging, and the velocity of issue by the method of the parabola given above ( 8). The discharge, divided by SV y gave the product nn' and the observed velocity divided by V (= v / 2 pH)? " ave n/ - The series of the numbers nn' showed the discharge corresponding to each angle of convergence, and con- sequently the angle of maximum discharge, and the Flow of Water through Conical Adjutages. 4 1 series of n' 9 marked the progression by which the veloci- ties increased. 39. The same adjutage, under charges which varied from 0.69 feet to 9.94 feet, or from i to 14, always gave discharges proportional to A/H, and therefore the coeffi- cient, or nn' 9 has been, q, p, the same also. A very small increase may be observed with the higher charges. With respect to the coefficients of the velocity, they also should have been found constant but for the resistance of the air. Now, this resistance diminishing the throw of the jet, and that in proportion as the charge is greater, we should expect in the coefficients calculated from it a decrease augmenting with the charge although, at the same time, there was no actual diminution in the velocity with which the fluid issued, or tended to issue. TABLE showing the " Coefficients with Conical Converging Adjutages" the Angle of Convergence being that giving the maximum Discharge, as determined in the next Table. Charge. Coefficient of the Discharge = nri. Velocity = ri. Adjutage, . . . 0.0508 feet diameter. Feet. 0.705 1.584 3- 2 53 4-893 6.579 9-938 0.946 0.946 0946 0.947 0.946 0.947 0.963 0.966 0.963 0.966 0.956 Adjutage, . . . 0.0656 feet diameter. 0.692 1.584 3-263 4-9I3 6.586 9-938 0.956 0.957 -955 0.956 0.956 0.957 0.966 0.968 0.965 0.962 0.959 4 2 Flow of Water through Conical Adjutages. Let us, in the next place, compare together the coef- ficients both of the discharges and of the velocities ob- tained, with the different adjutages of one and the same series, adjutages which only differed in the angle of their convergence. Each coefficient is derived from a mean of five or six experiments taken with different charges, very nearly the same as those put down in the preceding Table. TABLE showing the Variation of the Coefficients of Discharge and Velocity with Conical Converging Adjutages at dif- ferent Angles. Angle of Coefficient of the Angle of Coefficient of the Con* Con- vergence. Discharge. Velocity. vergence. Discharge. Velocity. Diameter, . .0. 0508 feet. Diameter, . . o. 0656 feet. 0' 0.829 0.830 1 36' 0.866 0.866 3 10' 0.895 0.894 2 5 0' 0.914 0.906 4 io' 0.912 0.910 5 26 0.924 0.920 5 26' 0.930 0.928 7 5 2 ' 0.929 0.931 6 54 0.938 0.938 8 58' -934 0.942 10 20' 0.938 0.950 10 30' 0.945 -953 12 4' 0.942 0.955 12 10' 0.949 0.957 13 24' 0.946 0.962 1 3 40' 0.956 0.964 14 28' 0.941 0.966 I 5 2' 0.949 0.967 1 6 36' 0.938 0971 19 28' 0.924 0.970 18 10' o-939 0.970 21 0' 0.918 0.971 23 o' 0.913 0.974 23 4' 0.930 o-973 29 58' 0.896 0.975 33 5*' 0.920 0.979 40 20' 0.869 0.980 4 8 5 0' 0.847 0.984 The facts established in the first Table, Diameter, 0.0508 feet, are represented graphically in the two follow- ing engravings ; the upper referring to the discharges, Flow of Water through Conical Adjutages. 43 the lower to the velocities. In each the horizontal line indicates degrees, extending from o to 48, and having 0.3 inch equal to 4. On this line the degrees of con- vergence of the adjutages are laid off as abscissae from o, the coefficients corresponding being laid off on the dotted vertical lines as ordinates ; the scales for these, in which i .00 = 4 inches nearly, are given at the left-hand side commencing with 0.80, which is set to the horizontal line, as none of the coefficients were less, so that the datum line in each engraving is 3.2 inches below that given through 0.80. The curve line A, B, C, is drawn through the termination of the ordinates of discharge, and A 7 , B 7 , C 7 , through those of the velocity. DISCHARGES 1-001 0-95- B 0-90. ^/ ~^____^ 085 ~~~: 6 4 8 M lS VF 20 24 28 32 36 40 44 4-8 LOCITIES. roo- B' c' 0-95- 0'90- /{ 0'85- X 6 A Q o t 1/? 9/J 0/ 3* C An AA Afi Fig. 14. 40. It follows, both from the Table and the wood engravings, ist, That, for the same orifice, and with the same constant charge, the actual discharge, commencing with 0.83 of the theoretical, increases gradually, in pro- portion as the angle of convergence increases, up to 13^, near B, at which the coefficient of discharge is 0.95 : be- yond this angle it diminishes at first slowly, as do all variables about the maximum. At 20 the coefficient is 44 Flow of Water through Conical Adjutages. yet as high as 0.92 : subsequently the diminution becomes more and more rapid, and terminates as low as 0.65, which is the coefficient of discharge through a thin plate, this last being the ultimate position of converging adjutages, that, namely, in which the angle of conver- gence has attained its maximum, or 1 80. Thus, then, we have for the maximum discharge an angle of con- vergence of between 13 and 14. andly. In looking at the coefficients and ordinates of velocity, we see them also increasing from o, nearly as those of the discharge, up to i o ; after that they increase more rapidly ; and when beyond the angle of maximum discharge, while the coefficients of discharge diminish, they continue to augment and approach their limit of unity. They are very nearly equal to unity at 50, and even at 40, are not far from it. Thus, conical adjutages may, by varying the angle of convergence, be made to form a series or progression, whose first term is the cylindrical adjutage, and last the orifice in a thin plate ; the velocity of projection, increasing with the angle of convergence, will vary from that of the tube additional, up to that of the simple orifice in a thin plate ; that is to say, from 0.82 x v / 2 < ^ r H up to i x 3rdly. If we compare the coefficients of discharge with those of the velocity, that is, the successive values of n x n' and n', and divide the former by the latter, we shall have the values of n, or the coefficients of the exterior contraction. From the angle o up to 10, n is sensibly equal to i, and, consequently, no such contrac- tion was present in the experiment ; and, notwithstand- ing the convergence of the sides, the fluid particles issued, q, p, parallel to the axis of the cone. Beyond 10, however, the contraction becomes apparent ; it reduces more and more the section of the vein, and terminates Flow of Water through Conical Adjutages. 45 by bringing it to an equality with that of the orifice in the thin plate, as is shown here : TABLE showing the Value of n, the Coefficient of Exterior Contraction, with different Angles of Convergence. Angle. ' n, or p. n 8 i.oo 15 0.98 20 0.95 30 0.92 40 0.89 50 0.85 180 0.65 In these experiments the length of the conical ad- jutage was fixed at about 2 J times the exterior diameter, as shown in Fig. 14 ; thus it was 0.1312 feet for those of 0.0508 feet, and 0.1640 feet for those of 0.0656 feet, in order to avoid, as far as pos- sible, complicating the re- sults with the effect of the friction against the sides, in this following the analogy of the cylindrical adjutage, Fig. 15. in which experience proves that, with respect to dis- charge, they produce their full effects most certainly when the length equals 2^ times the diameter. 41. As to those very large conical adjutages, or rather truncated pyramidal tubes, which in some manu- factories on the Continent discharge water upon mill- wheels, three very valuable experiments, made at a mill on the canal of Languedoc, are given by the engineer, 46 Flow through Conical Adjutages Diverging. Lespinasse. They were, in this case, formed by the sides of a rectangular pyramid, whose length was 9.59 feet; rectangle of large end, 2.4 feet by 3.2 feet; at the smaller, 0.443 feet by 0.623 f eet - The opposite faces made angles of 1 1 38' and 15 18' ; the charge was 9.6 feet : TABLE showing the " Coefficient" of Discharge with very large Converging Mill-sluices. Discharge. Coefficient. Cubic Feet. 6.767 6.692 6.714 0.987 0.976 0.979 We see, then, how very little such adjutages diminish the discharge : that which they give is only one or two hundredths below the theoretic discharge. 42. Conical Adjutages Diverging. This adjutage, of all others, gives the largest discharge. It may be described as a truncated cone attached to a reservoir by its smaller diameter, and of which the exterior mouth is con- sequently greater than that of the entry of the water. Although not much in practical use, they present phe- nomena of such interest as to deserve some notice. The property they have of increasing the discharge was known to the ancient Romans : some of the citizens, to whom had been granted the privilege of having a cer- tain quantity of water from the public reservoirs, found, by using these adjutages, the means of increasing their supply ; and the fraud became so extensive that their use was forbidden by law, except when the distance from the reservoir was not less than about 52 feet. Venturi is the experimenter to whom we are chiefly indebted for information respecting this particular adjutage. Flow through Conical Adjutages Diverging. 4 7 43. Those which he made use of carried a mouth- piece, ABCD, not unlike the form of the contracted vein, AB being equal to 0.1332 feet, and CD equal to 0.1109 feet. The body of the adjutage varied in length and in its divergence : this last was measured by the angle con- tained between the sides EC and FD, supposed pro- Fig. 1 6. longed until they meet. These adjutages were attached to a reservoir maintained at a uniform level ; the flow took place under a constant charge of 2.89 feet ; and the time required to fill a vessel of 4.838 cubic feet was observed. The following Table gives the result of his principal observations, premising that the time corresponding to (unity as a coefficient, that is to) the theoretic velocity, was 25.49 seconds : TABLE showing the Variation of the "Coefficient" of the Discharge with Conical Diverging Adjutages at different Angles and Lengths. Adjutage. Time of Flowing. Coefficient. Angle. Length. Feet. Seconds. 3 30' 0.364 2 7-5 o-93 25.49 I.OO 4 38' 1.095 21 1. 21 4 38' 1.508 21 I.2J 4 38' 1.508 19 -34 5 44' 0.577 2 5 I. O2 5 44' 0.193 3 1 0.82 10 16' 0.865 28 0.91 10 16' 0.147 28 0.91 H H' 0.147 42 0.61 48 Flow of Water under very small Charges. Venturi has drawn the conclusion that the adjutages of maximum discharge should have a length of nine times the diameter of the smaller base, and an angle of divergence equal to 5 6' : it is represented in the wood- cut, Fig. 1 6. This, he adds, would give a discharge 2.4 times greater than the orifice in a thin plate, and 1.46 times greater than the theoretic discharge. 44. Flow of Water under very small Charges. When the charge over the centre of the orifice is very small com- pared with the vertical depth of the orifice itself, the mean velocity of the different threads of the fluid vein that is to say, the velocity which, being multiplied by the area of the orifice, gives the actual discharge is no longer that of the central thread. It differs from it in propor- tion as the charge is less : its true value will be about the hundredth part less when the charge is equal to the depth of the orifice, and by about the thousandth part when equal to three times that depth. Let us examine what theory teaches on this point ; and first, of the law which it assigns for the velocity of the fluid threads in proportion as their depth below the surface of the water increases. The italic capital H is used for the depth from the surface to the sill or bottom line ; the italic h for the depth to the top of any rectangular orifice ; and the ca- pital H for the mean depth, or H+h . Fig. 17. 2 45. Velocity of any Fluid Thread whatever. Let A A, Fig. 1 8, be the level of the surface of water in a vessel, and upon the face AB which, for greater simplicity, we sup- pose vertical let us imagine a series of very small ori- fices placed one below the other, and of which that at B is ! JL... 1.. Flow of Water under very smalt Charges. 49 the lowest, and putting H for the height AB, the velocity of the jet issuing from B will be expressed by V 2^-H ; making BC equal to this quantity, it will represent this velocity; for any other point P, taken at a depth equal to AP or x, the velocity of issue will be repre- sented by the line PM = A/ igx y and calling this jy, we shall havejy= V igx, orjy 2 = 2gx- Through all the points M so found drawing a curve line, it will, from the above equation, be a parabola having 2g or 64.4 feet for its parameter ; and thus we have this proposition : The velocity of a fluid thread issuing at any depth is equal to the ordinate of a parabola whose parameter is equal to 2g, and the depth the abscissa. 46. Let us next suppose that this series of orifices over each other was continuous, forming a rectangular slit, whose width was /, and seek now the Discharge in Fig. i 8. this case, omitting at present the "contraction." Sup- pose this opening divided into elementary rectangles by horizontal lines, the volume of water which will issue from each of them in one second will be equal to the 50 Flow of Water under very small Charges. volume of a prism whose base is the elementary rectangle, and height the velocity, or ordinate, corresponding. The sum of all the prisms will also be equal to a single prism whose base is the parabolic segment ABCMA, and height /, the width of the opening. From a property of the parabola, this segment is f rds of the rectangle ABCK, whose area AB x BC, as shown above, is equal to Hx\/2gH; thus the discharge for the rectangular opening, whose height is H and width /, is 47. Let us now seek to determine the discharge through a rectangular orifice opened in the same face, but only from B to D, and with the same width /. Let h = AD, then the discharge of the opening from A to D will be sy _ Q = - I x h x \/2gh = -l \/~2g x h \/ h. 3 3 Now the discharge from the rectangular orifice, BD x /, will be the difference of those from the openings AB and AD, each into /, and therefore That which has been established in 13 namely, Q = S V 2gH, is, substituting for S its value, S=l(H - h), on the supposition, very nearly correct, that the velocity at the mean depth was the mean velocity. 48. Mean Velocity. Let us now determine the mean /V^oV f&ITJYBRSXTTj ev Flow of Water under very small Charges. 5 i velocity ( 44), and first, that of the rectangular open- ing up to the surface. Let G be the point from which issues the thread with this mean velocity. If we make AG = 2, it will be expressed by \/ 2gz y and this being multiplied into the area of the opening, / x H, must give the total discharge, which we have already found equal to - . / . ffi/~ffl j Flow of Water over Waste-boards > Weirs, &c. 73 the discharge would be negative. In weirs short in pro- portion to the depth flowing over, the effect of the end contraction cannot be considered independently of the length. It is found by experiment that, when the length equals or exceeds three times the depth, the formula applies ; but in lengths less than this in proportion to the depth, it cannot be used with safety ; the error increasing as the relative length of the weir diminishes. The end convergence influences the discharge to a certain dis- tance from the end of the weir, as, suppose, from y to the line AB ; or from s to A'B', if the whole length of the weir is greater than twice this distance, the effect of the end contraction is independent of the length ; but if the length is less than twice this distance, the whole breadth of the stream is affected in its flow by the end contractions, and consequently the proposed formula would not apply. In practical construction it is nearly always possible so to proportion the weir, that the length may not be less than three times the depth upon it. In cases where the length of the weir is equal to the width of the channel, that is, / = Z, there is no end contraction, so that we have Z = /=/', and the proportion of the length to the depth is not material. The formula proposed by Francis, given above, de- ducts from the length / of the overfall a quantity pre- sumed to represent the end contractions, and gives the value of I' the effective length. It is, giving b and a their numerical values, At the end of his second set of experiments Francis gives The use of the fractional index 1.47, not being so con- venient, it has been increased 2 per cent., and the quan- 74 Flow of Water over Waste-boards, Weirs, &c. tity deducted from / doubled, giving results agreeing most nearly with all his experiments. If n = 2, that is, if no vertical posts, as R, are fixed on the crest, then the quantities to be deducted from /are o.i^ in the former, and Q.2h in the latter formula. 63. As to the Coefficients, first, for those having the same Widths. Resuming the consideration of Table, 60, since the discharges are for the same widths sensibly proportional to HVH, when we omit extreme cases, the coefficients ought to be very nearly constant, and we find they are so in the Table 60. In strictness, when we take the coefficients of some one vertical column of the Table, we see them commencing with the higher charges decrease by very small degrees, in most of the experiments, down to a certain charge, beyond which they increase rapidly ; thus we shall have at this parti- cular charge, which is generally about 0.328 feet, a mini- mum, given in the darker type. And secondly, with the same constant charge, we may observe that the discharges decrease, at first more rapidly than the widths of the overfalls, and afterwards less so, it follows that under the same constant charge (the widths, commencing at a first width equal to that of the channel itself, being diminished) the coefficients de- decrease down to a certain point, beyond which they augment. Thus, here also there is a minimum ; and it occurs when the width of the overfall is about the fourth part of that of the supplying channel ; and so in both the horizontal and vertical lines of the Tables of coefficients, pages 66, 67, there is a minimum ; we have, therefore, a general minimum. In its immediate neighbourhood, and for a certain distance, the variations are small ; the co- efficients in that part differ very little from one another, and may be regarded as constant. But in the other parts of the Table the differences rise to be considerable; Flow of Water over Waste-boards, Weirs, &c. 75 they exceed an eighth, or 1 2 per cent. ; so that the dis- charge by overfalls cannot be given exactly with a con- stant numerical coefficient, m, in an expression of the form H^/H. In practice, then, we would require the aid of very extended tables of coefficients, the prepara- tion of which would demand many hundred experiments. However, the study of the direction in which the co- efficients tend gives us the means of abridging this vast labour, and of determining a few simple rules suitable to the different cases which commonly present themselves. 64. Coefficients and Formula to be employed. We have seen ( 61) that the expression I* H*/H should not be employed on the one hand, when the charges were below 0.196 ft. (= 2f ins.) ; and on the other, when the transverse area that is, the depth multiplied by the width of the overfall exceeds the fifth part of the area of the section of the water in the supplying channel, the initial velocity becoming then considerable. Between these limits, the expression above given can be em- ployed with a coefficient, variable, it is true, but only varying with the width of the overfall. Commencing with a width equal to that of the canal itself, the coefficients diminish with the width of the over- fall until it has reached the fourth part of that of the channel, and then they increase, although the widths are still decreasing ; and what is very remarkable is, that the diminution of the coefficients follows the relative width of the overfall, compared with that of the canal, whilst the augmentation, which occurs afterwards, de- pends only on the absolute width. We have, con- sequently, four cases to distinguish relative to the coefficient to be used : First. In the neighbourhood of that common mini- mum we have mentioned, the variations of the coefficients are very trifling. According to the experiments made j6 Flow of Water over Waste-boards, Weirs, &c. by Castel, from a width of overfall nearly equal to the third of that of the canal, which is supposed to exceed 0.984 feet, down to an absolute width of 0.164 feet, the coefficients do not vary more than from 0.59 to 0.6 1. Taking the mean, and remarking that 5-35 ( = v^x 3 \ x .6o = 3.21, \ 3/ we shall have between the limits indicated above This formula furnishes the best mode of gauging small streams of water, as in the Examples appended. Secondly. When the width of the overfall is at its maximum that is, equal to that of the canal, extending from side to side, being then precisely similar to a weir, properly so called, the coefficients present a remarkable steadiness under the different heads. M. Castel, in his experiments upon the canal of 2.427 feet, with an over- fall or barrier of a height equal to 0.557 f ee t> had no dif- ference in the coefficients obtained with charges which varied from 0.098 feet to 0.262 feet, 60. With an over- fall of 0.738 feet high, the coefficients have only ranged from 0.664 to 0.666 for charges of o. 101 feet up to 0.242 feet : at the mean he had 0.665. And since 5.35x0.665 = 3.55775, we have, putting L for the width of the canal or length of the barrier <2 = 3.558 Zx^v^. This formula may be used with advantage in certain cases, even in large water-courses, and with charges of 0.131 feet and 0.098 feet; but to insure certainty in its Flow of Water over Waste-boards, Weirs^ &c. 7 7 use it is necessary that the charge be less than the third of the height of the barrier. Thirdly. For widths of overfall comprised between that of the channel itself and the fourth part of the same, the coefficient of the expression 5.35 IxH \/ If will vary with the relative width, that is to say, with the ratio of the width of the overfall to that of the canal of supply, and is given by the columns of this Table : TABLE showing the Variation of the " Coefficient" with the relative width of the Overfall in the Formula 55. Coefficients for Canal of Relative Widths. 2.427 Feet. 1. 1 80 Feet. 1. 00 0.662 0.667 0.90 0.656 0.659 0.80 0.644 0.648 0.70 0-635 0-635 0.60 0.626 0.623 0.50 0.617 0.613 0.40 0.607 0.609 0.30 0.598 0.600 0.25 0-595 0.598 They have been formed by taking proportionals to the coefficients deduced from direct experiments, as given in 60, a method which cannot here lead to any error. The coefficients determined by each canal have been given separately, in order to show that coefficients sensibly the same correspond to the same relative width, although the actual value of the widths was in one canal nearly double of that of the other, affording a proof that ^above 0.25, or the fourth of the width of the channel, the coeffi- cients depend on the relative, and not on the absolute 78 Flow of Water over Waste-boards, Weirs, &c. width of the overfall ; taking relative widths of 0.25, 0.6, and 0.8, and the mean coefficients, we have (0.596) (3-i88) <2=5-35 x 0.624 l*HVH= 3.338 '0.645) \3-45o; Fourthly. It is quite otherwise when this width falls below that of the fourth of the supplying canal : then, and when at the same time it is absolutely less than 0.262 feet or 0.196 feet, that of the canal has no further effect, and every particular width has its own coefficient. Thus in the canal of 1.180 feet, as well as in that of 2.427 feet, the widths, . . 0.164 ft., 0.098 ft., 0.065 ft., 0.033 ft., have given, . . 0.6 r, 0.63, 0.65, 0.67, as the respective coefficients in both canals. 65. Observations on Formula of 54. Having thus given in detail all that has reference to the simplest of the formulae for the discharge of overfalls, let us con- sider the two others, and first, that of in which h represents the quantity AD, p. 58, Fig. 20, the surface of the fluid having become curved before its arri- val at the overfall. A slight inspection of the last column of the Table given in 6 1 shows, that although the series of values viHVH-hVh does not differ much from the series of the corresponding discharges, yet that it follows them less closely than those of H^ H. Thus in this important point, the formula in the last column is not so well established as that which precedes it. It is also of more difficult application, containing an additional Flow of Water over Waste-boards, Weirs, &c. 79 term, and one whose exact determination is a matter of very great difficulty. 66. Observations on the Formula in 56. This formula, which involves a term that is a function of the velocity with which the water flowing in the canal arrives at the overfall, is well founded and of practical utility. It is evident that in the case of a high velocity, in which the flow takes place both from the charge H and from an initial velocity, w, taken at the surface, it is neces- sary to add to the charge a term depending on this ve- locity, which leads to the equation, 56, <2 = 5-35 m f l*HvH+ 0.035 w 2 . The experiments of M. Castel give the values of m' the coefficient. In these experiments the velocity w of the surface of the water in the canal has not been actu- ally measured, but it can be determined from the mean ve- locity, which is equal to the discharge Q y divided by the section of the running water in the channel of supply, which in this case is L x (H+ a), representing by L the width of the channel of supply, and by a the height of the sill of the overfall above the bottom of the channel ; and as Q vx{L(H+a}}=Q. Wehavez/ = L (H+ a)' It will hereafter be shown that the velocity of water at the surface is, on an average, a fourth part higher than the mean velocity ; so that we have " Even with this value of w which is the highest we may assume the coefficient m' differs from the coefficient m in the common formula only when the velocity in the 8o Flow of Water over Waste-boards^ Weirs, &c. channel is great enough to occasion the term 0.035 w<2 which is that which makes the difference between the two formulae to have a value comparable to that of H. As it will be in most cases very small, and as it is under the radical sign, it will only influence the value of m' by half its amount relatively to H; for example, if it is the 2, 4, or 6 hundredths of H y the coefficients, ceteris pari- bus, will only differ by the i, 2, or 3 hundredths. In these three cases the section of the sheet of water at the over- fall, or /x H y is found to be respectively equal to 0.1724, 0.244, an d -3> of the section of the canal of supply, or of L (H+ a)y whence may be deduced the conclusion of which we have already made use ( 61, 64), that when the for- mer of these two sections is less than the ith part of the latter, the coefficients m and m' will be the same within a hundredth part nearly. Such has been the case with the overfalls used by M. Castel, whilst their width has been less than the half of that of the channel. When it was greater, the term 0.035 w * nas na d more effect, and the differences became larger. But the use of this term is far from having brought to equality the coeffi- cients m and m 1 for various widths of overfall ; it has not reduced even by half the differences which occur in the values of m ; and neither the expression Q = 5-35 *^ x ffHx 0.035 nor that of can be employed with a constant coefficient, except in the case of a width of overfall equal to that of the canal of supply. In order to obtain the coefficient in this case, M. Castel dammed up the canal of 2.427 feet by means of barriers of copper, whose height was decreased succes- Flow of Water over Waste-boards, Weirs, &c. 8 1 sively from 0.738 feet to 0.104 f ee t : an d he has thus ob- tained the coefficients in this Table : TABLE showing the Variation of the Coefficient in the For- mula 56, due to circumstances affecting the Velocity of Approach. Height of sill j Coefficients m\ the Charge being of Dam above bottom of Channel. 0.26 Feet. 0.19 Feet. o.i 6 Feet. o.i 3 Feet. Feet. 0-738 0-557 0.651 0.640 0.655 0.647 0.657 0.650 o.66ol x- mean of 0.654 a n 0.426 0.650 0.649 0.652 0.656 >o.65o 0.305 0.635 0.642 0.646 0.650 1 0.246 0.647 0.652 0-655 o.66oj 0.134 0.667 0.664 0-665 0.668 0.104 0.676 0.676 0.676 0.680 Those of the five first barriers give nearly the same coefficient ; and although they do not present the same regularity which we had in ordinary overfalls, we may assume 0.650 as the mean. As to the last two barriers of 0.134 feet and 0.104 f eet > they are in a distinct class, they were very low, and in them the charges very much exceeded the height of the sill above the bottom ; so that the case was nearly as much one of water flowing in an unobstructed channel, as of passing over the sill of a waste-board. It may be remarked, also, that the near equality between the coeffi- cients for each barrier in the horizontal ^line of coeffi- cients speaks strongly in confirmation of the formula which has determined them. The experiments upon the canal of 1.180 ft., with its barrier of 0.557 ft-> have indicated coefficients whose mean was 0.654. Taking, then, the mean between this G 82 Flow of Water over Waste -boards, Weirs, and 0.650, that is, 0.652, and observing that 5.35 x 0.652 = 3.488, we shall have, finally, for the discharge, with a velocity of approach equal to w feet per second on the surface Q = 3.488 L H*/H+ 0-035 w *' The velocity w is to be determined by direct observation. 67. Overfalls, with Channels attached. We frequently find channels are adapted to overfalls : they may be con- sidered as the prolongation of their horizontal and ver- tical edges. The water discharged is now confined, and suffers a resistance from the friction of the bottom and sides, which retards the motion, and this retardation, re-acting on the water which arrives at the overfall, diminishes the discharge. The following experiments by MM. Poncelet and Lesbros exhibit the effects of this resistance. The additional channel was always 3 = 9.84 feet long, and of the same width as the overfall, o m .ao = 0.656 feet, and adjusted so as to be horizontal : TABLE showing the Effect upon the " Coefficients " produced by Channels added externally to an Overfall. Coefficient. Charge. Without added Channel. With added Channel. Loss per cent. Feet. 0.675 0.582 0.479 18 0-475 0.590 0.471 20 o-337 0.591 0-457 23 0.196 0-599 0.425 29 0.147 0.609 0.407 33 0.091 0.622 0.340 45 Mean, 0.430. The amount of diminution in the discharge from over- falls with the channel attached has therefore been so Flow of Water over Waste-boards^ Weirs, &c. 83 much the less, as the charge has been greater. From this we may infer, that with charges of 3 or 4 feet and greater, such as are often in operation at the head of large feeders and water-courses, the diminution of dis- charge due to the resistance of the bottom and sides of the channel is inconsiderable. With orifices the same experimenters arrived at results analogous to those of overfalls. They applied the additional channel, 9.84 feet long by 0.656 feet wide, mentioned above, to the exterior of the orifices from which had been derived the Table 21, where it is mentioned that the orifices were all 0.656 feet wide ; and from numerous experiments it was de- duced that when the charges, measured from the centre of the orifice, were from 2 to 2j times greater than the height of the orifice itself, the channel attached had no decided influence upon the discharge : it was the same in amount whether this was or was not present ; but with very small charges it diminished the discharge even a fourth or more. Further investigations as to the effect of inclining the channel attached to the orifice were undertaken. When the slope was i in 100, or 34 V the coefficients were found the same as when the channel was horizontal, but at i in 10, or 5 54', they were increased 3 or 4 per cent. Castel also experimented on overfalls, and on his canal of supply, 2.247 feet wide, he placed an overfall of 0.656 ft. wide, with a channel attached 0.669 ft- long"? inclined i "in 13.3 or 4 18': Fig. 25. Fig. 26. G 2 84 Flow of Water over Waste-boards ; Weirs, &c. TABLE showing the Result of Caste? s Experiments on Channels added externally to Overfalls. Charge. Coefficients. Feet. 0.364 0.526 0.311 0.527 0.249 0.196 0.164 0.527 0.528 0.530 The coefficients were obtained with the formula, They vary very slightly, although the charges were more than doubled. The mean is 0.527, and would probably have been 0.53 if the inclination had been i in 12, which is common in practice. With the simple overfall the co- efficient was 0.60 ; so that the additional channel di- minished the discharge about 1 2 per cent. 68. A particular case yet remains to be considered namely, that of the demi-deversoirs or deversoirs incomplets, as Dubuat has called them, or drowned weirs in English writers, so called when the tail water has risen above the level of the sill, Fig. 27. Dubuat divides the height of the water above the sill into two parts, Kb and 3C. In Fig. 27. the former, the flow takes place as in an ordinary overfall, in which Kb ( = H) is the charge ; so that Flow of Water over Waste-boards, Weirs, &c. 85 the discharge ( 66) is expressed by Q = 3.488 IH \/ H+ 0.035 w *' I n ^e second portion it may be as- sumed that the discharge is the same as in a rectangular orifice, of which bC is the height, and the charge equal to Kb the difference of level between the upper and lower surface of the water. bC is the height of this latter sur- face above the sill of the overfall, and if we call the height $D of the surface above the bottom of the canal, a, and the height CD of the sill above the same point, b y it will be equal to a - b. To the charge A.b or H is to be added, as in the case of closed orifices, the height due to the velocity u of the water in the canal, and the velocity of issue will be F= V '~2g(H+ 0.01553^) = V 2g(H+o. 01 ze> 3 ), For since the surface velocity is generally greater than the mean by one-fourth, that is w= 1.25 u, we w z have, squaring, w z = 1.5625 u z , and dividing = u* ; but at page 62, 56, it is shown that the additional head, h, due to antecedent velocity, is expressed by 0.01553 ^ 2 > so that h = ' ; = o.oi w z , as in the above equation. 1-5625 Thus we shall have for the discharge of this part ( 29) adding the two partial discharges, and putting Q for the total discharge Q = 3.488 IH ///+ 0.035 W* + 4.96 l(a-b] \/H+ o.oi w' z . 69. Arrangements preliminary to guaging. Perma- nent weirs constructed completely across the bed of a river may sometimes give the means of measuring the 86 Flow of Water over Waste-boards, Weirs, &c. discharge ; but it is in such cases necessary that the crown of the weir should have a well-defined edge, so that the water which flows over it may fall freely, and without meeting any check from the reaction of the body of water already passed over, as in the experiments with added channels ( 67). It is but seldom that they are so constructed ; however, we may, without great expense, adapt a weir with the usual rounded crest to purposes of guaging by raising on the crest some temporary struc- ture that shall have the necessary well-defined edge 2 and of a sufficient height, so that the discharge may not suffer from any such reaction. We must so regulate the length of this apparatus that the depth of the water, H y flowing over may not be less relatively than the fourth part of the depth of the river as it approaches the weir. In such cases the dis- charge will be given by the formula Q= 3.558 LH\/7l ( 64), L being the length of the tempory crest or edge. In case H should exceed the fourth part of the depth of the current of the stream, we must use the formula in 66, (7 = 3.488 LH\/H+ 0.03 5 w*, in which w is the velocity taken at the surface. 70. If the mode of gauging by overfalls be but seldom applicable to large rivers, it is, on the other hand, the most suitable for small streams and water-courses. We divide these into two cases : first, those in which the quantity of water discharged is at or under about 40 cubic feet per second ; and secondly, those discharging more than that quantity. A spot must be chosen where an overfall can readily be established; and in order that Q = 3.2 1 x l*H,/~H may be safely applied, it should have a length, /, greater ab- solutely than 0.3 feet, but less relatively than the third part of the width of the bed of the stream, and so dis- Flow of Water over Waste-boards, Weirs, &c. 87 posed that we may have a charge, H, greater than 0.1968 ft. ; all being, moreover, subject to the condition that the area / x H be not greater than the fifth part of the trans- verse section of the current immediately above the over- fall ; then, without fearing any error greater than the hundredth part, we may apply the formula Secondly, if the quantity discharged should exceed 40 cubic feet per second, we must pond up the water by a dam extending from bank to bank, and at each ex- tremity place vertical side-boards, so that the opening traversed by the water may be rectangular, the crest being truly horizontal, and using either of the formulae mentioned above, according to the conditions specified that is, according as there is velocity of approach or not. The two examples following will point out the manner of proceeding, and furnish an opportunity of adding some practical details, elucidating what has been laid down in 63 to 66. 71. First. Let us suppose it necessary to gauge the dis- charge of a small river or water-course ; we must search for a part suitable for the construction of an overfall. This will probably be found at a point where the bed has become contracted, and the banks are somewhat steep, and immediately below a wide part of the stream ; at such locality the width at the water surface is found to be, suppose, 1 1. 808 feet, and the greatest depth 2.62 feet. After a preliminary examination of the transverse sec- tion, and of the surface velocity, measured by means of some light body thrown into the current, we find, ap- proximately, multiplying the assumed section by the velocity, that the stream is discharging nearly 35 cubic feet per second. Since the width is 11.808 feet, we may give 3.936 feet to that of the overfall for gauging (that is, 11.808 -f- 3) ; 88 Flow of Water over Waste-boards^ Weirs, &*c. the charge .//will then be about 1.97 feet, for the formula =3.21 IH Weirs, &c. 95 rage coefficient, the overfall being of thin plate-iron and 10 feet long, is According to T. E. Blackwell, . . . '. . 0.667 D'Aubuisson ( 64. second case), 0.665. The average of the 3 feet overfall (less than one-third of the channel of supply), constructed of plate-iron 0.0052 feet thick, according to Blackwell, is~o.63 1, in the "first case; " in 64 we find 0.600 as being used by D'Aubuisson in analogous circumstances. It may, however, be observed, that the charges in Mr. Blackwell' s experiments, which give 0.631, not going higher than 0.5 feet, and the coeffi- cients decreasing up to that head, make it probable that they would have decreased much lower had the experi- ments been continued, and so reduced the coefficient 0.631 to a value nearer to 0.600. Again, in the experiments with the added channels or broad crests, we find the average of Mr. Blackwell, when the crest is horizontal, to be 0.473. The average of those in D'Aubuisson, 67, p. 82, is 0.430, but particu- lar experiments give a closer agreement in the coefficients : for instance, if we take out that derived from the charge 0.337 f eet i* 1 this last Table, the coefficient is 0.457 J an< i under the nearly equal charge of 0.333, we have the same identical coefficient 0.457 as the mean of the two given for the 3 feet and 6 feet overfalls. Castel's experiments for overfalls, with channels attached, sloping i in 13.3, give at the mean 0.527 ; the mean of those sloping i in 12, is 0.508. In these also, if we take out the par- ticular heads of 0.164 feet in the former, and 0.166 in the latter, we have the respective coefficients 0.530 and 0.532. 79. The overfalls having the sill or bar, in the first set of experiments of plank, and in the second set at Chew Magna, of cast-iron, each o. 1 66 feet thick, and with square top edges, represent a very common structure for waste weirs, tumbling-bays, &c., on artificial canals and g6 Flow of Water over Waste-boards ', Weirs, &c. feeders . The average of the first set, with charges from 0.083 ^0 i-i66, and lofeet length of overfall, gives 0.556; that of the second set is as high as 0.723. The plan and longitudinal section of the channel of approach is evidently more favourable in the latter case than the former; and the velocity of the approaching water must also have been considerable, from the cir- cumstances mentioned in 76. The very low coefficient 0.556 is not, however, readily to be accounted for; nor is it easy to assign a reason why, in the first set, the change from a thickness of 0.0052 feet to 0.166 feet in the over- fall bar should lower the coefficients to such an extent, every other circumstance being apparently the same. If we look to the coefficients of particular experi- ments we also find discrepancies, as, for instance Thickness of crest of Charge. Coefficient. Overfall. ft. in. ft. in. Length [O.OO52 = 0.166 = TV 2 0.083 = 0.083 I 0.808 1 0-435 J 10 ft. r 0.0052 [_ o.i 66 0.75 = 0-75 9 0-529 1 0-558 The overfall of T Vth inch thick, which had, compared with that of 2 inches, the nearly double coefficient the charge being one inch has, when the charge is 9 inches, one somewhat lower. Again, with the overfall 3 feet long and charge of 6 inches, the coefficient namely, 0.592, is the same with each of these thicknesses of crest. However unaccountable the above discrepancies, Blackwell's experiments, in other parts, are consistent, and confirm those of Castel, &c. ( 78), which is important, as the volume discharged was thirteen times larger than in the latter. 80. The following Table, compiled from various sources, exhibits at one view the results of different ex perimenters. Flow of Water over Waste-boards, Weirs, &c. 97 Overfall, 0.5 feet long. SMEATON AND BRINDLEY. Heads, .083 ,1042 .1146 1354 .1927 .2604 .4166 .4687 54*7 Coeffs- .713 .681 654 .638 .636 .602 .609 571 633 Overfall, 1.533 feet lon g- Du BUAT. Heads, ,1482 .2666 3887 .5627 Coeffs. .648 .624 .627 .630 Overfall, 0.656 feet long. D'AUBUISSON AND CASTEL. Heads, .098 131 . 164 . 196 .262 .328 393 459 524 590 Coeffs. .632 .624 .620 .617 .616 .617 . 620 .624 .628 .633 Overfall, 0.6458 feet long. PONCELET AND LESBROS. Heads, 033 .066 .099 1332 .1998 .2664 333 5 .666 75 Coeffs. .636 .625 .618 .611 .601 595 592 590 .585 577 First Set. Overfalls 3 feet and 10 feet long. SIMPSON AND BLACKWELL. Heads, .083 .166 25 33 .416 .50 .583 .666 75 Coeffs. .742 .738 .636 .635 .625 592 .580 529 Second Set. Overfall 10 feet long. SIMPSON AND BLACKWELL. Heads, .083 .166 25 33 .416 50 .583 .666 75 Coeffs. .608 .682 .725 745 .780 749 .772 .802 .781 Overfall, 10 feet long. J. B. FRANCIS. LOWELL EXPERIMENTS. Heads, ' 0.62 0.65 0.80 |o.83 0.98 I. 00 i. 06 !-25 1.56 Coeffs. 0.622 0.622 0.623 0.625 0.625 0-622 0.6270.623 0.62 The total volume of water which passed into the measuring tank in the Lowell experiments was between 1 1,000 and 12,000 cb. feet. In every case except the last three we may perceive that the coefficient decreases as the H ,tJII7IKSIT 98 Flow of Water over Waste-boards > Weirs, &c. charge increases. Another exception may be found by referring back to the experiments of the first set, over- fall being a plank 0.166 feet thick, in which the coeffi- cients increase with the increase of the charges ; the lengths being 3 feet, 6 feet, and i o feet ; and with the lengths of 6 feet and 10 feet attaining a maximum value at the charge of 0.583 feet from which they slightly in- crease. Experiments on the smaller differ from those on the greater "heads" in this, that they can be, and generally are, continued for a longer period of time : a measuring tank of a definite capacity being always part of an ex- perimental apparatus, the smaller discharges may be much prolonged, and thus all errors, such as in the noting the commencement and end of an experiment, are relatively diminished. 8 1 . Method of determining the Coefficient from Experi- ments. Smeaton's experiments were conducted by making observations upon the time in which a vessel of 20 cubic feet capacity was filled by the water flowing over a notch 0.5 feet long, and with the different charges given in the Table above. Thus, with 0.1042 feet charge we should have, from the formula 2 2 - IHv 2gH= - x 0.5 x 0.1042 x v/o.1042 x 8.024 = 0.08995 o o cubic feet per second; but the experiment gives 20 cubic feet in 326 seconds, or (20 -^- 326 =) 0.06135 cubic feet in one second, and 0.08995 =0.06,35 i : ,i: 0.682 (= i x So also : Du Buat had a gauging reservoir to receive the discharged water, whose area was 108 square feet French ; the water discharged by a notch (reversoir) 1 7 Flow of Water over Waste- boards, Weirs, &c. 99 pouce 3 ligne long, with a head of i pouce 8 ligne, raised the surface of the reservoir above mentioned i pouce in three minutes : hence the discharge per second was I 555 2 x 5 -* I ^ = 43 2 cubic pouce, and the formula gives 2 ? = -x 17.25 x 1.666 v/i.666x o and as with this unit 2g is equal to 724, we have A/ ' 2g equal to 26.907, the resulting value is 665.3 cubic pouce, and 665.3: 432 :: i : 0.648; and in all experiments in general, the cubic quantity discharged in the observed time is to be reduced to the quantity per second, by di- viding the former by the time expressed in seconds ; and this, the actual discharge, being divided by the result of the formula expressed in numbers, gives the coefficient by which the formula must be affected to make its re- sults coincide with actual experimental results. Mr. T. E. Blackwell used in his first set a gauging tank of a capacity of 444.39 cubic feet: we find with an overfall 3 ft. long, and a head of 0.083' f eet > that in 757 se- conds the discharge is 137.91 cubic feet, i.e. -^-=0.182 757 cubic feet per second, now -x 3 x 0.083 V/o.o83 x 8.024 =0.3836 cubic feet, o and 0.3836 : 0.182 :: i : 0.4744 = m. The average of three experiments, of which the above is one, gives m = 0.466 ; With an overfall of 10 feet long, and a head of i foot, we have the discharge equal to 442.29 cubic feet in 15.5 seconds, i. e. - = 28.535 cubic feet per second, and j'd 2 - x 10 x i v/ i x 8.024 = 53-493 ; therefore H 2 loo Flow of Water over Waste-boards, Weirs, &c. 82. Mr. Beardmore, in his " Hydraulic Tables/' has used the formula (a] ...... Q in constructing the Table headed " Discharge of Weirs or Overfalls." This formula is very nearly identical with that in 64, second case ; for as this (a) gives the dis- charge, not for any length, /, but for one foot in length only, and per minute instead of per second, as all the formulae "given in the present work, we must, in order to compare them, divide [a] by 60, and multiply by / : hence, as 214 -5- 60 = 3.566, we have '-FF 2 Q = 3-566 IHVH = -mlHVigH, *3 and consequently as - x V~2g x m = 3.556, we have m = 3 X 3 '_ = .666 S4 ; 3 2 V 2g hence, we may write (a) thus, <2 = -x/x6ox 3 . 5 66^v / ^, 3 which is adapted for any length /, and per second of time, and not per foot of length of overfall, and per minute. This author also remarks, " That the constant 214 is liable to some variation under unfavourable circum- stances : for instance, when the weir is formed of a num- ber of short bays, divided by vertical beams, grooved for sliding down the horizontal waste-boards to regulate the surface-level of top water. In these cases, the water passing the edges assumes the vena contracta form in each bay, and/ consequently, the total width, L, of the opening should be reduced to obtain the true quantity of water passing. These, and other causes which may Flow of Water over Waste-boards, Weirs, &c. 101 render the observations liable to error, must be treated with judgment, according to circumstances/' .... "The best way of gauging for the value of H in weirs is to have a post with a smooth head, level with the edge of the waste-board or sill : to be driven firmly in some part of the pond above the weir which has still water. A common rule can then be used for ascertaining the depth, or a gauge, showing at sight the depth of water passing over, may be nailed with its zero at the level of the sill of the weir. Among the conditions essential to a cor- rect result are the absence of wind and current, a good thin-edged waste-board, the water having a free fall, and a weir not so long in proportion to the width above it as to wire-draw the stream ; for in this case the water will arrive at the weir with an initial velocity due to a fall, which is not estimated in the gauging, and the result will in all probability be too small, unless it be fully esti- mated for in the formula employed. CHAPTER II. FLOW OF WATER UNDER A VARIABLE HEAD. 83. Flow of Water when the L eve Us variable upon one or loth Faces of the Orifice of Discharge. When a reservoir, instead of being maintained constantly full, as we have supposed it to fce hitherto, receives no supply, or receives less than it discharges through an orifice in the bottom, the surface of the fluid gradually descends, and the tank or reservoir is at length emptied. The laws of the dis- charge are in this case different from those which have been stated in the first chapter, and the questions to be resolved are of a different character. The form of the vessels may be also, either prismatic that is, of identical sections at every height of the sur- face or having sides sloping at some known inclination. 84. Ratio between the Velocities at the Orifice and in the Vessel. Let us suppose that the fluid contained in a pris- matic vessel be divided into extremely small horizontal sections, and that they descend parallel to each other, the particles of the fluid in each of the sections must then have the same velocity. This is the hypothesis of the parallelism of the horizontal sections, admitted, and per- haps too much extended, by many hydraulicians. Let v be the velocity of the particles in the vessel ; V that which they have at the orifice ; A the horizontal sec- tion of the reservoir or vessel containing the water ; ,5*, or rather mS, that of the orifice ; m being the coefficient of contraction, the volume of water which flows out in the indefinitely small time r will be expressed by mSVr. Flow of Water under* a Variable Head. 1 03 During this same time the surface of the water de- scends by a quantity VT, and the corresponding value of the volume of water is Am = mSVr, or v : V \ : mS : A, giving an example of that hydraulic axiom namely, that the velocities are in the inverse ratio of the various transverse sections. 85. Head due to the Velocity of the Water at its Point of Discharge. The velocity V of the issuing fluid does not now maintain the same constant rate. It is uniform only for a given instant ; for, besides being due to the actual head at the given instant, the velocity V is a consequence of the velocity v acquired during the descent of the paral- lel sections above mentioned : the two velocities acting in the same direction, from above downwards, the result F 3 is equal to their sum. Thus, if H' = be the height due to the velocity of the water at its point of discharge, H being always the actual head in the vessel, we shall have 2g 2g whence we have H' = H . ^ When mS is small compared with A, as is generally the case, m*S 2 , with regard to^4 2 , may be neglected; so that H 1 = H, that is, the velocity of issue at any given in- stant is that due to the actual head at that same moment. In this chapter it is always assumed to be so, r although the hypothesis of the parallelism of the horizontal sec- tions, however admissible in their descent, does not hold good when they have arrived near the orifice, the cir- cumstances of the movement of the molecules of the fluid become then very complicated, and are indeed entirely unknown. IO4 Flow of Water under a Variable Head. & ~& -=--^^^^: J D...jP 86. Nature of the Motion. Let M (Fig. 30) represent a vessel of water filled up to AB ; let us divide the height from B to the orifice D into a great number of equal parts, B#, ab y be, &c. Suppose, then, that a body, P, were impelled from below upwards with a velocity such that it rises to the point H, PH being equal to DB, and let us divide PH into the same num- Fig. 30. ber of equal parts. In proportion as the body rises, its velocity will di- minish, in such a manner that when it shall have arrived successively at the points a' y b', c f , the velocities will be respectively V Ha', x/H^', \/fLc' ... o, as is shown in works on the Elements of Mechanics. Recurring to the fluid contained in the vessel M, in proportion as it flows out, the surface AB is lowered ; and when it shall have successively reached the points a y b, c, the respective ve- locities of the issuing water will be ( 85) as \/Da, \/D?, o, or, according to the construction, as their . . . o; so that, in proportion equals xHd/, \H', c . as the vessel is emptied, the velocity of the discharge will decrease down to zero, following the same law as the velocity of the body impelled from below upwards, each being an example of an uniformly retarded motion; consequently, the discharge also will be governed by the same law. It will be the same, also, in the descent of the surface of the water in the vessel, which will be uniformly re- tarded, its velocity being in a constant ratio to that at the orifice, namely, as the section of the orifice to the area of the surface of the water. 87. Volume discharged. According to the laws of an Flow of Water under a Variable Head. 105 uniformly retarded motion, when a body, starting with a certain velocity, loses it gradually until it is reduced to zero, it only describes one-half the space it would have traversed in the same time if it had moved uniformly with the velocity with which it commenced the motion. Now the volume of water which flows out from any vessel until it is all discharged may be regarded as a prism, whose base is the orifice, and height the space which the first issuing particles would describe, with a uniformly retarded motion identical with that by which the dis- charge takes place ; but if the same particles had always preserved their initial velocity (which is that due to the primary charge), the space described in the same time, or the height of the prism, and, consequently, the volume of water discharged, would have been doubled. Hence this theorem : The volume of water which passes through an orifice at the bottom of a prismatic vessel, receiving no supply, and therefore becoming empty, is only one-half of that which would be given during the time of complete discharge, if the flow had taken place under a constant charge equal to the primary. 88. Time which is required to empty a vessel. Let H be this charge ; A the horizontal section of the vessel ; T the time which it may require to be completely dis- charged. The volume of water discharged during this time that is to say, all the water the vessel contains (above the orifice) is Axff. The volume, according to the theorem above, which would have been discharged in that time under the constant charge H y would have been 2 ( A x H}. This same volume, or the discharge during the time T, is also equal to mSTVzgH. We may use the Italic capital H instead of the Ro- man H, conventionally applied hitherto in this formula ( 44), since the orifice is now supposed to be in the bot- io6 Flow f'f Water under a Variable Head. torn of the vessel, and therefore //and H are identical. Equating these two values, we have and solving for T, we have T- __ and dividing above and below by V H y we have, finally = 2 x mS If we represent by T r the time which the volume AH would take to flow out under the constant head H y we should have had ( 14) mST' V*> or T' = mS A/ 2g consequently, T = 2 T' ; that is to say, the time which a prismatic vessel takes to be completely discharged is double that in which the same volume would flow out, if the head had remained constantly the same as it was at the commence- ment of the discharge. 89. Time which the Surface of the Water takes to descend a given Depth. Let t be the time sought in which the level descends the given depth a : now the time in which the whole volume would be discharged is ( 88) the head at the commencement being H \ and putting H-a = h for the head at the end of /, we have the time in which Flow of Water under a Variable Head. 107 the volume hA would be entirely discharged equal to Now, the time /, that in which the surface descends a height equal to a y is evidently the difference between the two expressions given above, that is .... mSv 2g mSv 2g mSv ig 90. Volume discharged in a given time. The above expression for the time which the water requires to descend any given height, by a simple transformation, gives both the value of a y and also the volume of water discharged during the given time : thus we have from the equation (a) and tmS^ 2g 2A hence, squaring both sides of the former equation hence A * \ qA ) Hence, substituting for \fh its value given above, and io8 Flow of Water under a Variable Head. multiplying both sides by A, we have the discharge Q' for the given time - h} A = 9i. Mean hydraulic Charge. A prismatic vessel re- ceiving no supply, discharges 1 through an orifice S, during T seconds, having at the com- i mencement the head H, at jj the end h ; required the mean | hydraulic charge H', by which, I cceteris paribus, the same * quantity is discharged : we have ( 14) substituting this value of T in (b\ we have Clearing of fractions, and dividing, we have- H-h or \ 2 COR. If // = o, then H' = H ~- 92. Case of a prismatic Basin receiving a constant Flow of Water under a Variable Head. 109 Supply while discharging. Let q be the volume received per second (less than that discharged), and x the space the water surface lowers in the time t : then dx will be its descent during the indefinitely small interval dt, and thus Adx will express the volume flowing out during dt, if no supply entered ; but as it receives q in one sec., and, therefore, qdt in dt, the actual discharge will be Adx + qdt. From 14 we thus have (a) . . . Adx + qdt = mSdt\/ig(H - x] putting H - x = h y and therefore - dx = dh, we have (b) . . . qdt- Adh = mSdt^/ig V ' h, which gives -Adh /T v h - q In order to integrate this equation, we may put [d] . . . mS*/2g V ' h - q = y y and thus the integral of which is y + q hyp. Determining the value of C for the commencement of the motion, when t = o and x = o, and /7also being equal to h, we have, substituting for y its value above, C equal to j (mS-/*g /H- q + q hyp. log mS +/2g VH~ q] . 1 1 o Flow of Water under a Variable Head. Hence t is equal to mSV ' igh- <[) It is evident from this expression that when q = o, that is, when no supply is flowing in, it becomes identical with that in 89. If we had to determine the height which the level of the water would descend in a given time, the question would be reduced to this other namely, to find the charge h at the end of this time, and subtract it from H y the head at the commencement of the discharge. To obtain h we must substitute successive values of it; i. e. of (H-x), in the equation given above, and thus tenta- tively determine that which satisfies the equation. 93. Case when the Water is discharged over a Weir. In the case when the water issues from the basin by an over- fall, supposing that it receives no supply, we shall have, from what has been laid down in 46 and 55 fy J * -. Adx = ml {H x) dt \/2g yTT x, 3 whence, by a method analogous to that which has been used above, we have 94. Reservoirs not being prismatic. We have hitherto considered only the particular case of prismatic basins or reservoirs : the determination of the time of discharge for any other form is much more complicated, and is even Flow of Water under a Variable Head. \ 1 1 impossible in most cases which present themselves. The fundamental equation is, however, always = mSdt *g (H- from whence we have Adx dt VI. But here A is variable, and we must, in order to inte- grate, express A in terms of x, which can only be effected when we know the law by which A decreases, and in the cases where the basin itself is a solid of revo- lution, whose generatrix is known. In every other case it will be necessary to proceed by approximations and by parts. To this end, we must divide the basin or reser- voir into horizontal sections of small depth. Each of these may be taken as prismatic, and we can determine the time it takes to be discharged by the aid of the for- mula given above. The sum of these partial times will give the time that the surface of the water takes to descend a height equal to the sum of the heights of the prisms. 95. Flow of Water when it is discharged from one reser- voir into another. i st. In the case when the orifice is covered with water on both faces, the levels remaining constant, the quan- tity discharged is the same as if it had been into the air under a charge H- h, equal to the difference of the charges upon each face ; thus we have, representing by Q the discharge per second, 2ndly. Let the level remain constant in the upper I 12 Flow of Water under a Variable Head. basin, and the lower, of a given area, receive the dis- charge ; required the time in which it reaches the level of the upper basin or a given height. This problem is the inverse of that in 89, in which the surface of the water descended with a uniformly retarded motion. In the present case, the surface of the lower basin rises with a uniformly retarded motion. Let // represent the charge AC (Fig. 32) at the commencement, and h the charge AD at the end of the time /, A the horizontal sec- tion of the vessel being filled, and S and m as before, we shall have, for filling up to DE, and Fig .3*. mS for filling up to AF, m These latter formulae are of some importance : they serve- to determine the time in which canal locks, &c., may be filled, and to assign the area of sluice-way re- quired to fill a certain volume in a given time. 96. The Level of the Water being variable in each Ves- sel, We now come to the third case that can arise, namely, when two reservoirs of different level communi- cate with each other, each being limited in area and re- ceiving no supply, and thus one surface descends as the other rises. Such is the case of the two basins K and L (Fig. 33), communicating by a wide pipe EF, provided with a sluice-door or cock at G. Before the opening of this sluice-door the level of the water is at AB in the first reservoir, and CO in the second. At the end of a certain Flow of Water under a Variable Head. 113 time after the opening of the communication it has de- scended to MN in the first, and has ascended to PQ in the second. It is re- quired to determine the relation between these two heights at a given time, or, vice versa, from the given difference in the respective heights, to determine the time corresponding to a given discharge. Let / equal the given time, BE = H y CF = h, NE = x, PF =y, A = horizontal section of the first vessel, and B that of the second, ^ = section of the pipe of communi- cation : in the coefficient m we must include the resistance of the water passing through this pipe. Whilst the water has risen in the second basin by the quantity dy, during the instant dt y it will have lowered in the other by dx ; and remembering that x diminishes while y and / in- crease, we have Adx = - Bdy and ( 14), ~l_~Z_~IJ"~Z_n~-IL~Z_J7_~ N _ K P L E n F g- 33- (a) ... from whence Adx = - ms (x -y] . dt, Adx ms x -y) The first equation being integrated, remembering that when x = H y = hiv?Q have solving for y, we have A(H-x) and substituting this value of y in the preceding equation I 1 14 Flow of Water imder a Variable Head. (b\ integrating, and observing that // = x when / = o, we have (H-h)- If it were required to find the time in which the two surfaces would be at the same level, w r e should have from (c] AH+Bh *=y^w> and, this value of x being substituted in the above ex- pression for /, will give ,_ (d) mS\/ig(A+B} From whence it is evident that for the same value of (H - h} the time / is the same whether A be the horizontal section of the basin that lowers, and B that of the other, whose surface rises, or, vice versa, B that which falls, and A that which rises. EXAMPLES AND PRACTICAL APPLICATIONS ON CHAPTERS I. AND II. 97. THE following Rules, approximately true, may be found useful in every-day practice. It is important to know how they are derived, and thus be able to reproduce them, as no book may be at hand for reference, and the memory may fail. They all depend upon the volume and weight of water in relation to the weights and measures of the United Kingdom. The statical pressure, i. e. of still water, in any pipe, or on the bottom of a tank, is qp, equal to 3lbs. per square inch for every 7 feet head. Thus a main laid across a valley is, let us suppose, at the lowest part, 130 feet below the surface from which it is supplied. From the rule 130 -^ 7 = 18.57 a nd 3 x 18.57 = 55-7 Ibs. per square inch; this result is about the T ^th part too small, it should be 56.26 Ibs. If, on the other hand, we had a known pressure of water, suppose of 38.5 Ibs. per square inch, to determine the vertical head in feet ; by the Rule .38.5 -f~ 3 = 12.833 and 7 x 12.833 = 89.83 ft. The exact result is 88.956 ft, so that when the pressure is given, the result is about the T ntjth part too large. Since a cubic inch of water weighs 252.458 grains, a column one foot high and one square inch in base weighs 3029.496 grains, which, divided by 7000 to reduce it to I 2 i i 6 Examples and Ibs. av., is 9 = f th Ibs. nearly, the exact fraction 7000 being - , or 3.03 Ibs. for 7 feet, which is one per cent, greater. In the same manner the longitudinal bursting pres- sure of water in a pipe per inch of length is found by multiplying the diameter in inches into the pressure per square inch that is, f x H ft. x D ins. Thus, if the di- ameter of a pipe be 26 inches, and H, as above, we have (using the exact result) 26 x 56.26 = 1462.8 Ibs. When computing the resistance against the plunger of a forcing pump in motion, it is usual to take half the height in feet for the pressure per square inch that is, ths of a Ib. av. per ft. of height. Thus, to force water to the height of 47 feet we have 23 J Ibs. per square inch resistance ; this gives a fair allowance for friction, pass- ing through valves, &c. In pumping engines for mines it is useful to be able readily to compute the total weight of water in the ver- tical pipe at any lift, from that per yard or per fathom (= 6 feet). For this purpose ; Square the diameter in inches and the result is nearly equal to the Ibs. per yard vertical, and for the fathom multiply this by 2 ; or per foot use 3 as a divisor. Thus, in a pipe 13 inches in diameter and rising 40 fathoms we have 169 (= i3 2 ) x 2 = 338 Ibs. and 40 x 338 = 13520 Ibs. The exact multiplier is 2.0454, giving a re- sult a little more than i\ per cent, greater than the ap- proximate rule. In all these the number of gallons is found by cutting off, from the number expressing the Ibs. weight, one figure for decimals ; thus in the length of 40 fathoms of the above pipe we have 1352 gallons, to which, adding 2\ per cent., or 27!, we have 1380 gal- Practical Applications. \ 1 7 Ions. To prove the rule we have, putting d for the di- inches, and 1 ** x 3 2>5 ameter in inches, and for , 14 4 X 1728 weight in Ibs. per yard, or as the multiplier of d? is equal to - , which being divided out equals 1.023, we have d* x 1.023 = weight in Ibs. per yard. The numbers 62.5 and 6.25, the former the approxi- mate number of Ibs. in a cubic foot, and the latter the number of gallons in the same, may, for facility of computation, be written -r , and , the division ID 10 by 4 x 4 being very easy. Hence these Rules. First. To change cb. ft. of water into Ibs. Add three places and divide by 16. Thus the number of Ibs. in 347.7 cb. feet is 347700 ~ (4 x 4) = 21731.25 Ibs. And Secondly. To change cb. feet into gallons. Add two places and divide by 16. The number of gallons in 893.47 cb. feet is 89347 -f- (4 * 4) = 5584-2 gallons. A distributing reservoir contains 21,450,000 cubic feet. Compute the number of days it would supply a town requiring 1 1,000,000 gallons per diem : 2,145,000,000 (4x4)= 134,062,500, and this, divided by the required supply of 11,000,000 gallons, gives 12 days. Let it be required to calculate the number of cubic feet which an impounding reservoir should contain so as to be able to supply 25 gallons per diem to each person in a population of 85,000 for 200 days. For the above rate of supply we have this rule. Multiply the popu- 1 1 8 Examples and lation by the assigned number of days and by four 85000 x 200 x 25 x = 85000 x 200x4 = 68 millions cb. ft. If the number of days assigned were 250, then, add- ing three cyphers, we have the result. Thus 126,000 inhabitants require a reservoir containing 126 millions of cb. ft. to give 25 gallons per diem for 250 days. If, Thirdly, it were required to compute the number of cubic feet in, let us suppose, 337,489 gallons, multiply by , the reciprocal of 6. 25, that is, cut off two places as decimals and multiply twice by 4 ; 3374.89 x (4 x 4) = 53998.24 cubic feet. A tank contains 1,457,965 gallons, to what number of cubic feet is that equal ? 14579.65 x (4 x 4) = 233274.4 cubic feet. To change cubic feet per minute into gallons per diem. Multiply by 9000. For i cb. ft. x 60 x 24 x i x i5x6x 1 00 = 9000. The well-known definition of Horse-power, that is, 33,000 Ibs. raised one foot high in one minute, renders it easy to compute the power of any stream for mill work when once its discharge is known. It will be necessary to have a weight of about 45,000 Ibs. of water per minute falling one foot to develope one horse-power at the point of application of the power, and this requires 720 cb. feet per minute, or 1 2 cb. feet per second, falling one foot. 9 8 . Questions solved by means of the Formula mS\/2gH. = Q, the Charge on the centre being represented by H. (I.) In order to obtain a comparative view of the effects result- ing from the use of the different coefficients for the dis- charge through various orifices, given in 18 to 43, to Practical Applications. \ \ 9 which we first confine our attention, let us take a circular orifice of 0.25 ft. in diameter, the area S being therefore o.25 2 x 0.7854 = 0.04909 sq. ft., and determine :- First, the discharge through it in some given time, as 40 minutes, with a constant charge of, suppose, 9 ft. above the centre of the orifice ; and, secondly, with the same orifice and charge, seek the different intervals of time required to discharge a given volume of water, as 1000 cubic feet. As the charge is so great compared with the diameter in the above data, we may use the formula ( 14) in which H is the charge on the centre. In the first case mentioned above we calculate the value of which becomes 0.04909 sq. ft. x 40 min. x 60 x 8.024 (= \/2g) x 3 (= x/H) = 2836.067 cb. ft., and multiply it by the several values of m, as is done below. For the se- cond case we have / 7= x = T seconds ; SVig -s/H m the value of the first factor of the left-hand side is i ooo _ i ooo 0.04909 x 8.024 x 3 = IHiT7 = which must be divided also by the several values of m to obtain T, the time required to discharge the given quan- tity. Value of Q' in 40 min. Value of T, m x 2836.067. or 846.24 cb. ft! -f m min. sec. (i) w(i8) = o.5o . . . . 1418 cb. ft. . . 28 12 internal tube (2)m( i8) = o.62 . . . . 1758 .... 22 45 thin plate. (3) ( 34) = - 82 2 325 .... 17 1 2 cylindrical adjutage. (4) ( 4) = '95 ..... 2694 .... 14 51 conical converging adjutages. (5) ( 18) = i.oo . . . . 2836 .... 14 6 form of vena con- tracta and conl. con- verging. (6) m ( 43) = 1.46 . . . . 4140 .... 9 39 conical diverging ad- jutages. 1 20 Examples and (II.) Required the discharge in six minutes, through a rectangular sluice 3 ft. by i ft., the side 3 ft. long being horizontal, the depth to the sill from the surface being 7 ft., and m being equal to 0.62. Here 0.62 x 3 sq. ft. x 8.024 V / 6^5 = Q> and ^6.5 = 2.5495 may be taken equal to 2.55, hence Q = 38.06 cb. ft. per sec., and Q / = 6 x 60 x 38.06 = 13701.6 cubic feet. (in.) A reservoir having at full water a depth of 40 feet over the centre of the discharging sluice, whose area is 2 feet horizontal by i .5 ft. vertical when fully opened : Required the discharge at that depth, and also when the water has sunk to the heads, 30 ft., 20 ft., and 10 ft., the value of m being taken at 0.62 in each case, we have S = 1.5 x 2 = 3 sq. ft, and v/4O~, \/^o,^/2o, and v^o, being respectively 6.324,5.477, 4-47 2 > 3- l6 2. We must multiply these numbers successively by 0.62 x 8.024 x 3 = 14.92464, which is the same in each. Hence, for 40 ft. head the discharge is 94.384 per sec. ; for 30 ft., 81.742 cb. ft. ; for 20 ft., 66.743 cb. ft. ; and for 10 ft., 47.192, or the half of that for 40 ft. ; 3.162 being necessarily half 6.324, as they are the roots of numbers in the ratio of i to 4. This question points out the fact that leakages of sluices in lock-gates, &c., increase far less rapidly than the head, being, in fact, as the square roots of the charges. ( Vide Smeaton's Reports, vol. i., pp. 196-9)- Practical Applications. 1 2 1 (IV.) What is the discharge through a circular pipe 4 ft. diameter in the embankment of a reservoir, the head upon the centre being 90 ft., m being taken equal to 0.60 ? In this case S = (4) 2 x 0.7854 = 12.5664 and VQO = 9.487, hence 0.6 x 12-5664 x 8.024 x 9.487 = 573.9 cb. ft. per sec. (v.) A rectangular sluice, sides 4 ft. horizontal and 3 ft. vertical, having a charge of 20 ft. on the centre, is raised i .5 ft. : required the discharge per sec., and also when fully opened. We have the value of S in the first instance, one-half that in the second, but the heads to the centre of the orifice are 20.75 ft. and 20 ft. respec- tively ; and assuming that m = 0.62 answers the particu- lar circumstances of this orifice we have, first 0.62 x 8.024 x 6 x V 20.75 (=4-5552) = 135-97 cb. ft. ; and secondly, 0.62 x 8.024 x 12 x A/20 (= 4.472) = 266.97 cb. ft. The double of the former would be 271.94 cb. ft. (vi.) In cities in which water is supplied at high pressure, and constant service, it is sometimes usual to give the water to manufactories and works through a very small orifice, perforated in a disc, which is closed up and secure from any possibility of unfair interference. Calculate the discharge through an orifice 0.0089 in * di- ameter for 24 hours, the head being 129 ft. and m equal to 0.62 ; we have log. m -f log. ,5*+ \ log. [ig] + \ log. H +log. 86400" = log. Q' 9 the log. of S being 2 log. 0-0089 + log 1 - o -7^54, we have thus Q' = 303-655 cb. ft. 1 2 2 Examples and (vn.) Suppose the pressure on the mains to be mea- sured by a head of 1 50 ft. of water, and the diameter of the orifice 0.02 ft. : required the quantity delivered in 24 hours, the coefficient of discharge being 0.62. The V 1 50 being equal to 12.247, and *$*= (o.oa) 2 x 0.7854 = 0.00031416, we have ( T x m x ,5* x VTg x -/H Q'= J24 h x 3600" x 0.62 x 0.00031416 x 8.024 x 12.247 = 1653.7 cb. ft. (vin.) What must be the diameter of the orifice to give 600 cb. ft. per diem, the head on the main being 100 feet ? Here c 600 (= (20 600 S = - - - -r ~ - = - - - =0.0001396 sq. ft, 24x3600x0.62x8.024x10 4298300 and as S = d* x 0.7854, we have , 10.0001396 / = ^0.7854 = ^ - OOOOI 77744 = 0.004216 ft. which is a little more than ^th of an inch. As the exact adjustment of this diameter would be nearly impossible, the process is somewhat tentative. (IK.) In the sluices constructed in tidal harbours for scouring away at low water the silt that generally accu- mulates in them, we obtain examples on a very large scale of the discharge of water through orifices. This simple remedy for a defect that had rendered nearly useless some of the most important tidal harbours on the coast of England, which had not the advantage of any sufficient natural streams to keep them open, was Practical Applications. 123 introduced by the great Smeaton from his personal obser- vation of the practice in the Low Countries (vide Reports, vol. ii., p. 202-209). A bank thrown across some part, covered at high tide, impounded the water allowed to enter during the rise of the tide, and which at low water is discharged very rapidly through sluices constructed in this embankment, the sills of which are placed at low water of springs, or as low as possible. The practice subsequently fell into disrepute, as it was found that the area of the back-water was itself soon silted up ; but the same engineer adopted the simple and efficient remedy of dividing the back-water into two sepa- rate areas by a second bank at or about perpendicular to the first mentioned, and by occasionally using one of these to cleanse the other, they were both, as well as the harbour itself, kept clear. Ramsgate and Dover are well-known examples (vide Smeaton's Reports and Sir J. Rennie on Harbours) ; from which last-mentioned work we take an example from the description of Hartle- pool Harbour, on the coast of Durham. Each sluice was 3 feet wide and 6.33 feet high, having a charge estimated at 10 ft. on the average. From the detailed plans of these works given by Sir J. Rennie, we may consider the coefficient 0.600 applicable ; hence 0.600 x 3 x 6.33 x 8.024 x v/io = 289.14 cb. ft. per sec. is the discharge for each sluice ; and as it is also stated that the total area of the scouring sluices was 366sq. ft., of which 24 sq. ft. were given by four sluices, each 3 ft. by 2, in the lock-gates, which communicated with the back water or slake, we have 342 sq. ft. left for those through the embankment ; and each of these being 3 x 6.33 = 19 sq. ft., we have their number 18, i. e., 342 -r- 19 ; and the discharge for one being 289.14 cb. ft., 124 Examples and the total discharge is 18 x 289.14 = 5204.52 cb. ft. per sec., or 312,271 cb. ft. per minute. Now the back-water containing 1 5,420,000 cb. ft., it could be discharged in about 50 minutes (15420000 -7-312271). It is essential that the back-water should be discharged rapidly before the rising tide diminishes the force of the artificial scouring action. (x.) The widely different statements as to the effi- ciency of hydraulic prime movers, some being asserted to give as high as 80, and others 60 per cent, of the power used, may, perhaps, be traced to a false estimate of the actual discharge ; for unless this be gauged, it must be calculated, and some coefficient used. In the case of undershot wheels with sloping sluices, as in Poncelet wheels, the bottom and sides being in continuation of the channel of supply, the coefficient is 0.74 when the sluice is inclined i base to 2 height, and 0.80 when i to i (Claudel, Aide Memoire, p. 78, 100). If we had taken it 0.62, and with a six feet fall, the sluice being sup- posed 6 ft. wide and raised i ft., we have 0.62 x 6 x 8.024 x ^6 = 73 cb. ft. ; had the modulus been taken as 80 per cent, from this discharge : it would, in reality be but 67 per cent, found by the proportion 0.74 : 0.62 : : 80 : 67. (xi.) If the modulus of a water-wheel be estimated at 88 per cent, with a coefficient of discharge of 0.65, the wheel being 7 ft. broad, and the sluice, which slopes at i to i, raised 0.75 ft, the head being 5.5 ft. : required the true modulus. Here 0.65 x 0.75 x 7 x 8.024 x ^T-} - Q = 64.2 cb. ft., hence (as 0.80-0.65 = 0.15), Q (i + 0.15) : 64.2 : : 88 : 76.56 per cent., Practical Applications. 125 the true discharge on the wheel being 73.8, that is, 64.2 x 1.15. (xu.) Relative Level in two Vessels communicating by a submerged Orifice. Let a cistern, A, receive a constant supply of water, and discharge it into a vessel, B, through #, which finally discharges into the air : the orifice at b is i.o foot horizontal by Fig. 34- 0.2 feet, the charge H upon the centre 1.25 ft., and m=o.6i ; hence Q = o 62 x 8.024 x 0.2 x i.i 18 = 1.11238 cb. ft. per sec. which must equal the supply received by A, and transmitted through a to B. Now a is 0.8 ft. by o.i ft., the sluice being capable, however, of being raised to 0.5 feet ; and hence the charge upon it, reckoning from the surface of B, is equal to = 7.8o6ft; ^0.62 x 8.024 x o.o8 ( and as we should expect the square roots of these charges are inversely as the areas of the orifices ; that is, 2.794 : 1.118 : : 0.2 : 0.08. Hence, if we suppose the constant supply to be so increased as to raise the surface of the water in A one foot above its level in the last case, we may determine the corresponding rise in B, and also the additional quantity that has been supplied. The total vertical height above the centre of b is now 1.25 -t- 7.806+ i = 10.056, which has to be divided into parts whose square roots have the ratio 0.2 to 0.08, that is, of the areas of the orifices. Now (o.2) 2 : (o.o8) 2 being as 4.00 to 0.64, we have 4.64 : 0.64 : : 10.056 : 10.056 x 0.64 4.64 = 1-387 1 26 Examples and for the surface of B above 6, and 10.056 - 1.387 = 8.669 for tjie surface of A above that of B, and the quantity re- ceived in A is now i . 172 cb. ft. per sec. The rise of i foot in A corresponded, therefore, to one of (1.387 - 1.25 =) 0.137 feet in B. If we suppose the surface of A lowered i foot, then B descends 0.1388 ft., and the constant supply is now 1.05 cubic feet per second. Hence the total range of B is only 0.277 feet for tne corresponding change of 2 feet in A. (xin.) The time of filling a lock on a navigable canal consists of two distinct intervals : one, the time of fill- ing up to the centre of the sluices ; the second, that of raising the surface up to the level of the upper reach. The length of a lock being 115.1 ft., and breadth 30.44 ft., the horizontal area is 3503.6 square feet, and the ver- tical depth from centre of sluice to lower reach 1.0763 feet, the charge being 6.3945 feet; hence, the cubic con- tent of the lower portion, that is, the value of Q', is 3771 cubic feet; the area of the two sluices 2 x 6.766 sq. feet - T 3-53 2 SO L- feet; and the charge on centre, as above, 6.3945 feet; the value of m, assumed by D'Aubuisson, being 0.548. From some experiments on the Canal of Languedoc, it was found that when two sluices were opened in the gates, the discharge was not double that given when only one was used : it was found, in fact, to be about an eighth part less, which reduces m from 0.625 ( 24) to 0.548. We have therefore ( 377 *cb. ft. m . S. Vig. V/H o.548x 13.532x8.024x^6.3945 99. Determination of the Charge necessary to give a cer- tain Quantity with a given Value of S. To determine the head necessary to give a certain discharge, we have but Practical Applications. 127 to solve Q = mS v/2 A/ETfor H ; and hence s Vigt (xiv.) Required the head necessary to give 7.85283 cb. ft. per sec. through an orifice 0.5 feet square, m being equal to 0.625. Here /'7.8c283\ 2 , r - - - = 6.324 2 = 40 feet, Vi. 24175; or 2 (log. 7.85283 - log. 1.24175) = log. H ; that is 2 (0.8950245 - 0.0940167) = 1.6020156 - log. of 40. If the orifice had been 0.75 feet square, determine the charge necessary to give the same discharge as in the last example, namely, 7.85283 cubic feet per second. Here and 2 log. 2.7837 = 2 (0.4446224) = 0.8892448, giving H = 7.748984 feet = 7.749 ft. What additional head would each orifice require to discharge 10 cubic feet per second, the coefficient re- maining 0.625 ? Here 7.85283 : 10 : : v/^o": Q 3 ' 2 Q = log. 63. 24 -log. 7.85283 7.8528 (= 1.8009919 -0.8950245) =0.9059674 = log. of 8.053, which is the square root of the charge required, whose value is therefore 64.85 ft., and, deducting 40, we have the increase of head equal to 24.85 feet. And, 7.85283 : 10 : : ^7.748984 : 128 Examples and - log. 7.85283 (= .1.4446224 - 0.8950245) - Q-5495979 = log. 053.5448, which is the square root of the charge sought, 2 x 0.5495979 = 1,0991958 = log. of 12.575, from which deducting 7.748984, we have the additional head in this case equal to 4.826 feet. (xv.) Calculate the head that is equivalent to the difference between the coefficients 0.600 and 0.950 ; that is, havingthe discharge under certain data, with ^ = 0.950 ; determine what additional head would be required to give the same discharge when m = 0.600. Thus, let the charge on the centre be 8.55 feet, the orifice circular and 0.045 feet diameter, and so nearly the form of the vena contracta that the coefficient rises to 0.950 ; we have there- fore S = (o.o45) 2 x 0.7854 = 0.00159, also \/8.55 = 2.924 and Q** 0.95 x o.ooi59x 8.024 x 2.924 = 0.03544 cb. ft., and the head necessary to give this discharge with m = 0.6 is found (as mS */ ig = 0.6 x 0.00159 x 8.024 = 0.007655) by- = H ' or > ^ logarithms, \m .SVig) Vo.007655 03544 V _ _ (2.5494937 ~ 3-8839452) x 2 - 1-331097 = log. 21.43 feet. Thus 21.43 - 8.55 = 12.88 feet is the additional head or pressure required to discharge the same volume of water through the orifice in a thin plate that was discharged with 8.55 feet pressure through an orifice nearly of the true form. Thus, the accelerating force due to this form, when compared with the thin plate, is measured by a pressure equal to more than one-third of the weight of the atmosphere. Practical Applications. 129 i oo. Results of the Suppression of the Contraction on part of the Perimeter, 25 to 26. A sluice 3 feet square, and with a charge on the centre of 1 2 feet, has, from the thickness of the frame, the contraction suppressed on all sides when fully open ; but when partially opened, the contraction exists on the upper edge, that is, against the bottom of the gate, which is formed of a thin plate of metal. Required the discharge when opened i foot, and also 2 feet, and when fully opened. (xvi.) When opened i. foot high, the total perimeter is 8 ft., and the part on which the contraction is sup- pressed is 5 feet : hence _ = 5 / 8' Hence, from the formula ( 26) _ f AA \ m . S . V 2^- an d / x H ** 2.789 sq. ft. common to each ; and thus For No. i, we have 3-315 x 2.789 x ^1.6702 = i2.oocb.ft. No. 3, 3.315x2.789x^2.505 =14.63 No. 5, 3.315x2.789x^6.6808 = 23.90 No. 6, 3.315x2.789x^10.02 =29.26 Or generally, the discharge varying as / . H . \/ffj and / . If being constant, it is evident that it increases as \/~H\ so that, by increasing the depth indefinitely at the ex- pense of the width /, we increase the discharge. Thus, let If = 1 6, the log. of f /^//7\/2^ that is, of (3.3 15 x 2.789=) 9.2455 being 0.9659304, we must add to it half the log. of H for the log of the discharge: half the log. of 16 is 0.6020600, and adding, we have 1.5679904 = log. of 36.982 cb. ft. 102. Questions on | ml ^2g . (H \/7l '- h\/~ti] = <2, the Italian dimensions being all reduced to English mea- sures. (xvill.) Ignazio Michellotti having determined to modify the mode of measuring a ruota introduced by his father, F. D. Michellotti, which had the upper edge level with the surface of the supplying canal, and was estimated to give a discharge equal to 11.83 c ^- ft- P er sec - defined Practical Applications. 133 the uncia or inch of water to be that flowing through an orifice 0.5567 feet high, 0.41755 ft, wide, and having a pressure on the upper edge of 0.5567 ft. This he sup- posed would give the twelfth part of 11.832 cb. ft., or 0.986. Calculate its true value : m being 0.600, we have then H= 0.5567 + 0.5567 = 1.1134 feet, and /= 0.41755 feet. Ans. 1.02 cb ft. (xix.) The measure used on the Canal Lodi was defined to be 1.12 ft. by o.i24i6 / ft. wide, with a charge on the upper edge 0.32 ft., and these dimensions were supposed to give 0.77 cb. ft. per sec. Here H = 1.12 +0.32 = 1.44 ft., and /= o.i 2416' ft. Ans. 0.6165 cb. ft. per sec. (xx.) That used on the canal of Cremona was 1.31816' ft. high by 0.131 ft. wide, having a head also 0.131 ft., and estimated to discharge 0.88 cb. ft. Hence fo.6 x 0.13 1 (i . 449 v' 449 0.131 ^0.131)8.024=0.715 cb.ft. (xxi.) That ofCrema was 1.2766:. high, 0.1275 ft. wide, a charge of 0.255 ft. : calculate the discharge. Ans. 0.7225 cb. ft. per sec. (xxn.) The Sardinian Civil Code determines the unit in which all grants of water should be expressed thus : " The measure or modulo (Fig. 35) is that quantity of water which, under simple pressure, and with a free fall, issues from a rectangular quadri- lateral opening, so placed that two of its sides shall be vertical, having a breadth of 0.6562 ft. (English mea- sure), and a height also of Fig. 35. 0.6562 ft. It shall be opened in a thin wall (or platc- parete\ against which the water stands, with its upper 134 Examples and surface perfectly free, at a constant height of 1.3124 ft. (= 2 x 0.6562) above the lower edge of the outlet." It is required to calculate the value of this unit in cubic feet per second. We have therefore / = 0.6562, and H and h being 1.3124 and 0.6562 respectively | x 0.6 x 0.6562 (1.3124 .3124-0.6562^0.6562) 8.024 = 2.046 cb. ft. per sec. When grants are made for more than one module, the only dimension which varies is the breadth of the outlet, the height and pressure remaining in all cases invariable : two modules would have a breadth of outlet of 1.3124 ft., three would have 1.9686 ft., and so on. 103. DescriptionofaPiedmonteseOiitlet("\\.dX\zn Irri- gation/' pp. 21, 22, vol. ii.). " AB (Fig. 36) is aportion of the transverse section of the supplying canal ; the first part of the measuring apparatus is the sluice, which Fig. 36. consists of masonry side-walls, and a gate of wood, work- ing vertically. The dimensions of this primary outlet are not rigidly fixed, its object being merely to admit a larger or smaller supply into the chamber CD. The sluice is established in the bank of the canal, at such point as may be fixed upon by the canal authorities, or most convenient for the land-owner. Its sill is sometimes on the same level as the canal bed, sometimes above it, and very frequently as represented in the diagram. Practical Applications. 1 3 5 There is a fall in front of the outlet, so as to draw the water towards it. For a length of from 40 to 50 feet from the sluice, the bed of the channel is made perfectly horizontal, paved with masonry or cut stone, the upper surface of which is on the same level as the sill of the sluice. At a distance from the outlet, ranging from 1 6 to 32 feet, is fixed the partition or slab of stone cd in which the regulating or measuring outlet ef is cut, the height of which is fixed at 0.56 ft., while the breadth varies with the number of units or inches to be given, each inch being represented by 0.42 ft. of breadth. The lower edge of the measuring outlet is ordinarily placed at 0.819 feet above the level of the flooring of the cham- ber CD. A small return cut in the inner face of the slab, at a height of 0.28 ft. above the upper edge of the outlet, indicates the constant level of the water necessary to insure the established pressure. This height is main- tained by the raising or lowering, as may be requisite, of the sluice at the entrance of the chamber. (xxm.) Calculate the value of a grant of three inches of water from this structure. We have ^=0.56 + 0.28 = 0.84 ; hence 3 x | x 0.6 x 0.42 x (0.84 \/o.84 - 0.28 1/0.28) 8.024 = 2.514 cb. ft p / = 2.514 cb. ft per sec. 104. Description of the Modulo Magistrate of Milan. This module, as applied upon the Naviglio Grande, which in a course of 3 1 miles from its head on the River Ticino to the city of Milan, distributes 1851 cb. ft. per second, is in its principle identical with that already described ( 103). For the interesting history of this canal, and the gradual improvements in the management of the grants of water, we refer to " Italian Irrigation," vol. i., pp. 203, 228 ; vol. ii., pp. 36, 56. The honour of the dis- covery is due to Soldati, of Milan, about the year 1571, i 3 6 Examples and who invented it in answer to an invitation from the ma- gistracy of that city to architects and engineers to design a measuring apparatus. The unit fixed upon, called the oncia magistrale, had the following dimensions (Fig. 37) : Height, 0.655 ft. ; breadth, 0.3426' ft. ; with a constant pressure of 0.32944 22 INCHES. I FEET 100 Fig. 37- ft. above the upper edge of the outlet. When one out- let is designed for the discharge of several water-inches, the breadth only varies, in the proportion of 0.3426' ft. for each additional water-inch, the height and pressure remaining constant, as in Fig 38, which shows an outlet for six water-inches. The outlet is cut with care in a K j 2' 0556 * st/ 2 i i <~0-3k26-* \ i Fig. 38. single slab of stone. To preserve it from being tam- pered with, an iron rim is fixed upon it, of the exact di- Practical Applications. 137 mensions corresponding to the discharge. They ought invariably to be cut in a simple plate, with no arrange- ment of any kind to increase the volume beyond that due to pressure alone. The thickness of the slab varies somewhat with the dimensions of the outlet ; but in a rigidly exact module this dimension should be fixed as well as all the others. These are the conditions appli- cable to the measuring outlets, the discharge from which is | x 0.6 x 0.34266' (0.98444 \/o. 98444 - 0-32944 ^0.32944) 8.024 = 0.866 cb. ft. To illustrate the other arrangements of the modulo, the horizontal and vertical sections (Figs. 39 and 40) are given from the same work. PLAN. Fig. 39- The sluice AB (Fig. 39) is placed on the bank of the canal of supply, with the sill CD (Fig. 40) on the same SECTION. Fig. 40. level as the bottom of this canal. It is formed of two 1 3 8 Examples and side-walls or cheeks, of good masonry, in brick or stone, with a flooring generally of the latter material. To pre- vent erosive action, the bed of the canal, for such dis- tance as the force of the current may render necessary, is paved with slabs of stone or boulders, both above and below the head. The sluice gate is usually made of the same breadth as that of the measuring orifice GH (Fig- 39), while its height is regulated by that of the sluice itself. The sluice-gate or paratoja IK (Fig. 40) works in grooves, and is fitted with a rack and lever, by which it can be readily raised or depressed at pleasure. As the surface level of the canals of the Milanese varies com- paratively little, the upright of the sluice has a small catch in iron or wood attached to it, by which it is kept at a fixed height, corresponding to the requisite pressure on the original orifice GH (Fig. 40). This little catch is locally termed the gatello ; and as it is provided with a lock and key, the latter of which is intrusted to the guardian of the canal, the proprietor of the water-course supplied through the module is supposed to be restricted to his legitimate supply, and probably is so within reasonable limits, provided always that the guardian is incorruptible. In the rear of the sluice-gate, at the head, is placed the first chamber LM (Pigs. 39 and 40), called the tromba coperta, or covered chamber. Its length is equal to very nearly 20 feet, with a breadth varying ac- cording to the size of the head-sluice, which it exceeds by the fixed quantity of 0.82 ft. on each side, or 1.64 on the entire breadth. The bottom of the covered chamber DH (Fig. 40) is formed with a slope to the rere, the height H/i being 1.3125 ft. English : its object is to di- minish the velocity with which the water reaches the measuring outlet GH. Further to assist in effecting this object, the perfect modulo is provided with a horizon- tal top of stone slabs or planks, called the ciclo morte, Practical Applications. 1 3 9 the under surface of which is at precisely the same height as the water ought to have over the outlet GH, so as to secure the fixed discharge, that is, 0.32944 ft. above the upper edge of GH. It is found that this does reduce the irregular motion of the water, and so tends to secure the great object of the modulo, that the discharge should take place under simple pressure, and without antecedent velocity. To admit of ready inspection of the height of the water within the covered chamber, the following ar- rangements are made : The entrance to the chamber is covered with a stone slab of convenient thickness, shown in section at E (Fig. 40), the lower surface of which is precisely on the same level as the upper edge of the out- let GH. The height of the slope H^ being 1.3125 ft., and that of the outlet GH being 0.655, the surface of the slab at E should be 1.9675' ft. above the sill of the head CD. An open groove LD is made in the masonry, large enough to admit a graduated rod or measure ; and when the water stands at a height of (1.9675' + 0.3234=) 2.297 ft- above the sill at D, it is known that the proper head of pressure exists at GH. As it is found to be greater or less, the sluice is depressed or raised, so as to adjust the pressure to the fixed standard. The slab of stone in which the measuring outlet is cut being fixed at GH (Figs. 39 and 40), immediately in rere of it there is placed the tromba scoperta, or open chamber. Its breadth at N (Fig. 39) is two local inches (0.3275 ft. English), greater on each side than that of the measuring outlet, or on both sides 0.6550 ft. Its total length NO is very nearly 17.75 ft. English. Its side-walls, which are perpendicu- lar, like those of the covered chamber, have a splay out- wards, so that the breadth at O is 0.9825 ft. greater than at N, or 1.31 ft. in excess of that of the regulating outlet 140 Examples and GH, being the same as that of the covered chamber throughout. To insure the free run of the water from GH, the flooring of the open chamber has a drop or fall of 0.1633 ft. at H, and an equal quantity distributed uni- formly between H and O (Fig. 40). There is therefore a total fall from the under edge of the measuring outlet to the end of the open chamber of 0.3275 ft. or, as the length is 17.72 ft., of i in 54. When the water reaches O, it enters the channel of distribution for the use of the consumers : generally the point O, and the bed of the channel, which is carried on at the usual inclination, are upon the same level, but sometimes there is a fall. 105. From the preceding details, it appears that the modulo magistrate has a total length of nearly 37.75 ft. English, and a breadth variable according to the quan- tity of water it is intended to measure. If a single water- inch, for instance, be granted, the breadth of the covered chamber would be 2.12835 feet, and that of the open chamber 1.145835 feet at its upper, and 2.12835 at its lower extremity. The flooring of the former rises 1.31 ft. to the rere, while that of the latter falls 0.3275 ft. in the same direction. It is essential to the effective ope- ration of the regulating sluice in the modulo magistrate that there should be a difference of level between the water in the canal and in the apparatus of at least 0.655 ft. ; and as the height of water in the latter must be 2.297 ft., the depth of water in the canal of supply must neces- sarily be not less than the sum of these numbers, or 2.952 ft., very nearly 3 ft. It is curious to reflect that this apparatus was invented empirically by Soldati, in 1571, so many years before the discovery of the Toricellian theorem, which must be placed in the year 1643, when that philosopher showed that the laws of running water were identical with those of falling bodies, the foundation of all our know- Practical Applications. 1 4 1 ledge of Hydraulics. This is not the only instance in which the practical sagacity of the engineer has antici- pated the discoveries of theory. Two essential objects are supposed to be fulfilled by these arrangements : ist. To maintain on the measur- ing outlet a constant pressure ; and 2nd. To make this pressure as much as possible the sole force influencing the discharge, that is, that the water have no velocity antecedently. The first is secured by the mechanical arrangements at the head, the sluice with its rack, lever, &c., and to a certain extent the cielo morte. By raising or lowering this sluice the level of the water in the covered chamber is maintained, independent of the vari- ations in the surface of the external canal. The second by the interior arrangements, the covered chamber with its fixed top, and floor sloping up to the outlet ; while the free passage of the water is secured by the open chamber, with its small fall at the head and con- tinued inclination at the bottom. 1 06. The differences in the estimates of the quantity of water discharged by the modulo magistrate, as given by different Italian engineers, are very remarkable, consider- ing the great attention that has been paid to the theory and practice of Hydraulics in that country. De Regi gives it as i .42 cb. ft. per sec. ; Breschetti states the average re- sult of experiments on the Muzza Canal to give 1.57 cb. ft. per sec. ; Mazzeri estimates it as low as 1.21 cb. ft. ; Brunacci at i .46 ; while the Department of Public Works in Lombardy considers it equal to 1.64 cb. ft. per sec. The extremes, we see, are 1.21 and 1.64 cb. ft. per sec., a difference of 0.43 cb. ft., between a third and fourth of the total discharge. Captain Smith accounts for this great difference by stating " That the estimate of the Govern- ment is founded on the experience of the results on the great canals, where the outlets are almost uniformly of 142 Examples and large dimensions. " (pp. 222, 223, vol. i.) Now it is cer- tain that, all other circumstances being alike, the quan- tities of water discharged from large are proportionally greater than those discharged from small outlets. Hence the oncia magistrate, as determined by experiments with the former, has a decidedly higher value than when de- termined by the latter. The cause of this is clear. To give a discharge of, say, six water-inches, the breadth of the outlet is made six times that for one inch, the height and the pressure remaining in both cases the same. The proportion be- tween the sectional area and perimeter of the outlets becomes, however, materially altered, and the influence of the perimeter in effecting the contraction of the vein diminishes gradually as the size of the outlet increases ; and in a similar proportion the discharge becomes greater. To elucidate this, it may be remarked, that in an outlet for one oncia magistrate the ratio of the section to the perimeter is as i to 23.33 5 f r two, as i to 1 6.66 ; for four, as i to 13.33; f r eight, as i to 1 1.66; for ten, as i to 1 1.33, or about half what is for one oncia ; for twenty oncia, as i to 10.66, and so on ; and there are real differences of discharge due to the variable ratios now given. Very serious pecuniary loss may consequently be the result to the proprietors of the canal or the consumers of the water. It appears (vol. i., pp. 226, 227) that for summer irrigation each cubic foot per second is capable of irrigating 61.8 acres, and that the annual rent of this quantity, summer and winter, is 1 3 $s. ; the difference of 0.43 cb. ft. between the highest and lowest estimate of the discharge of the modulo magistrate is worth 5 135., and would irrigate 26 acres at the above rate. The recognition of the differences between the dis- charges of large and small outlets was very early made in Lombardy. In the module of Cremona, invented in Practical Applications. 143 1561, no single outlet was allowed to exceed 1.31 ft. high by 3.18 ft. broad, equal to about 12 or 13 water-inches. In the Milanese single outlets have been restricted for nearly three centuries and a half to discharges of from 9 to 1 2 once. In Piedmont they have been more careful, and have there limited single outlets to 6 once, which, by general consent, seems to be the most approved size for diminishing to the utmost the error due to the in- equality of discharges from large and small openings. For practical purposes, therefore, and taking the mean of the various estimates of the value of the oncia magis- trate just adverted to, it may be considered as equal to very nearly i \ cb. ft. per sec. 107. Another mode of insuring a constant discharge through an orifice having a charge subject to variation has been brought into use by the late Mr. Thorn,- an hy- draulic engineer of great eminence. It attains this Fig. 41. object by mechanical means chiefly. Fig. 41 repre- sents a vertical section ' of the regulator at the Gorbals Waterworks, near Glasgow. The discharge pipe from 144 Examples and the reservoir is on the right-hand side. If the quantity drawn off by the town or mill to be supplied should in- crease, then the level of the surface /, / will descend ; and the apparatus must be such that it may permit a larger quantity to pass through the pipe, and vice versa. Again, if the level of /, / should remain constant, and, from an increased or diminished rainfall, that of the reservoir rise or fall, then this apparatus should be so constructed as to adjust the orifice of the discharging main pipe that it deliver only that constant quantity carried off from the receiving basin, and needed for the town or mill- works. Fig. 41 gives a longitudinal section of the detail of the regulator : d is a moveable cast-iron cylinder or float attached at top to a chain passing over the pulley orwheel , and surrounded by a fixed cylinder of a diameter slightly larger, containing water, and represented in sec- tion at e. The other end of this chain is fixed to the bent lever b y working freely on a stud carried by two cast iron brackets screwed to the extremity of the pipe pass- ing through the base of the embankment of the reser- voir, and terminating in a square mouth-piece, faced to receive a square hinged flap-valve, a y which is retained in any desired position by the lower and shorter arm of the bent lever which works against the back of the valve by an anti-friction roller at v ; the inner cylinder d must be loaded with weights sufficient to keep the flap-valve quite closed when the outer cylinder e is empty. Now if we suppose the water in the outer cylinder e to stand at the level ss, the cast-iron float being immersed to a certain depth below this surface, part only of its weight, acting by the chain upon the bent lever b y will press against the square flap-valve and thus partially open the mouth of the main-pipe, restricting the discharge through it to the desired quantity. Suppose, then, that from any Practical Applications. 145 circumstances this discharge should become too small, and therefore the surface /, / descend, it will then be ne- cessary that the self-acting apparatus should be such as to permit the valve to open, and therefore, also, the cast- iron float to rise, which it will do if the water-level in the outer cylinder be made to rise ; for then the cast-iron float becomes specifically lighter, and presses with a less force upon the valve a, which immediately yields to the pressure of the water issuing through the discharge-pipe, and thus permits a greater quantity to escape. If, on the other hand, the quantity discharged had been too great, and thus the surface /, / rise, it will be necessary that the cast-iron float descend, and thus press the flap-valve closer upon the square face of the dis- charge-pipe. This it will do if the water in the outer cylinder be made to fall ; for thus the float becomes specifically heavier, and sinks, closing the flap-valve a : so that we have to devise such mechanical arrangements that when the discharge is too small, the water surface in the cylinder e shall rise, and when too great that it shall descend. This is effected in the following manner : A small closed cistern, g, is placed at the side of the portico of the entrance door of the building ; this is supplied with water by a horizontal pipe, r, in communication with the vertical pipe, h, placed on the discharging main for the escape of air, which would otherwise collect within it, and greatly impede the discharge. In all cases of discharge of water through pipes, care must be taken that the air which may collect be readily let off. Vide Buck's Account of the Montgomeryshire Canal Lock; Simms on Public Works in England, p. 8. The pipe, h, must necessarily be carried up the slope of the embankment, and communicate with the air above the level of the highest water in the reservoir. The cis- L 146 Examples and tern, g, is thus kept constantly supplied with water, and a communication is formed by the pipe k between it and the cylinder e. In the vertical part of this pipe are fixed two double-beat valves described below whose common spindle is fixed to the float n, placed in the re- ceiving basin /, / ; now if the surface of the water upon which n rests should rise beyond the proper level, then this float, n, also rises, and, forcing up the spindle, closes up the upper or discharge valve from the cistern, g, and, as the valves are fixed on one spindle, of course simul- taneously opens the lower one, so that the water which buoys up the float d, in the cylinder e, begins to flow out, and the consequent depression of the surface s, s, causing d to descend, partially closes the flap-valve, a ; and therefore the surface /, / begins to descend, and with it the float n, which necessarily opens the valve which had shut off the water from the cistern g, and it, again receiving a supply, d, rises, and consequently the flap- valve opens, and thus very soon arrives at a position giving nearly perfect equality between the -supply and consumption of water. In cases when the pressure upon a sluice is not great, the float n may be directly connected with the lever which works the sluice. Fig. 42 represents this simple apparatus : a, a is the transverse section of the conduit, in which the sluice b moves vertically, and is con- nected by an adjust- able link with an oscil- lating beam c, jointed to the top of the short pillar d. The other extremity of this beam Fig. 42. is similarly connected to a hollow wrought-iron float Practical Applications. Of IH1 147 e, which is acted upon by the water whose surface is intended to be preserved at the same constant level, and the supply of which is derived from the conduit a ; if then the surface at e rise, the sluice is depressed, and the discharge by the conduit lessened, and vice versa. This arrangement is evidently only suited to an open conduit, in which no great pressure can be brought upon the sluice ; if ap- plied to the mouth of a closed pipe with a great head of water pressing on it, the friction in the grooves of the sluice-frame would be so great as to require an enormous float e y and the action could not fail to be of an irregular character. . The double-beat valve, invented by Hornblower (Pole on the Cornish En- gine, pp. 85-88), is repre- sented fully opened in trans- verse section, at D, C, Fig. 43, and shut in Fig. 44. It is intended that the water or steam should pass from A to B when the valve is opened, and that the com- munication between them be intercepted when it is shut. The dark lines at Fig. 43. D, D represent the movable parts of the valve ; those at C, C indicate the parts that are fixed. The value of its peculiar construction may be best appreciated by L 2 148 Examples and considering the tests of a good valve, which should, in the first place, evidently afford a large passage to the steam or water, with a small displacement; and, secondly, should be capable of being opened with a small force. These conditions are fulfilled in the double-beat valve, which consists of a fixed part or seat C, formed by five partitions, which radiate from a central axis, and are joinedbelowtoaring,#,Fig.43, and closed on top by a circu- lar disc, in one piece with the partitions, and covering the spaces between them, and al- so by a movable part, D, the valve proper, which is a sort of case surrounding the seatC, and having a vertical motion, sliding up and down the ex- terior edges of the partitions in C ; this case is open on the top, and connected with its Fig. 44. actuating rod n, by the arms r, r. When it is at the lowest point of -its stroke, and shut, it bears upon the bevilled or conical surfaces a and a', which have but a very small breadth ; when, on the con- trary it is raised, as in Fig. 43, it permits the passage of the water through the different openings shown by the bent arrows. It is evident that by this arrangement it is not necessary to raise the valve through any great height in order to afford a large passage to the water, thus satisfying the first test mentioned above; on the other hand, the valve D, being pierced on its upper part by a circular opening nearly as great as that on the lower part, the force required to raise it is the excess of the pressure of the water or steam per square inch in A over that in B, multiplied into the difference of the Practical Applica lions. 1 4 9 circular areas above mentioned, this difference being evidently the annulus formed by the sum of the horizon- tal projections of the upper and lower conical surfaces, a and a', shown at E in Fig. 43, projected down from the transverse section. If this valve or case D should have been a simple disc with bevilled edges, as in the lower part of Fig. 43, we should have required to lift or start it a force equal to the excess of pressure in A over that in B, multiplied into the whole circular area of the top of the disc #, v ; and this would not only have to be provided by the prime mover, but a very greatly increased size and strength given to the rods, joints, &c., which actuate the valve. In a large disc-valve, as, suppose, 1 2 inches diameter, the area being 113.1 sq. inches, and with an excess of pres- sure in A above that in B of 1 5 Ibs. per square inch, it would require a force of 1696.5 Ibs. to lift it. If in an equal double-beat valve each annulus was j inch broad in the horizontal projection, the sum of their areas would be (i2 2 - 9 2 ) x 0.7854 = 49.48 sq. inches ; thus the force required for the starting of such a double-beat valve is less than half that necessary for an equal disc-valve, being 49.48 x 15 = 742 Ibs., or 954 Ibs. in favour of the double-beat valve, and so in proportion for pressures other than 1 5 Ibs. In the particular case of the valves raised by the float n. Fig. 41, it may be, moreover, remarked that the force necessary to raise them has to be applied but for a very short time, the instant it is raised, the pressure on each side is brought to a state nearly that of equilibrium ; the less, then, the resistance to the float at the moment of raising the valve is, the more sensitive it becomes to any alteration in the surface of the water in /, /, with an absence of any irregular or jerking motion. The flap-valves is consequently retained more steadily at its proper adjust- ment. 5 o Examples and The woodcut, Fig. 45, illustrates a somewhat simpler arrangement to effect the same object, as in Fig. 41, which has been adopted by Mr. Gale at the Kilmarnock Water Works. It represents a vertical section through the centre line of the valve house, showing half the roof and the end walls, the entrance door being at P, and, at the opposite end, the pipe A from the reservoir enters ; N and Fig- 45- N being the section of the foot of the external slope of the embankment impounding the water, and B the culvert conveying it to the filter beds ; the plane of the section is taken transversely to the embankment and perpendicular to its length. The lever L, its support, and the flap-valve are of the same construction as those described above. The weight H is sufficient by its leverage to close the flap-valve and prevent any discharge taking place : all the supply, therefore, must be given by a reduction of the pressure so produced. A chain attached to the outer end of the lever L Practical Applications. \ 5 1 passes over the pulley E suspended at F from a trans- verse beam shown in section. The other end of the chain is fixed to a float D, working in a cast-iron circular well C; on the cover of it is bolted a tube K, which rises above the highest water level in the valve house. The bottom of the well is in communication with the distributing reservoir by the horizontal pipe G, so that the water stands at the same level in both : if, therefore, the con- sumption in the town were such as to cause that surface to descend, the float D, becoming less supported, pulls upon the chain, and, lifting the end of the lever L, in- creases the discharge through A : this increased volume passing down the culvert B to the filter beds soon arrives at the distributing reservoir, and tends to restore its level ; on the other hand, if the surface were to rise, the float D, becoming more immersed, loses a portion of its weight, and, therefore, the valve at the end of the pipe A is proportionably closed, and the discharge les- sened in correspondence with the lessened consumption. The depth of water in the reservoirs at these works was about 1 8 feet ; at the Gorbals reservoir, about 50 feet, it would not have been possible with this greater pres- sure to have adopted the simpler arrangement just de- scribed ; the dimensions of the float D and and the weight H would have been inconveniently increased. The moderateur lamp affords a most ingenious ex- ample, though on a very small scale, of a constant flow of the oil, though the " head," or pressure, varies widely. The annular wick, or Argand burner, is placed on the upper part of the lamp, and is fed with oil from a cylin- der which is placed at the lower part of it, and closed at the bottom. The oil is raised from this by the descent of a piston, forced down by the uncoiling of a spiral spring, which is compressed in winding up the piston from the bottom of its course after the former time of use; 152 Examples and an ascending pipe, passing through this, conveys the oil up to the wick. Now, not only is the spring weaker as it expands with the descent of the piston, but the verti- cal height it has to raise the oil also increases : thus, if were not for the contrivance about to be described, we should have the brilliancy of the light continually lessen- ing as the rate of supply of oil to the wick diminished. This difficulty the celebrated James Watt did "not quite surmount when he turned his attention to this subject ( Vide Life, pp. 462-465). A straight wire, or rod, is placed concentrically within the ascending pipe, of a diameter but little less than it at its upper and thickest part, and long enough to enter the movable pipe when at its lowest position ; the lower part is only a support. The oil in rising is compelled to pass through the narrow annular space between the interior of the moveable pipe attached to the piston and the rod (or moderateur] ; from this it results that it meets with a resistance which causes its upward movement to be very slow. Now as the moveable pipe and piston descend, the same length of the moderateur is not always engaged in the pipe : at first, when the spring is strongest, and the height that the oil has to rise is least, then also the annular passage is longest, and the resistance to the ascent of the oil greatest ; and again, when the piston has descended, and consequently the spring is weaker, and the height the oil has to be raised greater, so also the length of this annular space is less, and the resistance to the ascent of the oil diminished in pro- portion as the ascending force itself is diminished. By a tentative process in each particular case filing a portion into a flat surface the needle is adjusted so as to give a uniform supply of oil, and make the lamp burn with equable light as long as the spring acts. This principle is evidently applicable to the discharge of water by simple modifications. ELEVATION. Practical Applications. 153 The " Module " adopted on the canal of Isabella II. is shown in Figs. 46, 47, 48, 49, taken from the work of Lieut. Scott Moncrieff. It consists of a float, M, and a plug, N, suspended from it, which works in a circular orifice in a plate set at the level of the bottom of the channel. All being contained in a rectangular well of masonry 3.28 ft. by 3.94 and 4.16 feet deep, communicating with the main channel by a lateral opening having an iron grat- ing in front, and covered by a locked iron trap-door, to pre- vent all tampering with it. The float M, which is formed of brass plate, is shown in plan in Fig. 47, and in eleva- vation, with plug attached, in the upper woodcut, Fig. 46. The surface of the water, and, therefore, the apparatus which ' rises and falls with it, being supposed at its highest level, and one meter in depth. It is also shown in Fig. 48, at a lar- ger scale, in a vertical cen- tral section through AB in Fig. 47, and two of the three central supports are given, carrying -\ a central disc, through which passes the screwed end of the rod that the plug is suspended from ; a butterfly nut on the top enables the whole to be adjusted. The plug and the plate, in which is the orifice for the outlet of the water, are of bronze to avoid rust. Fig. 46. PLAN. '54 Examples and The water entering laterally from the canal passes down through the annular space between the plug and plate. From the form of the plug it is evident that this space increases as the level of the water is lowered, and if the area of the annular open- ing be inversely propor- tional to the square root of the "charge" or depth of the canal above the bronze plate, then we should have a constant discharge under all the variations in the level of the water flowing down the canal. In Fig. 49 we have repre- sented four horizontal sections taken at the corresponding numbers on the right-hand side in Fig. 48 ; if the water surface were to descend until the horizontal dot- ted line at 2 reached the level of the orifice, Fig. 48. then the area for the discharge would be the annular Fig. 49. space shown at the corresponding number in Fig. 49 ; Practical Applications. 1 5 5 as the outer circle, representing the circular orifice, is constant, the area for discharge evidently increases as the plug descends. The objection to this module is, that it involves a con- siderable loss of head, the level of the water in the cul- vert, flowing off to the irrigation channels, must be lower than that in the canal of supply by, at least, double the height of the plug, so that it would be inapplicable to the great plains of India, where every inch of level has to be economized. The diameter of the orifice in Fig. 48, which shows the float and plug at a larger scale, is 0.20 metre (= 7.874 ins.), and that of the plug at the same level, when in its highest position, is 0.1653 metre (= 6.51 ins.), so that the area of the annular space in this case is 15.436 sq. ins. ; and, as the coefficient of con- traction, from experiment, was found to be 0.63, we have 0.63 x 15.436 = 9.72 square inches = m S, or 0.0675 square feet, which is the opening at 4 in Fig. 49, and the depth of the water over it, or the charge, is i metre (=3.281 ft.). Had the water surface lowered until G, at 3 on the plug, coincided with the orifice at I, then the annular opening is that shown at 3, Fig. 49; at this point the diameter of the plug is 0.1554 ft- Let D represent the diameter of the orifice in the Tbronze plate fixed in the bottom of the cham- ber, And H the depth of water over it when the main channel is running full. Let ^ 4 , represent the diameter of the plug at the base, and d-to d* &c., the respective diameters at the several points so numbered, Fig. 48, between the base and vertex of the plug, when //4, h^ c., are the corresponding depths estimated from the lowered surface of the channel, these 156 Examples and two quantities being so related to each other that the increased area of the annular space may compensate for the diminished charge and give a constant discharge. From experiments it was found that the coefficient of contraction in this case was 0.63. Let Q represent the unaltered quantity which it is desired to discharge at every different level of the water ; then to compute d the diameter of the plug at the base when the charge is H, we have Q = m x x (D* - df} 8.024 4 the constant multipliers amount to 3.97, and thus -~- = (D* - 3-97 and - (D 3 - d{ 3-97 Hence, if we assign successive values to either h^ or Ej Fig- 55- its masonry grating, is shown at M in the longitudinal section, Fig. 54, and also in the transverse section through EF, Fig. 52, and again, in the general plan at M. The cast-iron overfall is shown in section, at an enlarged scale, in Fig. 56, and the upper step on which the water tumbles, all three of which are shown in Figs. 54 and 55. So long as perfect reli- ance can be placed on the honesty of the guards, the distribution will be effected with great regularity. The action of the instrument Fig. 56. for measuring the velocity of rivers, called Pitot's tube, helps to explain some of the subsequent Practical Ex- amples. It is shown in Fig. 57, and consists of a glass tube bent at a right angle, and having at one end a bell- mouth which is immersed in the current horizontally, and turned so as to face up stream, the other end being M 1 62 Examples and above the surface and vertical. It is found that the water immediately rises in the vertical part, and it must continue to rise until the column produces an outward pressure at the bell mouth equal and opposite to that caused by the motion of the stream . Oscillations are checked by the bulb on the vertical stem and by a diaphram with a small orifice placed across the mouth. If, there- fore, h represent, in feet, the height of ^^=^^ the vertical column above the surface, we have the velocity of the stream in ft. per sec. expressed by v= 8.024 \/~h\ suppose then it were desired to gradu- ate the tube so that the several num- bers on its scale should represent velocity of the stream in miles per hour, we have for one mile per hour (since 2 2 the multiplier , or 1.466, alters miles per hour into feet per second) I x 7 x 1 2 = 0.4 inches ; \i5/ 64.4 for one mile per hour, and for other rates, as follows : Miles per Hour, Inches from the sur- 6 6 6 5 ? 5 g \ face at which the num- bers i, 2, 3 are to be placed, It is only necessary that the upper part of the verti- cal tube be of glass, the lower part may be thin copper- plate or other suitable material. It was by the use of this hydrometer that Pitot over- threw the theory of the old Italian hydraulicians that the velocity of the several fluid threads in a river increased as the square root of the depth from the surface, and Practical Applications. proved, on the contrary, that the velocity diminished from the surface to the bed, as will be mentioned further on. In Figs. 58, 59 are shown Mr. Ramsbottom's excel- lent apparatus for filling the tenders of locomotive en- gines with water while running. It consists of an open trough of water, fixed longitudinally between the rails at about the rail level ; and a dip-pipe or scoop attached to the bottom of the tender, with its lower end curved forwards and dipping into the water of the trough, so as to scoop up the water and deliver it into the tender tank whilst running along. A part longitudinal section of Fig. 5 8. the tender and trough, and part elevation on the right hand, are given in Fig. 58, and a transverse section in Fig. 59- The water trough A, A, of cast-iron, 1 8 inches wide at the top, and 6 inches deep, is laid upon the sleepers between the rails, at such a level that, when full of water, the surface is two inches above the level of the rails, its depth being 5 inches. The scoop B (the same letters have the same reference in each Figure), for raising the water from the trough, is of brass, with an orifice 10 inches wide by 2 inches high ; when lowered for dipping into the trough, it has its bottom edge just level with M 2 1 64 Examples and SCALE YS& Fig. 59- the rails and immersed two inches in the water. The water entering the scoop B is forced up the delivery-pipe C, which discharges it into the tender tank, being turned over at the top so as to prevent the water from splashing over. The scoop is carried on a transverse centre bear- ing D, and when not in use is tilted up by the balance- weight E, Fig. 59, clear of the ground, as shown by dotted lines, Fig. 58 ; for dip- ping into the water trough it is depressed by means of the handle and rod, F, from the foot-plate, which requires to be held by the engine man as long as the scoop has to be kept down. At N is a fixed strong rod supporting the transverse bearing D, D. The upper end of the scoop B is shaped to the form of a circular arc, as is also the bottom of the fixed de- livery-pipe C, so that the scoop forms a continuous pro- longation to the pipe when in the position for raising water. The limit to which the scoop is depressed by the handle F is adjusted accurately by set screws, which act as a stop, and prevent the bottom edge of the scoop being depressed below the fixed working level. The orifice of the scoop is formed with its edges bevilled off sharp, to diminish the splashing, and the top edge is carried for- ward 2 or 3 inches and turned up with the same object. The principle of action of this apparatus consists in taking advantage of the height to which water rises in a tube, when a given velocity is imparted to it on entering the bottom of the tube the converse operation being carried out in this case, the water being stationary, and the tube moving through it at the given velocity. Practical Applications. 165 The theoretical height, without allowing for friction, &c., is that from which a heavy body has to fall in order to acquire the same velocity as that with which the water enters the tube. Hence, since a velocity of 32.2 feet per second is acquired by falling freely through 1 6. i feet ver- tical, a velocity of 32.25 feet per second, or 22 miles per hour, would raise the water 16.24 feet : and other velo- cities being proportional to the square root of the height, a velocity of 30 miles per hour would raise the water 30 feet very nearly (a convenient number for reference), and 1 5 miles per hour would raise the water 7 J feet ; half the velocity giving one quarter of the height. The following Table gives, in the first column, the number of miles per hour at which the train may be ad- vancing ; in the second, the equivalent number of feet per second, and the third, the height in feet through which a body must fall, from a state of rest, to acquire that velocity by the action of gravity. The second column is obtained from the first by multiplying the miles per hour by the number 1.466. The third column is the number in the second divided by 8 and the quo- tient squared : Miles per Hour. Equivalent Feet per Second. Height fallen ver- tically to acquire this Speed. 7! miles. 1 1.99 ft. 1.888 ft. '5 21.99 7.56 20 29.32 '3-43 22 32-25 16.24 25 36.65 20.98 28 41.05 26.32 30 43-9 8 3- 2 5 35 51-31 41.09 40 58.64 53-73 45 65-97 67.99 50 73-30 83.90 60 87.96 120.78 1 66 Examples and In the present apparatus the height that the water is lifted is yj feet from the level in the trough to the top of the delivery pipe in the tender, which requires a ve- locity of 1 5 miles per hour ; and this is confirmed by the results of experiments with the apparatus : for at a speed of 15 miles per hour the water is picked up from the trough by the scoop and raised to the- top of the delivery pipe, and is maintained at that height whilst running through the trough, without being discharged into the tender. The maximum quantity of water that the apparatus is capable of lifting is the cubical content of the channel scooped out of the water by the mouth of the scoop in passing through the entire length of the trough : this measures 10 inches wide by 2 inches deep below the surface of the water in the trough, and 441 yards in length, amounting to ( x 441 x 3 J x = 1 148 gal- \i44 / i 6 Ions, or 5 tons of water. The maximum result in raising water with the apparatus is found to be at a speed of about 35 miles per hour, when the quantity raised amounts to as much as the above theoretical total : so that in order to allow for the percentage of loss that must unavoid- ably take place, it is requisite to measure the eifective area of the scoop at nearly the outside of the metal, which is \ inch thick and feather-edged outwards, making the orifice slightly bell-mouthed and measuring at the outside loj inches by 2\ inches; this gives 1356 gallons for the extreme theoretical quantity. The result of a series of experiments at different speeds is that at 1 5 miles per hour, the total delivery is = o gals. 22 = 1060 33 = Io8 4 1 = 1*5 99 50 "= 10 7 Practical Applications. 167 Hence it appears that the variation in the quantity of water delivered is very slight at any speed above 22 miles per hour, at which nearly the full delivery is obtained ; the greater velocity with which the water enters at the higher speeds being counterbalanced by the reduction in the total time of action whilst the scoop is traversing the fixed length of the trough. Mr. Ramsbottom was led to the invention of this ap- paratus on the occasion of having to provide for the ac- celerated working of the Irish mail, which has now to be run through from Chester to Holyhead, a distance of 84! miles, without stopping, in ^ hours and 5 minutes. This necessitated either an increase in the size of the tender tanks beyond the largest size previously used, contain- ing 2000 gallons ; or else required the alternative of taking water half-way, at Conway, either by stopping the train for the purpose, or by picking up the water whilst running. A supply of 2400 gallons is found re- quisite for this journey in rough weather ; and, although 1 800 to 1 900 gallons only are consumed in fair weather, it is necessary to be always provided for the larger supply, on account of the very exposed position of the greater portion of the line, which causes the train to be liable to great increase of resistance from the high winds fre- quently encountered. An increase of the tender tanks beyond the present size of 2000 gallons would have in- volved an objectionable increase of weight in construc- tion, and alteration in the standard sizes of wheels and axles, &c., for tenders ; and would have also caused a waste of locomotive power in dragging the extra load along the line. By this plan of picking up 1000 gallons at the half-way point where the water trough is fixed, the necessity for a tender larger than the previous size of 1500 gallons is avoided, effecting a reduction in load carried equivalent to another carriage of the train. 1 68 Examples and For further details reference is made to the descrip- tion by the inventor in the " Proceedings of the Institu- tion of Mechanical Engineers " for 1861, from which the above has been selected. The tubes of the Britannia Bridge were constructed on the edge of the shore of the straits near the site of the bridge, and from thence floated to their destination on eight pontoons or barges specially constructed, and arranged in. two groups of four each near the ends of the tube which they sustained, to the foot of the abut- ments, whence they were afterwards raised vertically to their present final position. It was determined to select a tide which would give not more than a maxi- mum relative speed of the current of 9 feet per second ; this was too high a rate at which to permit such a mass more than a thousand tons to move, and ultimately be arrested at a given position, the limit intended being one foot per second. It became, therefore, a point of importance to ascertain the resistance which the guide cables, passing through stoppers on the pontoons, would have to sustain, and so provide them of sufficient strength, and place ample power at the capstans to check the force of the tide, which was the only motive power employed. The vertical area of the eight pontoons was 400 feet, and upon the principle that the pressure of a current of water per square foot is equal to the weight of a column of water one square foot in base, and of a height equal to that in which a body, falling freely by gravity, would acquire the velocity of the current. Now from v* = z> we have ^ = , and the pressure, P y per square foot, will V* therefore be, P = x w. In which w y the weight of a o cubic foot of water, is either 62 J Ibs. or 64 Ibs., according as it is fresh or salt water; in this example we must take Practical Applications. 1 69 the latter figure. Hence P = -^-- x 64. If we take the 04.4 divisor and multiplier as equal we have this approximate rule. The pressure of a current of sea water against a vertical surface of one square foot is equal in ibs. to the square of the velocity in feet per second. For fresh water we have -^ or instead of unity. 64 1024 Hence for i ft. per second we have a pressure of i Ib. per square foot, and in the case of 9 feet per second, 8 1 Ibs. on the same area, a result about \\ per cent, too small. More accurately we have = - = 1.258 ft., being the altitude from which a body would have to fall freely by gravity to acquire the velocity of 9 ft. per second. And 1.258 x 64.04 = 82.29 Ibs., which, multiplied by the area 82.20 x 400 and reduced to tons, gives = 14.7 tons. The 2240 strength of cables in tons is generally estimated by the square of the circumference in inches divided by i o. The result being about half the breaking weight. Two 12 inch cables were employed for the guide lines. The single acting Cornish pumping engine, not having any crank axle or other revolving parts on which to key eccentrics for working the valves, is actuated by a contrivance called the " Cataract ; " the name, how- ever well applied to the original form, has no connexion with that now used. It is shown in a vertical cen- tral section in Fig. 60, in which G is a circular cast-iron tank bolted down on a floor placed below the foot- plate for the engineman ; in this is fitted centrally a cylinder a in which works a circular plunger, b, shown in section in dark lines, guided in the vertical by double glands ; on the right-hand side the cylinder opens into a small box cast in one piece with it, having fitted, 170 Examples and in the bottom, a valve, c, opening inwards of the same construction as that in the lower part of Fig. 43 in the top it has a cover bolted on with a conical hole in which fits a conical plug, d y at the end of a rod, jointed at / to another vertical rod terminating above in a screwed end with an adjusting i I I i m wheel-handle so that the plug may be opened or closed by theengineman to any extent required, and kept fixed in that position. The plung- er is worked by the rod g, forked at each end and connected to to it below by an eye bolt which passes wa- ter tight through the bottom and is held by a nut and screw on the outside : at the Fig. 60. upper end it is united by a pin to a lever shown in end view at e. The conical plug d and valve c are shown closed, the plunger at half stroke. The action in regu- lating the number of strokes per minute is as follows : The piston descends under steam pressure, let in above it only, as it is single acting, and a rod from the main beam descends with it, having a tappet or projection which strikes the opposite end of the levers and thus raises the plunger ; the water in the cistern G follows in to sup- ply the partial vacuum thus formed, raising the valve c y which falls when the plunger b has ceased to rise. Now if the plug d were quite closed b could not descend at all, and the rate at which it can do so is regulated by Practical Applications 171 raising d more or less, and so the water which has entered through c is discharged through the annular opening around d. When the plunger has reached the lowest point, the lever strikes a detent, which frees a weight by which, and not by an eccentric, the steam valve is opened; thus the number of strokes per minute depends upon the length of time in which the plunger b is descend- ing: from 2 to 12 strokes per minute are about the limits. A branch of the Midland Railway (Ireland) crosses the Royal Canal at 2 feet above the water surface, and to provide for the passage of barges, the engineer, Mr. J. Price, has constructed a vertically lifting bridge ; this, when raised sufficiently, is, after the traffic has passed under, let down with a regulated descent by four plung- ers, one over each angle of the abutment ; these work in cast-iron cylinders firmly bolted to the masonry, having a small orifice opening into the water of the canal ; on the rise of the plungers the water follows them and they cannot descend faster than it can escape. The principle has been applied in numerous other instances. The native troops in India are accustomed to relieve guard, on the sinking of a perforated metallic cup in a vase of water. As a converse of this, the ancients, in- stead of a sand-glass, employed a cistern, from which the water trickled through a small hole at the bottom, under the name of a Clepsydra or water-clock, to measure time. In a cylinder the rate of de- scent of the surface diminishes, as the level of the water is succes- sively lowered. To obtain an uni- form descent of the water, it would be necessary to adopt the figure of a conoid of the parabolic kind, each circular section of which is proportional to the square root of the corresponding altitude, and since (if d and h Examples and be corresponding diameters and heights) d* is propor- tional to \/h y we have d* proportional to h. Let h at the commencement be 24 ft., and d = 13.28 ft., also let the uniform rate of descent be i ft. per hour to find x the diameter at the end of the fourth hour, then 24 : 20 20 13. 28 4 : x 13. 28* =2 59 17.5=^, whence x = 12.688 ft. At the end of the twentieth hour the diameter in like manner should be 8.46 ft. To com- pute the diameter of the orifice we have the velocity of descent at the surface to that at the orifice inversely as their areas, that at the surface being i ft. per hour, and at the bottom 8.024^24 = 39.2 ft. per second, which we must multiply by 3600 to bring them to the same unit of time, 3600 x 39.2 : i : : I3.28 2 : 2 2 , or, taking the square roots 60 x 6.261 : i :: 13.28 : z = 0.424 inches. A conoid of such dimensions would, as represented in Fig. 61, there- fore, answer correctly as a Clepsydra, the equable subsi- dence of a float marking the series of twenty-four hours in a natural day. This float' being fastened to a thread wound about a cylindrical barrel, one foot in circum- ference, would carry the index of a dial regularly round. 1 08. Examples on Weirs. Some older writers, as Hutton, &c., give a geometrical construction to repre- sent the discharge through notches and weirs different from that in pp. 49 and 52 namely, if ACDB, Fig. 62, be the transverse section of the sheet of water flow- ing over, then, with either side, as AC, for axis, and A the vertex, drawing a parabola passing through D ; the volume discharged is equal to ACDSA mul- tiplied into the line repre- senting, at the same scale, the velocity in the lowest line CD. Let ACDB represent in Fig. 62. elevation a rectangular Practical Applications. 173 notch through which water is flowing from a reservoir maintained at the constant level AB, then the quantity discharged will be |rds of the quantity flowing through, an equal orifice placed at the whole depth AC in the same time. For if we suppose the transverse section to be divided into an indefinite number of elementary rectangles, as EF, E'FVCD, by equidistant parallel lines, then the volumes of water discharged through each of these equal rectangles will be proportional to the square roots of the depths, that is, to v/AE, x/AE', \/AC, and the sum of all these discharges will be the total dis- charge through the whole area ACDB. But the dis- charge through each rectangle, as EF, with the velocity due to its particular depth, will be equal to that of a rect- angle of less width, as, suppose, ES, at the same depth, provided its velocity is increased inversely as the width is diminished. Now, if through A as vertex, a parabola be drawn through D, the vertical AC being the axis, and if we terminate these lesser rectangles in the curve so drawn, then the velocity in each will be identical and equal to that of the lowest lamina, for the volume dis- charged at CD is to that discharged at any other depth EF, as VAC is to i/AE, that is, as the line CD, or its equal EF, is to the line ES ; and the velocity in the rect- angle ES is to that in the rectangle EF, when they have equal discharges, as EF to ES. The total discharge through the notch, in one second, ACDB, is therefore equal to the volume of a prism having the parabolic segment ACD S' SA as base and the velocity at CD for its height, and as the parabola is frds of the rectangle, the discharge will be f rds of an equal area placed at the depth CD. Let us, as in p. 119, exhibit the effect of the diffe- rent values of m on the quantity discharged per se- cond, namely, 0.60, 0.665, an ^ between 0.662 and 0.595. Vtde 77, pp. 93-94- 1 74 Examples and 9 * Thus, suppose an overfall of i ft. in width, having a depth of i ft. passing over : required the discharge in one second; the formula - mlH V 2gH then becomes - x m x 8.024, or ^ x 5.35. 1. m = ( 64, p. 75) 0.60, value of Q = 3.21 cb. ft. 2. m = ( 64, p. 75) 0.665 = 3-558 3. m = ( 82, p. 100) 0.66654 = 3-566 4- * - ( 6 4> P- 77) -595 ,, - 3-183 In the following questions it is intended to show the effect of the function of the head or charge H V H, which occurs in the formula for the discharge over weirs. A certain length is taken, and the discharge with a given head determined, and then this discharge being increased by a given quantity (xxv.), the corresponding increase of Tis determined. In the same, the discharge being doubled, it is sought (xxvi.) to determine the relative increase of the value ofH. (xxiv.) Calculate the discharge over a weir nooft. long, the depth from the surface of still water to the crest of the weir being 0.75 ft., using 0.665 f r the value of m, (as in second case above), we have 8. 024 x -x 0.665 = 3. 558. Hence 3. 558 x noo x 0.75 x 0.866= 2541.3 cb. ft. per sec., as ^0.75 =0.866. In Beardmore's Table II. we find the discharge for i ft. of length of weir with 0.75 ft. head, 138.88 cb. ft., and this multiplied by 1 100, gives 152768; but as all his Tables are calculated for the discharge per minute, dividing 152768 by 60, we obtain 2546.13, differ- ing from that calculated above by 4. 83; the coefficient used by Beardmore being 0.6665 ( 82, p. 100), giving 3.566, instead of 3.558 used above. (XXV.) To what height upon the crest would the water rise if the discharge was increased to 3000 cb. ft. per sec. ? We have from these data, 3.558 x i ioo VH*= 3000. Practical Applications. , 175 Hence x II00 0>8 being an increase of 0.837 - -75 = 0.087 ft., the increase of H being only n.6 percent., and that of Q being 17.9 per cent. The least laborious method of finding cube roots, when no table of logarithms is at hand, is the follow- ing : Assume a number whose cube is nearly equal to the given number ; then, as twice this cube, plus the given number, is to twice the given number, plus the assumed cube, so is the assumed root to the true ; in this case, for \Xo.58y first assume 0.8, which gives 0.512; secondly, assume 0.84, which gives 0.593. Hence 2 x -593 + -5 8 7 : 2 x 0.587 + 0.593 : : 0.84 : : 0.837. And by logarithms wehavelog. of 0.587 = 1.768638 1, which, divided by 3, gives 3) 3.7686381 1.9228793 answering to 0.83728. (xxvi.) If the discharge in (xxiv.) had been doubled, calculate the depth of water flowing over the crest. The average discharge in (xxiv.) being doubled, gives (2 x 2544 =) 5088 cb. ft. per second on a length of 1 100 ft. Hence 3-558 x iioo , and log. of 1.69 = 0.2278867, which, divided by 3, gives 0.0759622, answering to 1.19131 ; deducting 0.75, we have 0.44 ft. for the rise, to be added to the first supposed 0.75, in order to obtain a double discharge, so that, instead of 176 Examples and 1.50, i. e. twice the original head, we have but 1.19 ft. on the crest of the weir for twice the original discharge ; it is, in fact, evident that 0.75 is multiplied by Z/z* = i .5866, instead of by 2. If the length of the weir in (xxiv.) had been reduced one-half, namely, to 550 ft., calculate the head to which the water would rise upon the crest, the discharge being the same, namely, 2544 cb. ft. per sec. We have now Q = 2544 = 3.558 x 550 x H*. Hence = 1.191 ft. (xxvil.) The construction of the weir at Killaloe ( " Selection of Specifications " ) has the peculiarity of not being level on part of the crest. The inclination being i in 214, and the rise 1.5 ft., the length with that slope must be 1.5 x 214 = 321 ft. ; we have therefore as the weir is 1 100 ft. long, 779ft. for the level portion, and 321 ft. at an inclination of i in 214. Calculate the total quantity discharged over this weir when the depth of water on the level part is 1.8 ft., so as to have 0.3 ft. on the highest part of the crest at the west abutment. If then we divide this sloping part into eight lengths, of 40 ft. each, and calculate the discharge over each length with a head equal to the arithmetic mean of the head at each extremity of the 40 ft. lengths, the discharges will be sufficiently near the truth. The increase of depth on each 40 ft. is evidently -% ft , equal to o.i 8691 ft., and as the depth over the highest point at the west abutment is, by the terms of the question, 0.3 ft., the mean depth for the first 40 ft. is -3 + -3 + 0-18691 r *- ~ = 0-393455 ft-; Practical Applications. \ 7 7 to obtain the second, third, &c., we have but to add to this successively 0.18691, and consequently obtain the following numbers: 0.393, 0.580, 0.767, 0.954, 1.141, 1.328, 1.702 1.795; which, being multiplied by their re- spective square roots, give 0.2468, 0.442, 0.672, 0.932, 1.219, 1.530, 1.864,2.220. Hence the eight several discharges through the 40 ft. lengths are found by multiplying the common part of the formula ( 55) - m, /, H V H . V 2g, that is, 3.558 x 40 o = 142.3 into the values of ff^ff given above, and, adding these, we have the total discharge over the sloping part of this weir 1299 cb. ft. per sec. And for the length of 780 ft. of level crest with 1.8 ft. head, we have 6700 cb. ft. per sec. Hence the total discharge is 7999 cb. ft. per sec. As 8 x 40 = 320 ft., and the length of sloping portion is 321 ft., we must add one foot to 779, the length of the level portion. (xxvin.) In the weirs on the Shannon constructed by the Commissioners, it was requisite that salmon-gaps should be constructed, so that the fish be able to migrate up stream at the weirs during such periods as might not afford sufficient depth of water if the whole quantity were uniformly distributed over the total length of the weir. These were 10 feet wide, and the crest 1.5 ft. below that of the weir. Calculate the quantity flowing down three of these salmon-gaps, the water on the level part of the crest being 0.6 ft. deep. Here H '= 1.5 + 0.6 = 2.1, and 10 x 2.1 N/2.I = 324.8 cb. ft. (xxix.) A feeder or water-course along the side of a valley is required to be augmented by the streams and springs above its level. It is required to determine their N 178 Examples and total volume. For this purpose the several courses are dammed up at convenient and suitable places, and a nar- row board provided, in which is cut an opening for the overfall i ft. long, and 0.5 feet deep; it being reasonably surmised that this would be sufficient to gauge the largest of the streams ; and another piece was prepared that, when attached to the former, would reduce the length to 0.5 ft. for the smaller. Calculate the total quantity delivered by the five following streams and springs : No. i, on being dammed up, flowed over the i ft. open- ing 0.37 ft. deep. Hence Q - 3.558 x 0.37 ^0.37 = 0.8 cb. ft. No. 2, at 0.5 ft. in length of overfall, rose to 0.41 ft. in depth. Hence Q = 3.558 x 0.5 x 0.41 \/o.4i = 0.467 cb. ft. per second. No. 3, at i ft. length, was 0.29 ft. high on the over- fall, and Q = 3'558 x 0.29 \/o.29 = 0.555 cb. ft. per sec. No. 4, at 0.5 in length, rose 0.19 ft. Hence we have Q = o-5 x 3-2i x 0.19 Vo.ig = 0.133 cb. ft. per sec. No. 5, being a small spring, was not measured by the overfall ; but being banked up, a pipe, 0.0416 ft. in diameter, was let through the dam, and when the surface had become stationary, and consequently the discharge through the pipe equal to the supply from the spring, it was gauged into a vessel marked for i and 2, &c., im- perial gallons ; the time required to reach the former was 32 seconds. Hence the spring gave 0.005 cb. ft. per sec., as 6.25 gallons make one cubic foot. ications. Practical Applica The total quantity, therefore, received by the aque- duct from the lateral springs and streams above its level amounted to 1.96 cb. ft. per second. (xxx.) On the Manchester water-works weirs are constructed across some of the lateral mountain streams which supply the reservoirs, so that the higher velocity which the water has when flowing over at the greater depths may separate the turbid water, unfit for the town supply, from the clear. In heavy or sudden rains these streams bring down very rapidly water dis- coloured by peat and earth, and unfit for domestic use ; but in fine weather the quantity is much reduced, and the water clear and suitable for the mains of the town. The wood engraving represents a transverse section of the water-course which is carried through the masonry of the weir, conveying clear water from other streams, across the valley in which the weir is placed, and so serving as an aqueduct ; at the top this is open, and when the water flows over at a small depth, that is, when it is clear, it falls into the channel, Fig. 63, and is con- Fig. 63. veyed by it eventually into the main which supplies the town ; but if it rise and discharge a greater body of water, the increased velocity projects it beyond the edge of the N 2 i8o Examples and opening, Fig. 64, and it thus passes over the longitudi- nal opening, and flows down to the compensation reser- Fig. 64. voir for the supply of water to the mills situated on the river. By referring to 8, we find the means of calculating the curve of any issuing jet of water. But in this case we have a different velocity, and therefore a different parabola for every lamina into which we may suppose the water divided. Fig. 65 represents the different paths taken by each, that for the mean velocity at -|ths of the depth being drawn in a full line ; hence those above will Fig. 65. tend to depress the curve, and those below, on the con- trary, to carry it more up towards the horizontal line ; we may therefore suppose the whole sheet of water to be carried out in a curve at top and bottom parallel to that Practical Applications. 81 of the mean velocity, Fig. 66. If therefore we put H\ for the depth of the _________________ .^^ water flowing over the weir, the mean velocity being |rds of that at the bot- tom, we have v~- x 8.024 x for this mean velocity, and the curve taken by the lowest lamina is that due to a head - H^ for if in the expres- sion ( 48, p. 52) _ = H-h we put h = o, the resulting value of z' is -H. Now in Fig. 67 let x = i ft, andjy = 0.83 ft. ; hence, from 8, p. 12, - = 0.1722 /. Hi = g x o.i 722 -f- 4 =0.3874 ft. -Hi = = 9 4-^ So then, when the water flowing over has a depth at or greater than 0.3874 ft., it is carried com- pletely over the longi- tudinal opening. We must, then, gauge the stream in wet seasons, and so proportion x to y that the volume of water, from the head necessary to dis- charge it, have velo- city sufficient to pass Fig. 67. over the opening mn ; at lesser depths it strikes against 182 Examples and the point, and in part enters the clear water-channel, and in part flows over the weir ; for this reason it is ne- cessary to have a cover of timber, that the attendant may turn down upon the opening during such period if, at the commencement or end of a flood, the water should be turbid at such a depth as would not com- pletely pass over the opening m n. Calculate at what depth the water all flows in. If we suppose in Fig. 68 that nr = H ly which we may do, though Fig. 68. it be not normal to the axis of the sheet of water, then 4 y"' y + HI = 0.83 and HI = 0.83 - y, also - HI = in this sub- 9 4 stitute for HI its value above, we have 4. V^ 4- -(o.83-jy) = J - and -x 9 V 4 9 V 2 4- ^- + - 49 or \/2.266 -y + -, or 1.5 - 0.888 - y ; hence y = 0.612 and HI = 0.833 - 0.612 = 0.221 ft. Thus we see that, when the opening is constructed so that x = i ft., and n' n = 0.83 ft., a depth of water on the crest of 0.3874 ft., or more, carries all the water over the opening, and a depth of 0.221 ft., or less, admits all. If, then, we observe in ordinary seasons a stream dis- charging 26.6 cb. feet per sec., the water then being clear, Practical Applications. 1 83 and the most convenient length of the crest of the over- fall being 60 ft., we may, having selected some conveni- ent depth as n^ m, so adjust the opening mn that the whole of the clear water may fall into it. As a first step, we must calculate H^ for the length 60 feet, and a discharge of 26.6 cb.ft., if the coefficient //z be taken equal to 0.6665 (82); we have, therefore, HI = ^1 - J =0.25 (71), and the vertical depth of n or of n^ (Fig. 68) below the crest at m being given, we may calculate, first, the value of y, that is n n y so that the curve of the zipper surface of the sheet of water flowing over the crest may fall within the point n, and the whole stream be carried down the clear water aqueduct. Let mn^ be taken equal to i ft., then from ,. x iv \/ 'x y* = - - ( 8), we have y = = , and substituting for v its value in this particular case where HI = 0.25, we have v= - V 2gH l = - * 8.024 1/0.25 ( 48), o 3 and also for x its value, which is mn^ + //i = 1.25 ft. Hence 2 > 2 x -x8. 024V 0.3125 ^ = -77- - = 7 x 0.559 = 0.745 ft- And secondly, we may calculate at what amount of dis- charge and head HI the curve of the lower parabola of the sheet of water will pass completely over the opening mn y and so the stream, now turbid, be all carried over the clear water aqueduct into the settling and compen- sation reservoirs. Call the sought depth D, and as x is 2x- x 8.024 \/D now i ft., we have y = 0.745 = 3 = - VT> and 1 84 Examples and f ~ \2 D = I ^ 0.745 J = 0.31248, the discharge being about 37 cb. ft. per sec. (xxxi.) To determine generally the relation between the length and depth of the water on a weir having the same discharge, put 2 2 x m x A x h L *v h\ = m x / 2 x 3 3 hence, A it '.h?-k ::A, and 2 log ^i = log ^2 + - (log 4 - /j). O Calculate the height to which the water upon a weir 545 ft. long will rise when it is flowing down from another weir higher up upon the same river, whose length is 750 ft., and on which it rises 0.68 ft., it being supposed that no additional supply has been received in the inter- vening part of the course. Here log / 2 = 2.8750613 log /, = 2.7363965 0.1386648 x -= 0.0924432 O and log ^ 2 = 1.8325089 0.0924432 1.9249521 and hi = 0.84 ft. By the above method, namely, by discharging the same quantity of water over weirs of different length and measuring ..jthe depths, may be determined experi- Practical Applications. mentally the value of the index of H to which the dis- charge is proportional, on the supposition that m is constant, and that the discharge is directly as the length, for then Q = | . m . 4 . hf = | . m . 4 . ^ 2 fl , and therefore, /M fl / 2 (j-\ = -, or a (log hi - log. /z 2 ) = log / 2 - log 4, a = lQ g 4 - log 4 log ^! - log /V and The following Table is arranged from Series VII., Table X. of J. B. Francis, " Lowell Experiments," al- ready mentioned, 61, p. 68 : TABLE showing the Results of Experiments to determine the Index of H. Total length of Weir in feet. Or, very nearly, Depth on crest of Weir in feet. Average Index. I 16.980 17.0 0-51837 2 13.978 14.0 0.595H j 3 4 10.489 8.489 10.5 8-5 0.72733 0.83614 . 1.478 5 6.987 7.0 0.95882 6 5.487 5-5 1.13087 / (xxxii.) In the construction of reservoirs it is neces- sary to have a weir whose crest is on the level of the in- tended top-water line, with reference to which line the height of the embankment and of the puddle-wall must also be designed. Its length must be such that the water of a maximum rain-fall shall not rise on it above i\ 1 86 Examples and certain height. We may take the greatest available rain- fall at 2 inches in 24 hours ; this depth must be multiplied into the area of district which drains into the reservoir. We thus have, first, the total volume of water ; and se- condly, supposing the rain to have fallen at a uniform rate during the 24 hours, or at least to have been delivered by the water-courses into the reservoir at a uniform rate, we thence obtain the quantity per minute or per second which this weir must discharge. We then assign a cer- tain depth upon the crest, to which the water must be limited, and consequently, from the depth H and dis- charge Q we obtain L. Thus, suppose the area of the rain-basin or district draining into the reservoir were 6536 acres, and the maximum depth of rain in 24 hours to be 2 inches, we reduce both to the same unit of feet. The acre contains 10 square chains of 66 feet each, 66 2 x 10 = 43560 sq. ft., and 2 inches = 0.1666' ft. : hence, 43560 x 6536 x 0.1666' = 47,45 1,360 cb. ft. in 24 hours, which, reducing to seconds, we have 24 x 60 x 60 = 86400; and dividing 47,451,360 -f- 86400 = 549.2 cb. ft. per second, entering the reservoir, the length of weir to discharge this with a rise on the crest of 1.5 ft. is found - x 0.66 x 1.5 Vi. 5 x 8.024 As, however, the valves for discharging the storage would be opened, the rise upon the crest could be readily kept down to one foot. Practical Applications. 187 CHAP. II. FLOW OF WATER UNDER A VARIABLE HEAD. 109. In 88 we have the formula T~ ^ > mS \/^g (xxxiii.) This has been used to determine the value of m. A tube i inch in diameter is filled 9 inches in depth with mercury ; at the bottom is an orifice ^ inch in diameter ; the observed time of its total discharge was 140 seconds. Solving for m, we have m = - T .S . V 2g Changing the measures from inches into feet, we have lA = o.o83 2 x 0.7854 x 2 = 0.0109 sq. ft. and v/o.75 = 0.866 ft. S = - x 0.00545 a 0.0000136 sq. ft. o.oioo x 0.866 0.0094394 And m = -- ^ -- - - - = - _zz2zz = 0.62. 140x0.0000136x8.024 0.0152777 From the vortex motion of the fluid at small depths, no formulae which give the time for complete exhaustion are quite exact. Mercury is, probably, less affected by this motion than water, with which a funnel-shaped vor- tex is formed over the orifice ; this drawing in the air renders the discharge irregular, and reduces the orifice, so that the formula for partial exhaustion t mS V 2g gives more exact results, as in the following experiment. (xxxiv.) A prismatic vessel, having a diameter of 5.747 inches, has an orifice of 0.2 inch at the bottom, and 1 88 Examples and its surface is observed to sink from 1 6 inches to i foot of depth in 53 seconds* Transposing as before, we have / x S H being 1.3 3 ' ft., and h = i ft., the value of (Vi.33 - \/T) is 1.153 - i = 0.153 ft. The diameter of the vessel being 5.747 inches, or 0.4783 ft., the value of A will be 0.4783* x 0.7854 = 0.1797 sq. ft.: also 6* = o.o 1 66* x 0.7854 =0.000218 sq. ft. Hence m = a x 0.1797 x (i. .53-1=) 0.153 = ^055. = Q>6 A 53 x 0.000218 x 8.024 0.0927 (XXXV.) A prismatic basin, whose horizontal section is a square of 3 ft. in the side, has at the bottom an ori- fice 0.09 ft. in diameter ; it is filled up to a depth of 6 ft. above the centre of the orifice. Calculate the time re- quired for the surface to descend 3.5 ft., counting from the moment of opening the orifice. Here ^4=3x3 = 9 sq. ft., S = o.09 2 x 0.7854 = 0.00636 sq. ft. ; H '= 6, and ^ = 6 - 3. 5 = 2. 5, m being 0.6 1 ; therefore from the for- mula t 3x9(1.449 -..58.) = J _ = 0.61X0.00636x8-024 0.03II3 (xxxvi.) With the same dimensions calculate the time required for the surface to descend 2 ft. Here h = 6 -2 = 4, and V H - V ' h = 0.449 ft- > therefore 1 8 x 0.449 t = - izz = 259.6 = \' 20". 0.03113 (XXXVII.) Again, suppose the descent of the surface to be 5 ft., calculate the time, h = 6 - 5 = i, and A//7- */Ji = 1.449, so tnat g/ , 0.03113 0.03113 Practical Applications. 1 89 (xxxvin.) 91. Mean Hydraulic Charge. Let us suppose in any prismatic vessel receiving no supply, that the head, at the instant of opening the orifice of discharge, was 6 ft. = H y and at closing it had decreased to 5 ft. = h, calculate the mean constant charge at which, in the same time, the orifice would discharge the same volume of water; the vessel being now, necessarily, supposed to receive that same constant quantity which it discharges with a uniform velocity. The formula is If h be taken equal to 4, then H' = 4.96 ; if equal to 3, H' = 4.376 ; if h = 2, then H' = 3.732 ; and when h = o, we If in 10" we observe the surface to fall 2 ft., determine the coefficient of discharge. IfA = 6 ft, S = o.o i, and T= 10", then H being = 6, and ^ = 4, we have Q' = 12 cb. ft., and Q= 1.2, per sec., H' = 4.96, and V^H 1 = 2.227 ft. Hence 1.2 1.2 m = - - -- = - = 0.67. 0.1x8.024x2.227 1.787 (xxxix.) 92, p. 109. A reservoir, half an acre in area, with sides nearly vertical, so that it may be con- sidered prismatic, receiving a stream which yields 9 cb. ft. per second, discharges through a sluice 4 ft. wide, which is raised 2 ft. ; calculate the time required to lower the surface 5 ft., the charge upon the centre of the sluice, when opened, being 10 ft. From the formula given at the end of 92, we have, substituting the numerical values, A =21780 sq. ft. the acre, being 43560 sq. ft. ; S = 8 sq. ft., m being found 0.70, and h = 10 - 5 = 5, also Examples and q = 9 sq. ft. per second 2X21780 x 8 X8 ' 02 I0 - * 2.303 x 9 x log 0-7* 8x8.24 x/^-9 _ 0.7 x 8 x 8.024 *v 5 ~ 9 In this we have 0.7 x 8 x 8.024 = 44-9, and A/IO - t/5 = 3.162 - 2.236= 0.926. Hence 2 * 7 = 21.607 (41.6 + 3.37) = 972" = 16', 12". If ^, the constant supply received by the reservoir, had been 20 cb. ft, per second, then (44.9 X 3.162) - 20 121.97 (44.9 x 2.236)- 20 ~ 80.40 the log. of which 150.1809856 (in the former case sub- tracting 9 we had = 1.455, the log. being 0.1628630), 90.4 and the value of /is now 2 1.607 {41.6 + 2.303 x 20 x 0.181} = 1079" =17' 59" to lower the surface 5 ft. (XL.) Referring to the latter part of 92, in order to determine the depth which the surface would descend in a given interval of time, the formula must be arranged so as to separate the factors of */H from \/ h, then transposing, so as to make the left-hand side = o, we have * Practical Applications. 1 9 1 Let us suppose all the letters to have their former values, / being taken at 20 minutes, calculate the value of h (/=) 1200"- [44-9 x 3-i6 2 + 20.73 x log 133} = 1200 - 4020 = - 2820, and thus we have 21.61 x [44.9 V ' h + 20.73 x log (44.9 V~h - 9)] - 2820 = o, when the true value of h is substituted. To further pre- pare this last expression for the tentative determination of h, we multiply out by 21.61, hence 970.3 V ' h + 448 log (44.9 V h - 9) - 2820 = o. If we take at at first \/h = 2, the equation becomes - 25 = 0, = 2.4 + 422 = o, = 2.1 + 82.7 = o, = 2.03 + 7.44= o, \/h = 2.023 o. i = o, and h - 4.09. The surface, therefore, descends 5.9 feet in 20'. (X'LI.) A "pond, whose area is 1 2000 square feet, has an overfall outlet 3 feet wide, which at the commence- ment of the discharge has a head of 2.8 feet; calculate the length of time required for the surface to descend i foot, it being supposed that no supply is received. We have then H = 2.8, and h = 2.8 - i = 1.8, the value of m being taken at 0.6 1. The formula, 93 VH 192 Examples and being put into numbers for this question, we have t _ 3 X 12000 / 'I I \ / i I \ p.6i X3X8.024WI.8 VT8 ) ~ * 452 V^34 ~ 1-673; = M51_ M5= 1 830 -1466 = 364" =6' 4"- 1.34 1.673 Calculate the time in which the surface descends o. 5 feet. In this case h = 2-8 - 0.5 = 2.3, and . _ = * v 2.3 1.516' Hence 2_452 ^ =i /x> 1.516 1.673 Again, if we suppose the depth descended to be 1.5, and all the other quantities remain the same, we shall thus have h = 2.8 - 1.5 = 1.3, and _ = - , so that vi. 3 I - I 4 the depths then being 0.5, i, 1.5 feet ; the corresponding intervals are 2' 31", 6' 4", 1 i' 5". If h = o, it is evident that /becomes infinite, as - = infinity, and so also of any finite number in the numerator, arising from any other data. If the depth sunk had been nearly equal to the whole charge at the commencement, as, suppose 2.4, so that h = 2.8 - 2.4 = 0.4, then . _ = _ \ _ and vo.4 0.6324 g 66 = 24II // = 40 / ,!//. 0.6324 1-673 (XLII.) In question xni., 98, p. 126, taken from D'Aubuisson, the time of filling the lower part of a canal lock on the Canal du Midi, is calculated, i. e up to the level of the centre of the sluices, placed in the Practical Applications. i g 3 upper pair of gates ; we can now, by the second case of 95, calculate the time of filling up to the level of the upper reach, from the centre of the sluice doors, which, added to the 25", as determined in xiil., will give the total time. Substituting in the formula A mS feral numerica have /nri v \-* v 0.548 x 13-532 x 8.024 mS V ' 2g the several numerical values given at p. 91, we shall have ~ 2 x 3503.6 that is 7007.2 x 2.53^298 = 4' 58", 59-5 to which adding 25", we have 5' 23" as the total time of filling a lock of such dimensions. (XLIII.) The locks on the Montgomeryshire Canal have a length of 81 and width of 7.75 feet ; and at one, named the Upper Belun Lock, the lift or rise was 7 ft. A pipe leads the water from the upper level, and dis- charges below the surface of the lower level in the lock- chamber, the diameter of which is 2 feet. As the mouth of this pipe is a square, 2 feet in the side, gradually al- tered into a circular pipe, 2 ft. in diameter, we may take m- i, a result which is justified by comparing the ob- served time of filling this lock with that calculated by the formula when m is put equal to unity, for 2 x8i x 7.75 I x 2 2 x 0.7854 x 8.O24 the observed time being 2' 10". 2.6 4 5 =132". 2 12", CHAPTER III. FLOW OF WATER THROUGH PIPES, ARTIFICIAL CHANNELS, AND RIVERS. 1 1 o. GRAVITY is the sole force that acts upon a mass of water left to itself in a bed of any form ; it produces all the motion which takes place, the inclination of the surface of the water in the channel is the immediate cause of motion, being that which enables gravity to act : and thus the measure of this force is in feet per second, g x sin z', in which g represents the measure of the force of gravity at the earth's surface, being the rate of motion at which a body is moving at the end of one second when falling freely in vacuo, or 32.1908 ft. ; and /is the num- ber of degrees, &c., of inclination of the surface of the water in the channel to the horizon ; and sin i the ratio of the height fallen in any length to that length, or the height fraction, - ~r length Thus, if in one mile the surface was lowered 12 ft., we should have sin i= , or , and the constant dy- 5280' 440' namic force producing motion is measured by ^sin t = 32. 1 908 x = 0-073 1 6 ft. per sec. The angle of inclination being that which has the natural sine 0.0022727, or o 7' .45". If, then, water flow- Flow of Water through Pipes, &c. 195 ing in a channel or pipe, and subject to this constant accelerating force, meet with no resistance, it will de- scend with an increasing velocity which would never be found uniform. But observation and experience show that in open channels and pipes, even those of very great inclination, the rate of motion very soon becomes uniform. Bossut made the following experiment to prove this truth di- rectly: Having constructed a canal in wood, 650 ft. long, with a slope of i in 10, and marked off equal spaces of 1 08 ft. each, it was found that the water traversed each space, except the first, in equal times. There must then exist a retarding force, which destroys at each instant the effect of the accelerating force, and which, when the velocity has become uniform, is necessarily equal to it. But in pipes, channels, &c., there can be no retard- ing force but that which arises from the resistance of the sides or bed : and of its existence we cannot doubt, for the simple experiment of the measurement of the dis- charge through a tube in a certain time, and again when the tube has been lengthened all else remaining the same proves that the time required to yield a certain volume of water has been increased also ; and this can only arise from the fact that the tube, or other channel, by reason of its increased length, offered a greater resist- ance to the velocity. The surface thus opposed motion. To these retarding forces the name of Friction has been applied : though, from the difference between the laws of friction of water flowing over its resisting bed, and the friction of solid bodies sliding upon each other, we must look upon it as the application of an old word in a new sense, in preference to adding a new term to ex- press this peculiar resistance. It may be useful to state here briefly the laws of friction of solid bodies, with the view of showing this contrariety. O 2 196 Flow of Water through Pipes, & c. in. First Law. Experiment has shown that the friction or resistance to motion of bodies, sliding upon their surfaces of contact, is directly proportional to the force or weight pressing the two surfaces together, and differs only with the nature of the sliding surfaces, as wood, brass, iron, &c. Second Law. The amount of friction is independent of the extent of the surface pressed, provided the whole amount of the pressure remains the same, and that the substance of the surface pressed is the same. Third Law. The friction of a body, when in a state of continuous motion, bears a constant ratio to the pres- sure upon it, which is the same, whatever may be the velocity of the motion, it is, in other words, indepen- dent of the velocity. Thus the first only of these laws can be expressed algebraically. 112. In the case of fluids, it has been shown that the resistance to motion, which we observe, and which has been called friction also, is, on the contrary First Law. Independent of the pressure, that is, that the resistance to motion in a pipe with a head or pres- sure of, suppose, 100 ft., is the same as if the head were but 50 ft., or any other height, the velocity being H the same, Dubuat had proved this by experiments on the oscillation of water in syphons, which has been thus modified : Two vessels ABCD, abed (Fig. 69), were connected by the bent pipe EFG^/Jf, which turned round in the short tubes E and e y without Fi s- 6 9- allowing any water to escape; the axis of these tubes Flow of Wafer through Pipes, &c. 197 being in one right line. The vessels were about 10 inches deep, and the branches FG,y^ of the syphon were about 5 feet long. They were then set on two tables of equal height, and (the hole e being stopped) the vessel ABCD, and the whole syphon, were filled with water, which was also poured into the vessel abed till it stood at a certain height LM. The syphon was then turned into a horizontal position, and the plug drawn out of , and the time carefully noted which the water employed in rising to the level HKM in both vessels. The whole apparatus was now inclined so that the water ran back into ABCD. The syphon was now put in a vertical position, and the experiment re- peated : no sensible or regular difference was observed in the time ; yet in this experiment the pressure on the part Gg of the syphon was more than six times greater than before. As it was thought that the friction on this small part (only 6 inches) was too small a portion of the whole resistance, various additional obstructions were put into this part of the syphon, and it was even lengthened to 9 feet ; but still no remarkable difference was observed. It was even thought that the times were less when the syphon w^as vertical ; nor has any variation ever been observed in the friction of water in these different positions when the surface was glass, lead, iron, wood, &c. (Principes d'Hydraulique, tome i., 34 and 36, Dubuat.) Second Law. The resistance is, at any one velocity, proportional to the surface exposed to the action of the flowing water. In order to obtain an expression for this law, we may remark, in the first place, that in any chan- nel or pipe the resistance arising from the surface is shared by all the particles in the volume of water flowing down, those nearest the sides being most retarded, and each in succession less and less influenced. This is 1 98 Flow of Water through Pipes, proved by the result of observations shown in the en- graving, Fig. 70, which represents the transverse section of a trapezoidal channel, with lines of equal velocity plotted upon it, as given in the recent work of M. Darcy and M. Bazin. The width of this experimental chan- Fig. 71, Fig. 72. nel at the water surface was 2 metres, qp, and its depth 0.540 metre, with side slopes about 45. The measured discharge was 1.236 cubic metre per second (= 44.5 cubic feet), and the mean velocity 1.497 metres (= 5 ft. nearly) per second ;' obtained by dividing the discharge by the area of the transverse section, which was equal to very nearly 0.824 square metre. By improvements on Pitot's tube (p. 162) this instrument was adapted by them to the accurate measurement of the velocity in any Flow of Water through Pipes, &c. 199 part of the transverse section, and from the observations thus taken the lines of equal velocity were plotted (by a method described further on). The darker line, No. 3, shows the points in the flowing water at which the mean velocity of 1.497 metres per second was found. The line, No. i, which returns upon itself, shows continu- ously the points of highest velocity plotted ; No. 2 being also greater than the mean, while lines, Nos. 4, 5, and 6, show the successively decreasing velocities below the mean, the least being that nearest the surface of the sides and bottom. It would be easy to interpolate by hand any number of intermediate lines of equal velocity, and thus divide the whole mass of moving water into successive lamina, each suffering less resistance than the previous one as we proceed from the wetted surface of the bottom and sides inwards. The point of maximum velocity was situated on the central dotted line about one-third of the depth from the surface, and was equal to 1.82 metre per second. The greater, then, that surface is, the greater is the resistance. But the greater the volume upon which this retarding action of the surface has to act, the less reduced will be the velocity of the first, and therefore of each successive lamina : and thus we have the resistance directly proportional to the surface and inversely as the volume, i. e. proportional to area of sides and bottom . Now let us suppose the chan volume of moving water nel, Fig. 70 which is identical in every section through- out its length, and- having a uniform flow to be cut by two parallel planes perpendicular to the axis of the stream; and in the plan, Fig. 71, let a a 1 and A A' be the horizontal traces of these two planes, and let the base of the section, Fig. 70, be produced on each side until the produced part, CN and C'N 7 , equal the sum of 2OO Flow of Water through Pipes, &c. the sloping sides and short vertical portions, BA and B'A'. If, then, from the extremities N and N x of this line perpendiculars be let fall on the traces, Fig. 71, the rectangle a A A' a' so formed is evidently equal to the wetted surface of the channel between the two planes, that is, to the product of the distance between them, aA and AA', = NN X ; also the volume of water between the same planes is equal to the product of aA into the trans- verse section of the channel. Hence the ratio given above is equal to aA x NN ' aA x transverse section Striking out from each the length aA. of the channel common to both, we have the resistance directly pro- portional to the border or wetted perimeter, and in- versely as the area of the transverse section perpen- dicular to the axis of the stream. If, then, we put C for the contour of the border, and S for the area of section, we have the resistance proportional to ^. o Third Law. The resistance is proportional to the square of the velocity nearly, the border being constant. For the number of particles drawn in one second from their adhesion to the sides of the channel or pipe is pro- portional to the number of feet per second with which the water is moving, that is, to the velocity. And the force with which they are drawn is also as the same number of feet per second, or the same velocity : and thus the passive resistance of the wetted border to the flow of the water is proportional to the product of the velocity into the velocity ; this pa.rt, then, of the expres- sion for the resistance is represented by a?? y a being a constant, determined hereafter. Flow of Water through Pipes, &c. 201 Experimenters have shown that this gives the resist- ance a very little too high, and that with velocities increased in the ratio 2, 3, 4, &c., it is not represented by a x 4, a x 9, a x 16, &c., but more nearly by adding the simple power of the velocity, thus a (v 2 + bv\ the series of numbers V* + v not increasing so fast as v z . Fourth Law. In gases and elastic fluids we also have the friction proportional to the specific gravity or den- sity. In order to obtain from these laws a formula for the discharge of water through pipes and channels, we must make use of the well-known principle, that when any body is moving with a uniform velocity, the accelerating are necessarily equal to the retarding forces : for if the accelerating forces be supposed greater than the retard- ing, the velocity must increase; and if they should become less, then the velocity must, on the other hand, decrease. We must now, as in Chapters I. and II., find a general expression for the mean velocity, for this multiplied into the transverse area gives the discharge with a given inclination : and we can thus solve the questions that arise in practice, such as the requisite dimensions of pipe or channel to convey a given quantity of water, &c., &c. Now in any pipe or channel, whose length is /, and whose height, from the surface of the supply to the point of discharge or extremity of /, is represented by h, we have the accelerating force expressed by j x g y or sine of inclination of surface into gravity. The retarding forces are, from the second and third laws above given, neglecting bv t proportional to (0 -o x ^ 2O2 Flow of Water throu-gh Pipes, &c. and therefore we have h C (2) ..... "x - = -y a *v\ Each side of this equation represents an equal number of feet per second. The left-hand being 32.1908 ft. per sec. reduced, by being multiplied by a fraction whose value depends on the inclination of the surface, that is - ^ . And the right-hand side being the square of length the number of feet per second with which, at a mean, the water is moving when the motion has become uni- form, reduced by the constant multiplier a, and also by a quantity depending on the figure of the transverse sec- tion of the channel, a being some constant quantity to be determined by experiment. If the formula be correct, all good experiments will give the same value for #, that quantity by which the right-hand side of equation (2) must be multiplied to produce the equality. We may, however, simplify the expression by dividing out by g, and thus we have h a C (3) ..... 7 = g*s**> and as g is constant, put - = a', which must be constant <3 if a be so ; solving, then, for a', we have h S i Substituting the data of experiments in the left-hand side, and deducing v from Q / -f- TS = v, we obtain a y and comparing different experiments, we find that it remains very nearly the same in all. The celebrated Smeaton has given in his Reports (vol. ii., p. 297) a series of experiments on the velocity of water flowing through pipes under pressure. One of Flow of Water through, Pipes, &c. 203 these had the following data : Diameter of pipe, 4^ inches, or 0.375 ft. ; length, 14637 ft; fall or head, 51.5 ft.; and v = 1.815 ft- Hence 0.7854 i ' x = nearty, o.oooi = a'. 14637 0.375 x The quantity discharged is given by Smeaton in Scotch pints, which he states contain 103.4 cb. inches, and there- fore, the number of cb. ft. in one pint is - ^ = 0.05984, and as 200 pints per minute were discharged, we have Q'= 1 1. 968 cb. ft. Hence as -^r=-^=v, we have- ^ ZXD 60XO.II0247 = 1.815 ft. per second... Mr. Provis has published in the " Transactions ot Civil Engineers/' vol. ii., p. 203, some experiments on the flow of water through pipes ij inch = 0.125 ft- in diameter; of these, No. 4, with a length of 100 ft. de- livered 2 cb. ft. per minute, with a head of 2.5 ft. (It is presumed that the orifice of entry of the water was of the best form.) Here the velocity will be (2 cb. ft. + 60 x o.i25 2 x 0.7854 =) 2.72 ft. per second: and hence the value of a' is found 2.5 ft. o.i25 2 x 0.7854 i T- x ' x = very nearly, o.oooi; looft. 0.125x3.1415 2.72* and as from these and many other experiments a 1 = o.oooi, i we have - = 10,000. Substituting, then, this value of a', and solving the equation (4) for #, we have the following expression for the mean velocity r >*' JT x -7^* 10000 ; 204 Flow of Water through Pipes, &c. or, taking the root of the factor 10,000, and placing it outside, (5) ' From this expression it is evident that many geo- metrical questions arise in designing the best form ot channel, whether circular, rectangular, or trapezoidal, to convey given quantities of water: a given area having, with the same condition as to ratio of slopes, a great number of different borders, and one a minimum, and, vice versa, a given border, having a number of different sectional areas, and one a maximum. g The quantity -^ has been called the hydraulic mean depth or mean radius ; it is, in every form and section of channel, represented by a line AE, Fig. 72 ; the rectangle under which, and the border ABCC'B'A', (extended into one right line, AA X = NN X ), is equal to the area of the section ; the greater it is, the less the relative resistance of the surface to the volume of water passing over it. It is important, therefore, to have a clear idea of the influence of the figure of the trans- verse section of the channel upon the magnitude of this quantity, on which, other things being the same, the mean velocity depends, being directly proportional to its square root. As a simple form, let us take a channel whose transverse section is a rectangle, and, first, suppose the border to be constant, secondly, the area. Now when the border is constant, it is evident that there are two extreme positions of the figure : one, when the depth becomes zero, in which case the bottom width must equal the constant border, as suppose 200 ft., or yards or metres, and coincide with the line of water sur- Flow of Water through Pipes, &c. 205 face. This is shown in Fig. 73, in which the line ACB is the level of the surface of the water of the several transverse sections, and CD the vertical central line with reference to which they are all symmetrically ar- ranged. The line ACB represents 200 ft., and is, as has been stated, the limiting figure when the depth Fig. 73- becomes zero. The other extreme position of the figure of the rectangular section having a constant wetted bor- der is when the bottom width becomes zero, and then the depth must be equal to half the border, or 100 ft., and coincide with the vertical central line C D. Now if we take any other rectangle, as A'DT^Bi, having the same border, and its vertical central line coinciding with CD, the water-surface line also coin- ciding with ACB, we shall have AA' = AT)', and BBj = BiDj, and therefore the points D' and D! are in the right lines joining the point D and the points A and B : and thus all possible rectangles having this constant border may be inscribed in the isosceles right-angled triangle ADB. The following Table gives the dimensions of those 206 Flow of Water through Pipes, &c. transverse sections which are drawn in Fig. 73 within the triangle ADB : TABLE showing the Value of the Hydraulic Mean Depth, Area, &c., the Channel having a Rectangular Transverse Section, the Border being constant. Depth in Feet. Bottom Width in Feet. Area in square Feet. Hydraulic Mean Depth or Area 4- 200. Square Roots of Hydraulic Mean Depths. 200 o.oo O.OO i 198 198 0.99 0.99 2 196 392 1.96 1.40 3 194 582 2.91 1.84 5 190 950 4-75 2.18 10 180 1800 9.00 3.00 20 1 60 3200 16.00 4.00 25 r 5 3750 18.75 4-33 30 140 4200 2I.OO 4.58 40 120 4800 24.00 4.90 50 IOO 5000 25.00 5.00 60 80 4800 24.00 4.90 70 60 4200 21.00 4-58 75 50 3750 18.75 4-33 80 40 3200 16.00 4.00 90 20 1800 9.00 3.00 95 IO 950 4-75 2.18 97 6 582 2.91 1.84 98 4 392 1.96 1.40 99 2 198 0.99 0.99 IOO o.oo 0.00 It will be perceived at once that the area and hydraulic mean depth increase progressively up to that figure in which the depth is equal to half the bottom width, the rectangle being half of the square whose side is the bottom width : and although the ratio of the hydraulic mean depth to the depth is continually decreasing, yet the former quantity increases until this ratio has be- come J, after which point it diminishes, having the same Flow of Water through Pipes, &c. 207 value at depths equidistant from the maximum. The inner curve CND, Fig. 74, giyes a diagram representa- tion of the results of the Table. The depths are seve- rally plotted on the line CD from C as zero, (about larger than in Fig. 73 for the sake of clearness), and the hydraulic mean depths as vertical ordinates at the points in CD, corresponding to the depths. Joining Fig. 74- the termination of these ordinates, we have the curve line CND, and by it we can obtain the hydraulic mean depth due to any particular depth by drawing a verti- cal line up to the curve at the point. If, instead of a constant border, we assume the rect- angular transverse section to have a constant area, and for readier comparison with the foregoing take 5000 sq. ft., that area, namely, which was a maximum with the border of 200 ft., we now find that, with a depth indefi- nitely small, the bottom width must be indefinitely great ; and when we assume a bottom width indefinitely small, then the depth must be indefinitely great, as the product in each case is a given quantity : and rect- angles, intermediate between these extremes, being placed, as before, with respect to the lines AB and CD, Fig. 73, we shall find the points corresponding to D' U1U7BBSITY 208 Flow of Water through Pipes, and Dj lie in a curve, well known as the hyperbola, and represented in Fig. 73 by d, D x , d' y and d^ D 1? d% ; the points D x and D L being common to the rectangle of constant border and constant area. TABLE showing the Value of the Hydraulic Mean Depths, &c. y &c. y the Channel having a rectangular Transverse Section; Area constant. Depth. Bottom Width. Hydraulic Mean Depth. Square Roots of Hydraulic Mean Depth. Border. o 00 00 i 5000 0.9994 0.9997 5002 2 2500 1.997 I-4I3 2504 3 1666.6 2.989 1.726 1672.6 5 IOOO 4-95 2.225 1010 IO 500 9.604 3.878 520 20 250 17.24 4.152 290 25 2OO 20 4.472 250 30 166.6 22.065 4.700 226.6 40 125 24-39 4-939 205 50 IOO 25 5 200 60 83-33 24.564 4.956 203.33 70 71.4 23.65 4.863 211.4 75 66.6 23.084 4.800 216.6 80 62.5 22.47 4-74 222.5 90 55-5 21.23 4.607 235-5 95 52.6 20.61 4-54 242.6 98 51.02 20.24 4-50 247.02 99 50-505 20.12 4.48 248.505 100 5o 20 4-472 250 IOOO 5 2.494 1.58 2005 All possible rectangles having this constant area may be inscribed in the space formed by the two branches of the curve and the right line A, C, B, pro- duced each way indefinitely. The Table gives the di- mensions of some of those drawn in Fig. 73 within this space, the sides having dotted lines. Flow of Water through Pipes, &c. 209 An inspection of this Table shows that when the area of the rectangle is constant, the hydraulic mean depth increases with the increase of the depth, being at cor- responding depths somewhat greater than in the former Table, except at the maximum value, which is, by con- struction, the same in each : and at this point, as before, the hydraulic mean depth is half the depth. The outer curve CNR, Pig 74, gives a diagram of the results, the hydraulic mean depths being plotted, as before, at the depths of channel, from which they were calculated. The curve, commencing at zero at the same point C, also passes through a common point N, but from this it di- verges, and soon becomes convex to the line CD pro- duced, which it never can reach, the hydraulic mean depth having always a finite value, in the case of a con- stant area, as long as the depth is finite. In every part this last curve is exterior to that representing the re- sults of a rectangle with constant border, coinciding only at the points C and N. If we produce MN, so that MN = NO, and draw the line CO, producing it indefinitely, then the ordinates, as mn or m'ri, being produced to cut this line in o and o', we have the ratio of the hydraulic mean depth to the depth at each point, taking n and ri either on the inner or outer curve. It will be proved generally in 115 that the best form of channel, whether the transverse section be rect- angular, polygonal, or circular, is when half the depth of the water at the centre line is equal to the hydraulic mean depth ; a proposition which has appeared in the above Tables for the particular numbers chosen, which are mainly intended to illustrate the importance of the figure of the transverse section of a rectangular chan- nel in regard to the velocity and discharge. Let us suppose that, with the same inclination, we p 2 1 o Flow of Water through Pipes, &c. had two rectangular channels of equal transverse area ; but in one the depth and bottom width were 5 ft. and 1000 ft. respectively, and in the other 50 ft. and 100 ft., or numbers in those ratios ; then, from the second Table, we perceive that the square roots of the hydraulic mean depths are as 2.214 to 5, and therefore the mean velocity, which is proportional to this quantity, is more than double, and the volume of water flowing down, which is the product of the mean velocity into the transverse area, also more than twice as great. When the border is constant, the comparison gives results still wider. Thus, if from the first Table we take rectangular chan- nels, whose depth and bottom width are 10 and 180, and again 50 and 100 respectively, or any numbers in those ratios, we find the square roots of the hydraulic mean depths are as 3 and 5, and, multiplying each into the area, we have the volumes of water carried down as 5400 to 25000, the inclinations being supposed the same in both channels. We may also gather from these Tables, that in wide rivers and channels in which the depth is small com- pared with the width, the mean velocity- is very nearly proportional to the square root of the depth, for in such cases the hydraulic mean depth is nearly equal to the depth, as in the upper lines of each Table. It is also evident that on each side about the maximum value the mean velocity does not diminish very rapidly ; thus, in the second Table, the rectangles, 40 by 125, and 60 by 83.3, differ but very little in their mean velocity, and therefore in volume discharged from that of 50 by 100. In the first Table, in like manner, the mean velocities, for depths intermediate between 40 and 60, being nearly identical with the maximum at 50, the volumes dis- charged will only vary from the maximum discharge in proportion to the areas, and thus, in practice, the advan- Flow of Water through Pipes , &c. 2 1 1 tages of the best form of channel may, in a great de- gree, be obtained by others chosen within a consider- able range on each side of it. In the case of tubes having a uniform circular sec- S d** 0.7854 d tion. -^ = -j - ?- = -, the formula (5) becomes then C d* 3-t4i5 4 in the case of pipes flowing full ,,\ \h d \h (o) . . v = 100 /- x - = 50 Jj x d ft. per sec. We have seen, in speaking of the second law of friction, that each successive couche or lamina, into which we may suppose the fluid in motion to be di- vided, is less and less retarded from the border towards the centre of the section : the highest velocity being consequently near the centre and in open channels a little below the surface. The volume of water which tra- verses the section of which we speak, in one second, is due to these different velocities ; and the velocity, the expression for which has now been determined, is that one of these various velocities with which, if the whole section moved as one solid mass, the discharge would be the same : it is then the mean velocity, and is found in any actual experiment by dividing the volume dis- charged in one second by the section, as has been done in the two experiments used for the determining the value of a'. In order, then, to determine the discharge by any channel or pipe, for which we have deduced the value of v from the given inclination and hydraulic mean depth, we multiply the expressions (5) or (6) by the area. Thus from (5) we have (7) - - Q-S* 10^ *-; P 2 2 1 2 Flow of Water through Pipes, c or if we put H y for -^, the hydraulic mean depth, Q = S x ioo L x H y cb. ft. per sec. And again, from (6) we have for pipes running full, Q = 0.7854 d* x 50 / x d y or (8) ... Q - 39.27 JL x d* cb. ft. per sec. In many works and reports the discharge is spoken of per minute, instead of per second : and for this unit of time we have 60 x 39.27 = 2356.2 as the factor outside; hence (8a) . . . Q' = 2356 \-r x d* cb. ft. per minute, which may be written thus, (83) .... <2'=*356 x^J, being the formula used by Beardmore in calculating Table 5, in his work. If, as is not unusual, the diameter of the pipe be given in inches, which call di, the above equation becomes (8.) , . , . Q' = 4.72 J-V- cb. ft. per minute, for this change in the units of the !diameter is equivalent to multiplying the right-hand side of the equation by \/72*, or ^248832 =498.83. In order, therefore, that Q' remain unaltered, we must divide the factor 2356 by this, and consequently, 2356 -=- 498.83 = 4.72, as above. Flow of Water through Pipes, &V. 2 1 3 When the length of the pipes is given in yards, which call / as is sometimes done in practice, we have the right-hand side of the last equation multiplied by ^/3 = i-732, by which, in order that Q may remain the same, we must divide the numerical coefficient 4.72, which therefore becomes 2.725, and .... (>= 2.725 My- cb. ft. per minute. A/ /! And if in gallons per minute, which call Q" ' = Q' x 6.25, then both sides must be multiplied by 6.25, and we have (Be) .... Q" = 17.03 J-7 gallons per minute. Again, if we find (though it cannot be said to be very common in practice) that the discharge is expressed in gallons per hour, we have, making G = Q" x 60, and multiplying by 60, we find, 60 x 17.03 = 1021.8 . . . or G = J v y^-, nearly, as 1021. 8 2 = i6 5 . * *i In some works we find the above stated as (i5^ 6 ) which gives a less result. An approximate practical rule of very easy appli- cation can be derived from equation (8/), by multiply- ing by 1000 and adding 2 per cent. All these expressions from (8) have reference to pipes flowing full under pressure. Other formulae for the mean velocity, generally ex- pressed in words, are in use amongst engineers, which are derived from (5) and those above given, namely, that the mean velocity of water in any pipe or channel 214 Flow of Water through Pipes, that.has attained a uniform velocity is nine-tenths of the square root of the product of twice the fall per mile into the hydraulic mean depth ; or sometimes thus ex- pressed, 0.92 into a mean proportional between twice the fall per mile and the hydraulic mean depth. These, which would not be given in words but to obviate any disadvantage arising from the student meeting with them so expressed, are consequences of equation (5) ; for the numerator and denominator of the fraction j may be replaced by any numbers having the same ratio. If, then, we make / = 5280, i. e. the number of feet in a mile: the numerator, which we may call/, expresses the fall per mile thus, j = -~- ; and from (5) we have, therefore, and as 1.38 = 0.92 \/2, we have, by substituting this value in (9), (10) v-- 0.92 VifHy ft. per sec. This is sometimes written in the nearly identical form (100) v = \/ifH y ft. per sec. And if the velocity be expressed in feet per minute, we have, since 0.92 x 60 = 55.2 (io3) . . . v' = 55 \/ifH y ft. per minute, the decimal being neglected. Again in Dr. Young's " Natural Philosophy " we find this rule : " The square of the velocity in any Flow of Water through Pipes^ <5rV. 215 measures per sec. is equal to the product of the fall in 2800 yards into the hydraulic mean depth, all in the same units. For if /i be the fall in 2800 yards, and/ that in 2800 one mile, as above, then, since = 1.59, we have 1700 /! = i .59 x f also A/i-59 = i .26. If, now, we take the coeffi- cient in equation (10) as being 0.9 instead of 0.92, we may, since 0.9 v/2 = 1.27 express it thus : v = 1.27 VfJJy. In this changing /the fall per mile into/i the fall in 2800 yards, we have v = 1.27 x H or v = The quotient of these numbers is so nearly unity that we may assume the equation (\QC} ..... v=lfiHy y which being squared gives the above rule, and shows that this eminent author used the same coefficient of resistance as has been de- duced in equation (5), p. 204. 113. From the formula (8) to (8/) for the discharge of pipes running full under pressure, we can, being given any two of the three quantities Q, the inclination y, or i/ d y determine the other. Let it be required to find the di- ameter of the pipe, which, with a given inclination, shall convey a given quantity of water. Dividing equation (8) by 39.27, and squaring both sides, we have 39-3? and dividing by j, or multiplying both sides by - , and extracting the fifth root, 121 2 1 6 Flow of Water througli Pipes, &c. The requisite inclination is found from (n) by dividing both sides by d* y and if we multiply both sides by /, we obtain h : so that if the length the water has to be conveyed be also amongst the data, we obtain the head or pressure ne- cessary to force the given quantity along a pipe of known length and diameter (-4) . . , . . .f--Yx' *fL V39-27/ <*' We cannot, however, fully determine the figure of a rectangular or trapezoidal channel from (7) ; solving it for we have Q In this we require, in addition, to be given either ,5* or C, and also the ratio of the slopes of the sides if it be a trapezium ; moreover, S and C are so related that, with given slopes, there is a maximum value of S to every given value of C ; if S exceed this maximum, the solution is impossible. 114. It is found in practice that certain soils, in every excavation for whatever purpose, require a rate of slope in the sides adapted to the degree of cohesion of the ground, to obviate the danger of slips, which occur when they are too steep : this slope of the banks is, therefore, always found amongst the requisite data in the design- ing of channels being trapeziums in transverse section. In order that the side slopes of channels, intended to be permanent, may stand without any masonry or dry stone pitching, they should have a slope between the rates of Flow of Water through Pipes, &c. 217 i } horizontal to i vertical, and 2 to i : being made flatter according as the soil has less tenacity. In some cases even 2\ to i has been adopted; the half regular hexagon has slopes of 0.58 to i ; in channels for temporary use we may have i to i . And so also must the velocity be given ; and, for the same reason, some kinds of earth being worn away, and the form of channel destroyed, by a rate which carries down the particles of the soil through which it is exca- vated, a velocity must therefore be assigned within this rate of motion. It has been determined experimentally for many kinds of earth. The effect of the velocity of the water, in carrying down the particles of the ground through which the channel is excavated, depends jointly upon their tena- city and size. As to the size, we know that the cubical quantities or weights of any similar bodies decrease faster than their superficial areas ; and the pressure or force urging a body down stream being, ceteris paribus, proportional to the surface, is relatively greater the less the volume ; the smaller the particles, therefore, the less is the velocity required to move them. Mr. Beardmore* in Table 3 gives the following statement of the limit of bottom velocities in different materials in feet per minute : 30 ft. will not disturb clay with sand and stones. 40 will move along coarse sand. 60 fine gravel, size of peas. 120,, rounded pebbles, i in. diameter. 1 80 angular stones, about if in. do. The beds of rivers, protected by aquatic plants, however, bear higher velocities than this Table would assign. * Hydraulic Tables, by N. Beardmore. 2 1 8 Flow of Water through Pipes, &c. Such being the natural limitations in the choice of any particular rectangle or trapezium, the engineer must proceed to determine the figure of the transverse area without violating the conditions they impose. 115. When it is desired to convey the greatest pos- sible quantity of water in an open channel with a given area of transverse section, then the volume discharged being directly proportional to the area, and inversely as the wetted border, we must select the figure which for a given area has the least border, and for a given border has the greatest area. Geometry informs us that the circle has this property : the semicircle, and therefore the semicircular channel, has the same property; the ratio between the area of the semicircle and semi-circumference being the same as that between the circle and the entire circumference. Then follow the regular demi-polygons, with less and less advantage as the number of their sides is less ; and among the more practible forms are the demi-hexagon, and finally the half-square. As the transverse sections of artificial open channels are, when without masonry, trapezoidal, the question as to the form of greatest discharge is reduced to taking among all the trapeziums with sides of a determinate slope, that which gives the greatest section for a given wetted border ; or, in other words, which has the great- est hydraulic mean depth ; and every different area and ratio of slopes has its particular maximum trapezium. Let /be the depth of the trapezium BF (Fig. 75), B<- -b- ->C Fi g- 75- and b the bottom width BC, and n : i the ratio of the Flow of Water through Pipes, &*c. 219 slopes, or AF : FB ; then the general values of ,5* and C are (16) . . . . S = (b + np] x p = bp + np\ and C= b +"2 17) + i. Since, then, ,5" in the expression -^ , with slopes of n : i , is a maximum, its differential will be zero, and we have (18) ..... pdb + bdp + 2npdp = o ; and as the border is constant, its general value being differentiated, gives db + zdp Vn* + i = o. Hence db = - idp \/n^ + i ; this being substituted in (i 8), gives I = 2p ( \/72 2 + i - n) ; with which value of b we have (19) Therefore in all trapezoidal channels of the best form, with certain given slopes and area, the hydraulic mean depth is half the depth of the water : and hence we de- rive a construction for the cross section of a maximum r / discharging channel ; remarking that as -^ = -, we have C 2 S C x i Let the trapezium ABCD (Fig. 76) be the B C Fig. 76. channel sought ; from the middle point E of the top 22O Flow of Water through Pipes, &c. width draw lines EB and EC dividing the figure into three triangles, of which AEB and CED are identical; let EP be the perpendicular from E upon AB ; then EP AB + BC + CD x - = AB+CD x +BC x^J 2 22 EP and therefore - = . Hence from E as centre, and with 2 2 p as radius describing a circle, it will touch the two sides AB and CD. If, therefore, conversely, we describe a circle (Fig. 77) with any radius, and draw a tangent, parallel to a horizontal diameter, produced on each side indefinitely, and then between these lines draw tangents having the given inclination, we obtain a figure similar to that re- s' B c c' Fig. 77- quired, from which, by proportion, we find the trans- verse section of the channel sought : a construction given by Mr. Neville in his Hydraulic Tables. Other properties of the trapezium of greatest dis- charge, Figs. 76 and 77, are, First, that the line of sur- face of water AD of A'D', Fig. 77, is equal to the sum of the slopes AB and CD, or A'B 7 and C'D 7 , and conse- quently the wetted border is. equal to the sum of the top and bottom widths, or the mean breadth equal half the border. Secondly, the triangle BEC, Fig. 76, is similar to the triangles EAB and EDC ; the vertical Flow of Water through Pipes, &c. 2 2 1 angle BEC being equal to the angle of inclination of the sides to the horizon. Thirdly, the angle between the perpendiculars from E upon the sides AB and CD is double the angle of inclination of the sides, and the angle PEB half of the same, that is, of the angle BEC. This gives another construction when/ and the angle of inclination of sides are given. On a vertical line lay off /, and from the upper point E (Fig. 76), on each side, lay off the angle of inclination, bisect each of them by EB and EC, and through the lower point of /draw a perpen- dicular to intersect EB and EC, which gives the base BC, then from B draw BA perpendicular to EP, to intersect the horizontal line through E at A, and in like manner on the opposite side, giving the required trapezium ABCD. From the second property we obtain an expression for the area of the trapezium of greatest discharge in terms of the depth and angle of inclination /3 of the side slopes with the horizon, for the area of the triangle BEC is equal to / x p tan J/3, as half BC is the tangent of half the vertical angle to radius p ; also the sum of the areas of the triangles EAB and EDC is equal to EP x AB, but EP is equal to /, and AB is the cosecant of /3 to radius /, as is evident if from B, Fig. 75, we draw the perpendi- cular BF, the angle ABF being the complement of BAF, that is )3 ; thus the area of the trapezium is / 2 (tan Jj3 + cosec. /3). We may deduce from the Table that very large channels formed in any kind of earth cannot be de- signed so as to be of the best discharging form, as the depth of excavation would be too. great ; the ratio of the depth to the mean width must rather resemble that ob- served in large rivers. 222 Flow of Water through Pipes, TABLE giving the Values of tan Jj3 + cosec /3, and the Top and Bottom Widths , in Trapeziums of best Form and ordinary Slopes. Slopes. Angle /3. Tani/S + cosec /3. Top Width, Bottom Width. o to i 90 2.000 p X 2.OOO P X 2.0OO 4 " 75 58' i.Siz 2.O62 1.562 i ,, 63 26 1.736 2.236 1.236 ?" 60 1.732 2.309 1.^55 J M 53 8 1-750 2.500 I.OOO I tO I 45 i.88 2.828 0.828 'i M 38 39 1.952 3-202 0.702 '5 36 53 2.000 3-333 0.666 ij 33 4i 2.106 3.606 0.606 if 29 44 2.282 4.032 0-532 2 tO I 26 34 2.472 4.472 0.472 2 i M 23 58 2.674 4.924 0.424 2 2 21 48 2.885 5-385 0.385 2* 19 58 3.104 5.854 0-354 3 to r 18 26 3-3 2 5 6.325 0.325 The several numbers in the third column express the areas of the trapeziums of best form to the depth unity, and also the mean widths : multiplied by p* they give the area, and by / the mean width for the depth p. The numbers under the fourth and fifth columns are the top and bottom widths to the depth unity, and for any depth p give the top and bottom width respectively, by being multiplied into/; 'they are obtained by adding, for the fourth column, the numbers in the first and third, and for the fifth column, by subtracting the same, for in every trapezium the top width is equal to the mean width, plus the depth into the ratio of the slopes, and the bottom width equal to the mean, minus the depth into the ratio Flow of Water through Pipes, &c. 223 of the side slopes, and with a depth unity the value of these lines is derived by the addition and subtraction above mentioned. In all the different trapeziums of best discharge formed, as shown in Fig. 78, by drawing tangents to the same semicircle having the radius p, the hydraulic mean depths are evidently the same, namely, J/, what- ever the side slopes may be, and, therefore, with the same inclination of the bed of the channel, they all have the same velocity, and consequently the discharge, which will then be as the area, is proportional to the mean width, the depth / being constant. If through the middle point of/ we draw the indefi- nite line mn parallel to aa', then the areas of all thes9 figures formed by the several tangents to the semicircle will be proportional to the length of this line cut off by the tangents or side slopes ; this consideration serves to explain what maybe observed in the last Table, namely, that the numbers in the third column decrease from the first number, corresponding to o to i, down to a certain point, and then, rising to the first value at the slope of 1 5 to i, afterwards increase continuously. In the wood- cut the vertical tangents at D and D x form the rectangle of best discharge with a given area, DV x VT, and the length of mn cut off, which is the measure of the area, is equal to twice the depth / ; now every other tan- gent between that from D and that from B, which passes 224 Flow of Water through Pipes, &c. through the middle point O of DV (and, therefore, forms a trepezium BRUB X , having the same mean width as the rectangle), cuts off a smaller part of the line mn, and so is of less area than the rectangle, the minimum being the trapezium formed by the tangents dC and d'C', which touch at the- points C and C x in which the line mn cuts the semicircle, and having therefore the inclina- tion of 60 with the horizon, the mean width, to the depth unity, being 1.732, and the trapezium a half regu- lar hexagon ; but the tangents cutting mn beyond the point O form trapeziums of a continually increasing mean width, and, therefore, increase in area in the same proportion. If through the point O we draw any in- clined lines whatever, the areas of the figures so formed are all equal to that of the rectangle, but that particular line which, drawn through O, also touches the semi- circle, forms a trapezium, whose wetted border, as well as area, is equal to that of the rectangle, for the slope BR is equal to the half top width BE, which is equal to the radius DE (that is/ or DV), together with BD, which is equal to VR, and, therefore, BR is equal to DV -f VR, and consequently BRUB' = DVTD'. Also the slope of BR is i| to i, or, which is the same thing, VR = ij x OV ; for the sum of the sides of the triangle OVR is evidently equal to DV + VS, or 2DV, or 4-OV ; take OV from both, and we have 3 OV = OR + RV, and also OR 2 - RV 2 = OV 2 , that is, OR + RV x OR - RV = OV% substituting sOV x OR - RV = OV 2 , hence OR-RV = iOV; subtracting this last from OR + RV = 3 OV, Flow of Water through Pipes, &c. 225 we have 2RV = 3 OV - JOV, and, therefore, RV= The three sides of the triangle OVR are consequently as the numbers 5, 4, 3 : and in this trapezium the bottom width is -}th of the top, but this last relation between the top and bottom widths is not needed to the simultaneous equality of areas and borders in a trapezium and rect- angle, for if the slopes of the former be i ^ to i and the vertical sides of the rectangle bisect them, then, how- ever great the bottom width RU may be, or even if it disappear and the slopes meet in a point, the condition holds, and the rectangle and trapezium have the same discharge, velocity, and hydraulic mean depth conse- quent on their identity of border and area. In the woodcut, Fig. 7 8, the tangents from a and a' are at a slope of 2 to i : those from c and c' are at i to i . It is remarkable how small relatively the bottom width be- comes as the slope of the sides becomes flatter, at i \ to i, being but two-thirds of the depth and one-fifth of the top width ; at 3 to i, being one-third of the depth and a tenth of the top width ; for flatter slopes than this last, the trapezium of best form may be considered practi- cally to merge into a triangle. If the top width with the flatter slopes be considered to involve too great an expenditure in land, and that the upper part of the ex- cavated ground be of a nature to bear a steeper slope than the lower, then tangents to the semicircle with that slope will give a figure of best discharging form for a given area, and with those conditions, economizing both land and excavation, instances of such as having been adopted in practice are given in a future page. The bottom, also, of the channel may be constructed curvili- Q 226 Flow of Water through Pipes, &c. near by adopting for it the arc of the semicircle between the points at which the side slopes touch it, and which arc, therefore, subtends an angle at the centre equal to twice the angle of inclination of the sides. In these two last modifications the hydraulic mean depth, and there- fore the velocity, are evidently the same as in the simple trapezium, and the discharge diminished only as the area of either is diminished by the omitted portions of the original trapezium. In order to compare a trapezoidal channel of best discharging figure with others having, first, the same constant border ; secondly, the same constant area, and in both the inclination of the side slopes identical, we may proceed as in pp. 206, 208, in which were tabulated the results upon the hydraulic mean depth and discharge of all the different figures which a transverse section of a rectangular form may assume when the border is con- stant, and again when the area is constant. Let us suppose a trapezium with side slopes of 30 with the horizon, that is, 1.732 to i (nearly if to i), and a constant border of 200 units. It is evident (Fig. 79) that when we diminish the depth, the bottom width in- creases, and that the limiting figure is with a depth equal to zero, a bottom width equal to 200. On the other hand, when the bottom width is equal to zero, and the depth the greatest possible, the trapezium becomes an isosceles triangle, whose base angles are equal to Flow of Water through Pipes, &c. the angle of inclination of the sides, and consequently unlike the former limit (and both limiting figures in the rectangle of constant border) having a finite area. TABLE showing the Value of the Bottom Width, Border y and Hydraulic Mean Depth, &c. y the Transverse Section being a Trapezium with Side Slopes 30 with Horizon, and a Constant Border equal 200" Units. Depth in Feet. Bottom Width. Area. Hydraulic Mean Depth. Square Root of Hydraulic Mean Depth. O 2OO o I 196 197.7 0.99 0.99 2 I 9 2 390-9 1.96 1-39 3 1 88 579.6 2.90 1.70 5 180 943-3 4.72 2.17 10 160 1773.2 8.87 2-97 20 I2O 3092.8 15.46 3-93 2 5 IOO 3582.5 17.91 4-23 30 80 3958.8 19.79 4-45 35 60 4221.8 21. 1 I 4-58 40 40 437'-3 21.86 4.67 4i 36 4387-6 21.94 4.68 42 32 4399-3 22.00 4.69 43 28 4406.6 22.03 4.694 44 24 4409.25 22.04 4-695 44.09 23.63 4409.27 22.05 4.696 45 20 4407-3 22.04 4.694 50 433o.i 21.65 4-653 Taking in the next place a constant transverse area for the trapezium, as in Fig. 80, whose side slopes are A E D Fig. 80. at 30 with the horizon, and supposing this area to be that which was the maximum in the above Table, 4409.27 228 Flow of Water through Pipes, &c. square units : we find the mean widths by dividing this number by the several depths we assume ; the quotient is the mean width of the trapezium, from which the top and bottom widths are deduced by adding and subtract- ing the product of the depth into the ratio of the slopes. From this it readily appears that there is a limit to the depth, for, when the product above mentioned is equal to the mean width, the difference is zero, and the figure becomes an isosceles triangle, whose base angles are equal the angle of inclination of the sides, and area equal the constant area chosen. But the bottom width increases without limit as the depth chosen di- minishes, and at a depth equal to zero becomes infinite. TABLE showing the Value of the Bottom Width, Border, and Hydraulic Mean Depth, &c., the Transverse Sec- tion being a Trapezium, with Side Slopes 30 with Hori- zon, and the Constant Area 4409.27 Square Units. Depth. Bottom Width. Border. Hydraulic Mean Depth. Square Root of Hydraulic Mean Depth. oo 00 I 4407-5 44 IJ -5 0.99 0.99 2 2201.2 2209.2 1.99 1.40 3 1464.6 1476.6 2.98 '73 5 873.2 893.2 4-93 2.22 10 423.6 463.6 9-5o 3.08 20 185.8 265.8 16.60 4.07 25 I33-I 233-1 18.92 4-35 30 95-o 215.0 20.50 4-53 35 65.3 2Q5-.3 21.46 4- 6 3 40 40.9 200.9 21.95 4.685 41 36.5 200.5 21.99 4.689 42 32.2 200.2 22.02 4.692 43 28.1 200.1 22.03 4.694 44 24.0 200. 22.04 4-695 44.09 23.6 2OO.O 22.05 4.696 45 20. o 2OO.O 22.04 4-695 So 1.6 201.6 21.88 4.67 50-455 0. 201.8 21.85 4.674 i Flow of Water through Pipes , &c. 229 In both the Tables it is remarkable how nearly equal the numbers in the fifth column are, on each side of the maximum, even for a wide range of values assumed for the depth and bottom width, and this has an important practical bearing, for if a depth of 44 and bottom width of 24 units were found inapplicable or expensive from the great depth the excavated earth would have to be raised, we may adopt a channel having a depth of 30 and bottom width of 95 units, and as the transverse area is constant, the volume discharged w T ill be influenced only by the alteration in the mean velocity, that is, in the value of the square root of the hydraulic mean depth, and this we perceive is only reduced from 4.695 to 4.53, which is less than 3 J per cent. ; the longitudinal inclination of the bed of the channel being the same in both cases. The land required for the wider channel would be about 1 2 per cent, greater than for that of best discharging form. In forming a new or an improved river channel, the excavated earth is almost always carried to spoil on each side, and not to a contiguous embankment, as in road or railway works, which makes the depth of the cutting of the greater importance with a view to economy. In all cases the top width spoken of is supposed to be the level of high water of the greatest floods, and should be 3 or 4 feet below the surface of the land on each side, in order that the thorough drainage may not be injuriously affected. 1 1 6. The mean velocity of water flowing in an open channel is about 4~5ths of the maximum velocity, which is generally at the centre and upon the surface, or a little below it ; and, conversely, the maximum velocity at the surface is found from the mean velocity by adding a fourth (Minard, "Cours de Construction," p. 6). If U be the mean velocity in feet per second, and V 230 Flow of Water through Pipes, &c* the observed maximum, we have, therefore, approxi- mately (i) ....:. V=o.8V. Thus, if the mean velocity be 3 ft. per sec., that at the surface is - x 3 = 3.75 ft. per sec. 4 Also, if the observed central velocity at the surface were found to be 5.2 ft., the mean velocity is 4.16 ft. per sec. For many purposes of civil engineering this is suffi- ciently accurate, but MM. Darcy and Bazin have ob- jected to it as not taking any account of the inclination of the surface, or of the hydraulic mean depth which de- pends on the greater or less dimensions of the transverse profile, and have given for U, the mean velocity, this formula which is adapted to measures of English feet V being, as above, the observed central surface velo- city (2) .... 7=F--2 5 . It is stated that this was found by them to check very well with the mean velocity deduced by an independent method. At first sight this formula may seem, by com- parison, to lack simplicity ; but to apply the former in estimating the discharge of large rivers, it would, in the end, require the very same measurement of the trans- verse section as in equation (2). Dubuat has given an empiric formula, (3) on which have been founded Tables by many authors. 117. In the formulae for the velocity and discharge of open channels and pipes given in this Chapter, the Flow of Water through Pipes, &c. 23 i direction of both the pipe and channel is supposed to be nearly a right line : when they have quick curves, an addi- tional resistance is occasioned, which diminishes the discharge, or demands an increased head to give any required charge. This resistance is said to depend con- jointly upon the square of the velocity of the water, upon the number of bends, and on the square of the sine of angle they make with the straight line of direction ; and Mr. Beardmore has added, inversely, as the square root of the hydraulic mean depth ; but experimenters have not been consistent in the results obtained (D'Aubuis- son, 196-198). In the cases of pipes running full, the bends may occur in the vertical plane also ; and in this case the air is found to collect rapidly at the summit of such bends : air-valves must, therefore, be left to free the pipe, which may be in some cases self-acting, but are generally worked by hand at stated times. It was formerly thought necessary to proportion the diameter of the main pipe in the different parts of its course, so as to make it, at the termination, discharge the quantity due to its diameter. Thus, at the enlargement of the Edinburgh Waterworks, as designed by Mr. Jardine, the main for the first 18,300 ft. had a fall of 65 ft. ; and the diameter, commencing at 20 inches, decreased to 1 8 inches : the remainder of the distance, 27,900 ft., had a fall of 286ft., nearly three times the former and a diameter of only 15 inches. The discharge into the Castle Hill distributing reservoir is only that due to the smaller diameter, laid the whole distance with a uniform fall. The present and more correct practice is to give the main a uniform diameter throughout ; but at no point in the line of pipe track must the main at any of the vertical bends rise above the line of the average descent on which the discharge was calculated. 232 Flow of Water through Pipes, &c. 1 1 8. On this main were placed, at fourteen differ- ent points, the summits of bends in the vertical plane cast-iron vessels to receive the compressed air as it col- lected. Fig. 8 1 shows a vertical section of one of them, 4 ft. high, and 1.5 ft. wide, with the cock for let- ting off the air, which must be done every three or four ErfHI- 'f^=^^ days. The neglect of this "" n[[J ~ ~\\T^ precaution has, in former Fig. 81. days, been the cause of great disappointment upon the first opening of waterworks. At the present time a self-acting apparatus has been adopted, which will be described in Part II., with other details which are re- quired on the line of a main pipe track for its safe and efficient working. UIITIRS-ITY 39 PATERNOSTER Row, E.C. LONDON, August 1875. GENERAL LIST OF WORKS PUBLISHED BY MESSRS. LONGMANS, GREEN, AND Co. 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