Hydraulics

HYDRAULICS (Gr. iS&~,p, water, and a~Xs, a pipe), the branch of engineering science which deals with the practical applications of the laws of hydromechanics.

I. THE DATA OF HYDRAULICS

f. Properties of Fluids.The fluids to which the laws of practical hydraulics relate are substances the parts of which possess very great mobility, or which offer a very small resistance to distortion independently of inertia. Under the general headng Hydromechanics a fluid is defined to be a substance which yields continually to the slightest tangential stress, and hence in a fluid at rest there can be no tangential stress. But, further, in fluids such as water, air, steam, &c., to which the present division of the article relates, the tangential stresses that are called into action between contiguous portions during distortion or change of figure are always small compared with the weight, inertia, pressure, &c., which produce the visible motions it is the object of hydraulics to estimate. On the other hand, while a fluid passes easily from one form to another, it opposes considerable resistance to change of volume.

It is easily deduced from the absence or smallness of the tangential stress that contiguous portions of fluid act on each other w-ith a pressure which is exactly or very nearly normal to the interface which separates them. The stress must be a pressure, not a tension, or the parts would separate. Further, at any point in a fluid the pressure in all directions must be the same; or, in other words, the pressure on any small element of surface is independent of the orientation of the surface.

2. Fluids are divided into liquids, or incompressible fluids, and gases, or compressible fluids. Very great changes of pressure change the volume of liquids only by a small amount, and if the pressure on them is reduced to zero they do not sensibly dilate. In gases or compressible fluids the volume alters sensibly for small changes of pressure, and if the pressure is indefinitely diminished they dilate without limit.

In ordinary hydraulics, liquids are treated as absolutely incompressible. In dealing with gases the changes of volume which accompany changes of pressure must be taken into, account.

3. Viscous fluids are those in which change of form under a continued stress proceeds gradually and increases indefinitely. A very viscous fluid opposes great resistance to change of form in a short time, and yet may be deformed considerably by a small stress acting for a long period. A block of pitch is more easily splintered than indented by a hammer, but under the action of the mere weight of its parts acting for a long enough time it flattens out and flows like a liquid.

All actual fluids are viscous. They oppose a resistance to the relative motion of their parts. This resistance diminishes with the velocity of the relative motion, and becomes zero in a fluid the parts of which are relatively at rest. When the relative motion of different parts of a fluid is small, the viscosity may be neglected without introducing important errors. On the other hand, where there is considerable relative motion, ________________ the viscosity may be ex ~ ~~b pee ted to have an influence / too great to be neglected.

T,, Measurement of Viscosity.

/ Co effi c sent of Viscosity.

Suppose the plane ab, fig. I

~ of area w, to move with the Ff0 1 velocity ~ relativ~lv to the - surface cd and parallel to it.

Let the space hetw-een be filled with liquid. The layers of liquid in contact with ab and d adhere to them. The intermediate layers all offering an equal resistance to shearing or distortion, the rectangle of fittid abcd will take the form of the parallelogram abcd. Further, the resistance to the motion of ab may be expressed in the form R=~~V, (1)

where e is a coefficient the nature of which remains to be determined.

hxcept where other units aie given, the units throughout this article are feet, pounds. pounds per sq. ft., feet per second.

If we suppose the liquid between ab and cd divided into layers as shown in fig. 2, it will be clear that the stress R acts, at each dividing face, forwards in the direction of motion if we consider the upper layer, backwards if we consider the lower layer. Now suppose the original thickness of the layer T increased to nT; if the bounding plane in its new position has the velocity nV, the shearing at each dividing face will he exactly the same as before, and the resistance must therefore be the same. Hence, R=Ks(nV). (2)

But equations (I) and (2) may both be expressed in one equation if K and e are replaced by a constant varying inversely as the thickness of the layer. Putting ,c =u/T, ac =u/nT,

R uwVIT;

or, for an indefinitely thin layer, R =uwd\7/dt, (3)

an expression first proposed by L. M. H. Navier. The coefficient u is termed the coefficient of viscosity.

According to J. Clerk Maxwell, the value of u for air at 00 Fahr. in pounds, when the velocities are expressed in feet per second, is /L0OOO 000 025 6(4610+0);

that is. the coefficient of viscosity is proportional to the absolute temperature and independent of the pressure.

The value of a for Water at 770 Fahr. is, according to El. von Helmholtz and G. Piotrowski, /s=o-ooo 018 8,

the units being the same as before. For water u decreases rapidly with increase of temperature.

~. When a fluid flows in a very regular manner, asforinstance when it flows in a capillary tube, the velocities vary gradually at any moment from one point of the fluid ~_---2V---~ to a neighboring -i--- point. The layer adjacent to the sides of -~ the tube adheres toit 1v, and is at rest. The ~--v-~________________

I qft,/,qmI,/,gpg/,ftn/Oq/J/,yv/y/,/I# ~

layers more interior I - - -

than this slide on each: r ~ other. But the resist- T r ef~.~1

ance developed by ~ ~ ~ these regular move-, j, ~ d,lt ments is very small, if inlargepipesandopen FIG. 2.

channels there were a similar regularity of movement, the neighboring filaments would acquire, especially near the sides, very great relative velocities. V. J. Boussinesq has shown that the central filament in a semicircular canal of I metre radius, and inclined at a slope of only ooooi, would have a velocity of 187 metres persecond,2 the layer next the boundary remaining at rest. But before such a difference of velocity can arise, the motion of the fluid becomes much more complicated. Volumes of fluid are detached continually from the boundaries, and, revolving, form eddies traversing the fluid in all directions, and sliding with finite relative velocities against those surrounding them. These slidings develop resistances incomparably greater than the viscous resistance due to movements varying continuously from point to point. The movements which produce the phenomena commonly ascribed to fluid friction must be regarded as rapidly or even suddenly varying from one point to another. The internal resistances to the motion of the fluid do not depend merely on the general velocities of translation at different points of the fluid (or what Boussinesq terms the mean local velocities), but rather on the intensity at each point of the eddying agitation. The problems of hydraulics are therefore much more complicated than problems in which a regular motion of the fluid is assuMed, hindered by the viscosity of the fluid.

RELATION OF PRESSURE, DENSITY, AND TEMPERATURE

OF LIQUIDS

5. Units of VolumeIn practical calculations the cubic foot and gallon are largely used, and in metric countries the litre and cubic metre (= iooo litres). The imperial gallon is now exclusivel tised in England, but the Unted States have retained the old Englis wine gallon.

2 Journal de Al. Liouville, t. xiii. (1868); Mmo-ires de 1 Acodi~mi(des Sciences de iInstitui de France, t. xxiii., xxiv. (1877).

I cub. ft. = 6.236 imp. gallons = 7~48I U.S. gallons.

1 imp. gallon = 0.1605 cub. ft. = I ~200 U.S. gallons.

1 U.S. gallon o~I337 cub. ft. =0.8333 imp. gallon.

1 litre O~22Of imp. gallon = 0.2641 U.S. gallon.

Density of Water.Water at 53 F. and ordinary pressure contains 62.4 lb per cub. ft., or 10 lb per imperial gallon at 62 F. The litre contains one kilogram of water at 4 C. or 1000 kilograms per cubic metre. River and spring water is not sensibly denser than pure water. But average sea water weighs 64 lb per cub. ft. at 53 F. The weight of water per cubic unit will be denoted by G. Ice free from air weighs 57.28 lb per cub. ft. (Leduc).

6. Compressibility of Liquids.The most accurate experiments show that liquids are sensibly compressed by very great pressures, and that up to a pressure of 65 atmospheres, or about 1000 lb per sq. in., the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let Vi be the volume of a liquid in cubic feet under a pressure pj lb per sq. ft., and Vi its volume under a pressure pi. Then the cubical compression is (V,V1)/V1, and the ratio of the increase of pressure p,pj to the cubical compression is sensibly constant. That is, b=(p2p1)V1/(V,Vi) is constant. This constant is termed the elasticity of volume. With the notation of the differential calculus, k=dp/(-4) =-4.

_____________ Elasticity of Volume of Liquids. ___________

Colladon Canton. Oersted. and Strm. Regnault.

Water - - 45,990,000 45,900,000 42,660,000 44,090,000

Sea water - 52,900,000 - - - - -.

Mercury - 705,300,000 - - 626,100,000 604,500,000

Oil - - 44,090,000 - -.

Alcohol 32,060,000 23,100,000 -.

According to the experiments of Grassi, the compressibility of water diminishes as the temperature increases, while that of ether, alcohol and chloroform is increased.

7. Change of Volume and Density of Water with Change of Ternperature.Although the change of volume of water with change of temperature is so small that it may generally be neglected in ordinary hydraulic calculations, yet it should be noted that there is a change of volume which should be allowed for in very exact calculations. The values of p in the following short table, which gives data enough for hydraulic purposes, are taken from Professor Everetts System of Units.

Density of Water at Different Temperatures.

Temperature. p G Temperature. p G

Density of Weight of Density of Weight of cent. Fahr. Water. f cub.ft. Cent. Fahr. Wafer. I rub. ft.

0 320.999884 62.417 20 68o .998272 62-316

I 3~8.999941 62.420 22 71-6 997839 62289

2 356 999982 62.423 24 75.2 .997380 62261

3 374 1.000004 62.424 26 78.8 996879 62-229

4 39.2 I000013 62.425 28 82.4 996344 62I96

5 4P0 1.000003 62.424 30 86.995778 62161

6 42~8 999983 62.423 35 95 99469 62.093

7 44.6 .999946 62.42I 40 104.99236 61947

8 46.4 .999899 62.418 45 113.99038 61.823

9 48-2 999837 62.414 50 122 98821 61.688

10 50.0 999760 62.409 55 131.98583 61.540

Ii 51.8 999668 62.403 60 4.98339 61.387

12 53.6 .999562 62.397 65 149 98075 6I~222

13 55.4 .999443 62.389 70 158 97795 61.048

14 57.2 999312 62.381 75 167.97499 60.863

15 59.0 .999173 62.373 8o 176.97195 60.674

16 fio-8.999015 62.363 85 185.96880 60.477

17 626.998854 62.353 90 194 96557 6o275

18 64.4 998667 62.341 100 212.95866 59.844

19 66-2 998473 62.329

The weight per cubic foot has been calculated from the values of p, on the assumption that 1cub. ft. of water at 39.2 Fahr. is 62.425 lb. For ordinary calculations in hydraulics, the density of water (which wilhinfuturebe designated by the symbol G) wilibetakenat 6241b per cub. ft., which is its density at 53 Fahr. It may be noted also that ice at 32 Fahr. contains 57.3 lb per cub. ft. The values of p are the densities in grammes per cubic centimetre.

8. Pressure Column. Free Surface LevelSuppose a small vertical pipe introduced into a liquid at any point P (fig. 3). Then the liquid will rise in the pipe to a level 00, such that the pressure due to the column in the pipe exactly balances the pressure on its mouth. If the fluid is in motion the mouth of the pipe must be supposed accurately parallel to the direction of motion, or the impact of the liquid at the mouth of the pipe will have an influence on the height of the column. If this condition is complied with, the height h of the column is a measure of the pressure at the point P. Let is be the area of section of the pipe, h the height of the pressure column, p the intensity of pressure at P; then pw~~Ghcslb, p/G=h; that is, h is the height due to the pressure at p. The level 00 will be termed the free surface level corresponding to the pressure at P.

RELATION OF PREssuRE, TEMPERATURE, AND DENSITY OF GASES ~. Relation of Pressure, Volume, Temperature and Density in CompressIble Fluids.Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are Q._________________ conveniently treated together. It is necessary, therefore, to state as briefly as possible the properties of compressible fluids so far as knowledge of them is requisite in the solution of these +_..,

problems. Air may be P

taken as a type of these fluids, and the numerical data here given will relate to air., FIG

Relation of Pressure and Volume at Constant Tern perature.At constant temperature the product of the pressure p and volume V of a given quantity of air is a constant (Boyles law).

Let Po be mean atmospheric pressure (2116-8 lb per sq. ft.), Vi the volume of 1 lb of air at 32 Fahr. under the pressure Ps. Then ~o\To=262I4. (f)

If Gi is the weight per cubic foot of air in the same conditions, Go==f/Vs=2ff6.8/26214=.o8o75. (2)

For any other pressure p, at which the volume of 1 lb is V and the weight per cubic foot is G, the temperature being 32 Fahr.,

pV=p/G=26214; or G=p/26214. (3)

Change of Pressure or Volume by Change of Tern perature.Let Po, V0, G0, as before be the pressure, the volume of a pound in cubic feet, and the weight of a cubic foot in pounds, at 32 Fahr. Let 1~ V G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which differ a little from those of a mercurial thermometer). Then, by experiment, pV =-poVo(46o.6+t)/(46o.6+32) poVor/ro, (~)

where r, ?~s are the temperatures t and 32 reckoned from the absolute zero, which is 460 Fahr.;

p/C =po,-/Goro; (4a) C =p-z-oGo/por. (5)

If ~o2II68, Go=o8o75, ro=460~6+32=4926, then p/G=53.2r. (5a)

Or quite generally p/C = Rr for all gases, if R is a constant varying inversely as the density of the gas at 32 F. For steam R=85-5.

II. KINEMATICS OF FLUIDS

10. Moving fluids as commonly observed are conveniently classified thus:

(1) Streams are moving masses of indefinite length, completely or incompletely bounded laterally by solid boundaries. When the solid boundaries are complete, the flow is said to take place in a pipe. When the solid boundary is incomplete and leaves the upper surface of the fluid free, it is termed a stream bed or channel or canal.

(2) A stream bounded laterally by differently moving fluid of the same kind is termed a current.

(3) A jet is a stream bounded by fluid of a different kind.

(4) An eddy, vortex or whirlpool is a mass of fluid the particles of which are moving circularly or spirally. -

(5) In a stream we may often regard the particles as flowing along definite paths in space. A chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream.

11. Steady and Unsteady, Uniform and Varying, MotionThere are two quite distinct ways of treating hydrodynamical questions. We may either fix attention on a given mass of fluid and consider its changes of position and energy under the action of the stresses to which it is subjected, or we may have regard to a given fixed portion of space, and consider the volume and energy of the fluid entering and leaving that space.

If, in following a given path ab (fig. 4), a mass of water a has a constant velocity, the motion is said to be uniform. The kinetic energy of the mass a remains unchanged. If the velocity varies from point to point of the path, the motion is called varying motion. If at a given point a in space, the particles of Water always arrive with the same velocity and in the same direction, during any given time, then the motion is termed steady motion. On the contrary, if at the point a the velocity or direction varies from moment to moment the motion is termed ~ unsteady. A river which excavates its own bed is in 6unsteady motion so long as FIG 4 the slope and form of the bed is changing. It, however, tends always towards a condition in which the bed ceases to change, and it is then said to have reached a condition of permanent regime. No river probably is in absolutely permanent regime, except perhaps in rocky channels. In other cases the bed is scoured more or less during the rise of a flood, and silted again during the subsidence of the flood. But while many streams of a torrential character change the condition of their bed often and to a large extent, in others the changes are comparatively small and not easily observed.

As a stream approaches a condition of steady motion, its regime becomes permanent. Hence steady motion and permanent regime are sometimes used as meaning the same thing. The one, however, is a definite term applicable to the motion of the water, the other a less definite term applicable in strictness only to the condition of the stream bed.

12. Theoretical Notions on the Motion of WaterThe actual motion of the particles of water is in most cases very complex. To simplily hydrodynamic problems, simpler modes of motion are assumed, and the results of theory so obtained are compared experinsentally with the actual motions.

Motion in Plane Layers.The simplest kind of motion in a stream is one iii which the particles initially situated in any plane cross section of the stream con tinue to be found in plane ~ cross sections during the subsequent motion. Thus, if the particles in a thin plane layer ab (fig. 5) are found again in a thin plane layer ab after any interval 1G. 5 of time, the motion is said to he motion in plane layers. In such motion the internal work in deforming the layer may usually be disregarded, and the resistance to the motion is confined to the circumference.

Laminar MotionIn the case of streams having solid boundaries, it is observed that the central parts move faster than the lateral parts. fo take account of these differences of velocity, the stream may be conceived to be divided into thin laminae, having cross sections somewhat similar to the solid boundary of the stream, and sliding on each other. The different laminae can then be treated as having differing velocities according to any law either observed or- deduced from their mutual friction. A much closer approximation to the real motion of ordinary streams is thus obtained.

- Stream Line Motion.In the preceding hypothesis, all the particles in each lamina have the same velocity at any given cross section of the stream. If this assumption is abandoned, the cross section of the stream must be supposed divided into indefinitely small areas, each representing the section of a fluid filament. Then these filaments may have any law of variation of velocity assigned to them. If the motion is steady motion these fluid filaments (or as they are then termed stream lines) will have fixed positions in space.

Periodic Unsteady MotionIn ordinary streams with rough boundaries, it is observed that at any given point the velocity varies from moment to moment in magnitude and direction, but that the average velocity for a sensible period (say for 5 or 10 minutes) varies very little either in magnitude or velocity. It has hence (/~~4 ~

v~~P ~

FIG. 6.

been conceived that the variations of direction and magnitude 01 the velocity are periodic, and that, if for each point of the stream thi mean velocity and direction of motion were substituted for thi actual more or less varying motions, the motion of the streait might be treated as steady stream line or steady laminai mGI ion -

13. Volume of FlowLet A (fig. 6) be any ideal plane surface of area w, in a stream, normal to the direction of motion, and let \

be the velocity of the fluid. Then the volume flowing through the surface A in unit time is QwV. (1)

Thus, if the motion is rectilinear, all the particles at any instant in the surface A will be found after one second in a similar surface A, at a distance V, and as each particle is followed by a continuous thread of other particles, the volume of flow is the right prism AA having a base w and length V.

If the direction of motion makes an angle 8 with the normal to the surface, the volume of flow is represented by an oblique prism AA (fig. 7), and in that case Q=nV cos 0.

If the velocity varies at different points of the surface, let the surface be divided into very small portions, for each of which the 1111 1i FIG. 7.

velocity may be regarded as constant. If dw is the area and v, or v cos 8, the normal velocity for this element of the surface, the volume of flow is QJvdw, orfv cos 0 dw, as the case may be.

14. Principle of (~ontinuity.-If we consider any completely bounded fixed space in a moving liquid initially and finally filled continuously with liquid, the inflow must be equal to the outflow. Expressing the inflow with a positive and the outflow with a negative sign, and estimating the volume of flow Q for all the boundaries, = o.

In general the space will remain filled with fluid if the pressure at every point remains positive. There will be a break of continuity, if at any point the pressure becomes negative, indicating That the stress at that point is tensile. In the case of ordinary water this statement requires modification. Water contains a variable amount of air in solution, often about one-twentieth of its volume. This air is disengaged and breaks the continuity of the liquid, if the pressure falls below a point corresponding to its tension. It is for this reason that pumps will not draw water to the full height due to atmospheric pressure.

A pplication of the Principle of Continuity to the case of a Stream. If A,, Al are the areas of two normal cross sections of a stream, and V1, V1 are the velocities of the stream at those sections, then from the principle of continuity, = ViA2

Vi/Vi = A,/Af (2)

that is, the normal velocities are inversely as the areas of the cross sections. This is true of the mean velocities, if at each section the velocity of the stream varies. In a river of varying slope the velocity varies with the slope. It is easy therefore to see that in parts of large cross section the slope is smaller than in parts of small cross section. -

If we conceive a space in a liquid bounded by normal sections at A,, Ai and between A1, Af by stream lines (fig. 8), then, as there is no flow across the stream lines, Vi/Vf =Af/Af, as in a stream with rigid boundaries.

In the case of compressible fluids the variation of volume due to the difference of pressure at the two sections must be taken into FIG. 8.

account. if the motion is steady the weight of fluid between two cross sections of a stream must remain constant. Hence the weight flowing in most be the same as the weight flowing out. Let pi, Pm be the pressures, v,, vf the velocities, Gi, Cf the weight per cubic foot of fluid, at cross sections of a stream of areas Ai, Af. The volumes of inflow and outflow are Ajvi and Amen and, if the weights of these are the same, G1Aivi = GiAmv1

and hence, from (5a) 9, if the temperature is constant, p1A1t,1 = p2Afvi. (3)

i~. Stream Lines.The characteristic of a perfect fluid, that is, a fluid free from viscosity, is that the pressure between any two parts into which it is divided by a plane must be normal to the plane.

One consequence of this is that the particles can have no rotation impressed upon them, and the motion of such a fluid is irrotational.

A stream line is the line, straight or curved, traced by a particle in a current of fluid in irrotational movement. In a steady current FIG. 9.

each stream line preserves its figure and position unchanged, and marks the track of a stream of particles forming a fluid filament or elementary stream. A Current in steady irrotational movement may be conceived to be divided by insensibly thin partitions following the course of the stream lines into a number of elementary streams. If the positions of these partitions are so adjusted that the volumes of flow in all the elementary streams are equal, they represent to the mind the velocity as well as the direction of motion of the particles in different parts of the current, for the velocities ~ -!

FIG. f o. FIG. II. FIG. 12.

are inversely proportional to the cross sections of the elementary streams. No actual fluid is devoid of viscosity, and the effect of viscosity is to render the motion of a fluid sinuous, or rotational or eddying under most ordinary conditions. At very low velocities in a tube of moderate size the motion of water may be nearly pure stream line motion. But at some velocity, smaller as the diameter of the tube is greater, the motion suddenly becomes tumultuous. The laws of simple stream line motion have hitherto been investigated theoretically, and from mathematical difficulties have only been determined for certain simple cases. Professor H. S. Hele Shaw has found means of exhibiting stream I I T periments a thin sheet of fluid is caused to flow / I I I I between two parallel plates of glass. In the / I I I earlier experiments streams of very small air (..A \ \ ,, bubbles introduced into the water current \\\ \ \ rendered visible the motions of the water. By \% \ \ the use of a lantern the image of a portion of \ % \\ the current can be shown on a screen or photo ~N~\i\ graphed. In later experiments streams of / ~ colored liquid at regular distances were intro /f 7/ duced into the sheet and these much more /f / / clearly marked out the forms of the stream II I I stream lines were found to be stable at almost any required velocity. For certain simple Fio. 13. cases Professor Hele Shaw has shown that the experimental stream lines of a viscous fluid are so far as can be measured identical with the calculated stream lines of a perfect fluid. Sir G. G. Stokes pointed out that in this case, either from the thinness of the stream between its glass walls, or the slowness of the motion, or the high viscosity of the liquid, or from a combination of all these, the flow is regular, and the effects of inertia disappear, the viscosity dominating everything. Glycerine gives the stream lines very satisfactorily.

Fio. 9 shows the stream lines of a sheet of fluid passing a fairly shipshape body such as a screwshaft strut. The arrow shows the direction of motion of the fluid. Fig. 10 shows the stream lines for a very thin glycerine sheet passing a non-shipshape body, the stream lines being practically perfect. Fig. if shows one of the earlier air-bubble experiments with a thicker sheet of water. In this case the stream lines break up behind the obstruction, forming an eddying wake. Fig. 12 shows the stream lines of a fluid passing a sudden contraction or sudden enlargement of a pipe. Lastly, fig. 13 shows the stream lines of a ctirrent passing an oblique plane. H. S. Hele Shaw, Experiments on the Nature of the Surface Resistance in Pipes and on Ships, Trans. Inst. Naval Arch. (1897).

Investigation of Stream Line Motion under certain Experimental Conditions, Trans. Inst. Naval Arch. (1898); Stream Line Motion of a Viscous Fluid, Report of British Association (1898).

III. PFIENOMENA OF THE DISCHARGE OF LIQUIDS FROM ORIFICES AS ASCERTAINABLE BY EXPERIMENTS

16. When a liquid issues vertically from a small orifice, it forms a Jet which rises nearly to the level of the free surface of the liquid in the vessel from which it flows. The difference of level hr (fig. 14) is a so small that it may be ~, ~ at once suspected to be -==-~--= --------~- f~

due either to air resistance _

on the surface of the jet or to the viscosity of the liquid or to friction against the sides of the orifice.

Neglecting for the moment this small quantity, we may infer, from the eleva tion of the jet, that each - It, molecule on leaving the orifice possessed the velo city required to lift it against gravity to the height h. From ordinary dynamics, the relation Engels__~__.k between the velocity and height of projection is given by the equation - v~I2gh. (1)

As this velocity is nearly reached in the flow from FIG 14

well-formed orifices, it is sometimes called the theoretical velocity of discharge. This relation was first obtained by Torricelli.

If the orifice is of a suitable conoidal form, the water issues in filaments normal to the plane of the orifice. Let ~o be the area of the orifice, then the discharge per second must be, from eq. (1),

Qcev=w~,l2gh nearly. (2)

This is sometimes quite improperly called the theoretical discharge for any kind of orifice. Except for a well-formed conoidal orifice the result is not approximate even, so that if it is supposed to be based on a theory the theory is a false one.

Use of the term Head in Hydraulics.The term head is an old millwrights term, and meant primarily the height through which a mass of water descended in actuating a hydraulic machine. Since the water in fig. 14 descends through a height h to the orifice, we may say there are Ii ft. of head above the orifice. Still more generally any mass of liquid h ft. above a horizontal plane may be said to have h ft. of elevation head relatively to that datum plane. Further, since the pressure p at the orifice which produces outflow is connected with h by the relation p/G=h, the quantity p/G may be termed the pressure head at the orifice. Lastly, the velocity 2 is connected. with h by the relation vf/2g = h, so that v/2g may be termed the head due to the velocity v.

17. Coefficients of Velocity and ResistanceAs the actual velocity of discharge differs from ~ 2gh by a small quantity, let the actual velocity =v~=c,y~i~ii, (3)

where c, is a coefficient to be determined by experiment, called the coefficient of velocity. This coefficient is found to be tolerably constant for different heads with well-formed simple orifices, and it very often has the value 097.

The difference between the velocity of discharge and the velocity due to the head may be reckoned in another way. The total height h causing outflow consists of two partsone part h, expended effectively in producing the velocity of outflow, another hr in overcoming the resistances due to viscosity and friction. Let hr = Crltr, where Cr is a coefficient determined by experiment, and called the coefficient of resistance of the orifice. It is tolerably constant for different heads with well-formed orifices. Then = /2gh, = ~ (2gh(i +Cr)} (4)

The relation between c~ and cr for any orifice is easily found :

= C,~.f2gh ~ ~2gh/(I +Cr)I

c~=-.J ~f/(Ijcr)J. (5)

cr I/c,2I. (5a)

Thus if C,=o97, then Cr=Oo628, That is, for such an orifice about 63/4% of the head is expended in overcoming frictional resistances to flow.

Coefficient of ContractionSharp-edged Orifices in Plane Surfaces. When a jet issues from an aperture in a vessel, it may either spring 1 4(/I

FIG. 15.

clear from the inner edge of the orifice as at a or b (fig. 15), or it may adhere to the sides of the orifice as at c. The former condition will be found if the orifice is bevelled outwards as at a, so as to be sharp edged, and it will also occur generally for a prismatic aperture like b, provided the thickness of the plate in which the aperture is formed is less than the diameter of the jet. But if the thickness is greater the condition shown ___________

at c will occur. A

When the discharge occurs as at a or b, the filaments con- --c verging towards the orifice ,~ B ~

continue to converge beyond ~ - - - - - -

it, so that the section of the. -

jet where the filaments have V \l become parallel is smaller than the section of the orifice. The, inertia of the filaments opposes ,.~ ~

of motion at the edge of the ~ .~

orifice, and the convergence continues for a distance of about half the diameter of the orifice beyond it. Let w he the -

area of the orifice, and C,w the area of the jet at the point where convergence ceases; then c~ is a coefficient to be determined experimentally for each kind of orifice, called the coefficient of Contraction. When the orifice is a sharp-edged orifice in a plane surface, the value of C, is on the average 0-64, or the section of the jet is very nearly five-eighths of the area of the orifice.

Coefficient of DischargeIn applying the general formula Q =wv to a stream, it is assumed that the filaments have a common velocity v normal to the section w. But if / the jet contracts, it is at the con ,l ~/ tracted section of the jet that ii flie direction of motion is normal /: / -~ 1, a transverse section of the when cocn~tmn cc~(2gh)

___-~Ei ~ - or simply if c = c c Q = c~ (2gh)

- - ~ - ~a~ where c is called the cOefficient r - - - of discharge. Thus for a sharp- I- edged plane orifice C097X

-~ I~ ~-~ 0.640.62.

18. Experimental Determina tion of c~, c,, and c.The co efficient of contraction c, is FIG 16 directly determined by measur~

- ing the dimensions of the jet.

For this purpose fixed screws of fine pitch (fig. 16) are convenient. These are set to touch the jet, and then the distance between them can be nieasured at leisure.

The coefficient of velocity is determined directly by measuring the parabolic path of a horizontal jet.

Let OX, OY (fig. 17) be horizontal and vertical axes, the origin being at the orifice. Let h be the head, and x, y the coordinates of a point A on the parabolic path of the jet. If v~ is the velocity at the orifice, and t the time in which a particle moves from 0 to A, then x=v~t; y1/2gti.

Eliminating t, v,,=~ (gx2/2y).

Then cr=v~/iJ (2gh) =~ (xi/4yh).

In the case of large orifices suci- as weirs, the velocity can be directly determined by using a Pitot tube (f 144).

The coefficient of discharge, which for practical purposes is the most important of the three coefficients is best determined by tank measurement of the flow from the given orifice in a suitable time. If Q is the discharge measured in the tank per second, then c=Q/oii/ (2gh).

Measure ments of this kind though simple in principle are not free from some practical difficulties, and require much care.

In fig. 18 is shown an arrangement of measuring tank. FIG. 17.

The orifice is fixed in the wall of the cistern A and discharges either into the waste channel BB, or into the measuring tank. There is a short trough on rollers C which when run under the jet directs the discharge into the tank, and when run back again allows the discharge to drop -___rL

rriiir<~

~z~w I--, ~

FIG. 18.

into the waste channel. D is a stilling screen to prevent agitation of the surface at the measuring point, E, and F is a discharge valve for emptying the measuring tank. The rise of level in the tank, the time of the flow and the head over the orifice at that time must be exactly observed.

For well made sharp-edged orifices, small relatively to the water surface in the supply reservoir, the coefficients under different conditions of head are pretty exactly known. Suppose the same quantity of water is made to flow in succession through such an orifice and through another orifice of which the coefficient is required, and when the rate of flow is constant the heads over each I orifice are noted. Let h1, hi be the heads, wi, Wi the areas of the orifices, c,, cf the coefficients. Then since the flow through each orifice is the same Q =c~c~uV (2ghi) =c,wi~i (2ghi).

cf =cf (wI/Wf) -~l (hi/hf).

19. Coefficients for Belimouths and Belimouthed Oriflces.If an. orifice is furnished with a mouthpiece exactly of the form of the;~-- -D.~i.z~cL

O-~5D

~o.62.5d -

3~}~-6

d.o.8D~-~s4

FIG. 19.

contracted vein, then the whole of the contraction occurs within the mouthpiece, and if the area of the orifice is measured at the smaller end, c, must be put = I. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this is not done; and such a beilmouth may also have the form of the contracted jet. Fig. 19 shows the proportions of such a bellmouth or bellmouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose.

For such an orifice L. J. Weisbach found the following values of the coefficients with different heads.

Headoverorifice,inft.=h i~64 11.48 55-77 33793

Coefficient of velocity = C, - 959 967.975.994.994

Coefficient of resistance =c~ .087.069.052 ~oI2 012

As there is no contraction after the jet issues from the orifice, c5=i, c=c,; and therefore Q =c,u-.J (2gh) =C1% ~2gh/(I +c,)1.

20. Coefficients for Shar p-edged or virtually Sharp-edged Orifices. There are a very large number of measurements of discharge from sharp-edged orifices under different conditions of head. An account of these and a very careful tabulation of the average values of the coefficients will be found in the Hydraulics of the late Hamilton Smith (Wiley & Sons, New York, 1886). The following short table abstracted from a larger one will give a fair notion of how the coefficient varies according to the most trustworthy of the experiments.

Coefficient of Discharge for Vertical Circular Orifices, Sharp-edged, with free Discharge into the Air. Q =cce-.J (2gh).

Head Diameters of Orifice.

measured to ~02 ~04 fO ~20 4o ~6o IO

Centre of Orifice. Values of C.

0.3 - - -. 621 - - - - - -

0~4 - - 637 618 -. - - - - -.

o6.655 63o 613.601 596.588 -.

o~8 648 626 610 601.597.594 583

1.0 .644.623 608.600 598 595 591

2~0.632 614.604.599.599.597.595

4.0 .623.609 6o2.599 ~598 597 596

8~o 614 605 6oo .598.597 596 596

20O 601.599 ~596.596 596 596 594

At the same time it must be observed that differences of sharpness in the edge of the orifice and some other circumstances affect the results, so that the values found by different careful experimenters are not a little discrepant. When exact measurement of flow has to be made by a sharp-edged orifice it is desirable that the coefficient for the particular orifice should be directly determined.

The following results were obtained by Dr H. T. Bovey in the laboratory of McGill University.

Coefficient of Discharge for Sharp-edged Orifices.

Form of Orifice.

Rectangular Ratio Rectangular Raiio Square. of Sides 4:i. of Sides iS:,.

Headin ~ ~ -

Ii. cir- Long Tn cular. Sides Dia Long Long Sides angular.

vertical gonal Sides f r Sides honi sertical. vertical. zontal. vertical, zonial.

1 -620 627 -628 642 643 -663 664 636

2 613 620 628 634 636 650 651 628

4 608 616 6i8 628.629 -641, -642 623

6 -607 -614 616 -626 627 637.637 620

8 -6o6 613 614 623.625 634 -635 619

10 -605 612 -613 -622 -624 632 633 618

12 604 611 6I2 622 -623 63f -631 618

14 -604 -61o -612 621 -622 -630 -630 6,8

16 603 610 -6i1 620 622 630 630 617

i8 603 610 -611 -620 621 -630 629 616

20 -603 609 -6I1 -620 -621 -629 -628 6f6

The orifice was 0.196 sq. in. area and the reductions were made with g=32-176 the value for Montreal. The value of the coefficient appears to increase as (perimeter) / (area) increases. It decreases as the head increases, It decreases a little as the size of the orifice is greater.

Very careful experiments by J. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.) on the discharge from circular orifices gave the results shown on top of next column.

The edges of the orifices were got up with scrapers to a sharp square edge. The coefficients generally fall as the head increases and as the diameter increases. Professor W. C. Unwin found that the results agree with the formula c =o~6o75+ooo98/~ ho0o37d, where h is in feet and d in inches.

Coefficients of Discharge from Circular Orifices. Temperature 51 to 550

~Head in Diameters of Orifices in Inches (d).

feet fHHl1~2HI21/2~2*I3

Coefficients (c)

75 6,6 614 6i6 -6,0 -616 ~612 607.607 -609

~0 613 6i2 612 -6i1 -612 611 -604 -608 609

I25 613 -614 6fo -6o8 612.6o8 -605.6o5 -606

1.50 6,0 -612 6ff .6o6 ~6fo 607 -603.607 -6o5

1.75 612 611 611 605 6ff .605 -604 -607 -605

2-00 6o9 -613 609 -6o6 -609 -6o6 -604 -604.6o5~

The following table, compiled by J. T. Fanning (Treatise on Water Supply Engineering), gives values for rectangular orifices in vertical plane surfaces, the head being measured, not immediately over the orifice, where the surface is depressed, but to the stillwater surface at some distance from the orifice. The values were obtained by graphic interpolation, all the most reliable experiments being plotted and curves drawn so as to average the discrepancies.

Coefficients of Discharge for Rectangular Orifices, Sharp-edged, in Vertical Plane Surfaces.

Head to Ratio of Height to Width.

Cenireof Orifice. 4 2 Ii I f I

.0.0 .0.

.0. e ~o~i ~e ~

~a ~ ~2 ~oa Feet. 3s ~e .s ~s a,.: 0..: g.- ~? ~

~ ~-: T~- o~ f~. o~.

02 - - -.. -. - ... - - - 6333

3 -. - -.. -. -. - - .6293 6334

4 - - - - - - - - - - 6140.6306 6334

.5 - - -. - - - - .6050 6150 6313 6333

6 - - -. -. .5984 -6063 -6156 ~63f7 6332

.7 .. - - - - .5994 6o74 6162.6319 6328

8 - - - - -6130 -600o -6082 6165 6322 6326

.9 - - -.6134-60066o86 6,68 6323 6324

1~0 - - - - 6135 6010 6090 6,72.6320 6320

P25 - - ~6188 -6140.6018 -6095 6173 6317 -6312

1.50 - -6187-6144-6026 -6,00 6,72 6313 6303

f75 - - 6186-6145-60336103 6,68 6307 6296

2 .. 6183 6144.6036 6104 6,66.6302 629,

2.25 - - -6f8o6143-6029-6103 6163.6293 6286

2.50 6290 6176-6139-60436102 6157 6282 6278

2-75.6280.6173 6136 6046 6ioI -6155.6274 6273

3 6273 6,70 6132.6048 6,00 6153.6267 6267

35.625o 6160.6123-6050-60946146 6254 6254

4 ~6245 ~6150 6110 6047 6085 6136.6236 6236

45.6226 6138 6,00.6044 6074 6125 6222 6222

5 6208 6124 6o88 6038.6063 6114 6202 6202

6 -6158 6094 6063.6020 6044.6087 6154 6154

7 6124-6064-60386011 6032 6058 611o 6114

8.6090 6036 6022.60,06022-6033-6073 6087

9 -6o6o 6020 6014 -6oto .6015 6020 6045 6070

10.6035 6015 6oio -6010 -6010 -6010 6030 -6o6o 15 6040 -6o18 60,0.6oII -6012-6013-60336o66

20 6045 6024-6012-60126014 -6018 6036 -6074

25.6048 -6028 ~6oI4 -6012 6o16 6022 6040 6083

30 6054-6034-6017-6013 6018 6027 6044 6092

35 6o6o 6039 6021.6014 6022 6032.6049 6103

40 6066 6045 6025 -6oi5 6026 6037 6055 6114

45 6054 6052 6029.6016 6030 6043 6062 6125

50.6o86 6o6o 6034 6o18 6035.6o5o 6070 6140

2 t. Orifices with Edges of Sensible Thickness.\iVhen the edges of the orifice are not bevelled outwards, but have a sensible thickness, the coefficient of discharge is somewhat altered. The following table gives values of the coefficient of discharge for the arrangenieRts of the orifice shown in vertical section at F, Q, R (fig. 20). The plan of all the orifices is shown at S. The planks. forming the orifice and sluice were each 2 in. thick, and the orifices were all 24 in. wide. The heads were measured immediately over the orifice. In this case, Q .Cb(l~Jj),J t2g(H+h)/2~.

22. Partially Suppressed Contraction.Since the contraction of the jet is due to the convergence towards the orifice of the issuing streams, it will be diminished if for any portion of the edge of the orifice the convergence is prevented. Thus, if an internal rim or border is applied to part of the edge of the orifice (fig. 21), the convergence for so much of the edge is suppressed. For such cases G. Bidone found the following empirical formulae applicable:

Table of Coefficients of Discharge for R

Head Il Height of above _______________________ __________

upper 1.31 o66 edg of Oiifiae infect. P Q R P Q

0-328 o598 0.644 0.648 0-634 0-665 0-

.656 0-609 0-653 0.657 0.640 0-672 0-

787 o612 0-655 0-659 o64f o674 0-

.984 o-6f6 0-656 o66o 0641 0-675 0

1.968 0-618 0649 0-653 0.640 0-676 o 3-28 0-608 0.632 0.634 0638 0-674 0

4-27 0-602 0-624 0-626 0.637 0.673 0-

4-92 0.598 0-620 0-622 0-637 0-673 0-

5-58 0.596 o618 0-620 0-637 0-672 0

6-56 0-595 0.615 0-617 0.636 0-671 0

9-84 0-592 0.611 0-612 0-634 0-669 uFor rectangular orifices, C, =0-62(1 +O152n/P);

and for circular orifices, C, =0.62(1 +0128n/P);

when n is the length of the edge of the orifice over which the border extends, and p is the whole length of edge or perimeter of the orifice. The following are the values of c,, when the border extends over 1/8, 4 or 3/4 of the whole perimeter:

C, C,

n/p Rectangular Orifices. Circular Orifices.

0.25 0-643 -640

0-50 0-667 66o 0.75 0.691 680

For larger values of n/p the formulae are not applicable. C. R.

4 Bornemann has shown, ~ however, that these formulae for suppressed con- P traction are not reliable.

- 23. Imperfect Con- - tractionIf the sides of - e- ~ the vessel approach near to the edge of the orifice, ~ ~ they interfere aith the - _~ convergence of the streams ~~ ~ to which the contraction is then modified. It is generally stated that the ~ - .. influence of the sides a ~i3~, begins to be felt if their distance from the edge of the orifice is less than 2-7

times the corresoonding R~FTh ii g L-~ --~ \

~ :~ ::

FIG. 20. FIG. 21.

width of the orifice. The coefficients of contraction for this case are imperfectly known.

ctangular Vertical Orifices in Fig. 20.

)rifice, H - h, in feet.

0-16 0-10

P Q R P Q R

ffi8 0.691 0-664 o-666 0.710 0-694 0-696

>75 o.685 0-687 o-688 0.696 0704 0-706

77 o-684 0-690 0.692 0.694 0.706 0-708

>78 0.683 0.693 0.695 0-692 0-709 0.711

79 0.678 0-695 0.697 o688 0-710 0-712

)76 0-673 0-694 0-695 0.680 I 0.704 0-705

>75 0-672 0-693 0.694 0-678 0-701 0-702

)74 0.672 0.692 0.693 0.676 0.699 0.699

>73 0-672 0.692 0.693 0-676 o-b98 0-698

72 0-671 0-691 0-692 0-675 0-696 0-696

>70 o-668 0-689 0-690 0.672 0-693 0-693

24. Orifices Furnished with Channels of DischargeThese external borders to an orifice also modify the contraction.

The following coefficients of discharge were obtained with openings 8 in. wide, and small in proportion to the channel of approach (fig. 22, A, B, C).

to-h, in h, in feet.

feet. ~ ~32S ~56 328 492 656 ~84

A 480 511 542 -574 599 601 6o1 6oI -6oi B 0.656 480 510.538 -506.592 600 602 -602 -6o1

C .527 553 574 592 607 610 610 -609 6o8

A) 488 577 -624 631.625.624 619 613 -6o6

B ~.oI64 487.571.606.617 -626 628.627 623 -6i8

C) ~85.614 ~ ~ ~52~fli65oj65o ~49

25. Inversion of the Jet.When a jet issues from a horizontal orifice, or is of small size compared with the head, it presents no I marked peculiarity of form. But if the orifice is in a vertical surface, and if its dimensions are not small compared with the head, Sl~oje .1 zn~ 20

k---- - --

it undergoes a series of singular changes of form after leaving the orifice. These were first investigated by G. Bidone (1781-1839); subsequently H. G. Magnus (1802-1870) measured jets from different orifices; and later Lord Rayleigh (Proc. Roy. Soc. xxix. 71) investigated them anew.

Fig. 23 shows some forms, the upper figure giving the shape of the orifices, and the others sections of the jet. The jet first contracts as described above, in consequence of the convergence of the fluid ,trcams within the vessel, retaining, however, a form similar to that of the orifice. Afterwards it expands into sheets in planes perpendicular to the sides of the orifice. Thus the jet from a triangular orifice expands into three sheets, in planes bisecting at right angles the three sides of the triangle. Generally a jet from an orifice, in the form of a regular polygon of n sides, forms n sheets in planes perpendicular to the sides of the polygon.

Bidone explains this by reference to the simpler case of meeting streams. If two equal streams having the same axis, but moving in opposite directions, meet, they spread out into a thin disk normal to the common axis of the streams. If the directions of two streams intersect obliquely they spread into a symmetrical sheet perpendicular to the plane of the streams.

Let ai, a, (fig. 24) be two points in an orifice at depths hi, h, from the free surface. The filaments issuing at ai, ai will have the different r~- velocities -~ 2ghi and ~1 2ghi.

Consequently they will :~_t_~] tend to describe parabolic i-.J~,1 paths aicb1 and azcbi of TtL. ~ different horizontal range, and intersecting in the point c. But since two I filaments cannot simul - a1 taneously flow through the - - same point, they must e exercise mutual pressure, - 3~ and will be deflected out of the paths they tend to ~j describe. It is this mutual pressure which causes FIG. 24. the expansion of the jet into sheets.

Lord Rayleigh pointed out that, when the orifices are small and the head is not great, the expansion of the sheets in directions perpendicular to the direction of flow reaches a limit. Sections taken at greater distance from the orifice show a contraction of the sheets until a compact form is reached similar to that at the first contraction. Beyond this point, if the jet retains its coherence, sheets are thrown out again, but in directions bisecting the angles between the previous sheets. Lord Rayleigh accepts an explanation of this contraction first suggested by H. Buff (1805-1878), namely, that it is due to surface tension.

26. Influence of Temperature on Discharge of Orsfices.Professor ~V. C. Unwin found (Phil. Mag., October 1878, p. 281) that for sharp-edged orifices temperature has a very small influence on the discharge. For an orifice I cm. in diameter with heads of about I to 11/2 ft. the coefficients were:

Temperature F C.

205 - .594

62.598

For a conoidal or bell-mouthed orifice I cm. diameter the effect of temperature was greater:

Temperature F C.

190 0~987

130 0.974

600 0.942

an increase in velocity of discharge of 4% when the temperature increased I30.

J. G. Mair repeated these experiments on a much larger scale (Proc. Inst. Civ. Eng. lxxxiv.). For a sharp-edged orifice 23/4 in. diameter, with a head of 1.75 ft., the coefficient was 0.604 at 570 and 0-607 at 179 F., a very small difference. With a conoidal urifice the coefficient was 0.961 at 55 and 0.981 at 1700 F. The corresponding coefficients of resistance are oo828 and 0.0391, showing that the resistance decreases to about half at the higher temperature.

27. Fire hose Nozzles.Experiments have been made by J R. Freeman on the coefficient of discharge from smooth cone nozzles used for fire purposes. The coefficient was found to be 0.983 for 3/4-in. nozzle; 0.982 for 3/4 in.; 0.972 for I in.; 0.976 for 13/4 in.; and O-97i for 13/4 in. The nozzles were fixed on a taper play-pipe, and the coefficient includes the resistance of this pipe (Amer. Soc. f~iv. Eng. XXI.. 1889). Other forms of nozzle were tried such as ring nozzles for which the coefficient was smaller.

IV. THEORY OF THE STEADY MOTION OF FLUIDS.

28. The general equation of the steady motion of a fluid given under Hydrodynamics furnishes immediately three results as to the distribution of pressure in a stream which may here be assumed.

(a) If the motion is rectilinear and uniform, the variation of oressure is the same as in a fluid at rest. In a stream flowing in an open channel, for instance, when the effect of eddies produced by the roughness of the sides is neglected, the pressure at each point is simply the hydrostatic pressure due to the depth below the free surface. -

(b) If the velocity of the fluid is very small, the distribution of pressure is approximately the same as in a fluid at rest.

(c) If the fluid molecules take precisely the acceleeations which they would have if independent and submitted only to the external forces, the pressure is uniform. Thus in a jet falling freely in the air the pressure throughout any cross section is tiniform and equal to the atmospheric pressure.

(d) In any bounded plane section traversed normally by streams which are rectilinear for a certain distance on either side of the section, the distribution of pressure is the same as in a fluid at rest.

DI5TRIBUTfON OF ENERGY IN INCOMPRESSIBLE FLUIDS.

29. Application of the Principle of the Conservation of Energy to Cases of Stream Line Motion.The external and internal work done on a mass is equal to the change of kinetic energy produced. In many hydraulic questions this principle is difficult to apply, because from the complicated nature of the motion produced it is difficult to estimate the total kinetic energy generated, and because in some cases the internal work done in overcoming frictional or viscous rocistances cannot be ascertained; but in the case of stream line motion it furnishes a simple and important result known as Bernoulhs theorem.

Let AB (fig. 25) be any one elementary stream, in a steadily moving fluid mass. Then, from the steadiness of the motkn, AB is a fixed path in space through which a stream of fluid is constantly flowing. Let 00 be the free surface and XX any horizontal datum line. Let A A B .8

I ...,..... .,

K - K

FIG. 25.

i~ be the area of a normal cross section, v the velocity, p the intensity of pressure, and z the elevation above XX, of the elementary stream AB at A, and Wi, pi, vi, zi the same quantities at B. Suppose that in a short time I the mass of fluid initially occupying AB comes to AB. Then AA, BB are equal to vt, vit, and the volumes of fluid AA, BB are the equal inflow and outflowo=Qt=wvt=wivft, in the given time. If we suppose the filament AB surrounded by other filaments moving with not very different velocities, the frictional or viscous resistance on its surface will be small enough to be neglected, and if the fluid is incompressible no internal work is done in change of volume. Then the work done by external forces will be equal to the kinetic energy produced in the time considered.

The normal pressures on the surface of the mass (excluding the ends A, B) are at each point normal to the direction of motion, and do no work. Hence the only external forces to be reckoned are gravity and the pressures on the ends of the stream.

The work of gravity when AB falls to AB is the same as that of transferring AA to BB; that is, GQI (1li). The work of the pressures on the ends, reckoning that at B negative, because it is opposite to the direction of motion, is (PwXvt)(PiwiXvit)

Qt(ppi). The change of kinetic energy in the time t is the difference of the kinetic energy originally possessed by AA and that finally acquired by BB, for in the intermediate part AB there is no change of kinetic energy, in consequence of the steadiness of the motion. But the mass of AA and BB is GQt/g, and the change of kinetic energy is therefore (GQtfg) (vif/2 v(2). Equating this to the work done on the mass AB,

GQI(zzi)-}-Qt(ppi) =(GQt/g) (viV2 v2/2).

Dividing by GQt and rearranging the terms, v12g+ ~/G +1 =vi/2g+pi/G +Zi; (1)

or, as A and B are any two points, v12g+pIG+z=constant =H. (2)

Now v~f2g is the head due to the velocity v, p/G is the head equivalent to the pressure, and I is the elevation above the datum (see 16). Hence the terms on the left are the total head due to velocity, pressure, and elevation at a given cross section of the filament, 1 is easily seen to be the work in foot-pounds which would be done by 1 lb of fluid falling to the datum line, and similarly PIG and V/2g are the quantities of work which would he done by 1 lb of fluid due to the pressure p and velocity v. The expression on the left of the equation is, therefore, the total energy of the stream at thi section considered, per lb of fluid, estimated with reference to th~

datum line XX. Hence we see that in stream line motion, under the restrictions named above, the total energy per lb of fluid is uniformly distributed along the stream line. If the free surface of the fluid 00 is taken as the datum, and h, h, are the depths of A and B measured down from the free surface, the equation takes the form v2/2g+p/G h = v11/2g+pi/G hi; (3)

or generally V/2g +PIG h = constant. (3a)

30. Second Form of the Theorem of BernoulliSuppose at the two sections A, B (fig. 26) of an elementary stream small vertical pipes are introduced, which may be termed pressure columns - ~-~- I

1-6 B1~1

r-Engels__,jt - 4-__..

FIG. 26.

(~ 8), having their lower ends accurately parallel to the direction of flew. In such tubes the water will rise to heights corresponding to the pressures at A and B. Hence bp/G, and b=p/G. Consequently the topl of the pressure columns A and B will be at total heights btc=p/G+z and b+c=pi/G+zi above the datum line XX. The difference of level of the pressure column tops, or the fall of free surface level between A and B, is therefore and this by equation (1), 29 is (vii_v2)/2g. That is, the fall of free surface level between two sections is equal to the difference of the heights due to the velocities at the sections. The line AB is sometimes called the line of hydraulic gradient, though this term is also used in cases where friction needs to be taken into account. It is the line the height of which above datum is the sum uf the elevation and pressure head at that point, and it falls below a horizontal line AB drawn at H ft. above XX by the quantities a =12/2g and a =v,/2g, when friction is absent.

31. Illustrations of the Theorem of Bernoulli. In a lecture to the mechanical section of the British Association in 1875, W. Froude gave some experimental illustrations of the pnnciple of Bernoulli. He remarked that it was a common but erroneous impression that a fluid exercises in a contracting pipe A (fig. 27) an excess of pressure against the entire converging surface of pressure is experienced by the ~ entire diverging surface of the pipe.

Further it is commonly asstimed that ~ __-~- -when passing through a contraction A

C, there is in the narrow neck an excess of pressure due to the squeezing together of the liquid at that noint. These impressions are in no respect correct; the pressure is smaller as the section of the pipe is smaller and conversely.

Fig. 28 shows a pipe so formed that a contraction is followed by an enlargement, and fig. 29 one in which an enlargement is followed by a contraction. The A B vertical pressure columns .......-_-_.- show the decrease of _____~_. _.._.~,,, p pressure at the enlargement. The line abc in both figures shows the C variation of free surface ~___-- level, supposing the pipe ~ ~i~fl~vei~:1 actual FIG. 27. expended in friction against the pipe; the total head diminishes in proceeding along the pipe, and the free surface level is a line such as abici, falling below abc.

Froude further pointed out that, if a pipe contracts and enlarges again to the same size, the resultant pressure on the converging part exactly balances the resultant pressure on the diverging part so that there is no tendency to move the pipe bodily when water flows through it. Thus the conical part AB (fig. 30) presents the same projected surface as HI, and the pressures parallel to the axis of the pipe, normal to these projected surfaces, balance each other. Similarly the pressures on BC, CD balance those on GH, EG. In the same way, in any combination of enlargements and contractions, a balance of pressures, due to the flow of liquid parallel to the .__j.____.~__ ~

~.28.

axis of the pipe, will be found, provided the sectional area and direction of the ends are the same.

The following experiment is interesting. Two cisterns provided with converging pipes were placed so that the jet from one was exactly opposite the entrance to the other. The cisterns being filled ~ ~,,i-)

very nearly to the same level, the jet from the left-hand cistern A entered the right-hand cistern B (fig. 31), shooting across the free space between them without any waste, except that due to indirectness of aim and want of exact correspondence in the form of the orifices. In the actual experiment there was 18 in. of head in the right and 201/2 in. of head in the left-hand cistern, so that about FIG. 30.1~

21/2 in. were wasted in friction. It will be seen that in the open space between the orifices there was no pressure, except the atmospheric pressure acting uniformly thronghout the system.

32. Ventur Meter.An ingenious application of the variation of pressure a .d velocity in a converging and diverging pipe has been A B -

FIG. 31.

made by Clemens Herschel in the construction of what he terms a Venturi Meter for measuring the flow in water mains. Suppose that, as iii fig. 32, a contraction is made in a water main, the change of section being gradual to avoid the production of eddies. The ratio p of the cross sections at A and B, that is at inlet and throat, is in actual meters 5 to I to 20 to 1, and is very carefully determined by the maker of the meter. Then, if v and u are the velocities at A and B, ~upv. Let pressure pipes be introduced at A, B and C,

, L D~ ~ - _. L

FIG. 32.

and let Hi, Fl, Hf be the pressure heads at those points. Since the velocity at B is greater than at A the pressure will be less. Neglecting friction H1 +vf/2g = H +u2/2g, HiH = (u2v2)/2g = (piI)vf/2g.

Let Ii = HfH be termed the Venturi head, then u=~J {p2.2gh/(,p1I)~,

from which the velocity through the throat and the discharge of the main can be calculated if the areas at A and B are known and h observed. Thus if the diameters at A and B are 4 and 12 in., the areas are 12-57 and 113.1 sq. in., and p=9,

u=~8i/8o~ (2gh) =I~oO7V (2gh).

If the observed Venturi head is 12 ft.,

U =28 ft. per sec.,

and the discharge of the main is 28X12-57=351 cub. ft. per sec.

Hence by a simple observation of pressure difference,, the flow in the main at any moment can be determined. Notice that the pressure height at C will be the same as at A except for a small loss h1 due to friction and eddying between A and B. To get the pressure at the throat very exactly Herschel surrounds it by an annular passage communicating with the throat by several small holes, sometimes formed in vulcanite to prevent corrosion. Though constructed to prevent eddying as much as possiule there is some eddy loss The main effect of this is to cause a loss of head between A and C which may vary from a fraction of a foot to perhaps 5 ft. at the highest velocities at which a meter can be used. The eddying also affects a little the Venturi head h. Consequently an experimental coefficient must be determined for each meter by tank measurement. The range of this cuetficient is, however, surprisingly small. If to allow for friction, u =k~I pf/(p2_f)}~/ (2gh), then Herschel found values of k from 0-97 to 1-0 for throat velocities varying from _________ 8 to 28 ft. per Sec. The ~ meter is extremely con ve,uent. At Staines reser voirs there are two meters of this type on mains 94 in.

P in diameter. Herschel con trived a recording arrange ment which records the variation of flow from hour to hour and also the total - flow in any gven time. In ~ -~- Great Britain ~be meter is _______ constructed by G. Kent, who has made improvements Outlet D In/cl in the recording arrange- -=: - ~ nient.

In the Deacon Waste Water Meter (fig. 33) a different principle is used.

A disk D, partly counter ________ balanced by a weight, is suspended in the water flow ~ / ~ ing through the main in a conical chamber. The un Fiu. 33- balanced weight of the disk is supported by the impact of the water, If the discharge of the main increases the disk rises, hut as it rises its position in the chamber is such that in consequence of the larger area the velocity is less. It finds, therefore, a new positian of equilibrium. A pencil P records on a drum moved by clockwork the position of the disk, and from this the variation of flow is inferred, 33. Pressure, Velocity and Energy in Different S/ream Lines. The equation of Bernoulli gives the variation of pressure and velocity from point to point along a stream line, and shows that the total energy of the flow across any two sections is the same. Two other directions may be defined, one normal to the stream line and in the plane containing its radius of curvature at any point, the other normal to the stream line and the radius of curvature. For the problems most practically useful it will be sufficient to consider the stream lines as parallel to a vertical or horizontal plane. If the motion is in a vertical plane, the action of gravity must be taken into the reckoning; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will disappear.i Let AB, CD (fig. 34) be two consecutive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines psdp ~- -~-~

FIG. 34.

making an angle 4) with the vertical. Let F, Q be two particles moving along these lines at a distance PQ = ds, and let I be the height of Q abovethe horizontal plane with reference to which the energy is measured, v its velocity, and p its pressure. Then, if H is the total energy at Q per unit of weight of fluid, H =z+p/G-l-v/2g.

Differentiating, we get dH =dz+dp/G+vdv/g, (I)

for the increment of energy between Q and P. But dz=PQ cos 4)=ds cos 4);

.-. dH =dp/G+vdv/g+ds cos 4), (ia)

where the last term disappears if the motion is in a horizontal plane. Now imagine a small cylinder of section described round PQ

as an axis. This will be in equilibrium under the action of its centrifugal force, its weight and the pressure on its ends. But its volume is wds and its weight G~sds. Hence, taking the components of the forces parallel to PQ

= Gvlwds/gpGs, cos 4)ds, where p is the radius of curvature of the stream line at Q. Consequently, introducing these values in (I),

dH = v1ds/gp +vdvig = (r/g) (v/p +dv/ds)ds. (2)

CURRENTS

34, Rectili,f ear Current.Suppose the motion is in parallel straight stream lines (fig. 35) in a vertical plane. Then p is infinite, and from eq. (2), 33,

dH =vdv/g.

Comparing this with (I) we see that dz+dp/G=o; .. z+p/G=constant; (3)

or the pressure varies hydrostatically as in a fluid at rest. For two stream lines in a horizontal plane, I i5 constant, and there-, fore p is constant.

Radiating CurrentSuppose d~z water flowing radially between horizontal parallel planes, at -D

a distance apart = I. Conceive Q

two cylindrical sections of the FIG. 35.

current at radii ri and ri, where t1ie velocities are v1 and vf, and the pressures Pi and pi. Since the flow across each cylindrical section of the current is the same,,

Q = 2z-rilln = 2irrflvi nIh =rlvf 1 The following theorem is taken from a paper by J. H. Cotterill, On the Distribution of Energy in a Mass of Fluid in Steady Motion, Phil. Mag., February 1876.

Fhe velocity would be infinite at radius 0, if the current could be conceived to extend to the axis. Now, if the motion is steady, I-I = puG + Vif/2g = p1/G +v2/2g; = P2/G f-r12v12/r222g; (pfp1)/C =e,2(Ir,2/r22)/sg; (5)

P2!- Elri2ei~/rf22g. (6)

Hence the pressure increases from the interior outwards, in a way indicated by the pressure columns in fig. 36, the curve through the free surfaces of the pressure columns being, in a radial section, the quasi-hyperbola of the form xyf=c3. This curve is asymptotic to a horizontal line, H ft. above the line from which the pressures are measured, and to the axis of the current.

Free Circular VortexA free circular vortex is a revolving mass ,f water, in which the stream lines are concentric circles, and in which ~Tft~ft~I,J fi ,

tJL_.ll~1~~z~1~~

~i~r ~_~-r II ,

Ii ,

I ,

Fl G. 36.

the total head for each stream lir.e is the same. Hence, if by any slow radial motion portions of the water strayed from one stream line to another, they would take freely the velocities propel to their new positions under the action of the existing fluid pressures only.

For such a current, the motion being horizontal, we have for all the circular elementary streams I-I =P/G+v2/2g=constant; - .dH = dp/G +vdv/g = 0. (7)

Consider two stream lines at radii r and r+dr (fig. 36). Then in (2), 33, p=r and ds=dr, v2dr/gr~vdv/g = 0,

dy/v = dr/r, v=i/r, (8)

precisely as in a radiating current; and hence the distribution of pressure is the same, and formulae 5 and 6 are applicable to this case.

Free Spiral VortexAs in a radiating and circular current the equations of motion are the seine, they will also apply to a vortex in which the motion is compounded of these motions in any proportions, provided the radial component of the motion varies inversely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex, Then the whole velocity at any point will be inversely proportional tC the radius of the point, and the fluid will describe stream line having a constant inclination to the radius drawn to the axis of tl,c current. That is, the stream lines will be logarithmic spirals~ When water is delivered from the circumference of a centrifuga~ p~emp or turbine into a chamber, it forms a free vortex of this kind I he water flows soirallv outwards, its velocity diminishine and in pressure increasing according to the law stated above, and the head along each spiral streani line is constant. -

35, Forced VortexIf the law of motion in a rotating current is different from that in a free vortex, some force mpst be applied to cause the variation of velocity. The simplest case is that of a rotating current in which all the particles have equal angular velocity, as for instance when they are driven round by radiating paddles revolving uniformly. Then in equation (2), 33, considering two circular stream lines of radii r and r+dr (fig. 37), we have p=r, ds=dr. If the angular velocity is a, then v=ar and dv=edr. Hence dH = a2rdr/ga2rdr/g=2a2rdr/g.

Comparing this with (1), 33, and putting dz=.o, because the motion is horizontal, dp/G + a2rdr/g = 2afrdr/g, dp/G = a,2rdr/g, pfG =afr2/2g+constant. (9)

Let ~, r~, vi be the pressure, radius and velocity of one cylindrical section, P2, rf, vi those of anotocr toen pi/Ga2rii/2g pf/G--a2r22,!2g; (pfpi)/G a2(r22ri2)/sg = (z22v12)/2g. (10)

That is, the pressure increases from within outwards in a curve FIG. 37.

which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (~ig, 37).

DISSIPATION OF HEAD IN Snocn 36. Relation of Pressure and - Velocity in a Stream in Steady Motion when the Changes of Secijon of the Stream are Abrupt. When a stream changes section abruptly, rotating eddies are fosnsed which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. Suppose fig. 38 to represent a stream having such an abrupt change of section, Let AU, CD be normal sections at points where ordinary stream line motion has not been disturbed and where it has been re-established, Let me, p v be the area of section. pressure and velocity at AB, and wi Pm, v1 corresponding quantities at CD. Then if no work were expended internally, and assuming the stream horizontal, we should have ~)/G-I- V2 /21 = 15/G-I-v,f/2e, (r~

But if work is expended in producing irregular eddying motion, the head at the section CD will he diminished.

Suppose the mass ABCD comes in a short time I to ABCD. The resultant force parallel to the axis of the stream is pc, +Po(wiw)Piwi, where p,, is put for the unknown pressure on the annular space between AB and EF. The impulse of that force is ~ +po(wiw)piwi~t.

liie horizontal change of momentum in the same time is the difference of the momenta of C (CDCD and ABAB

because the amoun~

of momentum be ________ tween AB and CD

______ remains unchanged _____i~-it~--- if the motion is ~-~ ______ +--~- - steady. The volume -- ~T of ABAB or CDCD, ______ being the inflow and outflow in the time t, is Qtwvtwivjt, and the momentum of these masses is FIG. 38. (G/g)Qvland(G/g)Qvil.

The change of mo inentum is therefore (G/g)Qt(viv). Equating this to the impulse, {pw +Po(~i~)--Piwi }t = (G/g)Qt(viv).

Assume that Pi = p, the pressure at AB extending unchanged through the portions of fluid in contact with AE, BF which lie out of the path of the stream. Then (since Q w1v1)

(ppr) = (G/g)v~ (vfv);

p/GPi/G =v,(viv)/g; (2)

pIG +v12g = puG + Vif/2g+ (v_~viYu12g. (3)

This differs from the expression (I), 29, obtained for cases where no sensible internal work is done by the last term on the right. That is, (vvi)/2g has to be added to the total head at CD, which is P1IG+vu2/2g, to make it equal to the total head at AB, or (vvi)/2g is the head lost in shock at the abrupt change of section. But vi1 is the relative velocity of the two parts of the stream. Hence, when an abrupt change of section occurs, the head due to the relative velocity is lost in shock, or (zvi)/2g foot-pounds of energy is wasted for each pound of fluid. Experiment verifies this result, so that the assumption that po=p appears to be admissible.

If there is no shock, p1/G = p/G + (vvii)/2g.

If there is shock, p1/G = p/Gvj (viv)/g.

Hence the pressure head at CD in the second case is less than in the former by the quantity (tvi)2/2g, or, putting ~ivi =cev, by the quantity (v/2g) (Iw/wi). (4)

V. THEORY OF THE DISCHARGE FROM ORIFICES AND

MOUTHPIECES

3~. Minimum Coefficient of Contraction. Re-entrant Mouthpufl-c of Borda.ln one special case the coefficient of contraction can be determined theoretically, and as ~ it is the case wilere the convergence of the ~=:~=-~-

streams approaching 0 0 the orifice takes place through the greatest I possible angle, the co efficient thus deter- mined is the minimum coefficient.

Let fig. 39 represent it a vessel with vertical sides, 00 being the free water surface, at A c which the pressure is p~. Suppose the liquid issues by a horizontal mouthpiece, which is re-entrant and of the greatest length which permits the jet to spring clear from the p ---~-Engels

inner end of the orifice, without adher ing to its sides. With such an orifice the FIG. 39. velocity near the points,CD is negligible, and the pressure at those points may be taken equal to the hydrostatic pressure due to the depth from the free surface. Let ti be the area of the mouthpiece AB, w that of the contracted jet aa Suppose that in a short time t, the mass OOaa comes to the position OO aa; the impulse of the horizontal externai forces acting on the mass during that time is equal to the horizontal change of momentum.

The pressure on the side OC of the mass will be balanced by the pressure on the opposite side OE, and so for all other portions of the vertical surfaces of the mass, excepting the portion EF opposite the mouthpiece and the surface AaaB of the jet. On EF the pressure is simply the hydrostatic pressure due to the depth, that is, (p~+Gh)fl. On the surface and section AaaB of the jet, the horizontal resultant of the pressure is equal to the atmospheric pressure p~ acting on the vertical projection AB of the jet; that is, tile resultant pressure is p~t2. Hence the resultant horizontal force for the whole mass OOaa is (P~+Gh),tiPf2=Gliti. Its impulse in the time t is Gh~ t. Since the motion is steady there is no change of momentum between OO and aa. The change of horizontal momentum is, therefore, the difference of the horizontal momentum lost in the space 000O and gained ii~ the space aaaa. In the former space there is no horizontal momentum.

The volume of the space aaaa is wvt; the mass of liquid in that space is (G/g)wvt; its momentum is (G/g)wvft. Equating impulse to momentum gained, GhtJt = (G/g)wv2t; .~. f/ti=gh/vf.

But Vi =2gh, and ce/~1=c~

.~.w/tl=1/2=c~

a result confirmed by experiment with mouthpieces of this kind. A similar theoretical investigation is not possible for orifices in plane surfaces because the velocity along the sides of the vessel in the neighborhood of the orifice is not so small that it can be neglected. The resultant horizontal pressure is therefore greater than Ght~, and the contraction is less. The experimental values of the coefficient of discharge for a re-entrant mouthpiece are 0-5149 (Borda), 0.5547 (Bidone), 0.5324 (Weisbach), values which differ little from the theoretical value, 0.5, given above.

38. Velocity of Filaments issuing in a JeLA jet is composed of fluid filaments or elementary streams, which start into motion at some point in the interior of the vessel ~!_ __~_ from which the fluid is discharged, and -

gradually acquire ~

the velocity of the jet. Let Mm, fig. 40 be such a filament, the point M being taken where the velocity is insensibly small, and m at the most contracted section of the jet, where the filaments have he- FIG. 40.

come parallel and exercise uniform mutual pressure. Take the free surface AB for datum line, and let ~f, ri hi, be the pressure, velocity and depth below datum at M; p v, h, the corresponding quantities at m. Then 29, eq. (30),

vif/2g +p1/Ghi = V/2g +p/Gh- (1)

But at M, since the velocity is insensible, the presstire is the hydrostatic pressure due to the depth; that is, vi=o, p1=p~+Gh1. At m, p=p,~, the atmospheric pressure round the jet. Hence, inserting these values, o+p~/G+hihi =v/2g+p~/Gh; v/2g=h; (2)

or v=~J (2gb) 8025~Jh. (2a)

That is, neglecting the viscosity of the fluid, the velocity of filaments at the contracted section of the jet is simply the velocity due to the difference of level of the free surface in the I ~ reservoir and the orifice.

If the orifice is small in h, the filaments will all have nearly the same vel dimensions compared with 1 ~

ocity, and if h is measured to the centre of the orifice, the equation above gives the mean velocity of the 3 jet. - _.

Case of a Submerged Orifice.Let the orifice discharge below the level FIG. 41. of the tail water. Then using the notation shown in fig. 41, we have at Pd, vi=o,pi=Gh;+p. at m, p=Gh,+p~. Inserting these values in (i), 29,

o+hu +p~/Ghi = v/2g +h1hu+p~/G;

V/2g = hfha = h (3)

where Ii is the difference of level of the head and tail water, and may be termed the effective head producing flow.

Case where the Pressures are different on the Free Surface and at __________________ _________________ the Orifice.Let the fluid flow from a vessel in which the pressure ~__~c~_~_ -1~ - is Ps into a vessel in Jl, which the pressure is ~ p, fig. 42. The pres -, sure po will produce the same effect as a layer -----------~- A of fluid of thickness I p,/~ added to the head - water; and the pres I sure p ,will produce the same effect as a 4 - layer of thickness pIG

1~. added to the tail water. Hence the effective difference of level, or effective head producing flow, will __________________________________ be Fin. 42.

and the velocity of discharge will be v=V~2g~ho+(pop)/G}J- (4)

\Ve may express this result by saying that differences of pressure at the free surface and at the orifice are to be reckoned as part of the effective head.

Hence in all cases thus far treated the velocity of the jet is the veloc~ty due to the effective head, and the discharge, allowing for contraction of the jet, is Q cwv cw~I (2gb), (5)

where w is the area of the orifice, cn the area of the contracted section of the jet, and h the effective head measured to the centre of the orifice. If h and a, are taken in feet, Q is in cubic feet per second.

It is obvious, however, that this formula assumes that all the filaments have sensibly the same velocity. That will be true for horizontal orifices, and very approximately true in other cases if the dimensions of the orifice are not large compared with the head h. In large orifices in say a vertical surface, the value of h is different for different filaments, and then the velocity of different filaments is not sensibly the same.

SIMPLE ORIFICES----HEAn CONSTANT

39. Large Rectangular Jets from Orifices in Vertical Plane SurfacesLet an orifice in a vertical plane surface be so formed that it produces a jet having tracted section with ~ vertical and horizon tal sides. Let b (fig.

43) be the breadth of A B a rectangular con --6--H the jet, hi and h2 the ~ I depths below the free __________ surface of its upper ~ and lower surfaces.

- - - - - I Consider a lamina of the jet between the Its normal section is depths h and h+dh.

bdh, and the velocity The discharge pe; second in this lamina is therefore b~J~7f dh, and that of the whole jet is therefore Q (2gh)dh = lb~/~ ~h2~ h1~, (6)

where the first factor on the right is a coefficient depending on the form of the orifice.

Now an orifice producing a rectangular jet must itself be vt~ry approximately rectangular. Let B be the breadth, H1, H2, the depths to the upper and lower edges of the orifice. Put b(h2 ~hii)/B(H2i Hi~) =c. (7)

Then the discharge, in terms of the dimensions of the orifice, instead of those of the jet, is Q cB~s~~(H2i _Hii), (8)

the formula commonly given for the discharge of rectangular orifices. The coefficient c is not, however, simply the coefficient of contraction, the value of which is b(h2 h1) /B (Hf Hi),

and not that given in (7). It cannot be assumed, therelore, that c in equation (8) is constant, and in fact it is found to vary for different values of B1H2 and B/H1, and must be ascertained experimentally.

Relation between the Expressions (5) and (8).For a rectangular)rifice the area of the orifice is a, = B (Hf Hi),and the depth measured Lo its centre is 3/4(Hf+H1). Putting these values in (5),

Qi =cB(HiHi)~ ~g(Hi+Hi)~.

From (8) the discharge is = 3/4cB~~j(HilHif).

Hence, for the same value of c in the two cases, Qi/Q~ = 3/4(H2I Hi~)/~(Hf I-li) ~ (Hs+Hi)/211.

Let Hi/Hi = o-, then Qi/Qi 09427 (I ir3)/tf -o~ (i +a)}. (9)

If H1 varies from 0 to so, a(=Hi/Hf) varies from o to I. The following table gives values of the two estimates of the discharge for different values of a :- Hi/H2a. Qi/Qi. Hl/Hfa.j9i/Q~1

0~0 943 o8 999

02 979 09 999

0.5 995 1-0 1OOO

0-7 -998

Hence it is obvious that, except f or very small values of a, the simpler equation (5) gives values sensibly identical with those of (8). When a<05 it is better to use equation (8) with values of c determined experimentally for the particular proportions of orifice which are in question.

40. Large Jets having a Circular Section from Orifices in a Vertical Plane Surf ace.Let fig. 44 represent the section of the jet, 00 being FIG. 44.

the free surface level in the reservoir. The discharge through the horizontal strip aabb, of breadth aa=b, between the depths hi+y and hi+y+dy, is dQ =b~ ~2g(h1+y)~dy.

The whole discharge of the jet is Q=fdb,/ {2g(hi+y)~dy.

Butb=dsin4; y=3/4d(1cos4~); dy=1/2dsin4ddi. Lets=d/(2h1+d), then Q =1/2df~({2g(hl+d/2)~f sin 1~If s cos ~ d~.

From eq. (5), putting w.=ird2/4, hhi+d/2, c=i when d is the diameter of the jet and not that of the orifice, Qi = 3/4ird~I ~2g(hi+d/2)},

Q/Qi 217rf sin f~ {1 s cos For hi=oo, 5=0 and Q/Qi=1;

and for hi=o, s=i and Q/Q~=o96.

So that in this case also the difference between the simple formula (5) and the formula above, in which the variation of head at different parts of the orifice is taken into account, is very small.

NoTcHEs AND WEIRS

41. Notches, Weirs and Byewashes.A notch is an orifice extending up to the free surface level in the reservoir from which the discharge takes place. A weir is a structure over which the water flows, the discharge being in the same conditions as for a notch. The formula of discharge for an orifice of this kind is ordinarily deduced by putting H1 =0 in the formula for the corresponding orifice, obtained as in the preceding section. Thus for a rectangular notch, put Hi=o in (8). Then c B\!(2g)H1, (II)

where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 45 shows the mode in which the discharge occurs in the case of a rectangular notch or weir with a level crest. ~ the free surface level falls very sensibly near the notch, the head H should be measured at some distance back from the notch, at a point where the velocity of the water is very small. -

Since the area of the notch opening is BH, the above formula is of the form Q=cXBHXki! (2gH),

where k is a factor depending on the form of the notch and expressing the ratio of the mean velocity of discharge to the velocity due to the depth H. -

42. Franciss Formula for Rectangular Notches.The jet discharged through a rectangular notch has a section smaller than BH. (a) because of the fall of the water surface from the point where H

is measured towards the weir, (b) in consequence of the crest con- I traction, (c) in consequence of the end contractions. It may be pointed out that while the diminution of the section of the jet due to the surface fall and is proportional to the _____________________________ length of the weir, the ~ ~ to the crest contraction ~TI ~ end contractions have ~ ~ nearly the same effect whether the weir is wide L/~~ or narrow.

J. B. Franciss experiments showed that a / when the heads varied from 3 to 24 in., and ~ perfect end contraction, the length of the weir was nut less than three ______________________ / times thd head, dimin ~ ished the effective length of the weir by _____________ _______________ mately equal to onean amount approxi --- ~ tenth of the head.

__ -

Hence, if 1 is the length ~ of the notch or weir, and behind the wcir where - the water is nearly still, I- ~ H the head measured then the width of the - ~ jet passing through the -- ~-. notch wouLdbeto~2H,

- ~- ~-, allowing for two end -, contractions. In a weir ___________ - divided by posts there ____________ ~ - may be more than two - end contractions.

FIG. 45. - 1-lence, generally, tile width of the jet is 10.1 nH, where n is the number of end contractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if cH is the thickness of the stream over the weir, measured at the contracted section, the section of the jet will be c(loinH)H and (f 41) the mean velocity will be 3/4 ~ (2gH). Consequently the discharge will be given by an equation of the form Q=3/4c(1o-inH)H ~i5~Ti = 5-35c(l o. InHjHi.

This is Franciss formula, in which the coefficient of discharge c is much more nearly constant for different values of 1 and h than in the ordinary formula, Francis found for c the mean value 0-622, the weir being sharp-edged.

43. lriangular Notch (fig. 46)Consider a lamina issuing between the depths /t and h+dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is V (2gh). Hence the discharge for this lamina is bV~k dh.

But B/b=H/(l4/j); b=B(H/i)/H.

Hence discharge of lamina =B(Hh)d (2gh)dh/H;

and total discharge of notch nt =Q=BV (ag)J (Hh)hIdh/H

=1~1BV (2g)H~.

Coefficients for the Discharge over IVeirs, derived from the Ecperimeats of same head, and the results were pretty uniform, the resulting coeffi is very strongly marked.

Heads Sharp Edge. 1 Planks 2 in. thick, square on Crest.

I from still I wing- inches measured I I I0 It, Water in 3 It. long. In It. long. 3ff.long. 6 It. long. IoIt.long. making Reservoir. I of I i -677 809 -467 459 ~ ,7

2 -675 803.509* .561.585* 6

I 3 630.642* .563* .597* .569* I

4.617.656.549.575.602* I ~6

5 602.650* -588.6of* .609*

I 6.593 593* .608* .576*

7 -. .. .617* .608* 576* I

8 - - .581 6o65.590* .548~ I

I 9 - - .53(3.601) 569~ .558*

I 10 - - - .~I4* .539.534*

12 I. - -. .. 525.534*

~ 14. -. - -. .549* -.

The discharge per second varied from 461 to ~665 cub. ft. in two i or, introducing a coefficient to allow for contraction, Q=iact&d (2g)H~,

When a notch is used to gauge a stream of varying flow, the ratio B/H varies if the notch is rectangular, hut is constant if the notch is triangular, This led Professor James Thomson to suspect that the coefficient of dis charge, c, would 3

be much more constant with a-~a--~- -

different a ~ l notch, this --

mentally shown to be the case. FIG. 46.1-lence a triangular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notcn Professor J. Thomson found c =0.617. It will be seen, as in 4t, that since 1/2BH is the area of section of the stream through the notch, the formula is again of the form Q=1cXlBH~ki/(2gH),

where k = ~ is the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form.

44. Weir with a Broad Sloping Crest.Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a movement sensibly rectilinear and uniform (fig. 47). Let the inner ede be so rounded as to prevent a crest contraction. Consider a filament aa, the point a being so far back from the weir that the _~J_.~-.~

//~,/~//~ /~/// ~il~4

FIG. 47.

velocity of approach is negligible. Let 00 he the surface level in the reservoir, and let a be at a height h below 00, and h above a.

Let h be the distance from 00 to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is Gh; at a it is Gz. If v be the velocity at a, v2/2g~h-}-hz=h-e; Q=be V2g(he).

Theory does not furnish a value for e, but Q==o for e=o and for e = h. Q has therefore a maximum f or a value of e between o and h, obtained by equating dQ/de to zero. This gives e = 3/4h, and, inserting this value, Q0.385 bhsj2gh, as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonry T. E. Blackwell. When more than one experiment was made with the ients are marked with an (5). The effect of the converging wing-boards cresfs ~ If. wide, long, (lards ~ ft. long, 3 ft. long, 3 ft. long, 6 ft. long, ioft. long, ioff. iong, ii angle level, fall I in i 8, fall i 10 12. level, level, fall r in if 54 452.545.467 .. .381 ~467

75 482.546.533 - - .479* .495* -

- .44.537.539.492* - - -

36.419.431.455.497* - - ~515

71.479.516. -.. 518 - -

- .501*. - -53! .507 513 543

- 488 -513 -527 497 - - -

- .470.49f -. -. .468 507

- 476.492* 498.48n5 486 - -

- .465* ,455 -.

.467*

xperiments. The coefficient ~435 is derived from the mean value.

sluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction on the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent to the ordinary weir formula with c =0-577.

SPEcIAL CASES OF DISCHARGE FROM ORIFIcEs 45. Cases in which the Velocity of Approach needs to be taken into AccounL Rectangular Orifices and Notches.In finding the velocity at the orifice in the preceding investigations, it has been assumed that the head h has been measured from the free surface of still water above the orifice. In many cases which occur in practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the velocity in it can be disregarded.

Let h1, hi (fig. 48) be the heads measured from the free surface to the top and bottom edges of a rectangular orifice, at a point in the FIG. 48.

channel of approach where the velocity is u. It is obvious that a fall of the free surface, t~=uf/2g ha~ been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have been h1+l~ and h,+h. Consectuentlv the discharge, allowing for the velocity of approach, is ~ (I)

And for a rectangular notch for which h,=o, the discharge is Q=lcb~J2g~(hs +~)i_~. (2)

In cases where u can be directly determined, these formulae give the discharge quite simply. When, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let t~ be the sectional area of the channel where h, and hf are measured. Then u=Q/ t2 and l~=Q/2g f~i.

This value introduced in the equations above would render them excessively cumbrous. In cases therefore where f2 only is known, it is best to proceed by approximation. Calculate an approximate value Q of Q by the equation = 1/2cb~1 2g(hzl hfl}.

Then t~ = Q/2g~12 nearly. This value of f~ introduced in the equations above will give a second and much more approximate value of Q.

46. Partially Subnzerged Rectangular Orifices and Notches. When the tail water is above the lower but below the upper edge of the orifice, the flow in the two parts of the orifice, into which it is divided by the surface of the tail water, takes place under different condition. A filament M,mi (fig. 49) in the upper part of the oritice issues with a head h which may have any value between ~T~~T~TE ~

FIG. 49.

h1 and h. But a filament Mimi issuing in the lower part of the orihce has a velocity due to hh, or h, simply. In the upper part of the orifice the head is variable, in the lower constant. If Q~, Q2

are the discharges from the upper and lower parts of the orifice, b the width of the orifice, then Qi=3/4cb~J5~hihii} ~- ()

Qf=cb(hfh)~,f2gh -

In the case of a rectangular notch or weir, hi=o. Inserting this value, and adding the two portions of the discharge together, we get for a drowned weir Q =cb~J5~h(hfh/3), (4)

where h is the difference of level of the head and tail water, and hf is the head from the free surface above the weir to the weir crest (fig. 50).

From some experiments by Messrs A. Fteley and F. P. Stearns (Trans. Am. Soc. G.E., 1883, p. 102) some values of the coefficient c can be reduced hf/hf c hf/hi c O~f 0-629 0.7 0-578

0-2 0.614 o~8 0.583

0.3 0-600 0.9 0.596

04 0.590 0.95 0.607

0.5 0.582 f~o0 O~628

o6 0578

If velocity of approach istaken into account, let tj be the head due to that velocity; then, adding I) to each of the heads in the equations (3), and reducing, we get for a weir Q =cb-.J~~(hz+Ij) (h+l~)4 1/2(h+T~)i 1/2f~i]; (5) an equation which may be useful in estimating flood discharges.

Bridge Piers and other Obstructions in Streams.When the piers of a bridge are erected in a stream they create an obstruction to the flow of the stream, which causes a difference of surface- ~ -a~T level above and below the -~, pier (fig. 51). If it is neces- Ii.

sary to estimate this differ- .__~_______________ coca of level, the flow between the piers may be treated as if it occurred over a drowned weir. But the value of c in this case is //t/~/Zec~-~ ~

imperfectly known. FIG. 50.

4~. Bazins Researches on Weirs.H. Bazin has executed a long series of researches on the flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained in a series of papers in the Annales des Ponts et Chausses (October 1888, January 1890, November 1891, February 1894, December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The general plan of the experiments was to establish first the coefficients of discharge for a standard weir without end contractions; next to establish weirs of other types in series with the standard weir on a channel with steady flow, to compare the observed heads on the different weirs and to determine their coefficients from the discharge computed at the standard weir. A channel was constructed parallel to the Canal de Bourgogne, taking water from it through three sluices 03XI~0 metres. The water enters a masonry chamber 15 metres long by 4 metres wide where it is stilled and passes into the canal at the end of which is the standard weir. The canal has a length of I5 metres, a width of 2 metres and a depth of i6 metres. From FIG. 51.s this extends a channel 200 metres in length with a slope of 1 mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by floats actuating an index on a dial. Hook gauges were used in determining the heads on the weirs.

Standard Wcir.The weir crest was 3.72 ft. above the bottom of the canal and formed by a plate 3/4 in. thick. It was sharp-edged with free overfall. It was as wide as the canal so that end contractions were suppressed, and enlargements were formed below the crest to admit air under the water sheet. The channel below the weir was used as a gauging tank. Gaugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 0.5 metre in length, the end contractions being suppressed in all cases. Assuming the general formula Q=mlh~,l(2gh), (I)

Bazin arrives at the following values of m Coefficients of Discharge of Standard Weir.

Head h metres. Head h feet. m 1

005 164 0.4485

OIO 328 o4336

0.15 492 0.4284

020 656 0.4262

0.25 820 0.4259

0.30 984 0.4266

o~35 1.148 0.4275

0.40 1.312 0.4286

0-45 i.476 0.4299

050 1.640 0~43t3

0-55 I~8o4 0.4327 I

0-60 f-968 0.4341 j Bazin compares his results with those of Fteley and Stearns in 1877 and 1879, correcting for a different velocity of approach, and finds a close agreement.

Influence of Velocity of ApproachTo take account of the velocity of approach u it is usual to replace Ii in the formula by h+auf/2g where a is a coefficient not very well ascertained. Then Q p1(h+aui/2g).J ~2g(h~au2/2g)l plhj (2gh)(f~j~au2/2gh)i. (2)

The original simple equation can be used if m = u(s +aU1/2gh)l or very approximately, since uf/2gh is small, m =u(I +laUf/2gh). (3)

Now if p is the height of the weir crest above the bottom of the canal (fig. 52), u=Q/l(p+h).

- ~ Replacing Q by its value a k~\ Uu/2glz=Qu/{2ghli(p+h)9

p, so that (3) may be written ~ ~ rn=p~I+k{h/(p+hflfl. (5)

Gaugings were made with / /////////7/7,7~~ weirs of 0.75, 0.50, O35, and FIG. 52.0~24 metres height above the canal bottom and the results compared with those of the standard weir taken at the same time. The discussion of the results leads to the following values of m in the general equation (1)

m =u(+2.5u212gh)

p~I +oss{h/(P+h)Pl.

Values of 1-i Head h metres. Head h feet. p 0.05.164 O4481

0IO .328 0~4322

0~20.656, O4215

0.30.984 0.4174

0.40 1.312 0~4I44

0.50 1.640 o41I8

o6o 1.968 0~4092

An approximate formula for p is:

ff=0405+0003/h (kin metres)

u=o4o5+O0I/k (h in feet).

Inclined Weirs.Experiments were made in which the plank weir was inclined up or down stream, the crest being sharp and the end contraction suppressed. The following are coefficients by which the discharge of a vertical weir should be multiplied to obtain the discharge of the inclined weir.

Coefficient.

Inclination up stream - I to I o93

,, 3t02 0~94

,, 3 to I 096

Vertical weir 1~o0

Inclination down stream. - 3 to 1 1.04

3t02 1.07

1t01 1-10

,, 1 to 2 I~I2

I t04 1.09

The coefficient varies appreciably, if h/p approaches unity, which case should be avoided, In all the preceding cases the sheet passing over the weir is detached completely from the weir and its under-surface is subject to atmospheric pressure. These conditions permit the most exact determination of the coefficient of discharge. If the sides of the canal below the weir are not so arranged as to permit the access of air under the sheet, the phenomena are more complicated. So long as the head does not exceed a certain limit the sheet is detached from the weir, but encloses a volume of air which is at less than atmospheric pressure, and the tail water rises under the sheet. The discharge is a little greater than for free overfall. At greater head the air disappears from below the sheet and the sheet is said to be drowned. The drowned sheet may be independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a standing wave. In the latter case when the foot of the sheet is drowned the level ~

of the tail water influences ~ ~ the discharge even if it is below the weir crest. Weirs with Flat Crests. The water sheet may spring clear from the upstream edge ________________ _____________

or may adhere to the flat ~~W/////// /7//////4 crest falling free beyond the FIG. 53. downstream edge. In the former case the condition is that of a sharp-edged weir and it is realized when the head is at least double the width of crest. It may arise if the head is at least 11/2 the width of crest. Between these limits the condition of the sheet is unstable. When the sheet is adherent the coefficient m depends on the ratio of the head h to the width of crest c (fig. 53), and is given by the equation rnmi where mi is the coefficient for a sharpedged weir in similar con-, -_____

ditions. Rounding the upstream edge even to a small -~z~extent modifies the dis-, - -~: --

charge. If R is the radius -.~ i---. of the rounding the coefficient m is increased in the ratio I to I +R/h nearly. p The results are limited to R

less than 1/2 in.

Drowned Weirs.Let h ~iW~//~/~/ ///////////////////.~

(fig. 54) be the height of FIG

head water and h1 that of 54.

tail water above the weir crest. Then Bazin obtains as the approximate formula for the coefficient of discharge = I .oSmi~f +lhf/p] -~ { (h hf)/h},

where as before rni is the coefficient for a sharp-edged weir in similar conditions, that is, when the sheet if of the same height. ~

48. Separating ~

thwns derive thel water-supply from ~ --. streams in high moorland dis- FIG

tricts, in which the flow is extremely variable. The water is collected in large storage reservoirs, from which an uniform supply can be sent to the town. In ~ Plan- of Cn-st iron !~r~~ ~ i/A-

~ ~ ?ur - ~ -f~_i -~-~ i2~ -:~&. ~ -~a~ ~

~ ConrEngels ~

FIG. ,~6.

such cases it is desirable to separate the colored water which comes down the streams in high floods from the purer water of ordinary flow. The latter is sent into the reservoirs; the former is allowed to flow away down the origir.al stream channel, or is stored in separate reservoirs and used as compensation water. To accomplish the separation of the flood and ordinary water, advantage is taken of the different horizontal range of the parabolic path of the water falling over a weir, as the depth on the weir and, consequently, the velocity change. Fig. 55 shows one of these separating wcirs in the form in which they were first introduced on the Manchester Waterworks; fig. 56 a more modern weir of the same kind designed by Sir A. Binni~ for the Bradford Waterworks. When the quantity of water coming down the stream is not excessive, it drops over the iveir into a transverse channel leading to the reservoirs. In flood, the water springs over the motith of this channel and is led into a waste channel.

It may be assumed, probably with accuracy enough for practical purposes, that the particles describe the parabolas due to the mean velocity of the water passing over the weir, that is, to a velocity 3/4 ~ (2gh),

where h is the head above the crest of the weir.

Let cb=x be the width of the orifice and ac=y the difference of level of its edges (fig. 57). Then, if a particle passes from a to b in seconds, y=3/4gti, x=3/4-.I(2gh)t; .. y=i6x1/h, which gives the width x for any given difference 0f level y and head h, which the jet will just pass over the orifice. Set off ad vertically ~- TTlp~3\~j I \\ \ \~.,

\ \: ~

5 ~\, \_~

~ \ -.

I S \~

.. e +v~----

FIG. 57.

and equal to 3/4g on any scale; af horizontally and equal to 3/4~ (gh). Divide af, fe into an equal number of equal parts. Join a with the divisions on ef. The intersections of these lines with verticals from the divisions on af give the parabolic path of the jet.

MOUTHPIECESHEAD CONSTANT

49. Cylindrical iVlouthpieces.\Vhen water issues from a short cylindrical pipe or mouthpiece of a length at least equal to 13/4 times its smallest transverse dimension, the stream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without contraction, at the point of discharge. The discharge is found to he about one-third greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the action being something like that which occurs at an abrupt change of section.

Let fig. 58 represent a vessel discharging through a cylindrical mouthpiece at the depth /f from the free surface, and let the axis of the jet XX be taken as the datum with reference to which the head is estimated. Let t2 be the area of the mouthpiece, w the aiea of the stream at the contracted section EF. Let 2, p be the velocity and pressure at EF, and In, Pt the same quantities at Gil. If the discharge is into the air, ~i is equal to the atmospheric pressure p~.

The total head of any filament which goes to form the jet, taken at a point where its velocity is sensibly zero, is h+p,,/G; at EF the total head is v2/2g+P/G; at Gil it is v12/2g+pi/G.

Between EF and GH there is a loss of head due to abiupt change of velocity, which from eq. (3), 36, may have the value (v vi)/2g.

Adding this head lost to the head at Gil, before equating it to the heads at EF and at the point where the filaments start into motion, h +p~/G =v2/2g+p/G Vi/2g +Pi/G+ (v vi)/2g.

But rev =tjvi, and w=c,,t~, if c,, is the coefficient of contraction within the mouthpiece. Hence Supposing the discharge into the air, so that Pi =p~,

h+p,,/G=vl1/2g~p/G_f~(v11/2g) (iJc,i)f; (vi/2g) ~i f(i/c,,1)2} =h; ... vf= ,I (2gh)/~/~f~(I/c,,_I)2}; (1)

where the coefficient on the right is evidently the coefficient of velocity for the cylindrical mouthpiece in terms of _____________ the coefficient of con ==-I---=traction at EF. Let ~ C,, =0.64, the value for --r simple orifices, then the coefficient of velocity is A

~~--Gx c0=I/~lI+(I/c,,I)~

=o87 (2)

The actual value of c,,

found by experiment is o~82, which does not differ more from the might be expected if theoretical value than the friction of the FIG. 58. mouthpiece is allowed for. Hence, for mouthpieces of this kind, and for the section at Gil, c,,=0~82 c,,=ioo c=o82,

Q=o.82th (2gh).

It is easy to see from the equations that the pressure p at EF is less than atmospheric pressure. Eliminating Iii, we get (p,,p)/G=3/4h nearly; (3)

or p=p,,3/4GhIb per sq. ft.

If a pipe connected with a reservoir on a lower level is introduced into the mouthpiece at the part where the contraction is formed (fig. 59), the water will rise in. this pipe to a height KL = (p~ p)/~0 = 3/4/f nearly.

If the distance X is less than this, the water from the lower reservoir will be forced continuously into the jet by the atmospheric pressure, and discharged with it. This is the crudest form of a kind of pump known as the jet pump.

50. Convergent Mouth pieces.With convergent mouthpieces there is a contraction within the mouthpiece causing a loss of head and a diminution of the velocity of discharge, as with cylindrical mouthpieces. There is also a second contraction of the stream outside the mouthpiece. Hence the discharge is given by an equation of the form Q =c~c,,12~J (2gh), (4)

where P is the area of the external end of the mouthpiece, and c,,t~ the section of the contracted jet beyond the mouthpiece.

Convergent Mouthpieces (Castels Experiments).Smaliest diameter of orifice = 0.05085 ft. Length of mouth piece = 2 6 Diameters.

Angle of Coefficient of Coefficient of Coefficient of Contraction, Velocity, Discharge, Convergence.

C,, C,, C

0 0 999.830 829

f0 36 I~0O0 866 ~866

30 10 IoOI .894.895

4 I0 I002.910 ~9I2

5,, 26 1.004.920.924

7,, 52.998.931 929

8 58 992.942 934

10 20 987.950 ~938

12 4.986.955 ~942

I3 24.983 962 ~946

14 28 979 966.941

i6 36.969 971 ~938

19: 28 953 970 924

21 0 945 971 918

230 0 ~937.974 913

29 58 919 975 896

4~0 20.887 980 869

48 50 861 984 847

The maximum coefficient of discharge is that for a mouthpiece with a convergence of 13 24.

The values of C,- and c, must here be determined by experiment. The above table gives values sufficient for practical purposes. Since the contraction beyond the mouthpiece increases with the convergence,or, what is the same thing, c,, diminishes, and on the other hand the loss of energy diminishes, so that c,- increases with the convergence, there ~ -~~ is an angle for which the -1/2~ ~ product c,, c,,, and con- seq ucntly the discharge, ~f 51. Divergent Con- is a maximum.

X oidal Mouthpiece.Sup pose a mouthpiece so designed that there is - no abrupt change in the - section or velocity of the stream passing through it. It may have a form at the Ff0.59. inner end approxi mately the same as that of a simple contracted vein, and may then enlarge gradually, as shown in fig. 60. Suppose that at EF it becomes cylindrical, so that the jet may be taken to be of the diameter EF. Let si, 1, p be the section, velocity and pressure at CD, and 11, v1, ~, the same quantities at EF p~ being as usual the atmospheric pressure, or pressure on the free surface AB. Then, since there is no loss of energy, except the small frictional resistance of the ~-~i surface of the mouthpiece, h+p~/G =V2/2g+p/G

Vi/2g +p/G.

If the jet discharges into the air, Pi==P~ and Vi/2g=h; vi = -.1 (2gh);

or, if a coefficient is introduced to allow for friction, the mouthpiece is smooth Sf =c,-.,/(2gh);

where c~ is about 0.97 if and well formed.

Q = lie1 =c,,fnJ (2gh).

._--------------- 1-lence the discharge destream at EF, and not at pends on the area of the F all on that at CD, and the latter may be made as small as we please without FIG. 60. affecting the amount of water discharged.

There is, however, a limit to this. ,As the velocity at CD is greater than at EF the pressure is less, and therefore less than atmospheric pressure, if the discharge is into the air. If CD is so contracted that p=o, the continuity of flow is impossible. In fact the stream disengages itself from the ~ mouthpiece for some value of p greater than 0 (fig. 61).

From the equations, p/G =p~/G (v2 v11)/2g.

Let l2/w =rn. Then v=vim; p(G=p~(Gvi1(mI)(2g =p~/G(m----i)h; whence we find that pIG

will become zero or negative if ~V ~(h+p,-/G)/h}

~J{i+plGh};

FIG. 61. or, putting p~/G=34 ft., if ~/w~V{(h+34)/h~.

In practice there will be an interruption of the full bore flow with a less ratio of ~ because of the disengagement of air from the water. But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind is Q w~ ~2g(h+p~fG)~

that is, the discharge is the same as for a well-bellmouthed mouthpiece of area ~, and without the expanding part, discharging into a vacuum.

52. Jet PumpA divergent mouthpiece may be arranged to act as a pump, as shown in fig. 62. The water which supplies the energy required for pumping enters at A. The water to be pumped enters at B. The streams combine at DD where the velocity is greatest and the pressure least. Beyond DD the stream enlarges in section, FIG. 62.

and its pressure increases, till it is sufficient to balance the bead due to the height of the lift, and the water flows away ~iy the discharge pipe C.

FIG. 63 shows the whole arrangement in a diagrammatic way. A is the reservoir which supplies the water that effects the pumping; Fl G. 63.

B is the reservoir of water to be pumped; C is the reservoir into which the water is pumped.

DISCHARGE WITH VARYING HEAD

53. Flow from a Vessel when the Effective Head varies with the Time.Various useful problems arise relating to the time of emptying and filling vessels, reservoirs, lock chambers, &c., where the flow is dependent on a head which increases or diminishes during the operation. The simplest of these problems is the case of filling or emptying a vessel of constant horizontal section.

Time of Emptying or Filling a Vertical-sided Lock Chamber. Suppose the lock chamber, which has a water surface of li square ft., is emptied through a sluice in the tail gates, of area w, placed below the tail-water level. Then the effective head producing flow through the sluice is the difference of level in the chamber and tail bay. Let H (fig. 64) be the initial difference of level, h the difference Rea~L water teve,t Jf ____________

---=-= ~t~-ff _____ ____ -

FIG. 64.

of level after t seconds. Let dh be the fall of level in the chamber during an interval dl. Then in the time dt the volume in the chamber is altered by the amount t~dh, and the outflow from the sluice in the same time is cw~J (2gh)dt. Hence the differential equation connecting h and I is ,cw.J (2gh)dt+llh =o.

For the time 1, during which the initial head H diminishes to any c,ther value Il, {~/(cw~ 2g) }fh h dl.

= (f2/cw)IV (2H/g) .1 (2h/g)}.

For the whole time of emptying, during which 11 diminishes from H to 0,

T=(~~/cw)-~ (2H/g).

Comparing this with the equation for flow under a constant head, it will be seen that the time is double that required for the discharge of an equal volume under a constant head.

The tune of filling the lock through a sluice in the head gates is exactly the same, if the sluice is below the tail-water level. But if the sluice is above the tail-water level, then the head is constant till the level of the sluice is reached, and afterwards it diminishes with the time.

PRACTICAL USE OF ORIFICES IN GAUGING WATER

~4. If the water to be measured is passed through a known orifice under an arrangement by which the constancy of the head is ensured, the amotint which passes in a given time can be ascertained by the formulae already given. It will obviously be best to make the orifices of the forms for which the coefficients are most accurately determined; hence sharp-edged orifices or notches are most commonly used.

lValer InchFor measuring small quantities of water circular sharp-edged orifices have been used. The discharge from a circular orifice one French inch in diameter, with a head of one line above the too edge, was termed by the older hydraulic writers a water-inch. A common estimate of its value was 14 pints per minute, or 677 English cub. ft. in 24 hours. An experiment by C. Bossut gave 634 cub. ft. in 24 hours (see Naviers edition of Belidors Arch. Ilydr., p. 212).

L. J. Weisbach points out that measurements of this kind would be made more accurately with a greater head over the orifice, and he proposes that the head should be equal to the diameter ot the orifice. Several equal orifices may be used for larger discharges.

Pm Ferrules or Measuring Cocks.To give a tolerably definite supply of water to houses, without the expense, of,a meter, a ferrule with an orifice of a definite size, or a cock, is introduced in the service-pipe If the head in the Water main is constant, then a definite quantity of water would be delivered in a given time. The arrangement is not a very satisfactory one, and acts chiefly as a check on extravagant use of water. It ~ interesting here chiefly as an example of regulation of discharge by mc~fns of an orifice. Fig. 65

shows a cock of 1~-~1.Th l1 r~l ~ ~

UT ..,~ .-~ ~.;.,; of three cocks, the ~ 1. ~, \ middle one having - - cEngels the orifice of the 0 predetermined size ~ ~ .w~ / ~. Z - in a small circular ~ ~ - ~ plate, protected by - ~ pa~~y fr~s D purities in the water. The cock FiG 6- on the right hand - ~. can be used by the consumer for empt~ ing the pipes. The one on the left and the measuring cock are connected by a key which can be locked by a padlock, which is under the control uf the water company.

55. Measurement of the Flow in Streams,To determine the quantity of water flowing off the ground in small streams, which is available for water supply or for obtaining water power, small temporary weirs are often used, These may be formed of planks supported by piles and puddled to prevent leakage. The measurement of the head may be made by a thin-edged scale at a short distance behind the weir, where the water surface has not begun to slope down to the weir and where the velocity of approach is not high. The measurements are conveniently made from a short pile driven into the bed of the river, accurately level with the crest of the weir (fig. 66). Then it at any moment the head is h, the discharge is, for a rectangular notch of breadth b, Q=icbh..J2gh where C =0-62; or, better, the formula in 42 may be used.

Gauging weirs are most commonly in the form of rectangular notches; and care should be taken that the crest is accurately horizontal, and that the weir is normal to the direction of flow of the stream. If the planks are thick, they should be bevelled (fig. 67), and then the edge may be protected by a metal plate about ~1eth in. thick to secure the requisite accuracy of form and sharpness of euge. In permanent gauging weirs, a Cast steel plate is sometimes used to form the edge of the weir crest. The weir should be large enough to discharge the maximum volume flowing in the stream, and at the same time it is desirable that the minimum head should not be too small (say half a foot) to decrease the effects 05 errors of measurement. The section of the jet over the weir should not exceed one-fifth the section of the stream behind the weir, or the velocity of approach will need to be taken into account. A triangular notch is very suitable for measurements of this kind.

If the flow is variable, the head h must be recorded at equidistant intervals of time, say twice daily, and then for each 12-hour period Scale Wetr Fl G. 66.

the discharge must be calculated for the mean of the heads at the beginning and