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 end of the time. As this involves a good deal of troublesome calculation, E. Sang proposed to use a scale so graduatect as to read off the discharge in cubic feet per second. The lengths of the principal graduations of such a scale are easily calculated by putting Q~, 2, 3. .. in the ordinary formulae for notches; the intermediate graduations may be taken accurately enough by subdividing equally the distances between the principal graduations.

The accurate measurement of the discharge of a stream by means of a weir is, however, in practice, rather more difficult than might be inferred from the simplicity of ,L_

the principle of the ~ - e ~~?l~c~1

discharge, which / need not be serious if the form of the weir and the nature of its crest are pro perly attended to, FIG 6

other difficulties of measurement arise. The length of the weir should be very accurately determined, and if the weir is rectangular its deviations from exactness of level should be tested. Then the agitation of the water, the ripple on its surface, and the adhesion of the water to the scale on which the head is measured, are liable to introduce errors. Upon a weir 10 ft. long, with 1 ft. depth of water flowing over, an error of 1-1000th of a foot in measuring the head, or an error of 1-100th of a foot in measuring the length of the weir, would cause an error in computing the discharge of 2 cub. ft. per minute.

hook Gauge.For the determination of the surface level of water, the most accurate instrument is the hook gauge used first by U. Boyden of Boston, in 1840. It consists of a fixed frame with scale and vernier. In the instrument in fig. 68 the vernier is fixed to the frame, and the scale slides vertically. ______

The scale carries at its lower end a hook with a fine point, and the scale can be raised or lowered by a fine pitched FIG. 68

screw. If the hook is depressed below thewater surface and then raised by the screw, the moment of its reaching the water surface will be very distinctly marked, by the reflection from a small capillary elevation of the water surface over the point of the hook. In ordinary light, differences of level of the water of ooi of a foot are easily detected by the hook gauge. If such a gauge is used to determine the heads at a weir, the hook should first be set accurately level with the weir crest, and a reading taken. Then the difference of the reading at the water surface and that for the weir crest will be the head at the weir.

56. Modules used in Irrigation.In distributing water for irrigation, the charge for the water may be simply assessed on the area of the land irrigated for each consumer, a method followed in India; or a regulated quantity of water may be given to each consumer, and the charge may be made proportional to the quantity of water supplied, a method employed for a long time in Italy and other parts of Europe. To deliver a regulated quantity of water =f-i~=~-~

A~ Rg FIG. 69.

from the irrigation channel, arrangements termed modules are used. These are constructions intended to maintain a constant or approximately constant head above an orifice of fixed size, or to regulate the size of the orifice so as to give a constant discharge, notwith,tanding the variation of level in the irrigating channel.

5~. Italian ModuleThe Italian modules are masonry constructions, consisting of a regulating chamber, to which water is admitted by an adjustable sluice from the canal. At the other end of the chamber is an orifice in a thin flagstone of fixed size. By means of the adjustable sluice a tolerably constant head above the fixed orifice is maintained, and therefore there is a nearly constant discharge of ascertainable amount through the orifice, into the channel leading to the fields which are to be irrigated.

In fig. 69, A is the adjustable sluice by which water is admitted to the regulating chamber, B is the fixed orifice through which the water is discharged. The sluice A is adjusted from time to time by the canal officers, so as to bring the level of the water in the regulating chamber to a fixed level marked on the wall of the chamber. When - e r / // -

~::-~L~-~

- T ---/-

r / /; / / ~

/// adjusted it is locked Let Wi be the area of the ~ orifice through the sluice- at A, and w~ that of the fixed orifice at B; let h1 be the difference of level ~ ~j between the surface of the water in the canal and regulating chamber; h1 the head above the centre of the discharging orifice, when the sluice has been adjusted and the flow has become steady; Q the normal discharge in cubic feet per second. Then, ,ince the flow through the orifices at A and B is the same, Q = ciwj-.l (2ghi) = czwfI (2gh2),

where c1 and cf are the coefficients of discharge suitable for the two orifices. Hence Ciwf/ciwf = ~ (hf/hi).

If the orifice at B opened directly into the canal without any intermediate regulating chamber, the discharge would increase for a given change of level in the canal in exactly the same ratio. Consequently the Italian module in no way moderates the fluctuations of discharge, except so far as it affords means of easy adjustment from time to time. It has further the advantage that the cultivator, by observing the level of the water in the chamber, can always see whether or not he is receiving the proper quantity of water.

On each canal the orifices are of the same heiglit, and intended to work with the same normal head, the width of the orifices being varied to suit the demand for water. The unit of discharge varies on different canals, being fixed in each case by legal arrangements. Thus on the Canal Lodi the unit of discharge or one module of water is the discharge through an orifice 112 ft. high, 0.12416 ft. wide, with a head of 0.32 ft. above the top edge of the orifice, or .88 ft. above the centre. This corresponds to a discharge of about 0.6165 cub. ft. per second.

In the most elaborate Italian modules the regulating chamber is arched over, and its dimensions are very exactly prescribed. Thus in the modules of the Naviglio Grande of Milan, shown in fig. 70 the measuring orifice is cut in a thin stone slab, and so placed that the discharge is into the air with free contraction on all sides. The ~ I I e~v 7~ /z/e.

FIG. 71.

adjusting sluice is placed with its sill flush with the bottom of the canal, and is provided with a rack and lever and locking arrangement. The covered regulating chamber is about 20 ft. long, with a breadth 1.64 ft. greater than that of the discharging orifice. At precisely the normal level of the water in the regulating chamber, there is a ceiling of planks intended to still the agitation of the water. A block of stone serves to indicate the normal level of the water in the chamber. The water is discharged into an open channel 0.655 ft. wider than the orifice, splaying out till it is 1.637 ft. wider than the orifice, and about 18 ft. in length.

58. Spanish ModuleOn the canal of Isabella II., which supplies water to Madrid, a module much more perfect in principle than the Italian module is employed. Part of the water is supplied for irriga tion, and as it is very valuable its strict measurement is essential. The module (fig. 72) consists of two I chambers one above the other, the ~ upper chamber being in free communication with the irrigation canal, and the lower chamber discharging by a culvert to the fields. In the arched roof between the chambers there is a circular sharp-edged orifice in a bronze plate. Hanging in this there is a bronze plug of variable diameter suspended from a hollow brass float. If _________________ the water level in the canal lowers, the plug desceods and gives an enlarged onening, and conversely. Thus a per fectlv constant discharge with a vary ing l~ead can be obtained, provided no clogging or silting of the chambers pre vents the free discharge of the water or the rise and fall of the float. The theory of the module is very simple. Let R (fig. 71) be the radius of the fixed opening, r the radius of the plug at a distance h from the plane of flotation of the float, and Q the required discharge of the module. Then Q=cir(R2r1)~l (2gh).

Taking c =0.63,

Q==i5.88(Rr2)-~h; -

r=,I ~R5Q/I5.88~ h}.

Choosing a value for R successive values of r can be found for different values of Ii, an~ from these the curve of the plug can be drawn. The module shown in fig. 72 will discharge I cubic metre per second. The fixed opening is 0-2 metre diameter, and the greatest head above the fixed orifice is 1 metre. The use of this module involves a great sacrifice of level between the canal and the fields. The module is described in Sir C. Scott-Moncrieffs Irrigation in Southern Europe.

5~. Reservoir Gauging BasinsIn obtaining the power to store the water of streams in reservoirs, it is usual to concede to riparian owners below the reservoirs a right to a regulated supply throughout the year. This compensation water requires to be measured in such a way that the millowners and others interested in the matter can assure themselves that they are receiving a proper quantity, and they are generally allowed a certain amount of control as to the times during which the daily supply is discharged into the stream.

~~----a-EngelsI-~--- -~1

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Fig. 74 shows an arrangement designed for the Manchester water works. The water enters from the reservoir a chamber A, the object of which is to still the irregular motion of the water. The admission is regulated by sluices at b, b, Li. The water is discharged by orifices or notches at a, a, over which a tolerably constant head is maintained by adjusting the sluices at Li, b, Li. At any time the millowners can see whether the discharge is given and whether the proper head is maintained over the orifices. To test at any time the discharge of the orifices, a gauging basin B is provided. The Water ordinarily Flows over this, without entering it, on a floor of cast-iron plates. If the discharge is to be tested, the water is turned for a defimte time nto the gauging basin, by suddenly opening and closing a sluice at c. The volume of flow can be ascertained from the depth in the gauging hamber. A mechanical arrangement (fig. 73) was designed for ~ectrring an absolutely constant head over the orifices at a, a. The)rifices were formed in a cast-iron plate capable of sliding up and - ~/ / //////~ /7/ / / ~ // -/ -7 / /

B /~///~ /~ ~/ /z / ~/~// ~ / ,~

/Y/EngelsV~/// 7/ ~/ ~///~ // ~ ~>~ / -

/2L/~ ///Engels/z~/ /~/ 7/7/7

FIG. 73.Scale i1/2~.

down, without sensible leakage, on the face of the wall of the chamber. The orifice plate was attached by a link to a lever, one end of which rested on the wall and the other on floats f in the chamber A. The floats rose and fell with the changes of level in the chamber, and raised and lowered the orifice plate at the same time. This N ~ 6

Pia,n,. ~

~ }--~J A -EJ~

NJ7/~,//,/7 ,~/,/z,//., ,z /~,, / / / -/ z//// %~

Ff~. 74.Scale IlTimechanical arrangement was not finally adopted, careful watching of the sluices at Li, b, b, being sufficient to secure a regular discharge. The arrangement is then eqtnvalent to an Italian module, but on a large scale.

~ 60. Professor Fleeming Jenkins Constant Flow ValveIn the modules thtis far described constant discharge is obtained by varying the area of the orifice through which the water flows. Professor F. Jenkin has contrived a valve in which a constant pressure head is obtained, so that the orifice need not be varied (Roy. Scot. Society of ArEs, 1876). Fig. 75 shows a valve of this kind suitable for a 6-in. water main. The water arriving by the main C passes through an equilibrium valve D into the chamber A, and thence through a sluice 0, which can be set for any required area of opening, into the discharging main B. The object of the arrangement is to secure a constant difference of pressure between the chambers A and B, so that a constant discharge flows through the stop valve 0. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a diaphragm, separating A from a chamber B1 connected by a pipe B, with the discharging main B. Any increase of the difference of pressure in A and B will drive the plunger up and close the -~ eveR 0 C -~---- ~e<

Jjj~~c~

Fia. 75.Scale 214~

equilibrium valve, and conversely a decrease of the difference of pressure will cause the descent of the plunger and open the equilibrium valve wider. Thus a constant difference of pressure is obtained in the chambers A and B. Let w be the area of the plunger in square teet, p the difference of pressure in the chambers A and B in pounds per sqtiare foot, w the weight of the plunger and valve. Then if at any moment ~w exceeds w the plunger will rise, and if it is less than w the plunger will descend. Apart from friction, and assuming the valve D to be strictly an equilibrium valve, since w and w are constant, p must be constant also, and equal to w/~o. By making w small and w large, the difference of pressure required to ensure the working of the apparatus may be made very small. Valves working with a difference of pressure of 1/2 in. of water have been constructed.

VI. STEADY FLOW OF COMPRESSIBLE FLUIDS.

- 61. External Work during the Expansion of Air.If air expands without doing any external work, its temperature remains constant.

This result was first experimentally demonstrated by J. P. Joule. It leads to the condosion that, however air changes its state, the in ~ ternal work done is proportional to the change - of temperature. When, I) - in expanding, air does _______ _______ _______ d work against an external v resistance, either heat must be supplied or the temperature falls.

To fix the conditions, suppose 1 it) of air con fined behind a piston of ~i I sq. ft. area (fig. 76).

Let the initial pressure be Pi and the volume of the air v,, and suppose FiG. 76. this to expand to the pressure Ps and volume vi. If p and v are the corresponding pressure and volume at any intermediate point in the expansion, the work done on the piston during the expansion from v to v+dv is pdv, and the whole work during the expansion from vi to vf, represented by the area abcd, is fipdv.

Amongst possible cases two may be selected.

Case 1.So much heat is supplied to the air during expansion that the temperature remains constant. Hyperbolic expansion.

Then pv ~pivi.

Work done during expansion per pound of air =Jt~ipdv ~piv~~v2dv/v ~~ivi loge v1/Vi =pivi loge pi/pi. (1)

Since the weight per cubic foot is the reciprocal of the volume per pound, this may be written.

(puG1) loge Gf/G2. (Ia)

Then the expansion curve ab is a common hyperbola.

Case 2.No heat is supplied to the air during expansion. Then the air loses an amount of heat equivalent to the external work done and the temperature falls. Adiabatic expansion.

In this case it can be shown that pa-v =~ivi~.

where y is the ratio of the specific heats of air at constant pressure and volume. Its value for air is 1.408, and for dry steam f135.

Work done during expansion per pound of air. = f vipd = Pivif~dv/v Pi1I(YI)IlfIVf1 I/Vi~/hj = tPiviYI(y 1)11 i/vj)f I/v211~ =~Pivi/(y1)Uf (vf/v2)~~}. (2)

The value of Pivi for any given temperature can be found from the data already given.

As before, substituting the weights Gi, G, per cubic foot for the volumes per pound, we get for the work of expansion (p1/G1)(I/(~ I)} { I (G,/G1)Y1}, (2a)

=PivitII(7f)~ (1 ._.(pf/pf)(Yi)~y~ (2b)

62. Modification of the Theorem of Bernoulli for the Case of a. Corn pressible Fluid.In the application of the principle of work to a filament of compressible fluid, the internal work done by the expansion of the fluid, or absorbed in its compression, must be A ~

taken into account. Suppose, as before, that AB (fig. 77)

comes to AB in a short time t.

Let Pi, wi, Ci, Gi be the pres sure, sectional area of stream, velocity and weight of a cubic F1u. 77.

foot at A, and pi, us, vs, Gi the same quantities at B. Then, from the steadiness of motion, the weight of fluid passing A in any given time must be equal to the weight passing B:

Gfwivit Giccfvft.

Let Ii, zi be the heights of the sections A and B above any given datum. Then the work of gravity on the mass AB in t seconds is Giwivfl(zi z2) V~T(z1 zs)t, where W is the weight of gas passing A or B per second. As in the case of an incompressible fluid, the work of the pressures on the ends of the mass AB is Piwivit Psuivit, = (puG1 pf/Gi)Wt.

The work done by expansion of Wt lb of fluid between A and B is Wtf~pdv. The change of kinetic energy as before is (W/2g) (v22 vif)t. Hence, equating work to change of kinetic energy, W(zi z1)t+ (pi/Gf _piIGi)wt+wtx~:pdv (W/2g) (of1 vii)t; .~. zi+pi/Gl +vif/2g = 12 +p1IG2 +vs12g _f~2pdV. (i)

Now the work of expansion per pound of fluid has already been given. If the temp.erature is constant, we get (eq. Ia, 61)

If +pi/Gi +vi/2g = If +Pi/G2 +v12/2g (pi/Gi) loge (Gi/G1).

But at constant temperature Pi/Gi =pf/Gi; ~ If +v1112g = 11 +lif !2g (pi/Gi) loge (Pi/pf), (2) or, neglecting the difference of level, - (vffvi)/2g=(pi/Gi) loge (pu/ps). (2a) Similarly, if the expansion is adiabatic (eq. 2a, 61),

zi +PiIGi + Vi~i/2g = zs+piIGs +vif/2g (pu/Gi) { i/(y I)

ji (pf/pf)CYl)/~; (3)

or neglecting the difference of level (Vi1 _vii)/2g = (pi/Gi)~1 +1/(~ I)(1 (p2/pf)CY1)/u~}] p2/G2. (3a)

It will be seen hereafter that there is a limit in the ratio PuPs beyond which these expressions cease to be true.

63. Discharge of Air from a-n Orzfice.The form of the equation of work for a steady stream of compressible fluid is z1+pu/Gi + Vuf/2g = z1+pu/Gi +viu/2g (pi/G~){ I /(y 1)}

If (tf/pl)(~~_i)/~~},

the expansion being adiabatic, because in the flow of the streanis of air through an orifice no sensible amount of heat can be communicated from outside.

Suppose the air flows from a vessel, where the pressure is Pi and the velocity sensibly zero, through an orifice, into a space where the pressure is P2- Let vf be the velocity of the jet at a point where the convergence of the streams has ceased, so that the pressure in the jet is also P2. As air is light, the work of gravity will be small compared with that of the pressures and expansion, so that 1i12 may be neglected. Putting these values in the equation above puG1 =pa/Gl+v22/2g (p1/G1) if(y I)) { I (pf/p~)~~ ~~

vzf/2g pi/Gi p,/Gf + (PuG1) i/(~ I) { f (pf/p~)(7 i)/Y}

(p1/Gi)~-y/(~ 1) (P2!Pi)~ 1/-Y/(_t 1)1 pf/G2. But p1/Gi7 = p1/Gi7 .~. pf/Gf = (pi/Gi) (p2Ipi)~~ i)/y V2u12g = (pu/Gu){7/(-y 1) } { I (pf/pi)~ I)/Y}; (1) or vlf/2g

an equation commonly ascribed to L. J. Weisbach (Civilingenieur, 1856), though it appears to have been given earlier by A. J. C. Barre de Saint Venant and L. ~Vantzel.

It has already (~ 9, eq. 4sf) been seen that pi/Gf = (p,/G,) (Ti/TO)

where for air po2116.8, Go=~o8o75 and ro=4926.

v21/2g = { poTiY/GoTo(y I) I (pi/pu) (7_ i)Engels}; (2)

or, inserting numerical values, v22/2g = 1836ri(I (pi/pi) ffi(; (2a)

which gives the velocity of discharge vf in terms of the pressure and absolute temperature, ~u, ri in the vessel from which the air flows, and the pressure Pi in the vessel into which it flows.

Proceeding now as for liquids, and putting w for the area of the orifice and c for the coefficient of discharge, the volume of air discharged per secondat the pressure pi and temperature r2 is Q2 =C~Vi =cw ~I I)Gu)(I ~

= 108 7cw %(I (Pi/Pi)29(l. (3)

If the volume discharged is measured at the pressure Pt and absolute temperature i-i in the vessel from which the air flows, let Qi be that volume; then PuQi7 PfQf7

Qi = (Pi/pi)~Qi; Qi =cw ~)G1((pi/pi)2/7 (P2IPi)Engels i)/7}]

Let (P2/Pi)2~7 (P2/P1)~~ i)/7 = (P2/Pi)141 (Pup,)01 =~; then Qi ~co- -.1 f)GuI

Io8-7c \/ (r,~). (4)

The weight of air at pressure p1 and temperature Tf is G, =P,/53.2u-i lb per cubic foot.

Hence the weight of air discharged is G1Qi =cu, ~/ f)l 2043Cw~i .-,I (~/ri). (5)

\Veisbach found the following values of the coefficient of discharge C: Conoidal mouthpieces of the form of the~

contracted vein with effective pressures ~- c =

of .23 to f~I atmosphere - - - - J 097 tOO99

Circular sharp-edged orifices -. - O563,, 0.788

Short cylindrical mouthpieces -. oSi ,, 0.84

The same rounded at the inner end - 0-92 ,, 0.93

Conical converging mouthpieces - - - 0-90 ,, 0.99

64. Limit to the Application of the above Formulae.In the formulae above it is assumed that the fluid issuing from the orifice expands from the pressure Pi to the pressure ~f, while passing from the vessel to the section of the jet considered in estimating the area w. Hence ~f is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices. Till recently it was tacitly assumed that this pressure pi was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio P2/Pm exceeded a value which does not greatly differ from 0.5, this was no longer true. In that case the expansion of the fluid down to the external pressure is not completed at the time it reaches the plane of the contracted section, and the pressure there is greater than the general external pressure; or, what amounti to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area ~ of the contracted section of the jet, and may be greater than the area w of the orifice Napier made experiments with steam which showed that, so long a~ P1/Pu>O-5 the formulae above were trustworthy, when p1 was taker to be the general external pressure, but that, if pf/p~

It is easily deduced from Weisbachs theory that, if the pressure external to an orifice is gradually diminished, the weight of air discharged per second increases to a maximum for a value of the ratio P2/PI ={2/(-y+I)}Engels

=0527 for air =~0-58 for dry steam.

For a further decrease of external pressure the discharge diminishes, a result no doubt improbable. The new view of Weisbachs formula is that from the point where the maximum is reached, or not greatly differing from it, the pressure at the contracted section ceases to diminish. - -

A. F. Fliegner showed (Civilrngenieur xx., 1874) that for air flowing from well-rounded mouthpieces there is no discontinuity of the law of flow, as Napiers hypothesis implies, but the curve of flow bends so sharply that Napiers rule may be taken to be a good approximation to the true law. The limiting value of the ratio P2/Pu, for which Weisbachs formula, as originally understood, ceases to apply, is for air 0-5767; and this is the number to be substituted for P2/Pi in the formulae when pf/pi falls below that value. For later researches on the flow of air, reference may be made to G. A. Zeuners paper (Civilingenieur, I87f), and Fliegners papers (ibid., 1877, 1878).

VII. FRICTION OF LIQUIDS.

65. When a stream of fluid flows over a solid surface, or conversely when a solid moves in still fluid, a resistance to the motion is generated, commonly termed fluid friction. It is due to the viscosity of the fluid, but generally the laws of fluid friction are very different from those of simple viscous resistance. It would appear that at all speeds, except the slowest, rotating eddies are formed by the roughness of the sblid surface, or by abrupt changes of velocity distributed throughout the fluid; and the energy expended in producing these eddying motions is gradually lost in overcoming the viscosity of the fluid in regions more or less distant from that where they are first produced.

The laws of fluid friction are generally stated thus:

I. The frictional resistance is independent of the pressure between the fluid and the solid against which it flows. This may be verified by a simple direct experiment. C. H. Coulomb, for instance, oscillated a disk under water, first with atmospheric pressure acting on the water surface, afterwards with the atmospheric pressure removed. No difference in the rate of decrease of the oscillations was observed. The chief proof that the friction is independent of the pressure is that no difference of resistance has been observed in water mains and in other cases, where water flows over solid surfaces under widely different pressures.

2. The frictional resistance of large surfaces is proportional to the area of the surface.

3. At low velocities of not more than I in. per second for water, the frictional resistance increases directly as the relative velocity of the fluid and the surface against which it flows. At velocities of 1/2 ft. per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity.

In many treatises on hydraulics it is stated that the frictional resistance is independent of the nature of the solid surface. The explanation of this was supposed to be that a film of fluid remained attached to the solid surface, the resistance being generated between this fluid layer and layers more distant from the surface. At extremely low velocities the solid surface does not seem to have much influence on the friction. In Coulombs experiments a metal surface covered with tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when sand was scattered over the tallow the resistance was only very slightly increased. The earlier calculations of the resistance of water at higher velocities in iron and wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that differences of roughness of the solid surface very greatly influence the friction, at such velocities as are common in engineering practice. H. P. G. Darcys experiments, for instance, showed that in old and incrusted water mains the resistance was twice or sometimes thrice as great as in new and clean mains.

66. Ordinary Expressions for Fluid Friction at Velocities nof Extremely SmallLet f be the frictional resistance estimated in pounds per square foot of surface at a velocity of 1 ft. per second; ~ the area of the surface in square feet; and C its velocity in feet per second relatively to the water in which it is immersed. Then, In accordance with the laws stated above, the total resistance of the surface is R=fwvi (I)

where f is a quantity approximately constant for any given surface. If I=2gf/G,

R = 1/2Gwvu/2g, (2)

where E is, like f, nearly constant for a given surface, and is termed the coefficient of friction.

The following are average values of the coefficient of friction for water, obtained from experiments on large plane surfaces, moved in an indefinitely large mass of water.

Frictional Coefficient Resistance in of Friction, lb per sq. ft.

____________ f New well-painted iron plate - - - 00489 00473

Painted and planed plank (Beaufoy) .00350 00339

Surface of iron ships (Rankine) - .00362.00351

Varnished surface (Froucle) - - - ~oo258 00250

Fine sand surface .00418 00405

Coarser sand surface,,,... 00503 ~Oo488

The distance through which the frictional resistance is overcome is 1 ft. per second. The work expended in fluid friction is therefore given by the equation \Vork expended =fwvf foot-pounds per second (3).

= fGuvi/2g ,, ,,

The coefficient of friction and the friction per square foot of surface can be indirectly obtained from observations of the discharge of pipes and canals. In obtaining them, however, some assumptions as to the motion of the water must be made, and it will be better therefore to discuss these values in connection with the cases to which they are related.

Many attempts have been made to express the coefficient of friction in a form applicable to low as well as high velocities. The older hydraulic writers considered the resistance termed fluid friction to be made up of two partsa part due directly to the distortion of the mass of water and proportional to the velocity -

of the water relatively to the solid sur ________________ face, and another part due to kinetic .~_______________ energy imparted to the water striking the roughnesses of the solid surface and proportional to the square of the velocity. Hence they proposed to take in which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at velocities over about 1/2 ft. per second. Values of f expressed in this and similar forms will be given in connection with pipes and canals.

All these expressions must at present tie regarded as merely empirical ex- ~ pressions serving practical purposes. The frictional resistance will be seen to vary through wider limits than these expressions allow, and to depend on circumstances of which they do not take account.

67. Coulombs ExperimentsThe first direct experiments on fluid friction were made by Coulomb, who employed a circular disk suspended by a thin brass wire and oscillated in its own plane. His experiments were chiefly made at very low velocities. When the disk is rotated to any given angle, it oscillates under the action of its inertia and the torsion of the wire. The oscillations diminish gradually in consequence of the work done in overcoming the friction of the disk. The diminution furnishes a means of determining the friction.

Fig. 78 shows Coulombs apparatus. LK supports the wire and disk ag is the brass wire, the torsion of which causes the oscilla tions; DS is a graduated ~ K disk serving to measure the apparatus oscillates.

To this the friction disk the angles through which is rigidly attached hang- The friction disks were D ing in a vessel of water.

~ ~ from 47 to 77 in. dia meter, and they gener ally made one oscillation in from 20 to 30 seconds, through angles varying from 3600 to 6. When the velocity of the cir cumference of the disk was less than 6 in. per second, the resistance FI0.78. was sensibly propor tional to the velocity.

Beaufoas Expeairnents.Towards the end of the 18th century Colonel Mark Beaufoy (1764-1827) made an immense mass of experiments on the resistance of bodies moved through water (Nautical and H~druulic Experiments, London, 1834). Of these the only ones directly bearing on surface friction were some made in 1796 and 1798. Smooth painted planks were drawn through water and the resistance measured. For two planks differing in area by 46 sq. ft., at a velocity of 10 ft. per second, the difference of resistance, measured on the difference of area, was 0.339 lb per square foot, Also the resistance varied as the I 949th power of the velocity.

68. Froudes Experiments.The most important direct experiments on fluid friction at ordinary velocities are those made by William Froude (1810-1879) at Torquay. The method adopted in these experiments was to tow a board in a still water canal, the velocity and the resistance being registered by very ingenious recording arrangements. The general arrangement of the apparatus is shown in fig. 79. AA is the board the resistance of which is to be determined. B is a cut-water giving a fine entrance to the plane surfaces of the board. CC is a bar to which the board AA is attached, and which is suspended by a parallel motion from a carriage running on rails above the still water canal. U is a link by which the resistance of the board is transmitted to a spiral spring H. A bar 1 rigidly connects the other end of the spring to the carriage. The dotted lines K, L indicate the position of a couple of levers by which the extension of the spring is caused to move a pen M, which records the extension on a greatly increased scale, by a line drawn on the paper cylinder N. This cylinder revolves at a speed proportionate to that of the carriage, its motion being obtained from the axle of the carriage wheels. A second pen 0, receiving jerks at every second and a quarter from a clock F, records time on the paper cylinder. The scale for the line of resistance is ascertained by stretching the spiral spring by known weights. The boards used for the experiment L~IJ/7 ___

RrL~J-.

FIG. 79.

were ~t in. thick, 19 in. deep, and from 1 to 50 ft. in length, cutwater included. A lead keel counteracted the buoyancy of the board. The boards were covered with various substances, such as paint, varnish, Hays composition, tinfoil, &c., so as to try the effect of different degrees of roughness of surface. The results obtained by Froude may be summarized as follows:-

I. The friction per square foot of surface varies very greatly for different surfaces, being generally greater as the sensible roughness of the surface is greater. Thus, when the surface of the board was covered as mentioned below, the resistance for hoards 50 ft. long, at 1o ft. per second, was Tinfoil or varnish 0-25 lb per sq. ft.

Calico 0.47

Fine sand 0.405 ,,

Coarser sand 0488 ,,

2. The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 ft. long, For tinfoil the resistance varied as vf.u.

For other surfaces the resistance varied as ~ With boards 50 ft. long, For varnish or tinfoil the resistance varied as viS3. For sand the resistance varied as v15.

3. The average resistance per square foot of surface was much greater for short than for long boards; or, what is the same thing-, the resistance per square foot at the forward part of the board was greater than the friction per square foot of portions more sternward. Thus, Mean Resistance in lb per sq. ft.

Varnished surface - - 2 ft. long 0.41

50 ,, 0.25

Fine sand surface. - 2 ,, 081

50 ,, 0405

This remarkable result is explained thus by Froude: The portion of surface that goes first in the line of motion, in experiencing resistance from the water, must in turn communicate motion to the water, in the tlirection in which it is itself travelling. Consequently the portion of surface which sticceeds the first will be rubbing, not against stationary water, but against water partially moving in its owl1 direction, and cannot therefore experience so much resistance from it.

69. The following table gives a general statement of Froudes results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resistance at other speeds is easily calculated.

Length of Surface, or Distance from Cutwal ~-_,--~i_I_~t._

A B C A B C A B C

Varnish, - 200 41 390 1.85 325 264 1.85 ~278 ~240

Paraffin .. ~38.370 1.94 314.260 I93 271 237

Tinfoil - 2.16.30 295 1.99 278 263 1.90 262 244

Calico - - 1.93 87 725 1.92 -626 504 1.89 531.447

Fine sand 2OO ~8i 69o 2OO .583 -450 2OO 480 -3~84

Medium sand 2OO 90 730 2OO 625 488 2OO 534 465 [U~oarse sand - 200 1-10 ~88o 2~00 -714 520 2.00 ~588 ~9C~

Columns A give the power of the speed to which the resistance is approximately proportional.

Columns B give the mean resistance per sqtiare foot of the whole surface of a board of the lengths stated in the table.

Columns C give the resistance in pounds of a square foot of surface at the distance stcrnward f ruin the cutwater stated in the heading.

Although these experiments do not directly deal with surfaces ef greater length than 50 ft. they indicate what would be the resistances of longer surfaces. For at 50 ft. the decrease of resistance for an increase of length is so small that it will make no very great difference in the estimate of the friction whether we suppose it to continue to diminish at the same rate or not to diminish at all. For a varnished surface the friction at 10 ft. per second diminishes from 0.41 to 0.32 lb per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0-278 to 0-250 lb per square foot for an increase from 20 ft. to 50 ft.

If the decrease of friction sternwards is dfie to the generation of a Current accompanying the moving plane, there is not at first sight any reason why the decrease should nnt be greater than that shown by the experiments. The current accompanying the board might be assumed to gain in volume and velocity sternwards, till the velocity was nearly the same as that of the moving plane and the friction per square foot nearly zero. That this does nut happen appears to be due to the mixing up of the current with the still water surrounding it. Part of the water in contact with the board at any point, and receiving energy of motion from it, passes afterwards to distant regions of still water, and portions of still water are fed in towards the board to take its place. In the forward part of the board more kinetic energy is given to the current than is diffused into surroundine space, and the Current gains in velocity. At a greater distance back there is an approximate balance between the energy communicated to the water and that diffused. The velocity of the cfirrent accompanying the board becomes constant or nearly constant, and the friction per square foot is therefore nearly constant also.

I 70. Friction of Rotating Disks.A rotating disk is virtually a surface of unlimited extent and it is convenient for experiments on friction with different surfaces at different speeds. - Experiments carried out by Professor W. C. Unwirm (Proc. Inst. Civ. Eng. lxxx.) are useftil both as illtmstrating the laws of fluid friction and as giving data for calculating the resistance of the disks of turbines and centrifugal pumps. Disks of 10, I5 and 20 in. diameter fixed on a vertical shaft were rotated by a belt driven by an engine. They were enclosed in a cistern of water between parallel top and bottom fixed surfaces. The cistern was suspended by three fine wires. The friction of the disk is equal to the tendency of the cistern to rotate, and this was measured by balancing the cistern by a fine silk cord passing over a pulley and carrying a scale pan in which weights could be placed.

If ~ is an element of area on the tusk moving with the velocity 1, the friction on this element is fwi, where f and n are constant for any given kind of surface. Let a be the angular velocity of rotation, R the radius of the disk. Consider a ring of the surface between rand r+dr. Its area is 2,rrdr, its velocity arand the friction of this ring is f2im-rdrar. The moment of the friction about the axis of rotation is 2lra~frEngelsedr, and the total moment of friction for the two sides of the disk is M = ~ = 4lra/(fl +3) }fR~.

If N is the number of revolutions per sec.,

M = (2iir~+iN~/(n+3)}fR~+i, and the work expended in rotating the disk is ~ foot lb per sec.

The experments give directly the values of M for the disks corre sponding to any speed N. From these the values of f and n can he deduced, f being the friction per square foot at unit velocity. For comparison with Froudes results it is convenient to calculate the resistance at 10 ft. per second, which is F =jf 0.

The disks were rotated in chambers 22 in. diameter and 3, 6 and 12 10. deep. In all cases the friction of the disks increased a little as the chamber was made larger. This is probably due to the stilling of the eddies against the surface of the chamber and the feeding back of the stilled water to the disk. iIcimce the friction depends not only on the stirface of the disk but to some extent on the surface of the chamber in which it rotates. If the surface of the chamber is made rougher by covering with coarse sand there is also an increase of resistance.

Cr, in feet. For the smoother surfaces the friction varied - as the 1-85th power of the velocity. For the 3oft. rougher surfaces the power of the velocity to A B ~ which the resistance was proportional varied from 1.9 to 2~I. This is in agreement with Froudes results.

0 ~226 -

183 25 Experiments with a bright brass disk showed that the friction decreased with increase of I83 246.232 temperature. The diminution between 41

f87 474 and 130 F. amounted to 18%. In the general 2~06.405

2-00 488.456 equation M = eN for any given disk, C,=0.1328(i o-oosrt),

- I where c, is time value of c for a bright brass disk o-8,5 ft. in diameter at a temperature 1 F.

The disks used were either polished or made rougher by varnish or by varnish and sand. The following table gives a comparison of the results obtained with the disks and Froudes results on planks 50 ft. long. The values given are the resistances per square foot at 1o ft. per sec.

Froudes Experiments. I Disk Experiments.

Tinfoil surface - 0.232 Bright brass 0-202 to 0-229

Varnish 0.226 I Va-nish - - 0-220 to 0-233

Fine sand - - - 0-337 Fine sand, 0-339

Medium sand - - 0-456 Very coarse sand 0-587 to 0-715

VIII. STEADY FLOW OF WATER IN PIPES OF

UNIFORM SECTION.

71. The ordinary theory of the flow of water in pipes, on which all practical formulae are based, assumes that the variation of velocity at different points of any cross section may be neglected. The water is considered as moving in plane layers, which are driven through the pipe against the frictional resistance, by the difference of pressure at or elevation of the ends of the pipe. If the motion is steady the velocity at each cross section remains the same from moment to moment, and if the cross sectional area is constant the velocity at all sections must be the same. Hence the motion is uniform. The most important resistance to the motion of the water is the surface friction of the pipe, and it is convenient to estimate this independently of some smaller resistances which will be accounted for presently.

In any portion of a uniform pipe, excluding for the present the ends of the pipe, the water enters and leaves at the same velocity. For that portion therefore the work of the external forces and of the surface friction pj~

mtmst be equal. Let ~_~

fig. 80 represent a very tween cross sections at short portion of the pipe, of length dl, beI and z+dz ft. above any horizontal datum :

line xx, the pressures at the cross lections being ~-

p and p+dp lb per ~ X

square foot. Further, Fie. 80.

let Q be the volume of flow or discharge of the pipe per second, t~ the area of a normal cross section, and x the perimeter of the pipe. The Q cubic feet, which flow through the space considleredl per second, weigh GQ th, and fall through a heightdz ft. The work done by gravity is then GQdz; -

a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is p(p-l-dp)= dp lb per square foot of the cross section. The work of this pressure on the volume Q is Qdp.

The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is x dl.

The work expended in overcoming the frictional resistince per second is (see 66, eq. 3) ~Gxdlv3/2g or, since Q=~lv, rG(x/t7)Q(v2/sa)dl; the negative sign being taken because the work is done against a resistance. Adding all these portions of work, and equating the result to zero, since the motion is uniform, GQdz -~- Qdp IG (x/fJ)Q(v~i/2g)dl = o. Dividing by GQ,

dz+dP/G+l~(x/t2) (u1/2g)dl =0.

Integrating, z+p/G+r(x/c2) (vuf2g)l = constant. (I)

72. Let A and B (fig. 81) be any two sections of the pipe for which p, 1, 1 have the values pi, z~, ii, and P2, 12, if, respectively. Then zi +Pi!G + l(x/f~) (v2/2g)li zs+p2/G + 1(x/~1) (v2/2g)lf; or, if i2~li =L, rearranging the terms, lv2/2g = (i/L)t (zi+Pi/G) (z1+pfIG)jfl/x. (2)

Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights pi/G and p2/G due to the -~~J:~

Dat u,n Line Ff0.81.

pressures P and P2 at A and B. Hence (zi+pi/G)(12+p2/G) is the quantity represented in the figure by DE, the fall of level of the pressure columns, or virtual fall of the pipe. If there were no friction in the pipe, then by Bernoullis equation there would be no fall of level of the pressure columns, the velocity being the same at A and B. Hence DE or h is the head lost in friction in the distance AB. The quantity DE/AB=/1/L is termed the virtual slope of the pipe or virtual fall per foot of length. It is sometimes termed very conveniently the relative fall. It will be denoted by the symbol i.

The quantity ff/x which appears in many hydraulic equations is called the hydraulic mean radius of the pipe. It will be denoted by m.

Introducing these values, l~v2/2g = mh/L mi. (3)

For pipes of circular section, and diameter d, m = = 3/4IT-d/lrd = fd. -

Then ~v2/2g = 3/4dh/L = 1/2di; (4)

or h = 1(4L/d) (v/2g);, (4a)

which shows that the head lost in friction is proportional to the head due to the velocity, and is found by multiplying that head by the coefficient 4~L/d. It is assumed above that the atmospheric pressure at C and D is the same, and this is usually nearly the case. But if C and D are at greatly different levels the excess of barometric pressure at C, in feet of water, must be added to p2/G.

73. Hydraulic Gradient or Line of Virtual Slope.JoTh CD. Since the head lost in friction is proportional to L, any intermediate pressure column between A and B will have its free surface on the line CD, and the vertical distance between CD and the pipe at any point measures the pressure, exclusive of atmospheric pressure, in the pipe at that point. If the pipe were laid along the line CD instead of AB, the water would flow at the same velocity by gravity without any change of pressure from section to section. Hence CD is termed the virtual slope or hydraulic gradient of the pipe. It is the line of free surface level for each point of the pipe.

If an ordinary pipe, connecting reservoirs open to the air, rises at any joint above the line of virtual slope, the pressure at that point is less than the atmospheric pressure transmitted through the pipe. At such a point there is a liability that air may be disengaged from the water, and the flow stopped or impeded by the accumulation of air. If the pipe rises more than 34 ft. above tile line of virtual slope, the pressure is negative. But as this is impossible, the continuity of the flow will he broken.

If the pipe is not straight, the line of virtual slope becomes a curved line, hut since in actual pipes the vertical alterations of level ar2 generally small, compared with the length of the pipe, distances measured along the pipe are sensibly proportional to distances measured along the horizontal projection of the pipe. Hence the line of hydraulic gradient may be taken to be a straight line without error of practical importance.

7~. Case of a Uniform Pipe connecting two Reservoirs, when all the Resistances are taken into account.Let h (fig. 82) be the difference of level of the reservoirs, and v the velocity, in a pipe of ~length L and diameter d. The whole work done per second is virtually the removal of Q cub. ft. of water from the surface of the upper reservoir to the surface of the lower reservoir, that is GQh footpounds. This is expended in three ways. (i) The head vi/2g, corresponding to an expenditure of GQv2/2g foot-pounds of work, is employed in giving energy of motion to the water. This is ultiV2

(1+ ,~) 29

FIG. 82.

mately wasted in eddying motions in the lower reservoir. (2) A portion of head, which experience shows may be expressed in the form lovi/2g, corresponding to an expenditure of GQi~fdi/2g footpounds of work, is employed in overcoming the resistance at the entrance to the pipe. (3) As already shown the head expended in overcoming the surface friction of the pipe is ~(4L/d) (v/2g) corresponding to GQi~(4L/d)(v2/2g) foot-pounds of work. Hence GQh = GQv2/2g +GQ~ov1/2g +GQl.4L.Vf/d.2g; h= (I +l-o+1- 4L!d)vf/2g.

v=8o25%f ~hd/{(I+~o)d+4lL1l. 5

If the pipe is beilmouthed, is is about = 08. If the entrance to the pipe is cylindrical, io=O55. Hence f+ro=fo8 to 1.505.

In general this is so small compared with l4L/d that, for practical calculations, it may be neglected; that is, the losses of head other than the loss in surface friction are left out of the reckoning. It is only in short pipes and at high velocities that it is necessary to take account of the first two terms in the bracket, as well as the third. For instance, in pipes for the supply of turbines, v is usually limited to 2 ft. per second, and the pipe is bellmouthed. Then I.o8vu/2g =0.067 ft. In pipes for towns supply v may rasige from 2 to 41/2 ft. per second, and then I.5v2/2g=0.I to 0.5 ft. In either case this amount of head is small compared with the whole virtual fall in the cases which most commonly occur.

When d and v or d and h are given, the equations above are solved quite simply. When v and h are given and d is required, it is better to proceed by approximation. Find an approximate value of d by assuming a probable value for l as mentioned below. Then from that value of d find a corrected value for i and repeat the calculation.

The equation above may be put in the form h= (4l/d)(f+l0)d/41}+L]uiJ2g; (6)

from which it is clear that the head expended at the mouthpiece is equivalent to that of a length (1 +r0)d/4~

of the pipe. Putting I+1f= 505 and I=ooI, the length of pipe equivalent to the mouthpiece is 37.6 d nearly. This may be added to the actual length of the pipe to allow for mouthpiece resistance iii approximate calculations.

75. Coefficient of Friction for Pipes discharging Water.From the average of a large number of experiments, the value of l for ordinary iron pipes is ~=00o7567. (7)

But practical experience shows that no single value can be taken applicable to very different cases. The earlier hydraulicians occupied themselves chiefly with the dependence of i on the velocity. 1-laying regard to the difference of the law of resistance at very low and at ordinary velocities, they asstinied that i might be expressed in the form The following are the best numerical values obtained for I so expressed: a R. de Prony (from 51 experiments) - 0.006836 O~001116

J. F. dAubuisson de Voisins 0-00673 0-001211

J. A. Fytelwein 0.005493 0-00143

Weisbach proposed the formula 4~ = a-f~3/~f V =0.003598 +Ooo4289h/ v. (8)

76. Darcys Experiments on Friction in Pipes.All previous experiments on the resistance of pipes were superseded by the remarkable researches carried out by H. P. G. Darcy (18o3185&), the Inspector-General of the Paris water works. His experiments were carried out on a scale, under a variation of conditions, and with a degree of accuracy which leaves little to be desired, and the results obtained are of very great practical importance. These results may be stated thus: -

I. For new and clean pipes the friction varies considerably with the nature and polish of the surface of the pipe. For clean cast iron it is about 13/4 timesas great as for cast iron covered with pitch.

2. The nature of the surface has less influence when the pipes are old and incrusted with deposits, due to the action of the water. Thus old and incrusted pipes give twice as great a frictional resistance as new and clean pipes. Darcys coefficients were chiefly determined from experiments on new pipes. He doubles these coefficients for old and incrusted pipes, in accordance with the results of a very limited number of experiments on pipes containing incrustations and deposits.

3. The coefficient of friction may be expressed in the form r=a+fl/v; but in pipes which have been some time in use it is sufficiently accurate to take I=ai simply, where ai depends on the diameter of the pipe alone, but a and jI on the other hand depend both on the diameter of the pipe and the nature of its surface. The following are the values of the constants.

For pipes which have been some time in use, neglecting the term depending on the velocity; l~a(1+i~/d). (9)

For drawn wrought-iron or smooth cast- iron pipes .004973.084

For pipes altered by light incrustations. - .00996 084

These coefficients may be put in the following very simple form, without sensibly altering their value:

For clean pipes oos(I+I/I2d) a For slightly incrusted pipes - i= oI(I+f/I2d) 9

Darcys Value of the Coefficient of Friction I for Velocities not less than 4 in. per second.

Diameter _______ _________ Diameter _________________

of Pipe New Incrusted of Pipe New Incrusted in Inches. Pipes. Pipes. in Inches. Pipes. Pipes.

2 0.00750 0.01500 18.00528.01056

3.00667 01333 21 00524.01048

4 -00625 01250 24.00521.01042

5 -00600 01200 27.00519.01037

6 -00583 01167 30.00517.01033

7 00571 01 143 36.00514.01028

8 00563 01125 42 00512.01024

I 9 00556 ~0I1II 48 00510 ~0I02I

12 00542.0Io83 54 00509.01019

L 15.00533.01067 __________ _______ _________

These values of i are, however, not exact for widely differing velocities. To embrace all cases Darcy proposed the expression ~r~(a+ai/d)+($+th/d1)/v; (10)

which is a modification of Coulombs, including terms expressing th~ influence of the diameter and of the velocity. For clean pipes Darc) found these values a -004346

a,~OO03992

= 0010182

th 000005205.

It has become not uncommon to calculate the discharge of pipn by the formula of E. Ganguillet and W. R. Kutter, which will b discussed under the head of channels. For the value of c in th relation v=c.j (nzi), Ganguillet and Kutter take 41-6+I811/fl+.00281/i C I+~(4I6+o028I/i)(n/~m)]

where n is a coefficient depending only on the roughness of the pipe For pipes uncoated as ordinarily laid n=O013. The formula is ver~ cumbrous, its form is not rationally justifiable and it is not at al clear that it gives more accurate values of the discharge than simple formulae.

7~. Later Investigations on Flow in Pipes.The foregoing state ment gives the theory of flow in pipes so far as it can be put in simple rational form. But the conditions of flow are really moo complicated than can be expressed in any rational form. Takin1

even selected experiments the values of the empirical coefficient ~ range from o~I6 to 00o28 in different cases. Hence means of discriminating the probable value of ~ are necessary in using the equations for practical purposes. To a certain extent the knowledge that I decreases with the size of the pipe and increases very much with the roughness of its surface is a guide, and Darcys method of dealing with these causes of variation is very helpful. But a further difficulty arises from the discordance of the results of different experiments. For instance F. P. Stearns and J. M. Gale both experimented on clean asphalted cast-iron pipes, 4 ft. in diameter. According to one set of gaugings l= .oo51, and according to the other i .003 1. It is impossible in such cases not to suspect some error in the observations or some difference in the condition of the pipes not noticed by the observers.

It is not likely that any formula can be found which will give exactly the discharge of any given pipe. For one of the chief factors in any such formula must express the exact roughness of the pipe surface, and there is no scientific measure of roughness. The most that can be done is to limit the choice of the coefficient for a pipe within certain comparatively narrow limits. The experiments on fluid friction show that the power of the velocity to which the resistance is proportional is not exactly the square. Also in determining the form of his equation for ~ Darcy used only eight out of his seventeen series of experiments, and there is reason to think that some of these were exceptional. Barr de Saint-Venant was the first to propose a formula with two constants, dh/41 mV,

where m and is are experimental constants. If this is written in the form log m+n log v=log (dh/41),

we have, as Saint-Venant pointed out, the equation to a straight line, of which m is the ordinate at the origin and n the ratio of the slope. If a series of experimental values are plotted logarithmically the determination of the constants is reduced to finding the straight line which most nearly passes through the plotted points. SaintVenant found for n the value of 1.71. In a memoir on the influence of temperature on the movement of water in pipes (Berlin, 1854) by G. H. L. Hagen (1797-1884) another modification of the Saint-Venant formula was given. This is h/lmv/d, which involves three experimental coefficients. Hagen found fl1.75; x=I25; and m was then nearly independent,of variations of v and d. But the range of cases examined was small. In a remarkable paper in the Trans. Roy. Soc., 1883, Professor Osborne Reynolds made much clearer the change from regular stream line motion at low velocities to the eddying motion, which occurs in almost all the cases with which the engineer has to deal. Partly by reasoning, partly by induction from the form of logarithmically plotted curves of experimental results, he arrived at the general equation /z/l=c(v/d3)Pf, where n I for low velocities and n = 1.7 to 2 for ordinary velocities. P is a function of the temperature. Neglecting variations of temperature Reynolds formula is identical with Hagens if x=3fl. For practical purposes Hagens form is the more convenient.

Values of Index of Velocity.

Diameter Surface of Pipe. Atithority. of Pipe Values of n.

in Metres.

Tin plate. - Bossut J .036 I6971, I~ 2

- I 054 I73oj 7

Wrought iron (gas Hamilton Smith ~ ~}- I~75

I 014 1.8661

Lead -. -. Darcy - -. -~ 027 I-755~. I77

L 041 I760j Clean brass.. Mair -. - 036 I795 I~795

Hamilton Smith ~ o266 1.760

A h It d J Lampe - - - J 4185 1-850 I~8

sp a e - - 1 W. W. Bonn - 306 1.582

~ Stearns. .. ~I~2I9 i88o Riveted wrought ~ ~ .2776 1 8o4~

iron I Hamilton Smith -~ 3219 1.892 ~- I87

~ .3749 f.852J

Wroughtiron (gas t 0122 1.9015)

pipe) I Darcy - -. .~ 0266 I89g~ 1.87

L 0395 1.8381

1.0819 I95o~

New cast iron - Darcy -.. ~ :~g~ ~ ~5

~ 50 1-950j. I .0801 2OOO

Cleanedcastiron - Darcy - ~ .2447 2oOO 2OO

1.397 2.07

I 0359 f980~

Incrusted cast iron Darcy -.. -~ 0795 I990} 2OO

L 2432 I990J

IIHJ~IIII

= ii I I I I I

-8__

~-~.-

~iziiz:iiii;~:

-SEngels 03:22, 27 Mar 2006 (PST)i---~4_ i:~__ ,-~-~~___ ~, 5

-a_ _ -

~-9r-o~ ~2 151 E~E 171 E~E

In I886, Professor W. C. Unwin plotted logarithmically all the most trustworthy experiments on flow in pipes then available.i Fig. 83 gives one such plotting. The results of measuring the slopes of the lines drawn throtigh the plotted points are given in the table.

It will he seen that the values of the index n range from 1.72 for the smoothest and cleanest surface, to 2Oo for the roughest. The numbers after the brackets arc rounded off numbers.

The value of ii having been thus determined, values of m/d were next found and averaged for each pipe. These were again plotted logarithmically in order to find a value for x. The lines were not very regular, btit in all cases the slope was greater than I to I, so that the value of x must be greater than unity. The following table gives the results and a comparison of the value of x and Reynoldss value 3il.

Kindol Pipe. n 3n x Tin plate -.. 1-72 28 1.100

Wrought iron (Smith). I-75 1.25 I210

Asphalted pipes, 1-85 1.15 1.127

Wrought iron (Darey). 1-87 1.13 1.680

Riveted wrought iron. f87 1.13 1.390

New cast iron .,, 1.95 1.05 1.168

Cleaned cast iron,, 2-00 iOO 1168

Jucrusted cast irOn. 200 100 1-160

- With the eaception of the anomalous values for Darcys wrought. iron pipes, there is no great discrepancy between the values of x anc 3fl, but there is no appearance of relation in the two quantities For the present it appears preferable to assume that x is independeni of n.

It is now possible to obtain values of the third constant m, usinf the values found for n and x. The following table gives the results the values of m being for metric measures.

i Formulae for the Flow of Water in Pipes, Industries (Man chester, 1886).

I I I I I I ~ I I II 1 1 1

~ ! .4 ~-

-- __1_ - - -

-~----~e.4---~--

I I I I = = =

~-~5!~---------.--

- -~.- - - -

!~3T~ ~.r~k~t ~ ~ 181

Here, considering the great range of diameters and velocities in the experiments, the constancy of m is very satisfactorily close. The asphalted pipes give iather variable values. But, as some of these were new and some old, the variation is, perhaps, not surprising. The incrusted pipes give a value of in quite double that for new pipes but that is perfectly consistent with what is known of fluid friction in other cases.

Diameter Mean Kind of Pipe. in Value of Value Authority.

Metres. m. of m.

Tin plate - ~ ~ ~o1686 Bossut Wrought iron ~ ~~} ~0I3I0 Hamilton Smith (0.027 oI749-~ ~ Hamilton Smith Asphalted j 0.306 02107 1.018 1 ~ W. W. Bonn pipes 1 0.419 0165-0131 Lampe 1.219.01317 Stearns lJ2f9 o21o7J t Gale (0.278.01370

Riveted I 0~322.01440

wrought iron ~ ~ .01403 Hamilton Smith 0.657.01448

10.082.017251

New cast iron ~ ~ ~- 01658 Darcy 0.500.01745-

Cleaned cast 10080 oI979~

iron 1 0.245 02091 (01994 Darcy t,0297 oI9I3J

Incrustecl cast ~ .o~5~} -03643 Darcy ________ 0.243 03706 _______ ______________

General Mean Values of Constants.

The general formula (Hagens)h/l=mv/d.2gcan therefore be taken to fit the results with convenient closeness, if the following mean values of the coefficients are taken, the unit being a metre:

~ Kind of Pipe. m x n Wrought iron -. - .of3I f21 1.75

Asphalted iron - - 0183 1.127 1.85

Riveted wrought iron 0140 1.390 1.87

New cast iron, ,, 0166 I168 1.95

Cleaned cast iron,, 0199 1.168 2O

Incrusted cast iron, -0364 f~I60 20

The variation of each of these coefficients is within a comparatively narrow range, and the selection of the proper coefficient for any given case presents no difficulty, if the character of the surface of the pipe is known.

It only remains to give the values of these coefficients when the quantities are expressed in English feet. For English measures the following are the values of the coefficients:

Kind of Pipe. rn x n 1

Tin plate, - -0265 IIO 1.72

Wrought iron -, .0226 f21 1.75

Asphalted iron - 0254 1.127 1.85

Riveted wrought iron, .0260 1.390 1.87

New cast iron -,, 0215 1.168 1.95

Cleaned cast iron, 0243 1.168 2O

Incrusted cast iron - -0440 i-I6o 20

78. Distribution of Velocity in the Cross Section of a Pipe.Darcy made experiments with a Pitot tube in 1850 on the velocity at different points in the cross section of a pipe. He deduced the relation \v=f13(ri/R)~/i, where V is the velocity at the centre and 1 the velocity at radius r in a pipe of radius R with a hydraulic gradient i. Later Bazin repeated the experiments and extended them (Mirn. de lAcadimie des Sciences, xxxii. No. 6). The most important result was the ratio of mean to central velocity. Let b = Ri/U2, where U is the mean velocity in the pipe; then V/U=I+9o39jb. A very useful result for practical purposes is that at 0.74 of the radius of the pipe the velocity is equal to the mean velocity. Fig. 84 gives the velocities at different radii as determined by Bazin.

7~. Influence of Temperature on the Flow through Pipes.Very careful experiments on the flow through a pipe 0.1236 ft. in diameter -940 -

~~+1-~

FIG. 84.

and 25 ft. long, with water at different temperatures, have been made by J. C. Mair (Proc. Inst. Civ. Eng. lxxxiv.). The loss of head was measured from a point 1 ft. from the inlet, so that the loss at entry was eliminated. The 14 in. pipe was made smooth inside and to gauge, by drawing a mandril through it. Plotting the results logarithmically, it was found that the resistance for all temperatures varied very exactly as v795, the index being less than 2 as in other experiments with very smooth surfaces. Taking the ordinary equation of flow h=I(4L/D)(vf/2g), then for heads varying from 1 ft to nearly 4 ft., and velocities in the pipe varying from 4 ft. tu 9 ft. pei second, the values of ~ were as follows:

Temp. F. Temp. F. I

57 0044 to 0052 100 0039 to 0042

70 0042 to 0045 110 0037 to ~0o4I

80 0041 to 0045 120 0037 to ~oO4I

90 -0040 to -0045 130 0035 to .0039

160 -0035 to 0038

This shows a marked decrease of resistance as the temperature rises. If Professor Osborne Reynoldss equation is assumed h=n1LV/df, and n is taken 1.795, then values of m at each temperature are practically constant Temp. F. m. Temp. F. ni.

57 0.000276 100 0.000244

70 0.000263 110 0.000235

80 0.000257 120 0.000229

90 0.000250 130 0.000225

160 0000206

where again a regular decrease of the coefficient occurs as the temperature rises. In experiments on the friction of disks at different tempelatures Professor ,AT. C. (Jnwin found th.at the resistance was proportional to constant X (10-0021 t) and the values of m given above are expressed almost exactly by the relation m=o000311(fOo0215 I).

In tank experiments on ship models for small ordinary variations of temperature, it is usual to allow a decrease of 3% of resistance for 10 F. increase of temperature.

8o. Influence of Deposits in Pipes on the Discharge. Scraping Water Mains.The influence of the condition of the surface of a pipe on the friction is shown by various facts known to the engineers of waterworks. Jn pipes which convey certain kinds of water, oxidation proceeds rapidly and the discharge is considerably diminished. A main laid at Torquay in 1858, 14 m. in length, consists of 10-in., 9-in. and 8-in, pipes. It was not protected from corrosion by any coating. But it was found to the surprise of the engineer that in eight years the discharge had diminished to 51% of the original discharge.

J. G. Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and used under the direction of William Froude (see Incrustation of Iron Pipes, by W. Ingham, Proc. Inst. Mech. Eng., 1899). It was found that by scraping the interior of the pipe the discharge was increased 56%. The scraping requires to be repeated at intervals. After each scraping the discharge diminishes rather rapidly to 10% and afterwards more slowly, the diminution in a year being about 25%.

Fig. 85 shows a scraper for water mains, similar to Appolds but modified in details, as constructed by the Glenfield Company, at Kilmarnock. A is a longitudinal section of the pipe, showing the scraper in place; B is an end view of the plungers, and C, D sections of the boxes placed at intervals on the main for introducing or withdrawing the scraper. The apparatus consists of two plungers, packed with leather so as to fit the main pretty closely. On the spindle of these plungers are fixed eight steel scraping blades, with curved scraping edges fitting the surface of the main. The apparatus is placed in the main by removing the cover from one of the boxes shown at C, D. The cover is then replaced, water pressure is admitted behind the plungers, and the apparatus driven through the ~ ., ~ N

4: ____ ___

~0~$9~0$~N, ~

FIG. 85. Scale 211.

main. At Lancaster after twice scraping the discharge was increased 561/2%, at Oswestry 541/2%. The increased discharge is due to the diminution of the friction of the pipe by removing the roughnesses due to oxidation. The scraper can be easily followed when the ma,~ns are about 3 ft. deep by the noise it makes. The average speed of the scraper at Torquay is 24 m. per hour. At Torquay 49% of the deposit is iron rust, the rest being silica, lime and organic matter.

In the opinion of some engineers it is inadvisable to use the scraper. The incrustation is only temporarily removed, and if the use of the scraper is continued the life of the pipe is reduced. The only treatment effective in preventing or retarding the incrustation due to corrosion is to coat the pipes when hot with a smooth and perfect layer of pitch. With certain waters such as those derived from the chalk the incrustation is of a different character, consisting of nearly pure calcium carbonate. A deposit of another charactem which has led to trotible in some mains is a black slime containing a good deal of iron not derived from the pipes. It appears to he an organic growth. Filtration of the water appears to prevent the growth of the slime, and its temporary removal may be effected by a kind of brush scraper devised by G. F. Deacon (see Deposits in Pipes, by Professor J. C. Campbell Brown, Eroc. Inst. Civ. Eng., 1903-1904).

81. Flow of Water through Fire Ilose.The hose pipes used for fire purposes are of very varied character, and the roughness of the surface varies. Very careful experiments have been made by J. R. Freeman (Am. Soc. Civ. Rug. xxi., 1889). It was noted that under pressure the diameter of the hose increased sufficiently to have a marked influence on the discharge. in reducing the results the true diameter has been taken. Let v=mean velocity in ft. per sec.; r=hydraulic mean radius or one-fourth the diameter in feet; s~ hydraulic gradient. Then V=fl~1 (ri).

Diameter Inches. i V a Solid rubber 2.65 215.1863 12.50 123.3

hose ,, 344 4714 20OO 124.0

Woven cotton, 2-47 200 2464 13.40 119.1

rubber lined 2 ,, 299 ~5269 20OO 121.5

Woven cotton, (2-49 200 2427 13.20 117.7

rubber lined ,, 319 -5708 2I~0O I22I

Knit cotton, 2-68 I32 0809 7.50 1I16

rubber lined ,, 299 ~3931 17.00 114.8

Knit cotton, 2-69 204 2357 11.50 IO0~I

rubber lined ,, 3f9 ~5I65 18oo 1058

Woven Cotton, 2-12 154 3448 14Oo 113.4

rubber lined ,, 240 7673 2181 1184

Woven cotton, 2-53 54~8 ~o26I 3.50 94.3

I rubber lined ,, 298 8264 19.00 91O

Unlined linen 2-60 579 0414 350 739

______________ _____ 331 1.1624 20OO 79-6

82. Reduction of a Long Pipe of Varying Diameter to an Equivalent Pipe of Uniform Diameter. Dupuits Equation.Water mains for the supply of towns often consist of a series of lengths, the diameter being the same for each length, but differing from length to length. In approximate calculations of the head lost in such mains, it is generally accurate enough to neglect the smaller losses of head and to have regard to the pipe friction only, and then the calculations may be facilitated by reducing the main to a main of uniform diameter, in which there would be the same loss of head. Such a uniform main will be termed an equivalent main.

~,- I -~ 12 --si k---- 1

FIG. 86.

In fig. 86 let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, let ii, 1,. -. be the lengths, d,, df... the diameters, v,, v2.. the velocities, i1, i~... the slopes, for the successive portions, and let I, d, v and i be corresponding quantities for the equivalent uniform main B. The total loss of head in A due to friction is h=i,l1+ifl2l-- - -

i(vi1-~lj/2gdi) +l(vi41i/2gd2) + and in the uniform main il= I(v241/2gd).

If the mains are equivalent, as defined above, ~(v2.4lf2gd) = ~(vi2-4li/2gdj) +~(vf4li/2gds) +

But, since the discharge is the same for all portions, 3/4ircpv 3/4-,rdi1vi = 3/4irdfiv, ...

V1 = vdi/dif; vf = vdi/dzf Also suppose that I may be treated as constant for all the pipes. Then l/d (dud14) (l1/d1) --f- (d4/d14) (hid2) + - - -

1= (d5/di5)lf~(d/d25)l,f-...

which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge.

83. Other Losses of Head in Pipes.Most of the losses of head in pipes, other than that due to surface friction against the pipe, are due to abrupt changes in the velocity of the stream producing eddies. The kinetic energy of these is deducted from the general energy of translation, and practically wasted.

Sudden Enlargement of Section.Suppose a pipe enlarges in section from an area wo to an area di (fig.

87); then Vi/VO Wo/d1

or, if the section is circular, vi/vo=(do/di). ~ sesYi-s. wJ~_.. ~

The head lost at the abrupt change of velocity has already been shown to be the head due to the relative velocity of the two parts FIG. 87. of the stream. Hence head lost = (Vf vi)2/2g = (wi/do 1)2v12/2g = (dud0)1, }vff/2g or I),=i,vf1/2g, (1)

if l~ is put for the expression in brackets.

~ii~0=L.f 112115 1.7 1,.8 i.Q 20 12.513.013.514.01 5t6.o L 1.05! I.I0~ 1.22.3! f.341 I.38~ .4! i.581 I.73~ 1.87~ 2.00~ 2.24! 2.45! 2.651 2.83!

= I ~l ~I .25 ~~! .64! ~ 2.25! 4.oo~,~s! 9.ooIIo.olls.ool 36~h~j A brupt Contraction of SectionWhen water passes from a larger to a smaller section, as in figs. 88, 89, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe.

FIG. 88. FIG. 89.

Let w be the section and v the velocity of the stream at bb. At aa the section will be c,w, and the velocity (w/c,co)vv/ci, where c, is the coefficient of contraction. Then the head lost is = (v/c, v)2/2g = (f/c, I)1v2/2g; and, if c, is taken 0.64,

~mO3I6 v1/2g. (2)

The value of the coefficient of contraction for this case is, however, not well ascertained, and the result is somewhat modified by friction. For water entering a cylindrical, not bell-mouthed, pipe from a reservoir of indefinitely large size, experiment gives t),,=o.5o5 V/2g. (3)

If there is a diaphragm at the mouth of the pipe as in fig. 89, let ut be the area of this orifice. Then the area of the contracted stream is c,,w1, and the head lost is l~, = (w/C,w,) I ~1v/2g = ~,v1/2g (4)

if r, is put for { (w/C,wf) I ~. Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice: I wi/w = o.i I 0.2) 0.3 I 0.4) 0.5) o.6 I 0.7 I 0.8) 1I~1

I C1.6,6.614.622.6,o .6,7 I .6o~ .6o3 si I .1g8.596

= I 232.7 ~~1 ! 29.78 19.612 1525613077 lI876 2 i6~ 10734 0.4801

When a diaphragm was placed in a tube of tiniform section (fig. 90)

_-j-~_

Ff G. 90.

the following values were obtained, di being the area of the orifice and w that of the pipe:

10.1 10.2 10.3 ~ 105 I o.6)07 I o.8 0.9 ~f0

C, = .624 I .632 I .643.659.68, I .712.755 I .3i~ .8ei lao = J 225.9147.77130.831 7.80f 1~~l 2.7961.797.290 ~

Elbows.Weisbach considers the loss of head at elbows (fig.91) to be due to a contraction formed by the stream. From experiments with a pipe 1 ~ in. diameter, he found the loss of head lj,=l,vf/2g; (5)

l,=09457 sini1/2g~+2.o47 Situ 20 40 6o 8o 90 I0 iIO I 120 130 140

= ~ ~1 I 0.364 0,740 J 0.984 1.260 1.556 i.86x i,iss Hence at a right-angled elbow the whole head due to the velocity very nearly is lost.

Bends.Weisbach traces the loss of head at curved bends to a similar cause to that at elbows, but the coefficients for bends are not very satisfactorily ascertained. Weisbach obtained for the loss of head at a bend in a pipe ______ of circular section f~b=lbv1I2g; (6)

0~~ l-s=o13f+I847(d/2p)i, where d is the diameter of the pipe and p the FIG. 9I. radius of curvature of the bend. The resistance at bends is small and at present very ill determined.

Valves, Cocks and SluicesThese produce a contraction of the water-stream, similar to that for an abrupt ~ diminution of section already discussed. The loss of head may be taken as before to be = ~,v1I2g; (7)

where v is the velocity in the pipe beyond the valve and I~ a coefficient determined by experiment. The following are Weisbachs results.

Sluice in Pipe of Rectangular Section (fig. 92). FIG. 92 Section at sluice =wi itt pipe=w.

I I I I I I

10 10.9 o~8 I o~7 10.6 0-5 0.4 03 0~2

Joooj ~9I .39j .9512.o814.o218.f21 17.8144.5 193 Sluice in Cylindrical Pipe (fig. 93).

Rain of height 0f}I I I

opening to diameter 1.0111 1

of pipe ujw= I.00~ 0,9481.8~6.7401.609.466.3I5 1

= 0.07 0,26 0.81 2.06 5.52 J 17,0 97.8

FIG. 93. FIG. 94.

Cock in a Cylindrical Pipe (fig. 94). Angle through which cock is turned =0.

50 I 10 15 I 20 I 25 30 35

cross I -926 -850 772 692 -613.535.458

I Ratio of~ I I

sectionsj L ~ -05 j -29 75 1-56 3.10 5.47 9.68

r~. I 40 I 45 5e~ 55 606582

L:ioofll cross ~-I .385 I .3I5 250 190.137.091 0

sections ji I

f7~ 31.2, 52-6 j 106 2C16 486 00

Throttle Valve in a Cylindrical Pipe (fig. 95)

0= I 5 I 10 15 20 25 300 I 35 4G

(I I I ~ r~ I

l~ -24 52 ~90 1.54 25f 1 3~ 622 io~J

0= 45 50 55 60 65 I 70 90

l~ I87 32-6 588 118 256 751 no 84. Practical Calculations on the Flow of Water in Pipes.In the following explanations it will be assumed that the pipe is of so great a length that only the loss of head in friction against -, -

the surface of the pipe needs to be considered. In general r~ ~ ~ ~

it is one of the four quantities d, 1, v or Q which requires to be determined. For since the loss of head Ii is given by /=~/...=~ the relation h = ii, this need not be separately considered. Fl G. 95.

There are then three equations (see eq. 4, 72, and ~a, 76) for the solution of such problems as arise :

l~a(I+I/I2d); (I)

where a =0.005 for new and =001 for incrusted pipes.

Iv2/2g=1/2di. (2)

Q=~~rd2v. (3)

Problem 1. Given the diameter of the pipe and its virtual slope, to find the discharge and velocity of flow. Here d and i are given, and Q and v are required. Find l from (I); then v from (2); lastly Q from (s). This case presents no difficulty.

By combining equations (I) and (2), v is obtained directly: v=-~1 (gdi/2~) =-~! (g/2a)~/ +I/12d} ]. (4)

For new pipes.. - ~J (g/2a) 5672

For incrusted pipes. - =4oI3

For pipes not less than 1, or more than 4 ft. in diameter, the mean values of f are For new pipes 0.00526

For incrusted pipes 0.01052.

Using these values we get the very simple expressions v553I-%/ (di) for new pipes (4a)

=39II-~ (di) for incrusted pipes .1

Within the limits stated, these are accurate enough for practical purposes, especially as the precise value of the coefficient i cannot be known for each special case.

Problem 2. Given the diameter of a pipe and the velocity of flow, to find the virtual slope and discharge. The discharge is given by (3); the proper value of I by (I); and the virtual slope by (2). This also presents no special difficulty.

Problem 3. Given the diameter of the pipe and the discharge, to find the virtual slope and velocity. Find v from (3); l from (I); lastly i from (2). If we combine (I) and (2) we get = i(v/2g) (4/d) =2a{ + I/I2d~v1/gd; (5)

and, taking the mean values of I for pipes from I to 4 ft. diameter, given above, the approximate formulae are i=0.0003268 v1/d for new pipes j. (5a)

0.0006536 v/d for incrusted pipes Problem 4. Given the virtual slope and the velocity, to find the diameter of the pipe and the discharge. The diameter is obtained from equations (2) and (i), which give the quadratic expression d(2avu/gi) avf/6gi

d =avi/gi+-~I ~(av2/gi) (av1/gi+f/6)~. (6),

For practical purposes, the approximate equations d2av2/gi+f/I2. (6a)

0.00031 v/i+083 for new pipes

Problem 5. Given the virtual slope and the discharge, to find thE diameter of the pipe and velocity of flow. This case, which ofter occurs in designing, is the one which is least easy of direct solution From equations (2) and (~) we get di =32~Q1/g7r1i. (~)

If now the value of ~ in (I) is introduced, the equation becomes vr3

cumbrous. Various approximate methods of meeting the difficult)

may be used.

(a) Taking the mean values of i given above for pipes of I to ft. diameter we get d=V (32i~g7r1)li (Q2/i), (8)

022I61~f (Q/i) for new pipes

(b) A second method is to obtain a rough value ofdby assuming ~=a. This value is = V (32Q2/gir1i)V a ~o63 19 1~/ (Q/i)V a, Then a very approximate value of i is i=a(I+I/f2d);

and a revised value of d, not sensibly differing from the exact value, is d = ~J (32Q, giri)V I ~0063I9 ~J (Q/i)V ~.

(c) Equation 7 may be put in the form d=~i (32aQ/gir1)~J (I +1112d). (9) ~ ~ -.-Engels_--- Expanding the term in brackets, fi~E4~ -. - -

~ (I +I!I2d) 1 I/6odI/I800di... 2Q,s,.Neglectingthe terms after the second, d=41(32a!glr2)V (Q1/i).~I+I/6cdl V (32a/glI-2)V (Q/i) +o.o1667;(9a) and V (32a,lgiri) =0.219 for new pipes = 0.2 52 for incrusted pipes.

85. Arrangement of Water Mains for Towns .Supply.Town mains are ---- ~ usually suppliedby gravitation from a service reservoir, which in turn is supplied by gravitation from a storage reservoir or by pumping from a lower level. The service reservoir should contain three days supply or in important cases much more. Its elevation should he such that water is delivered at a pressure of at least about 100 ft. to the highest parts of the district. The greatest pressure in the mi~ins is usually about 200 ft., the pressure for which ordinary pifles and fittings are designed. Hence if the district supplied has _.~~:~Lc~i.Zoes__. -~ -- - -

iII1i~t~TT

FIG. 96.

great variations of level it must be divided into zones of higher and lower pressure. Fig. 96 shows a district of two zones each with its seriice reservoir and a range of pressure in the lower district from 100 to 200 ft. The total supply required is in England about 25 gallons per head per day. But in many towns, and especially in America, the supply is considerably greater, but also in many cases ~-~ 1~

_~- ~ -~L::~_

Fm. 97.

a good deal of the supply is lost by leakage of the mains. The supply through the branch mains of a distributing system is calculated froni the, population supplied. But in determining the capacity of th marns the fluctuation of the demand must be allowed for. It is usual to take the maximum demand at twice the average demand. Hencc if the average demand is 25 gallons per head per day, the mairs should be calculated for 50 gallons per head per day.

86. Determination of the Diameters of Different Parts of a Water Main.When the plan of the arrangement of mains is determined upon, and the supply to each locality and the pressure required is ascertained, it remains to determine the diameters of the pipes. Let fig. 97 show an elevation of a main ABCD..., R being the reservoir from which the supply is derived. Let NN be the datum line of the levelling operations, and H0, Hi.. - the heights of the main above the datum line, Hr being the height of the water surface in the FIG. 98.

reservoir from the same datum. Set up next heights AA1.. BB1, representing the minimum pressure height necessary for the adequate supply of each locality. Then AiB1C1Di... is a line which should form a lower limit to the line of virtual slope. Then if heights t)0, I)s, l)c... are taken representing the actual losses of head in each length Ia, lb ia... of the main, A,B0C0 will be the line of virtual slope, and it will be obvious at what points such as D0 and E0, the pressure is deficient, and a different choice of diameter of main is required. For any point z in the length of the main, we have Pressure height HrHz(ti~,+lji+.. .~).

Where no other circumstance limits the loss of head to be assigned to a given length of main, a consideration of the safety of the main from fracture by hydraulic shock leads to a limitation of the velocity of flow. Generally the velocity in water mains lies between 13/4 and 41/8 ft. per second. Occasionally the velocity in pipes reaches 10 ft. per second, and in hydraulic machinery working under enormous pressures even 20 ft. per second. Usually the velocity diminishes along the main as the discharge diminishes, so as to reduce somewhat the total loss of head which is liable to render the pressure insufficient at the end of the main.

J. T. Fanning gives the following velocities as suitable in pipes for towns supply Diameter in inches -.. 4 8 12 18 24 30 36

Velocity in feet per sec.. - 2.5 3.0 3.5 4.5 5.3 62 7.0

87. Branched Pipe connecting Reservoirs at Different Levels.Let A, B, C (fig. 98) be three reservoirs connected by the arrangement of pipes showni1, di, Qi, v1 if, d,, Qi, vf; if, df, Qi, v3 being the length, diameter, discharge and velocity in the three portions of the main pipe. Suppose the dimensions and positions of the pipes known and the discharges required.

If a pressure column is introduced at X, the water will rise to a height XR, meastiring the pressure at X, and aR, Rb, Rc will be the lines of virtual slope. If the free surface level at R is above b, the reservoir A supplies B and C, and if. ~ - R is below b, A and B supply C.

Consequently there are three cases:

I. Raboveb;Q1=Qi+Q1.

II. R level with b; Q1=Q3; Qi=o III. R below b; Qi+Qz=Qi.

To determine which case has to be dealt with in the given conditions, g suppose the pipe from X to B closed .D, _h..~ by a sluice. Then there is a simple ~ ~, main, and the height of free surface Do ~ h at X can be determined. For this - ,,-. I condition ~ ::;~~ =

32iQfli/gfrfdi5

Hence (li h)/(h.h~) i1d31/lidi5,

from which 11 is easily obtained. If then h is greater than hb. opening the sluice between X and B will allow flow towards B, and the case in hand is case I. If it is less than Its, opening the sluice will allow flow from B, and the case is case III. If h=h~, the case is case II., and is already completely solved.

The true value of h must lie between /r~ and Its. Choose a new value of h, and recalculate Q, Qf, Qi. Then if Qt>Qf+Qs in case I.,

or Qi+Qi>Qi in case Ill.,

the value chosen for Ii is too small, and a new value must be chosen. If Qi

or Qi+Q2

the value of h is too great.

Since the limits between which h can vary are in practical cases not very distant, it is easy to approximate to values sufficiently accurate.

88. Water Hammerif in a pipe through which water is flowing a sluice is suddenly closed so as to arrest the forward movement of the water, there is a rise of pressure which iii some cases is serious enough to burst the pipe. This action is termed water hammer or water ram. The fluctuation of pressure is an oscillating one antI gradually dies ut. Care is usually taken that sluices should only be closed gradually and then the effect is inappreciable. Very careful experiments on water hammer were made by N. J. Joukowsky at Moscow in 1898 (Stoss in Wasserleitungen, St Petersburg, 1900), and the results are generally confirmed by experiments made by E. B. \Veston and R. C. Carpenter in America. Joukowsky used pipes, 2, 4 and 6 in. diameter, from 1000 to 2500 ft. in length. The sluice closed in 0-03 second, and the fluctuations of pressure were automatically registered. The maximum excess presstire due to waterhammer action was as follows:

Pipe 4-in, diameter. Pipe 6-in, diameter.

Velocity Excess Pressure. Velocity Excess Pressure.

ft. per sec. lb per sq. in. ft. per sec. lb per sq. in.

05 31 o-6 43

2-9 i68 3-0 173

4-1 232 I 5-6 369

LEngels_ ~ L_~_~6_~

In sonic cases, in fixing the thickness of water mains, ioo lb per sq. in. excess pressure is allowed to cover the effect of water hammer. With the velocities usual in water mains, especially as no valves can be quite suddenly closed, this appears to be a reasonable allowance (see also Carpenter, Am. Soc. Meek. Eng., 1893).

IX. FLOW OF COMPRESSIBLE FLU1DS IN PIPES

89. Flow of Air in Long PipesWhen air flows through a long pipe, by far the greater part of the work expended is used in overcoining frictional resistances due to the surface of the pipe. The work expended in friction generates heat, which for the most part must be developed in and given back to the air. Some heat may be transmitted through the sides of the pipe to surrounding materials, but in experiments hitherto made the amount so conducted away appears to be ~-ery small, and if no heat is transmitted the air in the tube must remain sensibly at the same temperature during expansion. ln other words, the expansion may be regarded as isothermal expansion, the hear generated by friction exactly neutralizing the cooling due to the work done. Experiments on the pneumatic tubes used for the transmission of messages. by R. S. Culley and R. Sabine (Proc. Inst. Civ. Eng. xliii.), show that the change of temperature of the air flowing along the tube is much less than it would be in adiabatic expansion.

90. Diffe~ential Equation of the Steady Motion of Air Flowing in a Long Pipe of Uniform SectionWhen air expands at a constant absolute temperature r, the relation between the pressure ~ in pounds per square fool and the density or weight per cubic foot G is given by the equation p/G=cr, (1)

where c = 53-15. Taking r = 521, corresponding to a temperature of 6o Fahr.,

Cr = 27690 foot-pounds. (2)

The equation of continuity, which expresses the condition that in steady motion the same weight of fluid, W, must pass through each cross section of the stream in I, the unit of time, is C!~u=W=constant, (3)

~.. L.._gj ~ where If is the section of the pipe and u the velocity of the air. Combining (1) and (3),

i ffup/\V=ci=constant. (3a)

As A0 ~-X ~ Since the work done by Flu. 99 gravity on the air during its flow throtigh a pipe due to variations of its level is generally small compared with the work done by changes of pressure, the former may in many cases by neglected.

Consider a short length dl of thy pipe limited by sections Ai, A1 at a distance dl (fig. 99). Let p, u be the pressure and velocity at Ai, p+dp and u+du those at Ai. Further, suppose that in a very short time di the mass of air between AOAI comes to A0A1 so that AOAf = udi and AiAf = (u+du)dti. Let If be the section, and m the hydraulic mean radius of the pipe, and W the weight of air flowing through the pipe per second.

From the steadiness of the motion the weight of air between the sections A,A0, and A1Ai is the same. That is, Wdt = GIfudt GIf(u+du)dt.

By analogy with liquids the head lust in friction is, for the length dl (see 72, eq. 3), &(uf/2g)(dl/m). Let I-I =U2/2g. Then the head lost is i(H/m)dl; and, since Wdt lb of air flow through the pipe in the time considered, the work expended in friction is i(H/m)Wdl di. The change of kinetic energy in di seconds is the difference of the kinetic energy of A5Af and A1Ai, that is, (W/g)dt~ (u +du) u5}/2 (Wig) u du di = WdHdi.

The work of ~xpansion when Ifudt cub. ft. of air at a pressure p expand to If(u+dis)dt cub. ft. is Ifpdudt. But from (3a) u=crW/Ifp, and therefore du/dp = crW/12p.

And the work doiie by expansion is (crW/p)dp dl.

The work done by gravity on the mass between Ai and Ai is zero if the pipe is horizontal, and may in other cases be neglected without great error. The work of the pressures at the sections A0A1 is pIfudt (p+dp)If(u+du)dt = (pdu+udp)Ifdt But from (3a)

pu = constant, pdu+udp=o, and the work of the pressures is zero. Adding together the quantities of work, and equating them to the change of kinetic energy, WdHdt = (crW/p)dpdt~(H/m)Wdldi dH + (c-z-/p)dp+iiH/m)dl u, dH/H + (cr/H p)dp +(dl/m = o (4)

But u=cr\V/Ifp, and H =uf/2g=c~T2Vs2/2gt22p5,

.. dH/H + (2gtfip/crW2)dp +1d1/m = o. (4a)

For tubes of uniform section m is constant; for steady motion \V is constant; and for isothermal expansion r is constant. Integrating, log H + gIff pIW2cr +~l/m = constant; (5)

for l=o, let H=H0, and P=po; and fur 1=1, let H=Hi, and P=pi.

log (Hi/IL) I (gIff/Wfcr) (Pu Pu5) i1/ni o. (5a)

where ~o is the greater pressure and pi the less, and the flow is from Af towards A1.

By replacing W amid H,

log(po/pu) + (gcr/uop02) (Puf Psi) + li/rn = o. (6) Hence the initial velocity in the pipe is no = ~! ~gcr(poi Pi2)~l~pof(li/ni +log(pofpu)~ 1. (7)

When 1 is great, log Pu/Pt is comparatively small, and then u~=~I ~(gcrm/~)~(po1_pji)/po2}], (7a)

a very simple and easily used expression. For pipes of circtilar section m=d/4, where d is the diameter: no ~ 2Pi)/Pofij; (7b)

or approximately Us = (1.1319 0.72 64p, /po) ~! (gcrd,!4~) - (7c)

91. Coefficient of Friction for Air.A discussion by Professor Unwin of the experiments by Culley and Sahine on the rate of transmission of light carriers through pneumatic tubes, in which there is steady flow of air not sensibly affected by any resistances other than surface friction, furnished the value l= 007. The pipes were lead pipes, slightly moist, 21/2 in. (of87 ft.) in diameter, and in lengths of 2000 to nearly 6000 ft.

In some experiments on the flow of air through cast-iron pipes A. Arson found the coefficient of friction to vary with the velocity and diameter of the pipe. Putting he obtained the following values Diameter of Pipe in feet. a per second.

1.64 00129 00483.00484 -

I07 00972 00640 00650

83 u1525 00704 007i9

338 03604 00941 00977

266 03790 00959 00997

164 04518 01167 -01212

It is worth while to try if these numbers can be expressed in the form proposed by Darcy for water. For a velocity of ioo ft. per second, and without much erior for higher velocities, these number5 agree fairly with the formula ~0005(I+3/I0d), f,9)

which only differs from Darcys value for water in that the second term, which is always small except for very small pipes, is larger.

Some later experiments on a very large scale, by E. Stockalper at the St Gotthard Tunnel, agree better with the value i=O.0028(I +3/fod).

These pipes were probably less rough than Arsons.

When the variation of pressure is very small, it is no longer safe to neglect the variation of level of the pipe. For that case we may neglect the work done by expansion, and then Icli p0/G0 puG1 ~v1/2g) (i/rn) = 0, (10)

precisely equivalent to the equation for the flow of water, 10 and Zf being the elevations of the two ends of the pipe above any datum, Ps and pi the pressures, G0 and Gf the densities, and v the mean velocity in the pipe. This equation may be used for the flow of coal gas.

92. Distribution of Pressure in a Pipe in which Air is Flowing. From equation (7a) it results that the pressure p, at 1 ft. from that end of the pipe where the pressure is Ps, is ppo..flI~luo2/mgcr,~ (11)

which is of the form p=~ (al+b)

for any given pipe with given end pressures. The curve of free surface level for the pipe is, therefore, a parabola with horizontal axis. Fig. 100 shows calculated curves of pressure for two of Sabines experiments, in one of which the pressure was greater than atmo k- 4

0 21i3SFt -kff7Ft. 6340-JF4.. s~s~Ft.

FIG. 100.

spheric pressure, and in the other less than atmospheric pressure. The observed pressures are given in brackets and the calculated pressures without brackets. The pipe was the pneumatic tube between Fenchurch Street and the Central Station, 2818 yds. in length. The pressures are given in inches of mercury.

Variation of Velocity in the Pspe.Let ~o, hi be the pressure and velocity at a given section ot the pipe; p, u, the pressure and velocity at any other section. From equation (3a)

up = crW/t2 = constant; so that, for any given uniform pipe, up = u0po, u=uops/P; (12)

which gives the velocity at any section in terms of the pressure, which has already been determined. Fig. 101 gives the velocity f11j.51t. 42 27 Fi 634-o3F~. ~54F~

FIG. 101.

curves for the two experiments of Culley and Sabine, for which the pressure curves have already been drawn, it will be seen that the velocity increases considerably towards that end of the pipe where the pressure is least.

93- Weight of Air Flowing per Second The weight of air discharged per second is (equation 3a) ~uo ps/cr.

From equation (7b), for a pipe of circular section and diameter d = 3/4irl ~gdi(ps1pi) ftlcrl, =-611-.iIdi(pf-puf)ftlr}., (13) Approximately W (.6916P5 4438Pu) (di/Ilr)i. (13a) 94. A pplication to the Case cf Pneumatic Tubes for the Transmission of Messages.In Paris, Berlin, London,,and other towns, it has been found cheaper to transmit messages in pneumatic tubes than to telegraph by electricity. The tubes are laid underground with easy curves; the messages are made into a roll and placed in a light felt carrier, the resistance of which in the tubes in Loidon is only 3/4 oz. A current of air forced into the tube or drawn, through it propels the carrier. In most systems the current of air is steady and continuous, and the carriers are introduced or removed without materially altering the flow of air.

Time of Transit through the Tube.Putting t for the time of transit from o to 1,

t = ~u, From (4a) neglecting dl-I/Fl, and putting m =d/4, dl = gdt22pdp/2~\V2cr.

From (1) and (3)

u = Wcr/ptf dl/u = gdt23p5dp/2Vv3c1r1

t =J ~

= gdt~3(p01_puf)/6i~iVfc1r1. (14)

But W=pouot2/cr; ... I =gdcr(po3puf)/6lpo3uoi, = ~-ili(po3pi3)/6(gcrd)i(pspu1)I. (Is)

If r=52f, corresponding to 60 F.,

t .ooI4I2i.ili(po3_pif)fdi(pos_~pif)i (f5a)

which gives the time of transmission in terms of the initial and final pressures arid the dimensions of the tube.

Mean Velocity of Transmission.The mean velocity is l/t; or, for r521,

Ume~s o.7o8~/ ~d(p02 pu1)iftl(~o~ P11) ~. (16) The following table gives some results: Abcolute Pressures in Mean \ elocities for Tubes of a.

tb per sq. in. length in feet.

Ps Pi f000 2000 3000 4000 5000

Vacuum S. - 15 5 994 703 57.4 49.7 44.5

Working - - 15 10 67.2 47.5 38~8 344 30I

Pressure S - - 20 55 57.2 40.5 33O 286 256

Workin ~ - - 25 15 746 527 43.1 373 33.3

g 30 15 84.7 6oo 49O 42.4 379

Limiting Velocity in the Pipe when the Pressure at one End is diminished indefinitely-If in the last equation there be put Pm =0, then UmeinO708.I (d/tl);

where the velocity is independent of the pressure Ps at the other end, a result which apparently must be absurd. Probably for long pipes, as for orifices, there is a limit to the ratio of the initial and terminal pressures for which the formula is applicable.

X. FLOW IN RIVERS AND CANALS

~5. Flow of Water in Open Canals and Rivers.When water flows in a pipe the section at any point is determined by the form of the boundary. When it flows in an open channel with free upper surface, the section depends on the velocity due to the dynamical conditions.

Suppose water admitted to an tinfilled canal. The channel will gradually fill, the section and velocity at each point gradually changing. But if the inflow to the canal at its head is constant, the increase of cross section and diminution of velocity at each point attain after a time a limit. Thenceforward the section and velocity at each point are constant, and the motion is steady, or permanent regime is established.

If when the motion is steady the sections of the stream are all equal, the motion is uniform. By hypothesis, the inflow t~v is constant for all sections, and tl is constant; therefore v must be constant also from section to section. The case is then one of uniform steady motion. In most artificial channels the form of section is constant, and the bed has a uniform slope. In that case the motion is uniform, the depth is constant, and the stream surface is parallel to the bed. If when steady motion is established the sections are unequal, the motion is steady motion with varying velocity from section te section. Ordinary rivers are in this condition, especially where the flow is modified by weirs or obstructions. Short unobstructed lengths of a river may be treated as of uniform section without great error, the mean section in the length being put for the actual sections, In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for th flow of streams rest on a hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a plane layer abab (fig. 102) between sections normal to the direction of motion is treated as sliding down the channel to aabb without deformation. The component of the weight parallel to the channel bed balances the friction against the channel, and in estimating the friction the velocity of rubbing is taken to be the mean velocity of the stream. In actual streams, however, the velocity of rubbing on which the friction depends is not the mean V 0.3 o~5 07 I 11/2

~ 0.01215 0-01025 0.00944 0.00883 0.00836

velocity of the stream, and is not in any simple relation with it, for channels of different forms. The a 6 theory is therefore obviously based on an imperfect hypothesis. How- / / a ever, by taking variable values for -~1.~~i /~ F the coefficient of friction, the errors -~aar2h~~ /~ great extent neutralized, and they -az~ may be used without leading to FIG 102 practical errors. Formulae have been obtained based on less restricted hypotheses, but at present they are not practically so reliable, and are more complicated than the formulae obtained in the manner described above.

96. Steady Flow of Water with Uniform Velocity in Channels of Constant SectionLet aa, bb (fig. 103) be two cross sections normal to the direction of motion at a distance dl. Since the mass aabb moves uniformly, the external forces acting on it are in equilibrium. Let t~ be the area of the cross sections, x the wetted perimeter, FIG. 103.

pq+qr+rs, of a section. Then the quantity m=t~/~ is termed the hydraulic mean depth of the section. Let v be the mean velocity of the stream, which is taken as the common velocity of all the particles, i, the slope or fall of the stream in feet, per foot, being the ratio bc/ab.

The external forces acting on aabb parallel to the direction of motion are three:(a) The pressures on aa and bb, which are equal and opposite since the sections are equal and similar, and the mean pressures on each are the same. (b) The component of the weight \V of the mass in the direction of motion, acting at its centre of gravity g. The weight of the mass aabb is Gt~dl, and the component of the weight in the direction of motion is Gfldl X the cosine of the angle between \Vg and ab, that is, Gudl cos abc=Gtklt bc/ab= Gt~idl. (c) There is the friction of the stream on the sides and bottom of the channel. This is proportional to the area xdl of rubbing surface and to a function of the velocity which may be written f(v); f(v) being the friction per sq. ft. at a velocity v. Hence the friction is xdlf(v). Equating the sum of the forces to zero, c;t~i dlxdlf(v) o, f(v)/G~i/xmi. (1)

But it has been already shown (f 66) thatf(v) = IGIP/2g, .~. fv/2g=mi. (2)

This may be put in the form v = / (2g~-.,((mi) c~,/ (nii); (2a)

where c is a coefficient depending on the roughness and form of the channel. -

The coefficient of friction i varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity. It must also be made to depend on the absolute dimensions of the section, to eliminate the error of neglecting the variations of velocity in the cross section. A common mean value assumed for l~ is 0-00757. The range of values will be discussed presently. -

It is often convenient to estimate the fall of the stream in feet per mile, instead of in feet per foot. 1ff is the fall in feet per mile, f=528o i.

Putting this and the above value of l in (2a), we get the very simple and long-known approximate formula for the mean velocity of a stream v=fl~(2mf)., (3)

The flow down the stream per second, or discharge of the stream, is Q=v=&1c-%I (mi). (4)

97. Coefficient of Friction for O~en ChannelsVsrious expressions have been proposed icr tOe cuelflcient of friction icr channels as ~,i pipes. Weisbach, giving attention chiefly to the variation of the coefficient of friction with the velocity, proposed an expression of the form l~a(I+j3/v), (5)

and from 255 experiments obtained for the constants the values a0-o074o9; d=o.f920.

This gives the following values at different velocities:

2357 10 15

ooo812 0.90788 0-00769 000761 0.00755 0.00750

In using this value of l when 1 is not known, it is best to proceed by approximation.

98. Darcy and Bazins Expression for the Coefficient of Friction. Darcy and Bazins researches have shown that i~ varies very greatly for different degrees of roughness of the channel bed, and that it also varies with the dimensions of the channel. They give for i an empirical expression (similar to that for pipes) of the form l~a(I+f3/m); (6)

where m is the hydraulic mean depth. For different kinds of channels they give the following values of the coefficient of friction:

Kind of Channel. a I. Very smooth channels, sides of smooth cement or planed timber ..... 0.00294 OIO

II. Smooth channels, sides of ashlar, brick work, planks 0.00373 0.23

III. Rough channels, sides of rubble masonry or pitched with stone 0.00471 o~82

IV. Very rough canals in earth 0.00549 4.10

V. Torrential streams encumbered with detritus1 0.00785 574

- The last values (Class V.) are not Darcy and Bazins, but are taken from experiments by Ganguillet and Kutter on Swiss streams.

The following table very much facilitates the calculation of the mean velocity and discharge of channels, when Darcy and Bazins value of the coefficient of friction is used. Taking the general formula for the mean velocity already given in equation (2a) above, v=c-.J(mi),

where c=-~ (2gft), the following table gives values of c for channels of different degrees of roughness, and for such values of the hydraulic mean depths as are likely to occur in practical calculations: Values of c in v =c/ (mi), deduced from Darcy and Bazins Values.

- ..,~ .2 - - ,.a. .d~ f~. .~

~ l.~ ~ .~ ~ ~l 5-~ ~ .L~ ~

~II .se 5~ ~ ~ ~i EL) 5c 5~ L)~ ~~n ~.)h u~ .~.E 1.~.o fi~ L)h L)~ .a.E u~a >~-2 -~5 .a~ ~.2 .t~ Ea i~2 .a~ ..f~ iLl .~ .0 3 5 -~ ~ ~n-0 oa o.

i., a cc ~ ~ C C ~-5 ff g -~

> .a ~-

25 125 95 57 26 18.5 8-5 147 130 112 89 - -

5 135 110 72 36 25.6 9.0 147 130 112 90 71

75 139 116 81 42 30.8 9-5 147 130 112 90 - -

1.0 141 119 87 48 34-9 10-0 147 130 112 91 72

1.5 143 122 94 56 41.2 II 147 130 113 92 -

2-0 144 124 98 62 46O 12 147 130 113 93 74

2.5 145 126 101 67 - - 13 147 130 113 94. -

3.0 145 126 104 70 53 14 147 130 113 95 - -

3.5 146 127 105 73 - - 15 147 130 114 96 77

4.0 146 128 106 76 58 i6 147 130 114 97

45 146 f28 107 78 - - 17 147 130 114 97 -

5-0 146 128 108 8o 62 i8 147 130 114 98 - -

55 146 129 109 82 - - 20 147 131 ff4 98 80

6-o 147 129 ff0 84 65 25 148 1~1 115 100 - -

6-5 147 129 110 85 - - 30 148 131 115 102 83

7-0 147 129 110 86 67 40 148 131 116 103 85

7-5 147 129 II, 87 - 50 148 131 116 104 86

8-o 147 130 III 88 69 ~ 148 131 117 io8 91

99. Ganguillet and Kutters Modified Darcy Formuia.Starting from the general expression v=c-~jmi, Ganguillet and Kutter examined the variations of c for a wider variety of cases than those discussed by Darcy and Bazin. Darcy and Bazins experiments were confined to channels of moderate section, and ~o a 1imited variation of slope. Ganguillet and Kutter brcugnt into the discussion two very distinct and imtortant additional series of results. The gaugings of the Mississippi by A. A. Humphreys and H. L. Abbot afford data of discharge for the ease of a stream of exceoonally large section ano or very iow slope. ,Jn tOe otner nand, their own measurements of the flow in the regulated channels of some Swiss torrents gave data for cases in which the inclination and roughness of the channels were exceptionally great. Darcy and Bazins experiments alone were conclusive as to the dependence of the coefficient c on the dimensions of the channel and on its roughness of surface. Plotting values of c f or channels of different inclination appeared to indicate that it also depended on the slope of the stream. Taking the Mississippi data only, they found C~256 for an inclination 010-0034 per thousand, =I54 ,, ,, O02

so that for very low inclinations no constant value of c independent of the slope would furnish good values of the discharge. in small rivers, on the other hand, the valties of c vary little with the slope. As regards the influence of roughness of the sides of the channel a different law holds. For very small channels differences of roughness have a great influence on the discharge, but for very large channels different degrees of roughness have but little influence, and for indefinitely large channels the influence of different degrees of roughness must be assumed to vanish. The coefficients given by Darcy and Bazin are different for each of the classes of channels of different roughness, even when the dimensions of the channel are infinite. But, as it is much moreprobable that the influence of the nature of the sides diminishes indefinitely as the channel is larger, this must be regarded as a defect in their formula.

Comparing their own measurements in torrential streams in Switzerland with those of Darcy and Bazin, Ganguillet and Kutter found that the four classes of coefficients proposed by Darcy and Bazin were insufficient to cover all cases. Some of the Swiss streams gave results which showed that the roughness of the bed was markedly greater than in any of the channels tried by the French engineers. It was necessary therefore in adopting the plan of arranging the different channels in classes of approximately similar roughness to increase the number of classes. Especially an additional class was required for channels obstructed by detritus.

To obtain a new expression for the coefficient in the formula v=~J(2gft)V(mi)=c~J(mi),

Ganguillet and Kutter proceeded in a purely empirical way. They found that an expression of the form c=a/(I+~hJm)

could be made to fit the experiments somewhat better than Darcys expression. Inverting this, we get I/Crn I/a+iS/a/fli, an equation to a straight line having 1/~ m for abscissa, i/c for ordinate, and inclined to the axis of abscissae at an angle the tangent of which is a/a.

Plotting the experimental values of I/c and i/~!m, the points so found indicated a curved rather than a straight line, so that jI must depend on e. After much comparison the following form was arrived at c = (A+l/n)/(i +AnIV rn)

where n is a coefficient depending only on the roughness o~ the sides of the channel, and A and / are new coefficients, the value of which remains to be determined. From what has been already stated, the coefficient c depends on the inclination of the stream, decreasing as the slope i increases.

Let A=a+p/i.

Then c = (a+l/n +pIi)/k + (a+p/i)n/.I ml, the form of the expression for c ultimately adopted by Ganguillet and Kutter.

For the constants a, 1, p Ganguillet and Kutter obtain the values 23, 1 and 0.00155 for metrical measures, or 41.6, 1.811 and 000281 for English feet. The coefficient of roughness n is found to vary from 0-008 to 0.050 for either metrical or English measures.

The most practically useful values of the coefficient of roughness n are given in the following table:

Nature of Sides of Channel. Coefficient of Roughness n.

Well-planed timber o-oo9

Cement plaster 0~0Io Plaster of cement with one-third sand. - OoII

Unplaned planks Oo12

Ashlar and brickwork 0.013

Canvas on frames 0.015

Rubble masonry 0.017

Canals in very firm gravel o02o Rivers and canals in perfect order, free from stones 0-02 or weeds Rivers and canals in moderately good order, not 0.0 0 quite free from stones and weeds. -. - S 3

Rivers and canals in bad order, with weeds and detritus 0-035

Torrential streams encumbered with detritus - - 0.050

Ganguillet and Kutters formula is so cumbrous that it is difficult to use without the aid of tables.

Lowis DA. Jackson published complete and extensive tables foi facilitating the use of the Ganguillet and Kutter formula (Canai and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form: M =n(4r6+o.oos8i/i);

v=(~m/n)((M-f-f8II)/(M+~m)}~ (rn-i).

The following table gives a selection of values of M, taken from Jacksons tables: Values of M for n OOIO 0-012 0-015 0-017 O020 0.025 0.030

-00001 3.2260 3-8712 4-8390 5-4842 6-4520 8.0650 9.6780

00002 I8210 2-1852 273f5 3.0957 3.6420 4-5525 5.4630

00004 1.1185 1-3422 I6777 19014 2.2370 2.7962 3.3555

00006 0.8843 1-0612 I.3264 1.5033 1.7686 2.2107 2-6529

00008 0.7672 O9206 1.1508 1-3042 1.5344 1.9180 2.3016

00010 0.6970 0-8364 1-0455 fi849 1.3940 1.7425 2-0910

00025 0.5284 0-6341 0.7926 0-8983 1.0568 1.3210 1.5852

.00050 0.4722 0.5666 0-7083 0.8027 0-9444 118o5 1.4166

.00075 0-4535 0-5442 0-6802 0.7709 0.9070 f.f337 1.3605

00100 0-4441 0-5329 o6661 o7550 0-8882 I~II02 f3323

-00200 0-4300 o5I6o 0.6450 0.7310 0-8600 1.0750 1.2900

00300 O4254 0.5105 0-6381 0-7232 0.8508 1.0635 1.2762

A difficulty in the use of this formula is the selection of the coefficient of roughness. The difficulty is one which no theory will overcome, because no absolute measure of the roughness of stream beds is possible. For channels lined with timber or masonry the difficulty is not so great. The constants in that case are few and sufficiently defined. But in the case of ordinary canals and rivers the case is different, the coefficients having a much greater range. For artificial canals in rammed earth or gravel n varies from 00163 to 0.0301. For natural channels or rivers ii varies from 0020 to 0.035. In Jacksons opinion even Kutters numerous classes of channels seem inadequately graduated, and he proposes for artificial canals the following classification :

I. Canals in very firm gravel, in perfect order ii =o~02

II. Canals in earth, above the average in order n = oo225

III. Canals in earth, in fair order. .. - 11=0.025

- IV. Canals in earth, below the average in order n o0275

V. Canals in earth, in rather bad order, partially overgrown with weeds and obstructed by ~ n = 0.03 detritus. .

Ganguillet and Kutters formula has been considerably used partly from its adoption in calculating tables for irrigation work in India. But it is an empirical formula of an unsatisfactory form. Some engineers apparently have assumed that because it is complicated it must be more accurate than simpler formulae. Comparison with the results of gaugings shows that this is not the case. The term involving the slope was introduced to secure agreement with some early experiments on the Mississippi, and there is strong reason for doubting the accuracy of these results.

100. Bazins New Forrnula.Bazin subsequently re-examined all the trustworthy gaugings of flow in channels and proposed a modification of the original Darcy formula which appears to be more satisfactory than any hitherto suggested (Etude dune nouvelle formule, Paris, 1898). He points Out that Darcys original formula, which is of the form mi/v2 a+13/m, does not agree with experiments on channels as well as with experiments on pipes. It is an objection to it that if m increases indefinitely the limit towards which mi/v2 tends is different for different values of the roughness. It would seem that if the dimensions of a canal are indefinitely increased the variation of resistance due to differing roughness should vanish. This objection is met if it is assumed that -~ (mi/c2) =a+~J~/m, so that if a is a constant mi/v1 tends to the limit a when m increases. A very careful discussion of the results of gaugings shows that they can be expressed more satisfactorily by this new formula than by Ganguillet and Flutters. Putting the equation in the form ~v2/2g= mi, &oo02594(I+7R1rn), where ~ has the following values:

I. Very smooth sides, cement, planed plank, y = 0. Io9

11. Smooth sides, planks, brickwork. - - - 0.290

III. Rubble masonry sides 0.833

IV. Sides of very smooth earth, or pitching - - 1.539

V. Canals in earth in ordinary condition - - - 2353

VI. Canals in earth exceptionally rough. -. 3~f 68

101. The Vertical Velocity Curve.lf at each point along a vertical representing the depth of a stream, the velocity at that point is plotted horizontally, the curve obtained is the vertical velocity curve and it has been shown by many observations that it approximates to a parabola with horizontal axis. The vertex of the parabola is at the level of the greatest velocity. Thus in fig. 104 OA is the vertical at which velocities are observed; v,, is the surface; v~ the maximum and Cd the bottom velocity. B C D is the vertical velocity curve which corresponds with a parabola having its vertex at C. The mean velocity at the vertical is 1~,,, = 3/412v,+vd+ (djd) (v, Vd)1.

The Horizontal Velocity Curve.Similarly if at each point along a horizontal representing the width of the stream the velocities ar~

plotted, a curve is obtained called the horizontal velocity curve. In streams of symmetrical section this is a curve symmetrical about the centre line of the stream. The velocity varies little near the centre of the stream, but very rapidly near the banks. In un~q symmetrical sections the greatest velocity is at the point where the I stream is deepest, and the general form of the horizontal velocity curve - ~ is roughly similar to the section of / the stream.

/ 102. Curves or Contours of Equal / Velocity.If velocities are observed d / at a number of points at different I / widths and depths in a stream, it is I / possible to draw curves on the cross ~ _____________ section through points at which the velocity is the same. These represent contours of a solid, the volume FIG. I 04. of which is the discharge of the stream per second. Fig. 105 shows the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazins gaugings.

103. Experimental Observations on the Vertical Velocity Curve. A preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. It may be inferred that ..i ..:b S;..

Vertical Velocity Horizontal Velocity Ct,rves ~rticaI Velocity ~~I; ~

cfgh ef g h i j k tm1 10

Contours of E9ual Velocity FIG. 105.

though the velocity at a point fluctuates about a mean value, the fluctuations being due to eddying motions superposed on the general motion of the stream, yet these fluctuations produce effects which disappear in the mean of a series of observations and, in calculating the volume of flow, may be disregarded.

In the next place it is found that in most of the best observations on the velocity in streams, the greatest velocity at any vertical is found not at the surface but at some distance below it. In various river gaugings the depth d~ at the centre of the stream has been found to vary from 0 to o3d.

104. Influence of the WindIn the experiments on the Mississippi the vertical velocity curve in calm weather was found to agree fairly with a parabola, the greatest velocity being at -i5sths of the depth of the stream from the surface. With a wind blowing down stream the surface velocity is increased, and the axis of the parabola approaches the surface. On the contrary, with a wind blowing up stream the surface velocity is diminished, and the axis of the parabola is lowered, sometimes to half the depth of the stream. The American observers drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream like that due to the bottom and sides. If there were such a retarding action the position of the filament of maximum velocity below the surface would be explained.

It is not diffictilt to understand that a wind acting on surface ripples or waves should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the surface velocity of a stream is variable when the mean velocity of the stream is constant. Hence observations of surface velocity by floats or otherwise should only be made in very calm weather. But it is very difficult to suppose that, in still air, there is a resistance at the free surface of the stream at all analogous to that at the sides and bottom. Further, in very careful experiments, ~. P. Boileau found the maximum velocity, though raised a little above its position for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a velocity greater than that of the stream, and when the action of the air must have been an accelerating and not a retarding action. A much more probable explanation of the diminution of the velocity at and near the free surface is that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off in eddying masses and mingle with the rest of the stream. These eddying masses modify the velocity in all parts of the stream, but have their greatest influence at ,the free surface. Reaching the free surface they spread out and remain there, mingling with the water at that level and diminishing the velocity which would otherwise be found there. - -

Influence of the Wind on the Depth at which the Max-fmum Veloczty is found.In the gaugings of the Mississippi the vertical velocity curve was found to agree well with a parabola having a horizontal axis at some distance below the water surface, the ordinate of the parabola at the axis being the maximum velocity of the section. During the gaugings the force of the wind was registered on a scale ranging from 0 for a calm to 10 for a hurricane. Arranging the velocity curves in three sets(I) with the wind blowing up stream, (2) with the wind blowing down stream, (3) calm or wind blowing across streamit was found that an up-stream wind lowered, and a down-stream wind raised, the axis of the parabolic velocity curve. In calm weather the axis was at -~5ths of the total depth from the surface for all conditions of the stream.

Let h be the depth of the axis of the parabola, m the hydraulic mean depth, f the number expressing the force of the wind, which may range from+1o tob, positive if the wind is up stream, negative if it is down stream. Then Humphreys and Abbot find their results agree with the expression h/m=o.3f7 ~oo6f.

Fig. 106 shows the parabolic velocity curves according to the American observers for calm weather, and for an up- or down-stream wind of a force represented by 4.

7-4 1-2 7-C 17 7-8 7-9 8O 6.1

llth~~

FIG. 106.

It is impossible at present to give a theoretical rule for the vertical velocity curve, but in very many gaugings it has been found that a parabola with horizontal axis fits the observed results fairly well. The mean velocity on any vertical in a stream varies from 0.85 to 0-92 of the surface velocity at that vertical, and on the average if v0 is the surface and Vm the mean velocity at a vertical v,,, ~v0, a result useful in float gauging. On any vertical there is a point at which the velocity is equal to the mean velocity, and if this point were known it would be useful in gauging. Humphrcys and Abbot in the Mississippi found the mean velocity at o~66 of the depth; G. H. L. Hagen and H. Heinemann at 0.56 to 0.58 of the depth. The mean of observations by various observers gave the mean velocity at from 0.587 to o62 of the depth, the average of all being almost exactly 0-6 of the depth. The mid-depth velocity is therefore nearly equal to, but a little greater than, the mean velocity on a vertical. If V,,,d is the mid-depth velocity, then on the average v,,, = 098Vmd.

105. Mean Velocity on a Verticalfrom Two Velocity Observations.

A. J. C. Cunningham, in gaugings on the Ganges canal, found the following useful results. Let v0 be the surface, v,,, the mean, and v~s the velocity at the depth xd; then C,,, 3/4(Vo+3V2!sd)

=1/2(v,iid+v-sid)

106. Ratio of Mean to Greatest Surface Velocity, for the whole Cross Secton in Trapezoidal Channels.It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. In general in streams of fairly regtilar section the mean velocity for the whole section varies from 0.7 to 0.85 of the greatest surface velocity. For channels not widely differing from those experimented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let v, be the greatest surface velocity, v,,, the mean velocity of the stream. Then, according to Bazin, V,~Vs-254-~ (mi).

But tmsc~I(mi),

where c is a coefficient, the values of which have been already given in the table in 98. Hence vm~scvo/(c+25~4).

Values of Coefficient c/(c+254) in the Formula vm cv01(c+25.4)

Hydraulic Very Smooth Rough Very Rough Channeis Smooth Channels. Channels. channeis. encumbered Mean Depth channels. Ashlar or Rubble Canals la wifh cement. Brickwork. Masonry. Earth. Detritus.

0-25 83.79 ~69.51.42

0-5 84 81.74 58.50

0-75 84 82.76 63.55

1-0 85.. .77 65 -58

2O.. 83.79 71.64

3.0. .. 80 73 67

4,0 .... 81 75 70

5-0. .. .. - .76 71

6-0.. -84 -. 77.72

7-0. .. -.. 78 73

8-0 .. ., .. - -. -

9-0 .. .. ~82.74

10O .. ..

15-0. .. .. - 79 75

20O .. ... - 8o .76

30-0. ... ~82 .. .77

40-0 -. ... - -.. -

50-0 -. .. .. -. - -

- .. 79

107. River Bends.In rivers flowing in alluvial plains, the windings which alieady exist tend to increase in curvature by the scouring away of material from the outer bank and the deposition of detritus along the inner bank. The sinuosities sometimes increase till a loop is formed with only a narrow strip of land between the two I encroaching branches of the river. Finally a cut off may occur, I a waterway being opened through the strip of land and the loop left separated from the stream, forming a horse I, 4I ~1~8 11~~4r,1, shoe shaped lagoon or Thomson pointed out i~ (Proc. Roy. Soc., 1877,

that the usual supposi ~ ~. marsh. Professor James ~ p. 356; Proc. Inst. of ~ ~~tending to go forwards Mech. Eng., 1879, p. 456) tion is that the water in a straight line rushes against the outer bank and scours it, at the same time creating de _______ B:..i~ posits at the inner bank.

~o That view is very far ~ from a complete account of the matter, and Pro much more ingenious FIG. 107. account of the action at the bend, which he completely confirmed by experiment.

When water moves round a circular curve under the action of gravity only, it takes a motion like that in a free vortex. Its velocity is greater parallel to the axis of the stream at the inner than at the outer side of the bend. Hence the scouring at the outer side and the deposit at the inner side of the bend are not due to mere difference of velocity of flow in the general direction of the stream; but, in virtue of the centrifugal force, the water passing round the bend presses outwards, and the free surface in a radial cross section has a slope from the inner side upwards to the outer side (fig. 108). For the greater part of the water flowing in curved paths, this difference of pressure produces no tendency to transverse motion.

But the water im In.ncrBctnIc, OU.tWB&C mediately in contact - ~ torn and sides of the with the rough bot ~ ~e~~r /)i~1/2t channel is retarded ~ ~ and its centrifugal force is insufficient to ~ balance the pressure Secfion,at MN. due to the greater FIG. 108. depth at the outside of the bend. It there fore flows inwards towards the inner side of the bend, carrying with it detritus which is deposited at the inner bank. Conjointly with this flow inwards along the bottom and sides, the of water in I .01.05.10 115 1.20 125 13 I

I terms of radius -

I Hsdraulic mean depth ~ .032I .0523 .og6~ 1.12781.1574 .i85a I .214:

in terms of radius square of radius - - .ooi8~ 1.02111 .o599.1067.1651.228 1.294 1370

general mass of water must flow outwards to take its place. Fig. 107 shows the directions of flow as observed in a small artificial stream, by means of light seeds and specks of aniline dye. The lines CC show the directions of flow immediately in contact with the sides and bottom. The dotted line AB shows the direction of motion of floating particles on the surface of the stream.

108. Discharge of a River when flowing at different Depths. When frequent observations must be made on the flow of a river or canal, the depth of which varies at different times, it is very convenient to have to observe the depth only. A formula can be established giving the flow in terms of the depth. Let Q be the discharge in cubic feet per second; H the depth of the river in some straight and uniform part. Then Q=aH+bHf, where the constants a and b must be found by preliminary gaugings in different conditions of the river. M. C. Moquerey found for part of the upper Sane, Q=647H+82H2 in metric measures, or Q696H+26.8H2 in English measures.

109. Forms of Section of Chownels.The simplest form of section for channels is the semicircular or nearly semicircular channel (fig. 109), a form now often adopted from the facility with which it can be -- vii,, ,~. / ~

FIG. 109.

executed in concrete. It has the advantage that the rubbing surface is less in proportion to the area than in any other form.

Wooden channels or flumes, of which there are examples on a large scale in America, are rectangular in section, and the same form is adopted for wrought and cast-iron aqueducts. Channels built with brickwork or masonry may be also rectangular, but they are often trapezoidal, and are always so if the sides are pitched with masonry laid dry. In a trapezoidal channel, let b (fig. 110)

FIG. 110.

be the bottom breadth, b0 the top breadth, d the depth, and let the slope of the sides be n horizontal to I vertical. Then the area of section is t~=(b+nd)d~-(b0nd)d, and the wetted perimeter x=b+2d ~/(n2+I).

When a channel is simply excavated in earth it is always originally trapezoidal, though it becomes more or less rounded in course of time. The slope of the sides then depends on the stability of the earth, a slope of 2 to 1 being the one most commonly adopted.

Figs. III, 112 show the form of canals excavated in earth, the former being the section of a navigation canal and the latter the section of an irrigation canal.

110. Channels of Circular Section.The following short table facilitates calculations of the discharge with different depths of water in the channel. Let 1 be the radius of the channel section; then for a depth of water=Kr, the hydraulic mean radius is pr and the area of section of the waterway en, where ii, u, and v have- the following values:

.40.45.50.55.6o .61.70.75.8 .5~ .90.95.0

.242 .i61.293.320.343.365.387.408.429.449.466.484.500

.450.532.614.709.795.885 ~ I 1.075 1.175 1.276 1.371 1.470 1.571

III. ~gg-.~I1apet5 (/iannets or SewersIn sewers for discharging I storm Water and house drainage the volume or flow is extremely varrable; and there is a great liability for deposits to be left when the flow is small, which are not removed during the short periods I when the flow is large. The sewer in consequence becomes choked.

in,Ba~n,~k L

~--149---~8-O-~---.---- 33~0---- ~--~8-O-~--

FIG. If i.Scal~ 20 ft. = I in.

1~o.o---~----.8~oiEf 140-0----- -it as FIG. 112.Scale 80 ft. = 1 in.

To obtain uniform scouring action, the velocity of flow should be constant or nearly so; a complete uniformity of velocity cannot be obtained with any form of section suitable for sewers, but an approximation to uniform velocity ?s obtained by making the sewers of oval section. Various forms of oval have been suggested, the - simplest being one in which the radius of the ~. 4, crown is double the radius of the invert, and the greatest width is twothirds the height. The section of such a sewer -- is shown in fig. 113, the - -- numbers marked on the ,- figure being proportional ~-:--c~, +- numbers.

.~ ~-~ ,,, 1I2. Problems on ~, / Channels -in which the --. -~,a---< -- Flow is Steady and at Uniform Velocity.The / ~ /\ general equations given in f 96, 98 are - ~=a(I+~/m); (I)

~v2/2g =mi; (2)

FIG. 113. Q=tlv. (3)

Problem IGiven the transverse section of stream and discharge, to find the slope. From the dimensions of the section find Il and m; from (1) find I, from (3) find v, and lastly from (2) find i.

Problem 11.Given the transverse section and slope, to find the discharge. Find v from (2), then Q from (3).

Problem III.Given the discharge and slope, and either the breadth, depth, or general form of the section of the channel, to determine its remaining dimensions. This must generally be solved by approximations. A breadth or depth or both are chosen, and the discharge calculated. If this is greater than the given discharge, the dimensions are reduced and the discharge recalculated.

Since m lies generally between the limits m = d and m = 1/2d, where d is the depth of the stream, and since, moreover, the velocity varies as ~! (ni) so that an error in the value of m leads only to a much less error in the value of the velocity calculated from it, we may proceed thus. Assume a value for m, and calculate v from it. Let v, be this first approximation to v. Then Q/vi is a first approximation to tf, say th. With this value of f2 design the section of the channel; calculate a second value for m; calculate from it a second value of v, and from that a ~7 second value for tl. Repeat / the process till the succes / sive values of m approxi 7 mately coincide.

-m~-i--.>~ 113. Problem IV. Most Economical Form of Channel Fio 11 for given Side Slopes.Sup 4 pose the channel is to be trapezoidal in section (fig. 114), and that the sides are to have a given slope. Let the longitudinal slope of the stream be given, and also the mean velocity. An infinite number of channels could he found satisfying the foregoing conditions. 10 render the problem determinate, let it be remembered that, since for a given discharge tno ~ x, other things being the same, the amount of excavation will be least for that channel which has the least wetted perimeter. Let d be the depth and b the bottom width of the channel, and let the sides slope n horizontal to I vertical (fig. 114), then 214~--~ ~g1 Xd2+I)

, Both t~ and x are to be minima. 4~i~- Differentiating, and equating to 4~I1I~Iu51W102212/r~ ~ zero.

(db/dd+n)d+b+nd=o, 1~ ~ db/dd+2~(n2-~-I)=o; eliminating db/dd, lfl2~J (n+i)}d+b+nd =0;

b=2kf(n2+I)n~d.

tl/x= (b+nd)dI~b+2d~i (n+I)}.

I, Inserting the value of b, 120-0 in =~/X~2dV (n+f)ndlI

________________________ {4d~(n2+I)2ndl=1/2d.

-, That is, with given side slopes, - ~- - the section is least for a given discharge when the hydraulic mean depth is half the actual depth.

A simple construction gives the - form of the channel which fulfils this condition, for it can be shown that when m=1/2d the sides of the - channel are tangential to a semicircle drawn on the water line.

Since ~/x1/2d, therefore cz=1/2xd. (1)

Let ABCD be the channel (fig. 115); from E the centre of AD drop perpendiculars EF, EG, EH on the sides.

AB=CD=a; BC=b; EF=EH=c; and EG=d. fl= area AEB+BEC+CED,

=ac+1/2bd.

X =2a-f-b.

Putting these values in (I),

ac+1/2bd= (a+1/2b)d; and hence c=d.

\-::i,,,.j-~::

FIG. 115.

That is, EF, EG, EH are all equal, hence a semicircle struck from E with radius equal o the depth of the stream will pass through F and H and be tangential to the sides of the channel.

To draw the channel, describe a semicircle on a horizontal line with radius=depth of channel. ~-----b The bottom will be a FIG. 116.

horizontal tangent of that semicircle, and the sides tangents drawn at the required side slopes.

The above result may be obtained thus (fig. 116):

X=b+2d/sin~9. (1)

tl=d(b+dcot$); -

tl/d=-b+d cot$; (2) -

f~/di=b/d+cot 13., (3)

From (I) and (2),

xtI/dd cot 13+2d/sin 13.

This will be a minimum for dx/ddt2/d+cot /32/sin /30,

or ~2/d2 = 2 cosec. /3 cot /3. (4)

or d = {t2 sin 131(2 cos /3)}.

From (3) and (4),

b/d =2(1 cos /3)/sin /9=2 tan 1/2/3.

Proportions of Channels of Maximum Discharge for given A Tea and Side Slopes. Depth of channel = d; Hydraulic mean depth Area of section=1Z.

Inclination Ratio of Top width of Sides to Side Area f Bottom twice len~tb Horizon Slopes. Section 0. Widih. of each Side Slope.

Semicircle - -. - - - t.571di o Semi-hexagon - 60 0 3 :5 I.732di f~I55d 2310d Semi-square - 90 0 0: I 2d1 2d 2d 75 58 I :4 f~8I2d2 I 562d 2~O62d 630 26 I :2 I.736d2 f~236d 2.236d 53834 I75od d 2.500d 45 0 1: 1 I~828d2 0~828d 2828d 38 40 if: I I952d 0.702d 3202d 33 42 1f I 2~Io6d o6o6d 36o6d 29 44 if: I 2282d2 0532d 4.032d 26 34 2: I 2.472d2 o.472d 4.472d 23 58 23/4: I 2.674d O.424d 4924d 2f 48 24: I 2.885(1 O385d 5.385(1

19 58 23/4:1 3.Io4d o~354d 5~854d Half the top width is the length of each side slope. The wetted perimeter is the sum of the top and bottom widths 114. Form of Cross Section of Channel in which the Mean Velocity is Constant with Varying DischargeIn designing Waste channels from canals, and in some other cases, it is desirable that the mean velocity should he restricted within narrow limits with very different volumes of discharge. In channels of trapezoidal form the velocity increases and diminishes with the discharge. Hence when the discharge is large there is danger of erosion, and when it is small of silting or obstruction by weeds. A theoretical form of section for which the mean velocity would be constant can be found, and, although this is not very suitable for practical purposes, it can be more or less approximated to in actual channels.

Let fig. 117 represent the cross section of the channel. From the symmetry of the section, only half the channel need be considered.

Scale d.i IncltJ Foot.

FIG. 117.

Let obac be any section suitable for the minimum flow, and let it be required to find the curve beg for the upper part of the channel so that the mean velocity shall be constant. Take o as origin of coordinates, and let de, fg be two levels of the water above ob.

Let obo=b(2; de=y,fg=y+dy, od=x, of =x+dx; eg=ds. The condition to be satisfied is that v=c~ (mi)

should be constant, whether the water-level is at ob, de, orfg. Con. sequently m = constant = k for all three sections, and can be found from the section obac. Henri also -

Increment of section ydxk Increment of perimeter ds yfdxf = kfds = k(dx +dyf) and dx = kdy/ ~ (y1k1).

Integrating, x =k loge 1~+ ~!(y2k2)}-I--constant; and, since y=b/2 when x=o, ~ (3/4bf1e1)l].

Assuming values for y, the values of x can be found and the curvi drawn.

The figure has been drawn for a channel the minimum section 0 which is a half hexagon of 4 ft. depth. Hence k=2; b=9~2; tb rapid flattening of the side slopes is remarkable.

STEADY MOTION OF WATER IN OPEN CHANNELS oF VARYING

CRoss SECTION AND SLOPE

115. In every stream the discharge of which is constant, or ma be regarded as constant for the time considered, the velocity a different places depends on the slope of the bed. Except at certai, exceptional points the velocity will be greatsr as the slope of th bed is greater, and, as the velocity and cross section of the strear vary inversely, the section of the stream will be least where th velocity and slopeare greatest. If in a stream of tolerably uniform slope an obstruction such as a weir is built, that will cause an alteration of flow similar to that of an alteration of the slope of the bed for a greater or less distance above the weir, and the originally uniform cross section of the stream will become a varied one. In such cases it is often of much practical importance to determine the longitudinal section of the stream.

The cases now considered will be those in which the changes ci velocity and cross section are gradual and not abrupt, and in which the only internal work which needs to be taken into account is that due to the friction of the stream bed, as in cases of uniform motion. Further, the motion will be supposed to be steady, the mean velocity ateach given cross section remaining constant, though it varies from section to section along the course of the stream.

Let fig. ff8 represent a longitudinal section of the stream, A1A1 being the water surface, ~Bf Bi the stream bed. Let Af Bf, A1Bi be -~-.~_Q, o, ilA - Ii C~.

FIG. 118.

cross sections normal to the direction of flow. Suppose the mass of water AfB0A1B1 comes in a short time 8 to C0D0C1D1, and let the work done on the mass be equated to its change of kinetic energy during that period. Let 1 be the length A1A1 of the portion of the stream considered, and z the fall, of surface level in that distance. Let Q be the discharge of the stream per second.

Change of Kinetic Energy.At the end of the time 8 there are as many particles possessing the same velocities in the space Co D0A1B1 as at the beginning. The change of kinetic energy is i~

therefore the difference of ~ ~ the kinetic energies of A0B0C0D0 and AfB1C1D~. -

Let fig. 119 represent the cross section A0B0, and let - -.

w be a small element of its -- ~~_s-_ -

area ,at a point where the FIG II

velocity is v. Let ~h be the. 9.

whole area of the cross section and u0 the mean velocity for the whole cross section. From the definition of mean velocity we have no =~v/tZ0.

Let v=uo+w, where w is the difference between the velocity at the small element w and the mean velocity. For the whole cross section, = 0.

The mass of fluid passing through the element of section w, in 0

seconds, is (G/g)o,vO, and its kinetic energy is (G12g)wvO. For the whole section, the kinetic energy of the mass A0B0C0D0 passing in 9

seconds is -

(G0/2g)~wv1 = (G0/2g)~w(us +3uorW +3uoIo1 +10), = (G8/2g){uof12+~wwf(3uo+w)~.

The factor 3us+w is equal to 2Uo+V, a quantity necessarily positive. Consequently �wv> fnu0i, and consequently the kinetic energy of A0B0C0D0 is greater than (GO/2g)flouo or (GO/2g)Qu01,

which would be its value if all the particles passing the section had the same velocity u0. Let the kinetic energy be taken at a(G0/2g)tliuof = a(G0/2g)Quof, where a is a corrective factor, the value of which was estimated by J. B. C. J. Blanger at 1.1.1 Its precise value is not of great importance.

In a similar way we should obtain for the kinetic energy of A~B1C,D1 the expression a(G8/2g)t21u1 = a(C.8/2g)Qui1,

where f21, uf are the section and mean velocity at A~Bi, and where may be taken to have the same value as before without an~7 ion- - portant error.

Hence the change of kinetic energy in the whole mass A0B0A1B in 0 seconds is a(G0/2g)Q(uiuoI). (I)

Motive Work of the Weight and Pressures.Consider a smal filament aoai which comes in 0 seconds to cocj. The work done b) gravity during that movement is the same as if the portion a0c1 wero carried to aici. Let dQO be the volume of aoco or aic,, and Yo, yi th~

i depths of ai, a1 from the surface of the stream. Then the volumi dQO or GdQO pounds falls through a vertical height z+y, yo, alid the work done by gravity is GdQO(z+yi yo).

Putting lf~ for atmospheric pressure, the whole pressure per unit of area at avis Gyi+p~, and that at ai is (Gyi+p~)- The work of these pressures is G(yo +p~/G yi p~/G)dQO = G(yo yi)dQU.

Adding this to the work of gravity, the whole work is GzdQO; or, for the whole cross section, GIQO. (2)

Work expended in Overcoming the Friction of the Stream Bed. Let AB, AB be two cross sections at distances s and s+ds from A0B0. Between these sections the velocity may be treated as uniform, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between AB and ABis GQOi(u1/2g) (xIt2)ds, where u, x t~ are the mean velocity, wetted perimeter, and section at AB. Hence the whole work lost in friction from A0B0 to AiB1 will be GQOJI-(uh/2g)(x/cf)ds. (3)

Equating the work given in (2) and (3) to the change of kinetic energy given in (i),

a(GQO/2g) (uif a02) = GQ1O GQOfo~(u/2g) (x/t2)ds; .. I = a(uof usf)12g +f ,1(ui/2g) (x/i~)ds.

116. Fundamental Differential Equation of Steady VariedMotion. Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, a1bi, taken for simplicity normal to the stream bed (fig. 120). For that short length of stream the fall of surface level, or difference of level of ~ --~~ A

, ~._,,.._

FIG. I20.

a and a1, may be written dl. Also, if we write u for us, and u+du for hi, the term (usfui)/2g becomes udulg. Hence the equation applicable to an indefinitely short length of the stream is dz =udu/g+(x/ft)l(u2/2g)ds. (I)

From this equation some general conclusions may be arrived at as to the form of the longitudinal section of the stream, but, as the investigation is somewhat complicated, it is convenient to simplify it by restricting the conditions of the problem.

Modification of the For,nala for the Restricted Case of a Stream flowing in a Prismatic Stream Bed of Constant Slope.Let i be the constant slope of the bed. Draw ad parallel to the bed, and ac horizontal. Then dI is sensibly equal to ac. The depths of the stream, h and h+dh, are sensibly equal to ab and ab, and therefore dh=ad. Also cd is the fall of the bed in the distance ds, and is equal to ids. Hence dz=ac=cdad=idsdh. (2)

Since the motion is steady Q =t2u = constant.

Differentiating, ~tdu+udt~=o; .~. du = udll/tl.

Let x be the width of the stream, then d~1=xdh very nearly. Inserting this value, du = (ux/tt)dh. (3)

Putting the values of du and dz found in (2) and (3) in equation (i)~

ids dh = (ufx/g~)dh + (x/f2) l(ui/2g)ds.

dh/ds = i (xI~f) l~(uif2g) / 1 (uf/g) (x/fI) .} (4)

Further Restriction to the Case of a Stream of Rectangular Sectios and of Indefinite Width.The equation might be discussed in tho form just given, but it becomes a little simpler if restricted in th way just stated. For, if the stream is rectangular, xh=f~, and if is large compared with h, fl/x = xh/x h nearly. Then equation (4~ becomes dhIds=i(I Iu2/2gih)/(I ui/gh). (5) 117. General Indications as to the Form of Water Surface fur ni.shed by Equation (5)Let AfA1 (fig. 121) be the water surface B0B1 the bed in a longitudinal section of the stream, and ab any section at a distance s from B0, the depth ab being Jo. Suppose B0Bi, B0A5 taken as rectangular coordinate axes, then dh/ds is the trigonometric tangent of the angle which the surface of the stream at a makes with the axis B0B1, This tangent dh/ds will be positive, if the stream is increasing in depth in the direction B0Bi; negative, FIG. 121.

if the stream is diminishing in depth from B0 towards B1. If dh/ds =0, the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4)

dh/ds=o, if i(x/~2)I(u2/2g)=o; .~. 1(u/2g) = (fflx)i mi, which is the well-known general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied motion.

Consider the possible changes of value of the fraction (I Iuf/2gih)/(I ui/gh).

As Jo tends towards the limit 0, and consequently is is large, the numerator tends to the limitcc. On the other hand if h=cc, in which case u is small, the numerator becomes equal to 1. For a value H of Jo given by the equation H = iu2/2gi, we fall upon the case of uniform motion. The results just stated may be tabulated thus: For h=o,H,>H,cc.

the numerator has the value cc, 0, > 0, 1.

Next consider the denominator. If Jo becomes very small, in which case u must be very large, the denominator tends to the limit cc. As h becomes very large and u consequently very small, the denominator tends to the limit I. For h=u/g, or u=~ (gh), the denominator becomes zero. Hence, tabulating these results as before For h=o, u/g, >u1/g, cc, the denominator becomes cc, 0, > 0, 1.

118. Case 1.Suppose h>ui/g, and also h>FI, or the depth greater than that corresponding to uniform motion. In this case dh/ds is positive, and the stream increases in depth in the direction of flow. In fig. 122 let B0B, be the bed, CiCi a line parallel to the bed and at a height above it equal to H. By hypothesis, the surface FIG. 122.

AoAi of the stream is above C5Ci, and it has just been shown that the depth of the stream increases from Bf towards B1. But going np stream h approaches more and more nearly the value H, and therefore dh/ds approaches the limit o, or the surface of the stream is asymptotic to C0C1. Going down stream h increases and u diminishes, thenumeratorand denominator ofthefraction(i rui/2gih)/(I u2jgh) both tend towards the limit I, and dh/ds to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line DODf.

The form of water surface here discussed is produced when th~ flow of a stream originally uniform is altered by the construction. oI a weir. The raising of the water surface above the level C0Ci u termed the backwater due to the weir.

119. Case 2.Suppose h>uf/g, and also h

towarda Bi. Going up stream h apC~i proaches the limit ~ 1-I, and dh/ds tends to the limit zero.

That is, tip stream is asymptotic ~ tC,C1. Going down FIG 123 stream h diminishes - and u increases~ the inequality Ii> u2/g diminishes; the denominator of the fraction (1 lnt2/2gzIl)/(Ilf-/gh~) tends to the limit zero and consequently d/z/ds tends to ce. That is, down stream A,A1 tends to a direction perpendicular to the bed. Before, however, this limit was reached the assumptions on which the general equation is based would cease to be even approximately true, and the equation would cease to be applicable. The filaments would have a relative motion which would make the infitience of internal friction in the fluid too important to be neglected. A stream surface of this form may be produced if there is an abrupt fall in the bed of the stream (fig. 124).

On the Ganges canal, as originally constructed, there / ~ff were abrupt - falls precisely FIG. I24. of this kind, and it appears that the lowering of the water surface and increase of velocity which such falls occasion, for a distance of some miles up stream, was not foreseen. The result was that, the velocity above the falls being greater than was intended, the bed was scoured and considerable damage was done to the works. When the canal was first opened the water was allowed to pass freely over the crests of the overfalls, which were laid on the level of the bed of the earthen channel; erosion of bed and sides for some miles up rapidly followed, and it soon became apparent that means must be adopted for raising the surface of the stream at those points (that is, the crests of the falls). Planks were accordingly fixed in the grooves above the bridge arches, or temporary weirs were formed over which the water was allowed to fall; in some cases the surface of the water was thus raised above its normal height, causing a backwater in the channel above (Croftons Report on the Ganges Canal, p. 14). Fig. 125 represents in an exaggerated form what probably occurred, the diagram being intended L4c~

FIG. 125.

so represent some miles length of the canal bed abovc the fall. AA parallel to the canal bed is the level corresponding to uniform motion with the intended velocity of the canal. In consequence of the presence of the ogee fall, however, the water surface would take some such form as BB, corresponding to Case 2 above, and the velocity would be greater than the intended velocity, nearly in the inverse ratio of the actual to the intended depth. By constructing a weir on the crest of the fall, as shown by dotted lines, a new water surface CC corresponding to Case I would be produced, and by suitably choosing the height of the weir this might be made to agree approximately with the intended level AA.

120. Case 3.Suppose a stream flowing uniformly with a depth ~h1/2.

If such a stream is interfered with by the construction of a weir which raises its level, so that its depth at the weir becomes h1> u~g, then for a portion of the stream the depth h will satisfy the conditions 3 H, which are not the same as those assumed in the two previous cases. At some point of the stream above the weir the depth 3 becomes equal to u/g, and at that point dh/ds becomes infinite, or the surface of the stream is normal to the bed. It is obvious that at that point the influence of internal friction will be too great to be neglected, and the general equation will cease to represent the true conditions of the motion of the water. It is known that, in cases such as this, there occurs an abrupt rise of the free surlace of the stream, or a standing wave is formed, the conditions of motion in which will be examined presently.

It appears that the condition necessary to give rise to a standing wave is that i> l/2. Now ~ depends for different channels on the roughness of the channel and its hydraulic mean depth. Bazin calculated the values of l for channels of different degrees of roughness and different depths given in the following table, and the corresponding minimum values of i for which the exceptional case of the production of a standing wave may occur.

Slope below Standing Wave Formed.

which a Stand- Nature of Bed of Stream. ing Wave is impossible in Sfope in feet Least Depth feet per foot. per foot. in feet.

(0.002 O262

Very smooth cemented surface 0-00147 ~ 0.003 098

tooo4 065

(0.003.394

Ashlar or brickwork.. 0-00186 ~ 0.004 197

~ooo6 098

(0.004 I18I

Rubble masonry - - 0-00235 ~ ooo6.525

t0OfO ~262

(0.006 3478

Earth 0.00275 ~ 0OIO 1.542

l~0-015 919

STANDING WAvES

121. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of 0.023 ft. per foot, he admitted water till it flowed uniformly with a depth of 02 ft. He then placed a plank across the stream which raised the level just above the obstruction to 0.95 ft. He found that the stream above the obstrtiction was sensibly unaffected up to a point 15 ft. from it. At that point the depth suddenly increased from 02 ft. to 0.56 ft. The velocity of the stream in the part unaffected by the obstruction was 5-54 ft. per second. Above the point where the abrupt change of depth occurred 142=5.542=30.7, and gh=322Xo2=644; hence u was>gh. Just below the abrupt change of depth u=554Xo2/o56=I97; 142=388; and gh= 32-2X0-56=18-o3; hence at this point u

The change of level at a standing wave may be found thus. Let fig. 126 represent the longitudinal section of a stream and ab, cd 6 6 ~L: iL

Flu. 126.

cross sections normal to the bed, which for the short distance considered may be assumed horizontal. Suppose the mass of water abcd to come to abcd in a short time 1; and let Uf, u1 be the velocities at ab and cd fh, Qi the areas of the cross sections. The force causing change of momentum in the mass abed estimated horizont- ally is simply the difference of the pressures on ab and cd. Putting hi, h1 for the depths of the centres of gravity of ab and cd measured down from the free water surface, the force is G(hsfIehifli) pounds, anti the impulse in t seconds is G (hsc20h1t11) t second pounds. The horizontal change of momentum is the difference of the momenta of cdcd and abab; that is, (G/g) (fbuif?ou52)t.

Hence, equating impulse and change of momentum, G(hotlohcb)t = (G/g)(cliuii_ffsuoi)t; houo~h1tn=(thu10~c2ou02)/g. (I)

For simplicity let the section be rectangular, of breadth B and depths H, and H1, at the two cross sections considered; then h0=3/4H1, and hi=1/2H1. Hence H,H12 =(2/g)(HiuiHouof).

But, since usUo=t2iUi, we have hi1 = uoH51IH11,

H, I-li = (2u0/g) (Mu/Hi Ho). (2)

This equation is satisfied if Hf=H1, which corresponds to the case of uniform motion. Dividing by HiHf, the equation becomes (H1/H0)(H,+H1) = 2u0/g; (3)

.~. H1 = -~ (2uofHo/g+ 3/4 IIof) 1/2Ho. (4)

In Bidones experiment uo=554, and H, =0.2. Hence Hi=o52, which agrees very well with the observed height.

122. A standing wave is frequentiy produced at the foot of a weir. Thus in the ogee falls originally constructed on the Ganges canal a standing wave was observed as shown in fig. 127. The water falling over the weir crest A acquired a very high velocity on the FIG. 127.

steep slope All, and the section of the stream at B became very small. It easily happened, therefore, that at B the depth h-(u/g. In flowing along the rough apron of the weir the velocity u diminished and the depth h increased At a point C, where Ii became equal to nf/g, the conditions for producing the standing wave occurred. Beyond C the free surface abrtiptly rose to the level corresponding to uniform motion with the assignea slope of the lower reach of the canal.

A standing wave is sometimes formed on the down stream side of bridges the piers of which obstruct the flow of the water. Some interesting cases of this kind are described in a paper on the Floods in the Nerbudda Valley in the Proc. Inst. Civ. Eng. vol. xxvii. p. 222, by A. C. Ilowden. Fig. 128 is compiled from the data given in that paper. It represents the section of the stream at pier 8 of the Towah Viaduct, during the flood of 1865. The ground level is not exactly given by Howden, but has been inferred from data given ~- 7

on another drawing. The velocity of the stream was not observed, but the author states it was probably the same as at ~.* the Gunjal river during a similar flood, that is i~ c~ I658 ft. per second.

Now taking the depth on tl~e down stream face of the pier at 26 ft., the velocity necessary for the production of a standing wave would be u = ..J (gh)

=~l (32.2 X26) 29 ft.

FIG. 128. per second nearly. But the velocity at this point was probably from Howdens statements 16.58 Xiil = 25.5 ft., an agreement as close as the approximate character of thc data would lead us to expect.

XI. ON STREAMS AND RIVERS

123. Catchment Basin.A stream or river is the channel for th(discharge of the available rainfall of a district, termed its catchmeni basin. The catchment basin is surrounded by a ridge or watershet line, continuous except at the point where the river finds an outlet The area of the catchment basin may be determined from a suitablf contoured map on a scale of at least I in 100,000. Of the whole rain fall on the catchment basin, a part only finds its way to the stream Part is directly re-evaporated, part is absorbed by vegetation, par may escape by percolation into neighboring districts. The follow ing table gives the relation of the average stream discharge to thi average rainfall on the catchment basin (Tiefenbacher).

Ratio of average Loss by Evaporation, Discharge to &c,, in percent of average Rainfall. total Rainfall.

Cultivated land and spring- .3 to .33 67 ~o 70

forming declivities. -

Wooded hilly slopes ., 35 to 45 55 to 65

Naked unfissured mountains 55 to -60 40 to 45

124. Flood Discharge.-The flood discharge can generally only be determined by examiinng the greatest height to which floods have been known to rise. To produce a flood the rainfall must be heavy and widely distributed, and to prodtice a flood of exceptional height the duration of the rainfall must be so gaeat that the flood waters of the most distant affluents reach the point considered, sinsultaneously with those from nearer points. The larger the catchment basin the less probable is it that all the conditions tending to produce a maximum discharge should simultaneously occur. Further, lakes and the river bed itself act as storage reservoirs during the rise of water level and diminish the rate of discharge, or serve as flood moderators. The influence of these is often important, because very heavy rain storms are in most countries of comparatively short duration. Tiefenbacher gives the following estimate of the flood discharge of streams in Europe: Flood discharge of Stream, per Second per Square Mife of Catchrnent Basin.

In flat country 8-7 to 12-5 cub. ft.

In hilly districts 17.5 to 22.5

In moderately mountainous districts 36-2 to 45-0

In very mountainous districts -- 50-0 to 75-0 ,,

It has been attempted to express the decrease of the rate of flood discharge with the increase of extent of the catchment basin by empirical formulae. Thus Colonel P. P. L. OConnell proposed the formula y=M,/x, where M is a constant called the modulus of the river, the value of which depends on the amount of rainfall, the physical characters of the basin, and the extent to which the floods are moderated by storage of the water. If M is small for any given river, it shows that the rainfall is small, or that the permeability or slope of the sides of the valley is such that the water does not drain rapidly to the river, or that lakes and river bed moderate the rise of the floods. If values of M are known for a number of rivers, they may be used in inferring the probable discharge of other similar rivers. For British rivers M varies from 0-43 for a small stieam draining meadow land to 37 for the Tyne. Generally it is about 15 or 20. For large European rivers M varies from 16 for the Seine to 67~5 for the Danube. For the Nile M = ii, a low value which results from the immense length of the Nile throughout which it receives no affluent, and probably also from the influence of lakes. For different tributaries of the Mississippi M varies from 13 to 56. For various Indian rivers it varies from 40 to 303, this variation being dtie to the great variations of rainfall, slope and character of Indian rivers.

In some of the tank projects in india, the flood discharge has been calculated from the formula D=C~~n2, where D is the discharge in cubic yards per hour from n square miles of basin. The constant C was taken =61,523 in the designs for the Ekrooka tank, = 75,000 on Ganges and Godavery works, and = 10,000 on Madras works.

125. Action of a Stream on its Bed.If the velocity of a stream exceeds a certain limit, depending on its size, and on the size, heaviness, form and coherence of the material of which its bed is com- ~ posed, it scours its bed and carries forward the materials. -- --~~

The quantity of material which a given stream can carry in ~ suspension depends on the size a C

and density of the particles in FIG. 129. suspension, and is greater as the velocity of the stream is greater. If in one part of its course the velocity of a stream is great enough to scour the bed and the water becomes loaded with silt, ansi in a subsequent part of the rivers course the velocity is diminished, then part of the transported material must be deposited. Probably deposit and scour go on simultaneously over the whole river bed, but in some parts the rate of scour is in excess of the rate of deposit, and ______

in other parts the rate -

of deposit is in excess of the rate of scour. .

Deep streanis appear to - - - - b ~--- -

have the greatest scour- --

ing power at any given velocity. It is possible ~- c that the difference is FIG. 130.

strictly a difference of transporting, not of scouring action. Let fig. 129 represent a section ol a stream. The material lifted at a will be diffused through the mass o~ the stream and deposited at different distances down stream. Thi average path of a particle lifted at a will be some stich curve as abc and the average distance of transport each time a particle is liftec will be represented by ac. In a deeper stream such as that in fig. 130, the average height to which particles are lifted, and, since the rate of vertical fall through the water may be assumed the same as before, the average distance ac of transport will be greater. Consequently, although the scouring action may be identical in the two streams, the velocity of transport of material down stream is greater as the depth of the stream is greater. The effect is that the deep stream excavates its bed more rapidly than the shallow stream.

126. Bottom Velocity at which Scour commences.The following bottom velocities were determined by P. L. G. Dubuat to be the maximum velocities consistent with stability of the stream bed for different materials.

Darcy and Bazin give, for the relation of the mean velocity Vm and bottom velocity ri Cm 1i+IO87~~ (ml).

s1mi = lm~~I (i/2g);

.~. VmVi/(I_Io87~ (l/2g)).

Taking a mean value for i~, we get Vm = I 3I2Vo, and from this the following values of the mean velocity are obtained: Bottom Velocity Mean Velocity I. Soft earth :25 Vm.

2. Loam 0.50.65 .65

3. Sand 1.00 1.30

4. Gravel 2oo 262

5. Pebbles 3.40 446

6. Broken stone, flint.. 4.00 525

7. Chalk, soft shale.. 5.00 6.56

8. Rock in beds 6oo 787

9. Hard rock - - - 10OO 13.12

The following table of velocities which should not be exceeded in channels is given in the Ingenleurs Taschenbuch of the Verein Hutte: Surface Mean Bottom Velocity. Velocity. Velocity.

Slimy earth or brown clay .49.36 26

Clay .98.75.52

Firm sand 1.97 1.51 f~O2

Pebbly bed 4.00 3.15 2.30

Boulder bed 5oO 4.03 3.08

Conglomerate of slaty fragments 7.28 6io 4.90

Stratified rocks 8oo 7.45 6oo Hard rocks 14.00 12.15 10.36

127. Regime of a River Channel.A river channel is said to be in a state of regime, or stability, when it changes little in draught or form in a series of years. In some rivers the deepest part of the channel changes its position perpetually, and is seldom found in the same place in two successive years. The sinuousness of the river also changes by the erosion of the banks, so that in time the position of the rwer is completely altered. In other rivers the change from year to year is very small, but probably the regime is never perfectly stable except where the rivers flow over a rocky bed.

It a river had a constant discharge it would gradually modify its ,bed till a permanent regime was established. But as the volume ~m.--~

3.lf~ 7f~

discharged is constantly changing, and therefore ~ the velocity, silt is deposited when the velocity ~~--~ ~ decreases, and scour goes on when the velocity increases in the same place. When the scouring ,~ and silting are considerable, a perfect balance between the two is rarely established, and hence continual variations occur in the form of the river and the direction of its currents. In other cases, where the action is less violent,a tolerable balance may be established and the deepening of the bed by scour at one time is compensated b~ the silting at another. In that case the general regime is permanent though alteration is constantly going on. This is more likely tc happen ii by artificial means the erosion of the banks is prevented. If a river flows in soil incapable of resisting its tendency to scour it is necessarily sinuous (f 107), for the slightest deflection of the current to either side begins an erosion which increases progressively till a considerable bend is formed. If such a river is straightened it becomes sinuous again unless its banks are protected from scour.

128. Longitudinal Section of River Bed.The declivity of rivers decreases from source to mouth. In their higher parts rapid and torrential, flowing over beds of gravel or boulders, they enlarge in volume by receiving affluent streanis, their slope diminishes, their bed consists of smaller materials, and finally they reach the sea. Fig. 131 shows the length in miles, and the surface fall in feet per mile, of the Tyne and its tributaries.

The decrease of the slope is due to two causes. (1) The action of the transporting power of the water, carrying the smallest debris the greatest distance, causes the bed to be less stable near the mouth than in the higher parts of the river; and, as the river adjusts its slope to the stability of the bed by scouring or increasing its sinuousness when the slope is too great, and by silting or straightening its course if the slope is too small, the decreasing stability of the bed would coincide with a decreasing slope. (2) The increase of volume and section of the river leads to a decrease of slope; for the larger the section the less slope is necessary to ensure a given velocity.

The following investigation, though it relates to a purely arbitrary case, is not without interest. Let it be assumed, to make the conditions definite(1) that a river flows over a bed of uniform resistance to scour, and let it be further assumed that to maintain stability the velocity of the river in these circumstances is constant from source to mouth; (2) suppose the sections of the river at all points are similar, so that, b being the breadth of the river at any point, its hydraulic mean depth is ab and its section is cbf, where a and c are constants applicable to all parts of the river; (3) let us further assume that the discharge increases uniformly in consequence of the supply from affluents, so that, if 1 is the length of the river from its source to any given point, the discharge there will be ~ D X ki, where Ii is another constant applicable to all points in the course of the river.

Let AB (fig. 132) be C

the longitudinal section of the, river, whose ~ FIG. 132.

source is at A; and take A for the origin of vertical and horizontal coordinates. Let C be a point whose ordinates are x and y, and let the river at C have the breadth b, the slope i, and the velocity v.

Since velocity X area of section = discharge, vcbf = kl, or b = -SI (kl/cv). Hydraulic mean depth =ab=aI (kl/cv).

But, by the ordinary formula for the flow of rivers, ml -=

.i=)v1/m=(;ni/a)SI (c/k)).

But i is the tangent of the angle which the curve at C makes with the axis of X, and is therefore =dy/dx. Also, as the slope is small, l=AC=AD=x nearly.

.. dy/dx = ()vi/a)V (c/kx);

and, remembering that v is constant, y = (2fvi/a)~ (cx/k);

or = constant X x; so that the curve is a common parabola, of which the axis is horizontal and the vertex at the source. This may be considered an - ideal longitudinal section, to which actual rivers approximate more or less, with exceptions due to the varying hardness of their beds, and the irregular manner in which their volume increases.

129. Surface Level of River.The surface level of a river is a plane changing constantly in position from changes in the volume of water discharged, and more slowly from changes in the river bed, and the circumstances affecting the drainage into the river.

For the purposes oi the engineer, it is important to determine (i) the extreme low water level, (2) the extreme high water or flood level, and (3) the highest navigable level.

1. Low Water Level cannot he absolutely known, because a river reaches its lowest level only at rare inter ~ vals, and becatise alterations in the cultivation of the ~ land, the drainage, toe removal of forests, the removal ~ or erection of obstructions in the river bed, &c., gradu ~ ally alter the conditions of discharge. The lowest level of which records can be found is taken as the convenS- tional or approximate lo~ water level, and allowance i~

- made for possible changes.

2. High Water or Flood LevelThe engineer assumes as the highesi flood, level the highest level of which records can be obtained. Ir forming a judgment of the data available, it must be remembered thai the highest level at one point of a river is not always simultaneou, with the attainment of the highest level at other points and that the rise of a river in flood is very different in different parts of its course. In temperate regions, the floods of rivers seldom rise more than 20 ft. above low-water level, but in the tropics the rise of floods is greater.

3. Highest Navigable LevelWhen the river rises above a certain level, navigation becomes difficult from the increase of the velocity of the current, or from submersion of the tow paths, or from the headway under bridges becoming insufficient. Ordinarily the highest navigable level may be taken to be that at which the river begins to overflow its banks.

30. Relative Value of Different Materials for Submerged Works. That the power of water to remove and transport different materials depends on their density has an important bearing on the selection of materials for submerged works. In many cases, as in the aprons or floorings beneath bridges, or in front of locks or falls, and in the formation of training walls and breakwaters by pierres perdus, which have to resist a violent current, the materials of which the structures are composed should be of such a size and weight as to be able individually to resist the scouring action of the water. The heaviest materials will therefore be the best; and the different value of materials in this respect will appear much more striking, if it is remembered that all materials lose part of their weight in water. A block whose volume is V cubic feet, and whose density in air is w lb per cubic foot, weighs in air wV tb, but in water only (w62.L)

\ lb.

Weight of a Cub. Ft. in Ib.~

In Air. In Water.

Basalt ,., 187.3 124.9

Brick 130~0 67.6

Brickwork I 12O 496

Granite and limestone 170.0 107.6

Sandstone - - - 144.0 8i6

Masonry.. - - 116-144 53.6-81.6

131. Inundation Deposits from a River.When a river carrying silt periodically overflows its banks, it deposits silt over the area flooded, and gradually raises the surface of the country. The silt is deposited in greatest abundance where the water first leaves the river. It hence results that the section of the country assumes a peculiar form, the river flowing in a trough along the crest of a ridge, from which the land slopes downwards on both sides. The silt deposited from the water forms two wedges, having their thick ends towards the river (fig. 133).

FIG. 133.

This is strikingly the case with the Mississippi, and that river is now kept from flooding immense areas by artificial embankments or levees. In India, the term dettaic segment is sometimes applied to that portion of a river running through deposits formed by inundation, and having this characteristic section. The irrigation, of the countr in this case is very easy; a comparatively slight raising of the river surface by a weir or annit-ut gives a command of level which permits the water to be conveyed to any part of the district.

132. DeltasThe name delta was originally given to the L~shaped portion of Lower Egypt, included between seven branches of the Nile. It is now given to the whole of the alluvial tracts round river mouths formed by deposition of sediment from the river, where its velocity is checked on its entrance to the sea. The characteristic feature of these alluvial deltas is that the river traverses them, not in a single channel, but in two or many bifurcating branches. Each branch has a tract of the delta undet its influence, and gradually raises the surface of that tract, and extends it seaward. As the delta extends itself seaward, the conditions of discharge through the different branches change. The water finds the passage through one of the branches less obstructed than through the others; the velocity and scouring action in that branch are increased; in the others they diminish. The one channel gradually absorbs the whole of the water supply, while the other branches silt tip. But as the mouth of the new main channel extends seaward the resistance increases both from the greater length of the channel and the formation of shoals at its mouth, and the river tends to form new bifurcations AC or AD (fig. 134), and one of these may in time become the main channel of the river.

133. Field Operations preliminary to a Study of River ImprovementThere are required (1) a plan of the river, on which the positions of lines of levelling and cross sections are marked; (2) a longitudinal section and numerous cross sections of the river; (3) a series of gaugings of the discharge at different points and iii different conditions of the river.

Longitudinal Seclion.This requires to be carried out with great accuracy. A line of stakes is planted, following the sinuosities of the river, and chained and levelled. The cross sections are referred tc~ the line of stakes, both as to position and direction. The determination of the surface slope is very difficult, partly from its extreme smallness, partly from oscillation of the water. Cunningham recommends that the slope be taken in a length of 2000 ft. by four simultaneous observations, two on each side of the river.

134. Cross Sections A stake is planted flush with the water, and its level relatively to some point on the line of levels is determined.

Thea the depth of the water is determined at a series of points (if Sh~9 ~~j~toaL.

FIG. 134.

possible at uniform distances) in a line starting from the stake and perpendicular to the thread of the stream. To obtain these, a wire may be stretched across with equal distances marked on it by hanging tags. The depth at each of these tags may be obtained by a light wooden staff, with a disk-shaped shoe 4 to 6 in. in diameter. If the depth is great, soundings may be taken by a chain and weight. To ensure the wire being perpendicular to the thread of the stream, it is desirable to stretch two other wires similarly graduated, one above and the other below, at a distance of 20 to 40 yds. A number of floats being then thrown in, it is observed whether they pass the same graduation on each wire.

For large and rapid rivers the cross section is obtained by sounding in the following way. Let AC (fig. 135) be the line on which soundings are required. A base line AB is measured out at right angles to AC, and ranging staves are set up at AB and at D in line with AC. A boat is allowed to drop down stream, and, at the moment it comes in line with AD, the lead is dropped, and an observer in the boat takes, with a box sextant. ~_A D the angle AEB subtended by AB. The sounding line may /

have a weight of 14 lb of lead, ,

and, if the boat drops down I stream slowly, it may hang near / the bottom, so that the observation is made instantly. In extensive surveys of the MissisI sippi observers with theodolites were stationed at A and B. The theodolite at A was directed i / towards C, that at B was kept / on the boat. When the boat g came on the line AC, the ob- B server at A signalled. the sounding line was dropped, and the FIG. 135. observer at B read off the angle ABE. By repeating observations a number of soundings are obtained, which can be plotted in their proper position, and the form of the river bed drawn by connecting the extremities of the lines. From the section can be measured the sectional area of the stream tl and its wetted perimeter x; and from these the hydraulic mean depth m can be calculated.

135. Measurement of the Discharge of Rivers.The area of cross section multiplied by the mean velocity gives the discharge of the stream. The height of the river with reference to some fixed mark should be noted whenever the velocity is observed, as the velocity and area of cross section are different in differest states of the river. To determine the mean velocity various methods may be adopted; and, since no method is free from liability to error, either from the difficulty of the observations or from uncertainty as to the ratio of the mean velocity to the velocity observed, it is desirable that more than one method should be used.

INSTRUMENTS FOR MEASURING THE VELOCITY OF WATER

136. Surface Floats are convenient for determining the surface velocities of a stream, though their use is difficult near the banks. The floats may be small balls of wood, of wax or 01 hollow metal, so loaded as to float nearly flush with the water surface. To rendet them visible they may have a vertical painted stem. In experim~snts on the Seine, cork balls 13/4 in. diameter were used, loaded to float flush with the water, and provided with a stem. In A. J. C. Cunninghams observations at Roorkee, the floats were thin circular disks of English deal, 3 in. diameter and 1/2 in. thick. - For observations near the banks, floats 1 in. diameter and 3/4 in. thick were used. To render them visible a tuft of cotton wool was used loosely fixed in a hole at the centre.

The velocity is obtained by allowing the float to be carried down, and noting the time of passage over a measured length of the stream. If v is the velocity of any float, I the time of passing over a length 1, then v=1,1t. To mark out distinctly the length of stream over which the floats pass, two ropes may be stretched across the stream at a distance apart, which varies usually from 50 to25oft., according to the size and rapidity of the river. In the Roorkee experiments a length of run of 50 ft. was found best for the central two-fifths of the width, and 25 ft. for the remainder, except very close to the banks, where the run was made 121/2 ft. only. The longer the run the less is the proportionate error of the time observations, but on the other hand the greater the deviation of the floats from a straight course parallel to the axis of the stream. To mark the precise position at which the floats cross the ropes, Cunningham used short white rope pendants, hanging so as nearly to touch the surface of the water. In this case the streams were 80 to 180 ft. ill width. In wider streams the use of ropes to mark the length of run is impossible, and recourse must be had to box sextants or theodulites to mark the path of the floats.

Let AB (fig. 136) be a measured base line strictly parallel to the thread of the stream, and AA1, BB1 lines at right angles to AB

I marked out by ranging rods at Ai and B1. Suppose observers stationed at A

A ,~ A1 and B with sextants or theodolites, and i--- ~ --0 let CD be the path of any float down I stream. As the float approaches AAi, I / the observer at B keeps it on the cross wire / of his instrument. The observer at A

observes the instant of the float reaching the line AAi, and signals to B who then I reads off the angle ABC. Similarly, as the float approaches BB1, the observer at A keeps it in sight, and when signalled I to by B reads the angle BAD. The data so obtained are sufficient for plotting I I~ the path of the float and determining the distances AC, BD.

~ The time taken by the float in passing B .D B over the measured distance may be ob II ~ served by a chronograph, started as the float passes the upper rope or line, and FIG. 136. stopped when it passes the lower. In Cunninghams observations two chronometers were sometimes used, the time of passing one end of the run being noted on one, and that of passing the other end of the run being noted on the other. The chronometers were compared immediately before the observations. In other cases a single chronometer was used placed midway of the run. The moment of the floats passing the ends of the run was signalled to a timekeeper at the chronometer by shouting. It was found quite possible to count the chronometer beats to the nearest half second, and in sonic cases to the nearest quarter second.

137. Sub-surface Fioats.The velocity at different depths below the surface of a stream may be obtained by sub-surface floats, u.ed precisely in the same way as surface floats. The most usual arrangement is to have a large float, of slightly greater density than water, connected with a small and very light surface float. The motion of the combined arrangement is not ~ sensibly different from that of the large -- float, and the small surface float enables - an observer to note the path and velo I city of the sub-surface float. The in strument is, however, not free froni objection. If the large submerged float is made of very nearly the samc I density as water, then it is liable to lx thrown upwards by very slight eddies in the water, and it does not maintair its position at the depth at which it h intended to float. On the other hand if the large float is made sensibl3

jJ1/8~i% heavier than water, the indicating oi surface float must be made rather large and then it to scme extent influencn the motion of the submerged float FIG. 137. Fig. 137 shows one form of sub stirface float. It consists of a coupl of tin plates bent at a right angle and soldered together at the angle This is connected with a wooden ball at the surface by a very thu wire or cord. As the tin alone makes a heavy submerged float, it i better to attach to the tin float some pieces of wood to diminish it weight in water. Fig. 138 shows the form of submerged float use by Cunningham. It consists of a hollow metal ball connected to a slice of cork, which serves as the surface float.

138. Twin Floats.Suppose two equal and similar floats (fig. 139) connected by a wire. Let one float be a little lighter and the other a little heavier than water. Then the velocity of the comuiiied - a~a~-s~-

.. ___~~a/ /, ,~T~hff,~TL

3 c1ut,-s~ ~ ~)~ ~i~

s-3 diem.--,,

~.3diw,t,,i FIG. 138. FIG. 139.

floats will be the mean of the surface velocity and the velocity at the depth at which the heavier float swims, which is determined by the length of the connecting wire. Thus if v, is the surface velocity and va the velocity at the depth to which the lower float is sunk, the velocity of the combined floats will be v= l(v,+v5).

Consequently, if v is observed, and v, determined by an experiment with a single float, Vd2VV,.

According to Cunningham, the twin float gives better results than the sub-surface float.

139. Velocity Rods.Another form of float is shown in fig. 140. This consists of a cylindrical rod loaded at the lower end so as to float nearly vertical in water. A wooden rod, with a metal cap at the bottom in which shot can be placed, answers better than anything else, and sometimes the wooden rod is made in lengths, which can be screwed together ~, so as to suit streams of different depths. ~ ~ A tuft of cotton wool at the top serves - - to make the float more easily visible. / Such a rod, so adjusted in length that it / sinks nearly to the bed of the stream, gives directly the mean velocity of the whole vertical section in which it floats.

140. Revys Current Meter.No in- .ldia,n~. strument has been so much used in directly determining the velocity of a stream at a given point as the screw current meter. Of this there are a dozen varieties at least. As an example of the instrument in its simplest form, Revys meter maybe selected. This is an ordinary screw meter of a larger size than usual, more carefully made, and with its details carefully studied (figs. 141, 142). FIG. 140. It was designed after experience in gauging the great South American rivers. The screw, which is actuated by the water, is 6 in. in diameter, and is of the type of the Griffiths screw used in ships. The hollow spherical boss serves to make the weight of the screw sensibly equal to its displacement, so that friction is much reduced. On the axis aa of the screw is a worm which drives the counter. This consists of two worm wheels g and h fixed on a common axis. The worm wheels are carried on a frame attached to the pin 1. By means of a string attached to I they can be pulled into gear with the worm, or dropped out of gear and stopped at any instant. A nut m can be screwed up, if necessary, to keep the counter permanently in gear. The worm is two-threaded, and the worm wheel g has 200 teeth. Consequently it makes one rotation for 100 rotations of the screw, and the number of rotations up to 100 is marked by the passage of the graduations on its edge in front of a fixed ind~x, The second worm wheel has 196 teeth, and its edge is divided intc 49 divisions. Hence it falls behind the first wheel one division for a complete rotation of the latter. The number of hundreds of rota~ tions of the screw are therefore shown by the number of divisions or h passed over by an index fixed to g. One difficulty in the use of thc ordinary screw meter is that particles of grit, getting into the workinl parts, very sensibly alter the friction, and therefore the speed of thi meter. Revy obviates this by enclosing the counter in a brass bw with a glass face. This box is filled with pure water, which ensures i constant coefficient of friction for the rubbing parts, and prevents an~ mud or grit finding its way in. In order that the meter may place itseh with the axis parallel to the current, it is pivoted on a Vertical axi, and directed by a large vane shown in fig. 142. To give the yarn more directing power the vertical axis is nearer the screw than in ordinary meters, and the vane is larger. A second horizontal vane is attached by the screws x, x, the object of which is to allow the meter to rest on the ground without the motion of the screw being interfered with. The string or wire for starting and stopping the meter is flU)

~ -~~ ~

f ~i -L-Engels

FIG. 141.

carried through the centre of the vertical axis, so that the strain on it may not tend to pull the meter oblique to the current. The pitch of the screw is about 9 in. The screws at x serve for filling the dieter with water. The whole appar~tus is fixed to a rod (fig. 142), of a length proportionate to the depth, or for very great depths it is fixed to a weighted bar lowered by ropes, a plan invented by Revy. The instrument is generally used thus. The reading of the counter is noted, and it is put out of gear. The meter is o then lowered into the water to the required position from a platform between two boats, or better from a temporary bridge. Then the counter is put into gear for one, two or five minutes. Lastly, the instrument is raised and the counter again read. The velocity is o deduced from the number of rotations in unit time by the formulae given below. For surface velocities the counter may be kept permanently in gear, the screw being started and stopped by hand.

41. The Harlacher Current Meter.In this the ordinary counting apparatus is abandoned. A worm drives a worm wheel, which makes an electrical contact once for each 100 rotations of the worm. This contact gives a signal above water. With this arrangement, a series of velocity observations can be made, without removing the instrument from the water, and a number of practical difficulties attending the accurate starting and stopping of the ordinary counter are entirely got rid of. Fig. 143 shows the meter. The worm wheel I makes one rotation for I00 of the screw. A pin moving the lever x makes the electrical contact. The wires b, c are led through a gas pipe B; this also serves to ~ adjust the meter to any required position on the wooden rod dd. The rudder or vane is shown at WH. The galvanic current acts on the electromagnet m, which is fixed in a small metal box containing also the battery. The magnet exposes and withdraws a colored disk at an opening in the cover of the box.

O 142. Amsier Laffon Current Meter.A very convenient and accurate current meter is constructed by Amsler Laffon of Schaff FIG. 142. hausen. This can be used on a rod, and put into and out of gear by a ratchet. The peculiarity in this case is that there is a double ratchet, so that one pull on the string puts the counter into gear and a second puts it out of gear. The string may be slack during the action of the meter, and there is less uncertainty than when the counter has to be held in gear. For deep streams the meter A is suspended by a wire with a heavy lenticular weight below (fig. 144). The wire is payed out from a small winch D, with an index showing the depth of the meter, and passes over a pulley B. The meter is in gimbals and is directed by a conical rudder which keeps it facing the stream with its axis horizontal. There is an electric circuit from a battery C through the meter, and a contact is made closing the circuit every 100 revolutions. The moment the circuit closes a bell rings. By a subsidiary arrangement, when the foot of the instrument, 03 metres below the axis of the meter, touches the ground the circuit is also closed and the bell rings. It is easy to distinguish the continuous ring when the ground is reached from the short ring when the Counter signals. A convenient winch for the wire is so graduated that if set when the axis of the meter is at the water surface it indicates at any moment the depth of the meter below the surface. Fig. 144 shows the meter as used on ~ boat. It is a very convenient instrument for obtaining the velocity at different depths and can also he used as a sounding instrument.

143. Determination of the (~oefficients of the Current MeterSuppose a series of observations has been made by towing the meter in still water at different speeds, and that it is required to ascertain from these the constants of the meter. If v is the velocity of the water and n the observed number of rotations per second, let --

v=a+f~n (1)

where a and $ are constants. Now let the meter be towed over a measured distance L, and let N be the revolutions of the meter and the time of transit. Then the speed of the meter relatively to the water is Lit =v feet per second, and the number of revolutions per second is N/t=n. Suppose m observations have been made in this way, furnishing corresponding values of a and n, the speed in each trial being as uniform as possible, ~fl=fli+fli+ .

~V=Oi+vi+ - - -

�~flV=fliVi+fl2V,+ - - -

~n~=nl+nI+ -

l~nhf=~nl+n2+ -. -If a= m~n~~n]

m2nv ~v~n In a few cases the constants for screw current meters have been determined by towing them in R. E. Froudes experimental tank in FIG. 144.

which the resistance of ship models is ascertained. In that case the data are found with exceptional accuracy.

144. Darcy Gauge or modified Pitol Tube.A very old instruinent for measuring velocities, invented by Henri Pitot in 1730 (ilistoire de lAcadimje des Sciences, 1732, p. 376), consisted simply of a vertical glass tube with a right-angled bend, placed so that its mouth was normal to the direction of flow (fig. 145).

The impact of the stream on the mouth of the tube balances a column in the tube, the height of which is approximately h=vf/2g, where v is the velocity fl n at the depth x. Placed with its mouth parallel ~ L_~-_=~ to the stream the water ~ ~i~EEiEh ~ inside the tube is nearly ~j 1H ~ at the same level as the i~l~ ~, II] surface of the stream, v L~I ~ ~, I and turned with the x~i mouth down stream, the fluid sinks a depth F =vu/2g nearly, though ~j ,~ I ~ the tube in that case - - - - ~ interferes with the free A B C flow of the liquid and FIG. 145. somewhat modifies the result. Pitot expanded the mouth of the tube so as to form a funnel or bell mouth. In that case he found by experiment h = 1 -5V2 /2g.

But there is more disturbance of the stream. Darcy preferred to mak the mouth of the tube very small to avoid interference with the stream and to check oscillations of the water column. Let the difference of level of a pair of tubes A and B (fig. 145) be taken to be F = kvf/2g, then k may be taken to be a corrective coefficient whose value in well-shaped instruments is very nearly unity. By placing his instrument in front of a boat towed through water Darcy found F = I 034; by placing the instrument in a stream the velocity of which had been ascertained by floats, he found k = I 006; by readings taken in different parts of the section of a canal in which a known volume of water was flowing, he found 11=0.993. He believed the first value to be too high in con- I.? sequence of the disturbance caused g ~ by the boat. The mean of the other -. two values is almost exactly unity (Recherches hydrauliques, Darcy and Basin, 1865, p. 63). W. B. Gregory -~ ~ ~ used somewhat differently formed Pitot tubes for which the 11=1 (Am.

Soc. Mech. Eng., 1903, 25). T. E.

Stanton used a Pitot tube in deter mining the velocity of an air current, and for his instrument he found 11=1.030 to k=Io32 (On the Re ___________________________ sistance of Plane Surfaces in a Current of Air, Proc. Inst. Civ.

Eng., 1904, 156).

One objection to the Pitot tube in its original form was the great difficulty and inconvenience of reading the height F in the imme diate neighborhood of the stream surface. This is obviated in the Darcy gatige, which can be removed from the stream to be read.

Fig. i46 shows a Darcy gauge.

It consists of two Pitot tubes -, having their mouths at right angles.

In the instrument shown, the two tubes, formed of copper in the lower part, are united into one for strength, and the mouths of the tubes open vertically and horizon tally. The upper part of the tubes is of glass, and they are provided with a brass scale and two verniers b, b. The whole instrument is sup ported on a vertical rod or small pile AA, the fixing at B permitting the instrument to be adjusted to any height on the rod, and at the same time allowing free rotation, so that it can be held parallel to the current.

At c is a two-way cock, which can be opened or closed by cords. If this is shut, the instrument can be / lifted out of the stream for reading.

~. ~ ~ -L-- -~ - The glass tubes are connected at - top by a brass fixing, with a stop cock a, and a flexible tube and mouthpiece m. The use of this is as follows. If the velocity is re quired at a point near the surface of the stream, one at least of the water columns would be below the level at which it could be read. It would be in the copper part of the instrument. Suppose then a little air is sucked out by the tube m, and the cock a closed, the two columns will be forced up an amount corresponding to the difference between atmospheric pressure and that in the tubes. But the difference of level will remain unaltered.

When the velocities to be measured are not very small, this instrument is an adnurable one. It requires observation only of a single linear quantity, and does not require any time observation. The law connecting the velocity and the observed height is a rational one, and it is not absolutely necessary to make any experiments on the coefficient of the instrument. If we take v=k~I (2gh), then it appears fi-om Darcys experiments that for a well-formed instrument 11 does not sensibly differ from unity. It gives the velocity at a definite point in the stream. The chief difficulty arises from the fact that at any given point in a stream the velocity is not absolutely constant, but varies a little from moment to moment. Darcy in some of his experiments took several readings, and deduced the velocity from the mean of the highest and lowest.

145. Perrodil Hyd rod ynamometerT his consists of a frame abcd (fig. 147) placed vertically in the stream, and of a height not less than the streams depth. The two vertical members of this frame are connected by cross bars, and united above water by a circular bar, situated in the vertical plane and carrying a horizonta~ graduated circle ef. This whole system is movable round its axis being suspended on a pivot at g connected with the fixed support mn. Other horizontal arms serve as guides. The central vertical rod gr forms a torsion rod, being fixed at r to the frame abcd, and, nassing freely unwards thri-imich th~~ ei,idp,c it oarrios hn,-~nntnl needle moving over the graduated circle ef. The support g, which carries the apparatus, also receives in a tubular guide the end of the torsion rod gr and a set screw for fixing the upper end of the torsion rod when necessary. The impulse of the stream of water is received on a circular disk x, in the plane of the torsion rod and the frame abcd. To raise and lower the apparatus easily, it is not fixed directly to the rod inn, but to a tube ki sliding on mn.

Suppose the apparatus arranged so that the disk x is at that level the stream where the velocity is to be determined. The plane ~))>

FIG. f46.

abcd is placed parallel to the direction of motion of the water. Then the disk x (acting as a rudder) will place itself parallel to the stream on the down stream side of the frame. The torsion rod will be unstrained, and the needle will be at zero on the graduated circle. If, then, the instrument is turned by pressing the needle, till the plane abcd of the disk and the zero of the graduated circle is at right angles to the stream, the torsion rod will be twisted through an angle which measures the normal impulse of the stream on the disk x. That angle will be given by the distance of the needle from zero. Observation shows that the velocity of the water at a given point is not constant. It varies between limits more or less wide. When the apparatus is nearly in its right position, the set screw at g is made to clamp the torsion spring. Then the needle is fixed, and the apparatus carrying the graduated circle oscillates. It is not, then, difficult to note the ii mean angle marked by the needle.

Let r be the radius of the torsion rod, 1 its length from the needle 5,

over ef to r, and a the observed torsion angle. Then the moment e of the couple due to the molecular forces in the torsion rod is -

M rEjIa/l; ______ _______

where E, is the modulus of elas- =- -

ticity for torsion, and I the polar ~

moment of inertia of the section of ______

the rod. If the rod is of circular - -

section, I = 1/2irr4. Let R be the -- --- - ~- - - - radius of the disk and b its I J _T~IJI ~ leverage, or the distance of its - -- -__ --- - - -

centre from the axis of the torsion ---- - - -

rod. The moment of the pressure ~

of the water on the disk is - Fb = kb(Gf2g)frRfvf where Gis the heaviness of water ,.

and k an experimental coefficient. 2~ cT _d-_

Then EiIa/l~kb(G/2g)irRivf. - -~

For any given instrument, v=c~ja, FIG. 147.

where c is a constant coefficient for the instrument.

The instrument as constructed had three disks which could be used at will. Their radii and leverages were in feet 1st disk. .. 0.052 o~I6

2nd ,,. .. of05 0.32

3rd ,,. .. O2i0 o66

For a thin circular plate, the coefficient k=I~I2. In the actual instrument the torsion rod was a brass wire o~o6 in. diameter and 61/8 ft. long. Supposing a measured in degrees, we get by calculation V0.335-%f a; ofI5~a; 0.o42~f a.

Very careful experiments were made with the instrument. It was fixed to a wooden turning bridge, revolving over a circular channel of 2 ft. width, and about 7~6 fn circumferential length. An allowance was made for the slight current prodticed in the channel. These experiments gave for the coefficient c, in the formulav=c~a, 1st disk, C =0.3126 for velocities of 3 to 16 ft.

2nd ,, 0.1177 ,, ,, 13/4 to 33/4

3rd ,, 0.0349 ,, ,, less than 13/4

The instrument is preferable to the current meter in giving the velocity in terms of a single observed quantity, the angle of torsion, while the current meter involves the observation of two quantities, the number of rotations and the time. The current meter, except in some improved forms, must be withdrawn from the water to read the result of each experiment, and the law connecting the velocity and number of rotations of a current meter is less well-determined than that connecting the pressure on a disk and the torsion of the wire of a hydrodynamometer.

The Pitot tube, like the hydrodynamometer, does not require a time observation. But, where the velocity is a varying one, and consequently the columns of water in the Pitot tube are oscillating, there is room for doubt as to whether, at any given moment of closing the cock, the difference of level exactly measures the impulse of the stream at the moment. The Pitot tube also fails to give measurable indications of very low velocities.

PRocEssEs FOR GAUGING STREAMS

146. Gauging by Observation of the Maximum. Surface Velocity.The method of gauging which involves the least trouble is to determine the surface velocity at the thread of the stream, and to defluce from it the mean velocity of the whole cross section. The maximum surface velocity may be determined by floats or by a current meter. Unfortunately the ratio of the maximum surface to the mean velocity is extremely variable. Thus putting v0 for the surface velocity at the thread of the stream, and Vs~ for the mean velocity of the whole cross section, V,s/Vo has been found to have the following values: Vm/VO

Dc Prony, experiments on small wooden channels 0.8164

Experiments on the Seine 0~62

Destrem and De Prony, experiments on the Neva 078

Boileau, experiments on canals o~82

Baumgartner, experiments on the Garonne - o~8o Brunings (mean) 0.85

Cunningham, Solani aqueduct 0.823

Various formulae, either empirical or based on some theory of the vertical and horizontal velocity curves, have been proposed for determining the ratio vs/v0. Bazin found from his experiments the emplricai expression v5~V025.4~(mi);

where m is the hydraulic mean depth and i the slope of the stream. In the case of irrigation canals and rivers, it is often important to determine the discharge either daily or at other intervals of time, while the depth and consequently the mean velocity is varying. Cunningham (Roorkee Prof. Papers, iv. 47), has shown that, for a given part of such- a stream, where the bed is regular and of permanent section, a simple formula may be lound for the variation of the central surface velocity with the depth. When once the constants of this formula have been determined by measuring the central surface velocity and depth, in different conditions of the stream, the surface velocity can be obtained by simply observing the depth of the stream, and from this the niean velocity and discharge can he calculated. Let I be the depth of the stream, and v~ the surface velocity, both measured at the thread of the stream. Then v02~cz; where c is a constant which for the Solani aqueduct had the values 1-9 to 2, the depths being 6 to 10 ft., and the velocities 31/2 to 41/2 ft. Without any assumption of a formula, however, the surface velocities, or still better the mean velocities, for different conditions of the stream may be plotted on a diagram in which the abscissae are depths and the ordinates velocities. The continuous curve through points so found would then always give the velocity for any observed depth of the stream, without the need of making any new float or current meter observations.

147. Mean Velocity determined by observing a Series of Surface VelocitiesThe ratio of the mean velocity to the stirface velocity in one longitudinal section is better ascertained than the ratio of the central surface velocity to the mean velocity of the whole cross section. Suppose the river dhided into a number of compartments by equidistant longitudinal planes, and the surface velocity observed in each compartment. From this the mean velocity in each compartment and the discharge can be calculated. The sum of the partial discharges will be the total discharge of the stream. When wires or ropes can be stretched across the stream, the compartments can be marked out by tags attached to them. Suppose two such ropes stretched across the stream, and floats dropped in above the upper rope. By observing within which compartment the path of the float lies, and noting the time of transit between the ropes, the surface velocity in each compartment can be ascertained The mean velocity in each compartment is 0.85 to 0.91 of the surface velocity in that compartment. Putting k for this ratio, and v1, v~. .. for the observed velocities, in compartments of area fZi, lb.. - then the total discharge is Q=k(Qivf+lfzvi+ -. -).

If several floats are allowed to pass over each compartment, the mean of all those correspondingto one compartment is to be taken as the surface velocity of that compartment.

This method is very applicable in the case of large streams or rivers too wide to stret-h a rope across. The paths of the floats are then ascertained in this way. Let fig. 148 represent a portion of the river, which should be straight and free from obstructions.

Suppose a base line AB measured parallel to the thread of the stream, and let the mean cross section of the stream be ascertained either by A a ~ ------E sounding the terminal cross sections AE, BF, or by sounding a series of equidistant cross sections. The / cross sections are taken at right angles to the base line. Observers are placltl at A and B with theu dolites or box sextants. The floats are dropped in from a boat above ,~, AE, and picked up by another boat below BF. An observer with a, chronograph or watch notes the / \ time in which each float passes, ,, from AE to BF. The method of i proceeding is this. The observer B~ F A sets his theodolite in the direc tion AE, and gives a signal to drop a float. B keeps his instrument on the float as it comes down. At FIG. 148 the moment the float arrives at C in the line AE, the observer at A calls out. B clamps his instrument and reads off the angle ABC, and the time observer begins to note the time of transit. B now points his instrument in the direction BF, and A keeps the float on the cross wire of his instrument. At the moment the float arrives at D in the line BF, the observer B calls out, A clamps his Instrument and reads off the angle BAD, and the time observer notes the time of transit from C to D. Thus all the data are determined for plotting the path CD of the float and determining its velocity. By dropping in a series of floats, a number of surface velocities can he determined. When all these have been plotted, the river can be divided into convenient compartments. The observations belonging to each compartment are then averaged, and the mean velocity and discharge calculated. It is obvious that, as the surface velocity is greatly altered by wind, experimente of this kind should be made in very calm weather.

The ratio of the surface velocity to the mean velocity in the same vertical can be ascertained from the formulae for the vertical velocity curve already given (f 101). Exner, in Erbkams Zeitschrift for 1875,

gave the following convenient formula. Let a be the mean and V

the surface velocity in any given vertical longitudinal section, the depth of which is h v/V = (I +O147&V h)/(I +o22I6~/ h).

If vertical velocity rods are used instead of common floats, the mean velocity is directly determined for the vertical section in which the rod floats. No formula of reduction is then necessary. The observed velocity has simply to be multiplied by the area of the compartment to which it belongs.

148. Mean Velocity of the Stream from a Series of Mid Depth VelocitiesIn the gaugings of the Mississippi it was found that the mid depth velocity differed by only a very small quantity from the mean velocity in the vertical section, and it was uninfluenced by wind, If therefore a series of mid depth velocities are determined by double floats or by a current meter, they may be taken to be the mean velocities of the compartments in which they occur, and no formula of reduction is necessary. If floats are used, the method is precisely the same as that described in the last paragraph for surface floats. The paths of the double floats are observed aftd plotted, and the mean taken of those corresponding to each of the compartnients into which the river is divided. The discharge is the sum of the products of the observed mean mid depth velocities and the areas of the compartments.

149. P. P. Boileaus Process for Gauging Streams.Let U be the mean velocity at a given section of a stream, V the maximum velocity, or that of the principal filament, which is generally a little below the surface, W and w the greatest and least velocities at the surface. The distance of the principal filament from the surface is generally less than one-fourth of the depth of the stream; W is a little less than V; and U lies between W and w. As the surface velocities change continuously from the centre towards the sides there are at the surface two filaments having a velocity equal to U. The determination of the position of these filaments, which Boileau terms the gauging filaments, cannot be effected entirely by theory. But, for sections of a stream in which there are no abrupt changes of depth, their position can be very approximately assigned. Let zX and I be the horizontal distances of the surface filament, having the velocity W, from the gauging filament, which has the velocity U, and from the bank on one side. Then ~/l=c4-.,I {(W+2w)/7(W1v)~,

c being a numerical constant. From gaugings by Humphreys and Abbot, Bazin and Baumgarten, the values C =0-919, 0.922 and 0-925 are obtained. Boileau adopts as a mean value 0-922. Hence, if \V and w are determined by float gauging or otherwise, ~ can be found, and then a single velocity observation at ~ ft. from the filament of maximum velocity gives, without need of any reduction, the mean velocity of the stream. More conveniently W, w, and U can be measured from a horizontal surface velocity curve, obtained from a series of float observations.

150. Direct Determination of the Mean Velocity by a Current 7IIeter or Darcy Gauge.The only method of determining the mean velocity at a cross section of a stream which involves no assumption of the ratio of the mean velocity to other quantities is thisa plank bridge is fixed across the stream near its surface. From this, velocities are observed at a sufficient number of points in the cross section of the stream, evenly distributed over its area. The mean of these is the true mean velocity of the stream. In Darcy and Bazins experiments on small streams, the velocity was thus observed at 36 points in the cross section.

When the stream is too large to fix a bridge across it, the observations may be taken from a boat, or from a couple of boats with a gangway between them, anchored successively at a series of points across the width of the stream. The position of the boat for eah series of observations is fixed by angular observations to a base line on shore.

I 51. A. R. Harlachers Graphic Method of determining the Diicharge from a Series of Current Meter ObservationsLet ABC (fig. 149) be the cross section of a river at which a complete series of F1o. 149.

current meter observations have been taken, Let I., II., I~t.. be the verticals at different points of which the velocities were mensored.

Suppose the depths at I., II., III (fig- 14~), set off as vertical ordinates in fig. 150, and on these vertical ordinates suppose the velocities set off horizontally a.t their proper depths. Thus, if v is the measured velocity at the depth h from the surface in fig. 149, on vertical marked III., then at III. in fig. 150 take cd=h and ac~v Then d is a point in the vertical velocity curve for the vertical III., and, all the velocities for that ordinate being similarly set off, the curve can be drawn. Suppose all the vertical velocity curves I.

V. (fig. I5o), thus drawn. On each of these figures draw verticals corresponding to veloci I II Hf W ~ ties of X, 2X, 3X - - - ft.

rr~ rfl-l~ ~ ft~, per second. Then for ~P ~,J) instance cd at III. (fig.

150) is the depth at which a velocity of 2X

-~ ft. per second existed fig. 149 and if cd is set FIG. I5o~ off at Ill, in fig. 149 it gives a point in a curve passing through points of the Section where the velocity was 25 ft. per second. Set off on each of the verticals in fig. 149 all the depths thus found in the corresponding diagram in fig. 150. Curves drawn through the corresponding points on the verticals are curves of equal velocity.

The discharge of the stream per second may be regarded as a solid having the cross section of the river (fig. 149) as a base, and cross Left bank Ma,xtm~urn, surface veIo~

00 ~

- 4~08 4-80 666 7-30 9-24.9.80 11-8212-30 14-4~1480169211

Discha,rge per Secon.

Curves of ecpia Trartsforma-~ioim ra-tio 10:1

~L. ets:ct~i /~

~% FIG

sections normal to the plane of fig. 149 given by the diagrams in fig.

150. The curves of equal velocity may therefore be considered as contour lines of the solid whose volume is the discharge of the stream per second. Let ~ be the area 01 the cross section of the river, ta, the areas contained by the successive curves of equal velocity, or, if these cut the surface of the stream, by the curves and that surface. Let x be the difference of velocity for which the successive curves are drawn, assumed above for simplicity at 1 ft. per second. Then the volume of the successive layers of the solid body whose volume represents the discharge, limited by successive planes passing through the contour curves, will be 1/2x(f7o+P,), 3/4x(t~1+clf), and so on.

Consequently the discharge is Q=x(1/2(f~o+~2,,)+t21=f2i+. -. +s:i,_~l.

The areas ~ fl,. .. are easily ascertained by means of the polai planimeter. A slight difficulty arises in the part of the solid lyinf above the last contour curve. This will have generally a height which is not exactly x, and a form more rounded than the othei layers and less like a conical frustum. The volume of this may bi estimated separately, and taken to be the area of its base (the are~ (L.) multiplied by ~ to 1/2 its height.

Fig. i51 shows the results of one of Harlachers gaugings workec out in this way. The upper figure shows the section of the river and the positions of the verticals at which the soundings and gaugings were taken. The lower gives the curves of equal velocity, worked out from the current meter observations, by the aid of vertical velocity curves. The vertical scale in this figure is ten times as great as in the other. The discharge calculated from the contour curves is 14.1087 cubic metres per second. In the lower figure some other interesting curves are drawn. Thus, the uppermost dotted curve is the curve through points at which the maximum velocity was found; it shows that the maximum velocity was always a little below the surface, and at a greater depth at the centre than at the sides. The next curve shows the depth at which the mean velocity for each vertical was found. The next is the curve of equal velocity corresponding to the mean velocity of the stream; that is, it passes through points in the cross section where the velocity was identical with the mean velocity of the stream.

Categories: HUN-HYS | Engineering