(1)Fig. 58.where the coefficient on the right is evidently the coefficient of velocity for the cylindrical mouthpiece in terms of the coefficient of contraction at EF. Let cc= 0.64, the value for simple orifices, then the coefficient of velocity iscv= 1/√ {1 + (1/cc− 1)2} = 0.87(2)The actual value of cv, found by experiment is 0.82, which does not differ more from the theoretical value than might be expected if the friction of the mouthpiece is allowed for. Hence, for mouthpieces of this kind, and for the section at GH,cv= 0.82 cc= 1.00 c = 0.82,Q = 0.82Ω √(2gh).It is easy to see from the equations that the pressure p at EF is less than atmospheric pressure. Eliminating v1, we get(pa− p)/G =3⁄4h nearly;(3)orp = pa−3⁄4Gh ℔ 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 heightKL = (pa− p) / G =3⁄4h 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 Mouthpieces.—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 formQ = cvccΩ √(2gh),(4)where Ω is the area of the external end of the mouthpiece, and ccΩ the section of the contracted jet beyond the mouthpiece.Convergent Mouthpieces (Castel’s Experiments).—Smallest diameter of orifice = 0.05085 ft. Length of mouthpiece = 2.6 Diameters.Angle ofConvergence.Coefficient ofContraction,ccCoefficient ofVelocity,cvCoefficient ofDischarge,c0° 0′.999.830.8291° 36′1.000.866.8663° 10′1.001.894.8954° 10′1.002.910.9125° 26′1.004.920.9247° 52′.998.931.9298° 58′.992.942.93410° 20′.987.950.93812° 4′.986.955.94213° 24′.983.962.94614° 28′.979.966.94116° 36′.969.971.93819° 28′.953.970.92421° 0′.945.971.91823° 0′.937.974.91329° 58′.919.975.89640° 20′.887.980.86948° 50′.861.984.847The maximum coefficient of discharge is that for a mouthpiece with a convergence of 13°24′.Fig. 59.Fig. 60.The values of cvand ccmust 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, ccdiminishes, and on the other hand the loss of energy diminishes, so that cvincreases with the convergence, there is an angle for which the product cccv, and consequently the discharge, is a maximum.§ 51.Divergent Conoidal Mouthpiece.—Suppose 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 inner end approximately 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 ω, v, p be the section, velocity and pressure at CD, and Ω, v1, p1the same quantities at EF, pabeing 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 surface of the mouthpiece,h + pa/G = v2/2g + p/G = v12/2g + p1/G.If the jet discharges into the air, p1= pa; andv12/2g = h;v1= √(2gh);or, if a coefficient is introduced to allow for friction,v1= cv√(2gh);where cvis about 0.97 if the mouthpiece is smooth and well formed.Q = Ω v1= cvΩ √(2gh).Fig. 61.Hence the discharge depends on the area of the stream at EF, and not at all on that at CD, and the latter may be made as small as we please without 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 = 0, 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 = pa/G − (v2− v12) / 2g.Let Ω/ω = m. Thenv = v1m;p/G = pa/G − v12(m2− 1) / 2g= pa/G − (m2− 1) h;whence we find that p/G will become zero or negative ifΩ/ω ≥ √ {(h + pa/G) / h } = √ {1 + pa/Gh};or, putting pa/G = 34 ft., ifΩ/ω ≥ √ { (h + 34)/h}.In practice there will be an interruption of the full bore flow with a less ratio of Ω/ω, because of the disengagement of air from the water. But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind isQ = ω √ {2g (h + pa/G) };that is, the discharge is the same as for a well-bell-mouthed mouthpiece of area ω, and without the expanding part, discharging into a vacuum.§ 52.Jet Pump.—A 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, and its pressure increases, till it is sufficient to balance the head due to the height of the lift, and the water flows away by the discharge pipe C.Fig. 62.Fig. 63 shows the whole arrangement in a diagrammatic way. A is the reservoir which supplies the water that effects the pumping; B is the reservoir of water to be pumped; C is the reservoir into which the water is pumped.Fig. 63.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.Fig. 64.Time of Emptying or Filling a Vertical-sided Lock Chamber.—Suppose the lock chamber, which has a water surface of Ω square ft., is emptied through a sluice in the tail gates, of area ω, 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 of level after t seconds. Let −dh be the fall of level in the chamber during an interval dt. Then in the time dt the volume in the chamber is altered by the amount −Ωdh, and the outflow from the sluice in the same time is cω √(2gh) dt. Hence the differential equation connecting h and t iscω √(2gh) dt + Ωh = 0.For the time t, during which the initial head H diminishes to any other value h,−{Ω/(cω √2g) }∫hHdh/√h =∫0tdt.∴ t = 2Ω (√H − √h) / {cω √(2g)}= (Ω/cω) {√(2H/g) − √(2h/g) }.For the whole time of emptying, during which h diminishes from H to 0,T = (Ω/cω) √(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 time 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§ 54. 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 amount 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.Water Inch.—For 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 top 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 Navier’s edition ofBelidor’s Arch. Hydr., 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 of the orifice. Several equal orifices may be used for larger discharges.Fig. 65.Pin 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 is interesting here chiefly as an example of regulation of discharge by means of an orifice. Fig. 65 shows a cock of this kind used at Zurich. It consists of three cocks, the middle one having the orifice of the predetermined size in a small circular plate, protected by wire gauze from stoppage by impurities in the water. The cock on the right hand can be used by the consumer for emptying 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 of 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 if at any moment the head is h, the discharge is, for a rectangular notch of breadth b,Q =2⁄3cbh √2ghwhere 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 about1⁄10th in. thick to secure the requisite accuracy of form and sharpness of edge. 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 of 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.Fig. 66.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 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 graduated 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 = 1, 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.Fig. 67.Fig. 68.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 the principle of the operation. Apart from the difficulty of selecting a suitable coefficient of discharge, which need not be serious if the form of the weir and the nature of its crest are properly attended to, 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 screw. If the hook is depressed below the water 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 .001 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 shouldfirst 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 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, notwithstanding the variation of level in the irrigating channel.Fig. 69.§ 57.Italian Module.—The 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.Fig. 70.—Scale1⁄100.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 adjusted it is locked. Let ω1be the area of the orifice through the sluice at A, and ω2that of the fixed orifice at B; let h1be the difference of level between the surface of the water in the canal and regulating chamber; h2the 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, since the flow through the orifices at A and B is the same,Q = c1ω1√(2gh1) = c2ω2√(2gh2),where c1and c2are the coefficients of discharge suitable for the two orifices. Hencec1ω1/ c2ω2= √(h2/h1).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 height, 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 1.12 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.Fig. 71.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 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 Module.—On 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 irrigation, and as it is very valuable its strict measurement is essential. The module (fig. 72) consists of two 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 descends and gives an enlarged opening, and conversely. Thus a perfectly constant discharge with a varying head can be obtained, provided no clogging or silting of the chambers prevents 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. ThenQ = cπ (R2− r2) √(2gh).Taking c = 0.63,Q = 15.88 (R2− r2) √h;r = √ {R2− Q/15.88 √h}.Choosing a value for R, successive values of r can be found for different values of h, and from these the curve of the plug can be drawn. The module shown in fig. 72 will discharge 1 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-Moncrieff’sIrrigation in Southern Europe.§ 59.Reservoir Gauging Basins.—In obtaining the power to store the water of streams in reservoirs, it is usual to concede to riparianowners 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.Fig. 72.Fig. 74 shows an arrangement designed for the Manchester water works. The water enters from the reservoiratchamber A, the object of which is to still the irregular motion of the water. The admission is regulated by sluices at b, b, b. The water is discharged by orifices or notches at a, a, over which a tolerably constant head is maintained by adjusting the sluices at b, b, b. 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 definite time into 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 chamber. A mechanical arrangement (fig. 73) was designed for securing an absolutely constant head over the orifices at a, a. The orifices were formed in a cast-iron plate capable of sliding up and 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 mechanical arrangement was not finally adopted, careful watching of the sluices at b, b, b, being sufficient to secure a regular discharge. The arrangement is then equivalent to an Italian module, but on a large scale.Fig. 73.—Scale1⁄120.Fig. 74.—Scale1⁄500.§ 60.Professor Fleeming Jenkin’s Constant Flow Valve.—In the modules thus 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. Societyof Arts, 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 O, 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 O. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a diaphragm, separating A from a chamber B2connected by a pipe B1with the discharging main B. Any increase of the difference of pressure in A and B will drive the plunger up and close the 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 ω be the area of the plunger in square feet, p the difference of pressure in the chambers A and B in pounds per square foot, w the weight of the plunger and valve. Then if at any moment pω 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 ω and w are constant, p must be constant also, and equal to w/ω. By making w small and ω 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 of1⁄2in. of water have been constructed.Fig. 75.—Scale1⁄24.VI. STEADY FLOW OF COMPRESSIBLE FLUIDS.Fig. 76.§ 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 conclusion that, however air changes its state, the internal work done is proportional to the change of temperature. When, in expanding, air does work against an external resistance, either heat must be supplied or the temperature falls.To fix the conditions, suppose 1 ℔ of air confined behind a piston of 1 sq. ft. area (fig. 76). Let the initial pressure be p1and the volume of the air v1, and suppose this to expand to the pressure p2and volume v2. 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 v1to v2, represented by the area abcd, is∫v2v1p dv.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.Thenpv = p1v1.Work done during expansion per pound of air=∫v2v1p dv = p1v1∫v2v1dv/v= p1v1logεv2/ v1= p1v1logεp1/ p2.(1)Since the weight per cubic foot is the reciprocal of the volume per pound, this may be written(p1/G1) logεG1/G2.(1a)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 thatpvγ= p1v1γ,where γ 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 1.135.Work done during expansion per pound of air.=∫v2v1p dv = p1v1γ∫v2v1dv/vγ= −{p1v1γ/ (γ − 1)} {1/v2γ−1− 1/v1γ−1}= {p1v1γ/ (γ − 1)} {1/v1γ−1− 1/v2γ−1}= {p1v1/ (γ − 1)} {1 − (v1/v2)γ−1}.(2)The value of p1v1for any given temperature can be found from the data already given.As before, substituting the weights G1, G2per cubic foot for the volumes per pound, we get for the work of expansion(p1/G1) {1/(γ − 1)} {1 − (G2/G1)γ−1},(2a)= p1v1{1/(γ − 1)} {1 − (p2/p1)γ−1/γ}.(2b)Fig. 77.§ 62.Modification of the Theorem of Bernoulli for the Case of a Compressible 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 taken into account. Suppose, as before, that AB (fig. 77) comes to A′B′ in a short time t. Let p1, ω1, v1, G1be the pressure, sectional area of stream, velocity and weight of a cubic foot at A, and p2, ω2, v2, G2the 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:G1ω1v1t = G2ω2v2t.Let z1, z2be the heights of the sections A and B above any given datum. Then the work of gravity on the mass AB in t seconds isG1ω1v1t (z1− z2) = W (z1− z2) 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 isp1ω1v1t − p2ω2v2t,= (p1/G1− p2/G2) Wt.The work done by expansion of Wt ℔ of fluid between A and B is∫v2v1The change of kinetic energy as before is (W/2g) (v22− v12) t. Hence, equating work to change of kinetic energy,W (z1− z2) t + (p1/G1− p2/G2)Wt + Wt∫v2v1p dv = (W/2g) (v22− v12) t;∴ z1+ p1/G1+ v12/2g = z2+ p2/G2+ v22/2g −∫v2v1p dv.
(1)
where the coefficient on the right is evidently the coefficient of velocity for the cylindrical mouthpiece in terms of the coefficient of contraction at EF. Let cc= 0.64, the value for simple orifices, then the coefficient of velocity is
cv= 1/√ {1 + (1/cc− 1)2} = 0.87
(2)
The actual value of cv, found by experiment is 0.82, which does not differ more from the theoretical value than might be expected if the friction of the mouthpiece is allowed for. Hence, for mouthpieces of this kind, and for the section at GH,
cv= 0.82 cc= 1.00 c = 0.82,
Q = 0.82Ω √(2gh).
It is easy to see from the equations that the pressure p at EF is less than atmospheric pressure. Eliminating v1, we get
(pa− p)/G =3⁄4h nearly;
(3)
or
p = pa−3⁄4Gh ℔ 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 = (pa− p) / G =3⁄4h 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 Mouthpieces.—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 = cvccΩ √(2gh),
(4)
where Ω is the area of the external end of the mouthpiece, and ccΩ the section of the contracted jet beyond the mouthpiece.
Convergent Mouthpieces (Castel’s Experiments).—Smallest diameter of orifice = 0.05085 ft. Length of mouthpiece = 2.6 Diameters.
The maximum coefficient of discharge is that for a mouthpiece with a convergence of 13°24′.
The values of cvand ccmust 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, ccdiminishes, and on the other hand the loss of energy diminishes, so that cvincreases with the convergence, there is an angle for which the product cccv, and consequently the discharge, is a maximum.
§ 51.Divergent Conoidal Mouthpiece.—Suppose 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 inner end approximately 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 ω, v, p be the section, velocity and pressure at CD, and Ω, v1, p1the same quantities at EF, pabeing 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 surface of the mouthpiece,
h + pa/G = v2/2g + p/G = v12/2g + p1/G.
If the jet discharges into the air, p1= pa; and
v12/2g = h;
v1= √(2gh);
or, if a coefficient is introduced to allow for friction,
v1= cv√(2gh);
where cvis about 0.97 if the mouthpiece is smooth and well formed.
Q = Ω v1= cvΩ √(2gh).
Hence the discharge depends on the area of the stream at EF, and not at all on that at CD, and the latter may be made as small as we please without 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 = 0, 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 = pa/G − (v2− v12) / 2g.
Let Ω/ω = m. Then
v = v1m;
p/G = pa/G − v12(m2− 1) / 2g
= pa/G − (m2− 1) h;
whence we find that p/G will become zero or negative if
Ω/ω ≥ √ {(h + pa/G) / h } = √ {1 + pa/Gh};
or, putting pa/G = 34 ft., if
Ω/ω ≥ √ { (h + 34)/h}.
In practice there will be an interruption of the full bore flow with a less ratio of Ω/ω, 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 = ω √ {2g (h + pa/G) };
that is, the discharge is the same as for a well-bell-mouthed mouthpiece of area ω, and without the expanding part, discharging into a vacuum.
§ 52.Jet Pump.—A 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, and its pressure increases, till it is sufficient to balance the head due to the height of the lift, and the water flows away by 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; 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 Ω square ft., is emptied through a sluice in the tail gates, of area ω, 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 of level after t seconds. Let −dh be the fall of level in the chamber during an interval dt. Then in the time dt the volume in the chamber is altered by the amount −Ωdh, and the outflow from the sluice in the same time is cω √(2gh) dt. Hence the differential equation connecting h and t is
cω √(2gh) dt + Ωh = 0.
For the time t, during which the initial head H diminishes to any other value h,
−{Ω/(cω √2g) }∫hHdh/√h =∫0tdt.
∴ t = 2Ω (√H − √h) / {cω √(2g)}
= (Ω/cω) {√(2H/g) − √(2h/g) }.
For the whole time of emptying, during which h diminishes from H to 0,
T = (Ω/cω) √(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 time 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
§ 54. 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 amount 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.
Water Inch.—For 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 top 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 Navier’s edition ofBelidor’s Arch. Hydr., 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 of the orifice. Several equal orifices may be used for larger discharges.
Pin 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 is interesting here chiefly as an example of regulation of discharge by means of an orifice. Fig. 65 shows a cock of this kind used at Zurich. It consists of three cocks, the middle one having the orifice of the predetermined size in a small circular plate, protected by wire gauze from stoppage by impurities in the water. The cock on the right hand can be used by the consumer for emptying 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 of 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 if at any moment the head is h, the discharge is, for a rectangular notch of breadth b,
Q =2⁄3cbh √2gh
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 about1⁄10th in. thick to secure the requisite accuracy of form and sharpness of edge. 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 of 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 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 graduated 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 = 1, 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 the principle of the operation. Apart from the difficulty of selecting a suitable coefficient of discharge, which need not be serious if the form of the weir and the nature of its crest are properly attended to, 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 screw. If the hook is depressed below the water 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 .001 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 shouldfirst 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 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, notwithstanding the variation of level in the irrigating channel.
§ 57.Italian Module.—The 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 adjusted it is locked. Let ω1be the area of the orifice through the sluice at A, and ω2that of the fixed orifice at B; let h1be the difference of level between the surface of the water in the canal and regulating chamber; h2the 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, since the flow through the orifices at A and B is the same,
Q = c1ω1√(2gh1) = c2ω2√(2gh2),
where c1and c2are the coefficients of discharge suitable for the two orifices. Hence
c1ω1/ c2ω2= √(h2/h1).
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 height, 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 1.12 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 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 Module.—On 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 irrigation, and as it is very valuable its strict measurement is essential. The module (fig. 72) consists of two 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 descends and gives an enlarged opening, and conversely. Thus a perfectly constant discharge with a varying head can be obtained, provided no clogging or silting of the chambers prevents 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 = cπ (R2− r2) √(2gh).
Taking c = 0.63,
Q = 15.88 (R2− r2) √h;
r = √ {R2− Q/15.88 √h}.
Choosing a value for R, successive values of r can be found for different values of h, and from these the curve of the plug can be drawn. The module shown in fig. 72 will discharge 1 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-Moncrieff’sIrrigation in Southern Europe.
§ 59.Reservoir Gauging Basins.—In obtaining the power to store the water of streams in reservoirs, it is usual to concede to riparianowners 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.
Fig. 74 shows an arrangement designed for the Manchester water works. The water enters from the reservoiratchamber A, the object of which is to still the irregular motion of the water. The admission is regulated by sluices at b, b, b. The water is discharged by orifices or notches at a, a, over which a tolerably constant head is maintained by adjusting the sluices at b, b, b. 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 definite time into 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 chamber. A mechanical arrangement (fig. 73) was designed for securing an absolutely constant head over the orifices at a, a. The orifices were formed in a cast-iron plate capable of sliding up and 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 mechanical arrangement was not finally adopted, careful watching of the sluices at b, b, b, being sufficient to secure a regular discharge. The arrangement is then equivalent to an Italian module, but on a large scale.
§ 60.Professor Fleeming Jenkin’s Constant Flow Valve.—In the modules thus 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. Societyof Arts, 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 O, 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 O. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a diaphragm, separating A from a chamber B2connected by a pipe B1with the discharging main B. Any increase of the difference of pressure in A and B will drive the plunger up and close the 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 ω be the area of the plunger in square feet, p the difference of pressure in the chambers A and B in pounds per square foot, w the weight of the plunger and valve. Then if at any moment pω 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 ω and w are constant, p must be constant also, and equal to w/ω. By making w small and ω 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 of1⁄2in. 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 conclusion that, however air changes its state, the internal work done is proportional to the change of temperature. When, in expanding, air does work against an external resistance, either heat must be supplied or the temperature falls.
To fix the conditions, suppose 1 ℔ of air confined behind a piston of 1 sq. ft. area (fig. 76). Let the initial pressure be p1and the volume of the air v1, and suppose this to expand to the pressure p2and volume v2. 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 v1to v2, represented by the area abcd, is
∫v2v1p dv.
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 = p1v1.
Work done during expansion per pound of air
=∫v2v1p dv = p1v1∫v2v1dv/v
= p1v1logεv2/ v1= p1v1logεp1/ p2.
(1)
Since the weight per cubic foot is the reciprocal of the volume per pound, this may be written
(p1/G1) logεG1/G2.
(1a)
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
pvγ= p1v1γ,
where γ 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 1.135.
Work done during expansion per pound of air.
=∫v2v1p dv = p1v1γ∫v2v1dv/vγ
= −{p1v1γ/ (γ − 1)} {1/v2γ−1− 1/v1γ−1}= {p1v1γ/ (γ − 1)} {1/v1γ−1− 1/v2γ−1}= {p1v1/ (γ − 1)} {1 − (v1/v2)γ−1}.
= −{p1v1γ/ (γ − 1)} {1/v2γ−1− 1/v1γ−1}
= {p1v1γ/ (γ − 1)} {1/v1γ−1− 1/v2γ−1}
= {p1v1/ (γ − 1)} {1 − (v1/v2)γ−1}.
(2)
The value of p1v1for any given temperature can be found from the data already given.
As before, substituting the weights G1, G2per cubic foot for the volumes per pound, we get for the work of expansion
(p1/G1) {1/(γ − 1)} {1 − (G2/G1)γ−1},
(2a)
= p1v1{1/(γ − 1)} {1 − (p2/p1)γ−1/γ}.
(2b)
§ 62.Modification of the Theorem of Bernoulli for the Case of a Compressible 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 taken into account. Suppose, as before, that AB (fig. 77) comes to A′B′ in a short time t. Let p1, ω1, v1, G1be the pressure, sectional area of stream, velocity and weight of a cubic foot at A, and p2, ω2, v2, G2the 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:
G1ω1v1t = G2ω2v2t.
Let z1, z2be 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
G1ω1v1t (z1− z2) = W (z1− z2) 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
p1ω1v1t − p2ω2v2t,
= (p1/G1− p2/G2) Wt.
The work done by expansion of Wt ℔ of fluid between A and B is∫v2v1The change of kinetic energy as before is (W/2g) (v22− v12) t. Hence, equating work to change of kinetic energy,
W (z1− z2) t + (p1/G1− p2/G2)Wt + Wt∫v2v1p dv = (W/2g) (v22− v12) t;
∴ z1+ p1/G1+ v12/2g = z2+ p2/G2+ v22/2g −∫v2v1p dv.