DISCHARGE FROM ORIFICES]
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.
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, 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 ordinarily55
flows over this, without entering it, on a floor of castiron 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 castiron 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.
§g 6o. 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. Society
of Arts, 1876). Fig. 75 shows a valve of this kind suitable for a 6in. 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 O. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a diaphragm, separating A from a chamber B2 connected by a pipe BI with 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 w 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 pw 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/w. 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 s 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 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 lb of air con
fined behind a piston of
I sq. ft. area (fig. 76).
Let the initial pressure
be PI and the volume of
the air vi, and suppose
this to expand to the
pressure p2 and 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 vi to v2, represented by the area abcd, is
(v2 pdv. J vi
Amongst possible cases two may be selected.
Case I.—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
f vlpdv spiv f vidv/v
= plvl log, v2/vi =pith loge pi/p2. (1) Since the weight per cubic foot is the reciprocal of the volume per pound, this may be written
(PI/GI) loge GI/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
Pm' =pivi''+
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 1.135. Work done during expansion per pound of air.
= f vipdv = pivi f vidv/vY
=—{piETY/(Y—1)}{I/v2YI—I/viY—i} = { piviY/ ('y — Olt I /viYi — I /7121i }
={pivi/(7—I)}{I—(vi/v2)yi}. (2) The value of pivi for any given temperature can be found from the data already given.
As before, substituting the weights Gi, G2 per cubic foot for the volumes per pound, we get for the work of expansion
(pi/GI){ 1/(7 1)} {I—(G2/G1)7'1, (2a)
=pivi{I/('y—1)} [I —(p2/pi)tYi>/Y}• (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 pi, wi, vi, GI be the pressure, sectional area of stream, velocity and weight of a cubic foot at A, and p2, w2, v2, G2 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 :
Giwivit s G2w2v2t.
Let zi, z2 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
Giwivit (z1 —Z2) = W (zl — 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
pwivit — p2w2v21,
= (pi/GI —p2/G2) Wt.
The work done by expansion of Wt lb of fluid between A and B is Wt f pdv. The change of kinetic energy as before is (W/2g) (v22—vi2)t.
Hence, equating work to change of kinetic energy,
W(zi—z2)t+ (pi/Gi—p2/G2)Wt+Wt. vlpdv= (W//2g) (v22—vi2)t; z1+pt/G1 +v12/2g = z2 +p2/G2 +v22/2g —J ~Lpdv.
Now the work of expansion per pound of fluid has already given. If the temperature is constant, we get (eq. Ia, § 61)
zl+pl/G1+v12/2g=z2+p2/G2 +v22I2g—(pi/G1) loge (Gi/G2)• But at constant temperature pi/G1=p2/G2;
:. zl +v12/2g=z2 +v22/2g—(p1/GI) loge (p1/p2), or, neglecting the difference of level,
(v22 —v12)/2g = (pi/GI) loge (pi/p2) • (2a)
Similarly, if the expansion is adiabatic (eq. 2a, § 61),
zi+p1/Gi +vi2/2g = z2+p2/G2 +v22/2g — (p1/Gi) { I /(y  i) }
11 — (p2/pl)iY06} ; (3)
or neglecting the difference of level
(v22—v12)/2g=(pi/GI)[I+II(7 —I){—(p2/p1)PY1)/s}1—p2/G2• (3a) It will be seen hereafter that there is a limit in the ratio pi/p2 beyond which these expressions cease to be true.
§ 63. Discharge of Air from an Orifice.—The form of the equation of work for a steady stream of compressible fluid is
zi +Pi/Gi +vi2/2g = z2 +p2/G2 +v22/2g — (pl/GI) { I / (I, — 1) }
A A'
B B'
(I) been
(2)
the expansion being adiabatic, because in the flow of the streams 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 v2 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 ziz2 may be neglected. Putting these values in the equation above
p1/Gi =p2/G2+v22/2g–(pi/G1){1/(T–i)} {I—(p2/Pi)(Y–I)/y;
v22/2g=p1/Gi–p2/G2+ (p1/G1){I/(7–1)} {i –(p2/pi)(Y–1)/Y}
_ (pi/Gi){T/(T 1) —(p2Jpi)Y–1/Y/(T — I)} —p2/G2• Pi/ G17 = p2/G27 .. p2/G2 = (p1/Gi) (p2/pi)(v–1)/Y
v22/2g=(p1/Gi){7/(y — I)} {I—(p2/Pi)(1'—1)/Y}; (I)
or v22/2g={y/(y 1) } ((pi/G1) — (p2/G2));
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. Wantzel.
It has already (§ 9, eq. 4a) been seen that
Pi/Gi = (po/Go) (r1/To)
where for air po=2116.8, Go=•o8o75 and ro=492'6.
v22/2g={ poriy/Goro(1, — 1) } 11 — (P2/P1)(7 — 1)/Y}; (2)
or, inserting numerical values,
v22/2g = 183.6ri{ 1 — (p2/pi) °'29} ; (2a)
which gives the velocity of discharge v2 in terms of the pressure and absolute temperature, pi, ri, in the vessel from which the air flows, and the pressure p2 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 second at the pressure P2 and temperature r2 is
Q2 =cwv2 =cui J [(2gypi/(y– i)G1) (I – (p2/pi)(7–1)/7)] =108.7cw 'I [ri{ I — (p2/p1)°''}] • (3)
If the volume discharged is measured at the pressure pi and absolute temperature r1 in the vessel from which the air flows, let Qi be that volume; then
p1Qi7 =p2Q2Y;
Qi = (p2/pi)1/7Q2 ;
Qi=cw 1/ [{2gyp1/(yI)Gi} 1(p2/pi)2/Y–(p2/pi)(Y+
Let (p2/pi)2/7 —(p2/pi)(7—1)iY=(p2/Pi)1'41—(P2/Pi)1'2=>'; then
Qi 1o8•7cw V/ (ri>p). —I) Gil
(4) The weight of air at pressure pi and temperature r1 is
G1=pi/53.2r1 1b per cubic foot.
Hence the weight of air discharged is
W = G2Qi =cw d [2gtrp1GV/1/(y— I)]
=2'043cwpi J OP/Ti). (5)
Weisbach found the following values of the coefficient of discharge c :
Conoidal mouthpieces of the form of thel contracted vein with effective pressures c=
of •23 to 1.1 atmosphere . . . 0.97 to 0.99
Circular sharpedged orifices 0.563 „ 0.788
Short cylindrical mouthpieces . . o•81 „ 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 P2, while passing from the vessel to the section of the jet considered in estimating the area w. Hence p2 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 p2 was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio p2/pi exceeded a value which does not greatly differ from o.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 amounts to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area c.o., 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 as p2/pi>o•5, the formulae above were trustworthy, when p2 was taken to be the general external pressure, but that, if p2/pi 

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