pv^[gamma] = p1v1^[gamma],
where [gamma] 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.
_ _
/v2 /v2
= | p dv = p1v1^[gamma] | dv/v^[gamma]
_/v1 _/v1
= - {p1v1^[gamma]/([gamma] - 1)} {1/v2^([gamma] - 1) - 1/v1^([gamma] - 1)}
= {p1v1^[gamma]/([gamma] - 1)} {1/v1^([gamma] - 1) - 1/v2^([gamma] - 1)}
= {p1v1/([gamma] - 1)} {1 - (v1/v2)^([gamma] - 1)}. (2)
The value of p1v1 for any given temperature can be found from the data
already given.
As before, substituting the weights G1, G2 per cubic foot for the
volumes per pound, we get for the work of expansion
(p1/G1){1/([gamma] - 1)} {1 - (G2/G1)^([gamma] - 1)}, (2a)
= p1v1{1/([gamma] - 1)} {1 - (p2/p1)^([gamma] - 1)/[gamma]}. (2b)
[Illustration: 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, [omega]1, v1, G1 be the pressure, sectional area
of stream, velocity and weight of a cubic foot at A, and p2, [omega]2,
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:
G1[omega]1v1t = G2[omega]2v2t.
Let z1, 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
G1[omega]1v1t(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
p1[omega]1v1t - p2[omega]2v2t,
= (p1/G1 - p2/G2)Wt.
The work done by expansion of Wt lb. of fluid between A and B is Wt
[int][v1 to v2] p dv. The change of kinetic energy as before is (W/2g)
(v2² - v1²)t. Hence, equating work to change of kinetic energy,
_
/v2
W(z1 - z2)t + (p1/G1 - p2/G2)Wt + | p dv = (W/2g)(v2² - v1²)t;
_/v1
_
/v2 /
.: z1 + p1/G1 + v1²/2g = z2 + p²/G2 + v2²/2g - | p dv. (1)
_/v1
Now the work of expansion per pound of fluid has already been given.
If the temperature is constant, we get (eq. 1a, § 61)
z1 + p1/G1 + v1²/2g
= z2 + p²/G2 + v2²/2g - (p1/G1) log_[epsilon] (G1/G2).
But at constant temperature p1/G1 = p2/G2;
.: z1 + v1²/2g = z2 + v2²/2g - (p1/G1) log_[epsilon] (p1/p2), (2)
or, neglecting the difference of level,
(v2² - v1²)/2g = (p1/G1) log_[epsilon] (p1/p2). (2a)
Similarly, if the expansion is adiabatic (eq. 2a, § 61),
z1 + p1/G1 + v1²/2g = z2 + p2/G2 + v2²/2g
- (p1/G1){1/([gamma] - 1)} {1 - (p2/p1)^([gamma] - 1)/[gamma]}; (3)
or, neglecting the difference of level,
(v2² - v1²)/2g =
(p1/G1)[1 + 1/([gamma] - 1){1 - (p2/p1)^([gamma]-1)/[gamma]}] - p2/G2. (3a)
It will be seen hereafter that there is a limit in the ratio p1/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
z1 + p1/G1 + v1²/2g = z2 + p2/G2 + v2²/2g -
(p1/G1){1/([gamma] - 1)} {1 - (p2/p1^([gamma] - 1)/[gamma]},
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 p1 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 z1z2 may be
neglected. Putting these values in the equation above--
p1/G1 = p2/G2 + v2²/2g - (p1/G1){1/([gamma] - 1)}
{1 - (p2/p1)^([gamma] - 1)/[gamma];
v2²/2g = p1/G1 - p2/G2 + (p1/G1){1/([gamma] - 1)}
{1 - (p2/p1)^([gamma] - 1)/[gamma]}
= (p1/G1){[gamma]/([gamma] - 1) - (p2/p1)^([gamma] - 1)/[gamma]/([gamma] - 1)} - p2/G2.