Chapter 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 confined behind a piston
of 1 sq. ft. area (fig. 76). Let the initial pressure be p1 and the
volume of the air v1, 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 v1 to v2, represented by the area abcd, is
_
/v2
| p dv.
_/v1

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
_ _
/v2 /v2
= | p dv = p1v1 | dv/v
_/v1 _/v1

= p1v1 log_[epsilon] v2v1 = p1v1 log_[epsilon] p1p2. (1)

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

(p1/G1) log_[epsilon] G1/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

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)

§ 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.

But

p1/G1^([gamma]) = p2/G2^([gamma])
.: p2/G2 = (p1/G1)(p2/p1)^([gamma] - 1)/[gamma]

v2²/2g = (p1/G1){[gamma]/([gamma] - 1)} {1 - (p2/p1)^(([gamma] - 1)/[gamma]}; (1)

or

v2²/2g = {[gamma]/([gamma] - 1)} {(p1/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

p1/G1 = (p0/G0) ([tau]1/[tau]0)

where for air p0 = 2116.8, G0 = .08075 and [tau]0 = 492.6.

v2²/2g = {p0[tau]1[gamma]/G0[tau]0([gamma] - 1)}
{1 - (p2/p1)^([gamma] - 1)/[gamma]}; (2)

or, inserting numerical values,

v2²/2g = 183.6[tau]1 {1 - (p2/p1)^(0.29)}; (2a)

which gives the velocity of discharge v2 in terms of the pressure and
absolute temperature, p1, [tau]1, 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 [omega] 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 [tau]2 is

Q2 = c[omega]v2 = c[omega] [root][(2g[gamma]p1/([gamma] - 1)G1)
(1 - (p2/p1)^([gamma] - 1)/[gamma])]

= 108.7c[omega] [root][[tau]1 {1 - (p2/p1)^(0.29)}]. (3)

If the volume discharged is measured at the pressure p1 and absolute
temperature [tau]1 in the vessel from which the air flows, let Q1 be
that volume; then

p1Q1^[gamma] = p2Q2^[gamma];

Q1 = (p2/p1)^(1/[gamma]) Q2;

Q1 = c[omega] [root][{2g[gamma]p1/([gamma] - 1)G1}
{(p2/p1)^(2/[gamma]) - (p2/p1)^([gamma] + 1)/[gamma]}].

Let

(p2/p1)^(2/[gamma]) - (p2/p1)^([gamma] - 1)/[gamma] =
(p2/p1)^(1.41) - (p2/p1)^(1.7) = [psi]; then

Q1 = c[omega] [root][2g[gamma]p1[psi]/([gamma] - 1)G1]
= 108.7c[omega] [root]([tau]1[psi]). (4)

The weight of air at pressure p1 and temperature [tau]1 is

G1 = p1/53.2[tau]1 lb. per cubic foot.

Hence the weight of air discharged is

W = G1Q1 = c[omega] [root][2g[gamma]p1G1[psi]/([gamma] - 1)]
= 2.043c[omega]p1 [root]([psi]/[tau]1). (5)

Weisbach found the following values of the coefficient of discharge
c:--

Conoidal mouthpieces of the form of the \
contracted vein with effective > c =
pressures of .23 to 1.1 atmosphere / 0.97 to 0.99
Circular sharp-edged orifices 0.563 " 0.788
Short cylindrical mouthpieces 0.81 " 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 p1 to the pressure p2, while passing from
the vessel to the section of the jet considered in estimating the area
[omega]. 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/p1 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
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_c[omega] of
the contracted section of the jet, and may be greater than the area
[omega] of the orifice. Napier made experiments with steam which
showed that, so long as p2/p1 > 0.5, the formulae above were
trustworthy, when p2 was taken to be the general external pressure,
but that, if p2/p1 < 0.5, then the pressure at the contracted section
was independent of the external pressure and equal to 0.5p1. Hence in
such cases the constant value 0.5 should be substituted in the
formulae for the ratio of the internal and external pressures p2/p1.

It is easily deduced from Weisbach's 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/p1 = {2/([gamma] + 1)}^([gamma] - 1/[gamma])
= 0.527 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 Weisbach's
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 (_Civilingenieur_ xx., 1874) that for air
flowing from well-rounded mouthpieces there is no discontinuity of the
law of flow, as Napier's hypothesis implies, but the curve of flow
bends so sharply that Napier's rule may be taken to be a good
approximation to the true law. The limiting value of the ratio p2/p1,
for which Weisbach's formula, as originally understood, ceases to
apply, is for air 0.5767; and this is the number to be substituted for
p2/p1 in the formulae when p2/p1 falls below that value. For later
researches on the flow of air, reference may be made to G. A. Zeuner's
paper (_Civilingenieur_, 1871), and Fliegner's papers (_ibid._, 1877,
1878).