Page 368 - Bird R.B. Transport phenomena

350 Chapter 11 The Equations of Change for Nonisothermal Systems

When the momentum flux т and the heat flux q are zero, there is no change in entropy
following an element of fluid (see Eq. 11D.1-3). Hence the derivative d In p/d In T = 7/(7 - 1)
following the fluid motion has to be understood to mean (d In p/d In T) s = y/(y - 1). This
equation is a standard formula from equilibrium thermodynamics.

EXAMPLE 11.4-7 We consider here the adiabatic expansion 6 10 of an ideal gas through a convergent-divergent
nozzle under such conditions that a stationary shock wave is formed. The gas enters the noz-
One-Dimensional zle from a reservoir, where the pressure is p and discharges to the atmosphere, where the
0/
Compressible Flow: pressure is p . In the absence of a shock wave, the flow through a well-designed nozzle is vir-
a
Velocity, Temperature, tually frictionless (hence isentropic for the adiabatic situation being considered). If, in addi-
and Pressure Profiles tion, p /po is sufficiently small, it is known that the flow is essentially sonic at the throat (the
a
in a Stationary Shock region of minimum cross section) and is supersonic in the divergent portion of the nozzle.
Wave Under these conditions the pressure will continually decrease, and the velocity will increase in
the direction of the flow, as indicated by the curves in Fig. 11.4-4.
However, for any nozzle design there is a range of p /p 0 for which such an isentropic
a
flow produces a pressure less than p a at the exit. Then the isentropic flow becomes unstable.
The simplest of many possibilities is a stationary normal shock wave, shown schematically in
the Fig. 11.4-4 as a pair of closely spaced parallel lines. Here the velocity falls off very rapidly

IV = Po = Pa
Entering \v = v 0
gas ] p =

Nozzle

Pressure p
' Isentropic path

4
Isentropic path
Mach
number 1.0
Ma 2

Distance

Fig. 11.4-4. Formation of a shock wave in a nozzle.

H. W. Liepmann and A. Roshko, Elements of Gas Dynamics, Wiley, New York (1957), §§5.4 and 13.13.
6
J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York,
7
2nd corrected printing (1964), pp. 791-797.
8
M. Morduchow and P. A. Libby, /. Aeronautical Sci., 16, 674-684 (1948).
R. von Mises, /. Aeronautical Sci., 17, 551-554 (1950).
9
G. S. S. Ludford, /. Aeronautical Sci., 18, 830-834 (1951).
10