Page 99 - 48Fundamentals of Compressible Fluid Mechanics

4.4. ISENTROPIC ISOTHERMAL FLOW NOZZLE 61

sonic branch will be over prediction. The prediction of the Mach number are similar
shown in the Figure 4.7(b).
Two other ratios need to be examined: temperature and pressure. The

can be obtained via the isentropic model as

(4.85)

initial stagnation temperature is denoted as
. The temperature ratio of

While the temperature ratio of the isothermal model is constant and equal to one

%

(1). The pressure ratio for the isentropic model is

(4.86)

and for the isothermal process the stagnation pressure varies and had to be taken

%
into account as following

(4.87)

where the z is an arbitrary point on the nozzle. Utilizing equations (4.73) and the

isentropic relationship provides the sought ratio.

Figure 4.8 shows that the range between the predicted temperatures of the
two models is very large. While the range between the predicted pressure by the

two models is relatively small. The meaning of this analysis is that transfered heat
affects the temperature in larger degree but the effect on the pressure much less
significant.
To demonstrate relativity of the approached of advocated in this book con-
sider the following example.
Example 4.5:
Consider a diverging–converging nozzle made out wood (low conductive material)
with exit area equal entrance area. The throat area ratio to entrance area 1:4
and the stagnation temperature
. Assume that the back pressure is low enough to have supersonic flow

without shock and . Calculate the velocity at the exit using the adiabatic
model? If the nozzle was made from copper (a good heat conductor) a larger heat
transfer occurs, should the velocity increase or decrease? what is the maximum

respectively. The stagnation pressure is 5
possible increase?
is 27
/
SOLUTION
The first part of the question deals with the adiabatic model i.e. the conservation of
the stagnation properties. Thus, with known area ratio and known stagnation the
GDC-Potto provides the following table: