Page 98 - 48Fundamentals of Compressible Fluid Mechanics

60 CHAPTER 4. ISENTROPIC VARIABLE AREA FLOW

of the isothermal model are not constant. Again, the heat transfer is expressed as
(4.82)

For comparison between
the adiabatic model and the Comperison between the two models
isothermal a simple profile of k = 1 4
nozzle area as a function of the
1
distance is assumed. This pro-
file isn’t ideal profile but rather P / P 0 isentropic
0.8 T / T
a simple sample just to exam- P / P 0 isentropic
0 isothermal
ine the difference between the T/T
0.6 0 isothermal
two models so in actual situa-
tion can be bounded. To make
0.4
senses and eliminate unneces-
sary details the distance from
0.2
the entrance to the throat is nor-
malized (to one). In the same
0
fashion the distance from the 0 0.5 1 1.5 2
Distance (normalized distance two scales)
throat to the exit is normalized
(to one) (it isn’t mean that these Fri Apr 8 15:11:44 2005
distances are the same). In this
comparison the entrance area Fig. 4.8: Comparison of the pressure and temperature
ratio and the exit area ratio are drop as a function of the normalized length
(two scales)
the same and equal to 20. The
Mach number was computed for the two models and plotted in the Figure (4.7(b)).
In this comparison it has to be remembered that critical area for the two models are
different by about 3% (for ). As can be observed from the Figure (4.7(b)).
The Mach number for the isentropic is larger for the supersonic branch but the
velocity is lower. The ratio of the velocities can be expressed as

(4.83)

It can be noticed that temperature in the isothermal model is constant while tem-

perature in the adiabatic model can be expressed as a function of the stagnation

temperature. The initial stagnation temperatures are almost the same and can be
canceled out to obtain
(4.84)

Utilizing equation (4.84) the velocity ratio was obtained is plotted in Figure 4.7(b).

%

Thus, using the isentropic model results in under prediction of the actual

results for the velocity in the supersonic branch. While, the isentropic for the sub-