Page 85 - 48Fundamentals of Compressible Fluid Mechanics

4.2. ISENTROPIC CONVERGING-DIVERGING FLOW IN CROSS SECTION 47

and for perfect gas

(4.37)

Thus, the temperature varies according in the same way that Pressure does.

The relationship between the Mach number and the temperature can be

.
Differentiation of equation (4.9), the relationship between the temperature and the
stagnation temperature, yields
obtained by utilizing the fact that the process is assumed to be adiabatic
(4.38)

and simplifying equation (4.38) yields

(4.39)

%

4.2.1.2 Relationship Between the Mach Number and Cross Section Area
The equations used in the solution are energy (4.39), second law (4.37), state
1
(4.29), mass (4.26) . Note, equation (4.33) isn’t the solution but demonstration of
certain properties on the pressure.
The relationship between temperature and the cross section area can be
obtained by utilizing the relationship between the pressure and temperature (4.37)
and the relationship of pressure and cross section area (4.33). First stage equation
(4.39) is combined with equation (4.37) and becomes
(4.40)

Combining equation (4.40) with equation (4.33) yields

%

(4.41)

can be proved as

%

The following identify,

(4.42)

Utilizing the identify in equation (4.42) changes equation (4.41) into

(4.43)

1 The momentum equation is not used normally in isentropic process, why?
%