Problem 3.8

The test section of a supersonic wind tunnel is square in cross section with a side length of m. The Mach number in the test section is , the temperature is ^oC, and the pressure is kPa. Find the mass flow rate of air through the test section.

Given:
Supersonic wind tunnel with square cross-section with side, \( L = \) 1.22 m,
\( M = \) 3.5,
\( T = \) -100^oC = 173 K,
\( p = \) 20 kPa.

To calculate: \( \dot{m}=? \)

The problem description is schematically shown in Fig. 1.

Fig. 1: Schematic of the problem description

Using the definition of Mach number,

\[ V=M\,a \] \[ V=M\,\sqrt{\gamma\,R\,T} \]

The mass flow rate through the cross section is,

\[ \dot{m}=\rho\,A\,V \]

Using the equation of state for calculation of density and the area for the square cross-section,

\[ \dot{m}=\frac{p}{R\,T}\,L^{2}\times M\,\sqrt{\gamma\,R\,T} \]

\[ \dot{m}=\frac{20\times10^{3}}{287\times173}\times1.22^{2}\times3.5\times\sqrt{1.4\times287\times173} \]

\[ \boxed{\dot{m}= 553.245\,\text{kg/s}}\ . \]

Problem 3.8:

Solution: