Problem 3.12

A gas with a molar mass of and a specific heat ratio flows through a channel at supersonic speed. The temperature of the gas in the channel is ^oC. A photograph of the flow reveals weak waves originating at imperfections in the wall running across the flow at an angle to ^o to the flow direction. Find the Mach number and the velocity in the flow.

Given:
$ \hat{m} =$ 44 kg/kmol,
$ \gamma = $ 1.67,
$ T = $ 10^oC = 283 K,
Mach angle = $ \mu = $ 45^o.

To calculate: $ M=?, V=? $

The problem description is schematically shown in Fig. 1.

Fig. 1: Schematic of the problem description

The Mach angle is given by the equation,

\[ \sin\left(\mu\right)=\frac{1}{M} \]

\[ M=\frac{1}{\sin\left(\mu\right)}=\frac{1}{\sin\left(45\right)} \]

\[ \boxed{M=1.4142}\ . \]

The speed of sound in the channel, for the given temperature, can be calculated as,

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

where, the value of gas constant $R$ can be calculated as,

\[ R=\frac{R_{u}}{\hat{m}} \]

therefore,

\[ a=\sqrt{\gamma\,\frac{R_{u}}{\hat{m}}\,T} \]

The velocity can be calculated, using the universal gas constant $ R_{u}= $ 8314 J/kmol-K and the given molar mass of the gas $ \hat{m} =$ 44 kg/kmol,

as,

\[ V=M\,a=\frac{1}{\sin\left(\mu\right)}\times\sqrt{\gamma\,\frac{R_{u}}{\hat{m}}\,T}=\frac{1}{\sin\left(45\right)}\times\sqrt{1.67\times\frac{8314}{44}\times283} \]

\[ \boxed{V=422.6152\,\text{m/s}}\ . \]

Problem 3.12:

Solution: