Helium gas at 250 kPa and 100 °C flows through a rectangular duct of cross-section 40 cm × 50 cm, with a velocity of 500 ms^−1. Determine the Mach number, stagnation temperature, stagnation pressure, and the mass flow rate. Assume helium to be an ideal gas.

Question

Helium gas at 250 kPa and [latex]100^{\circ} C[/latex] flows through a rectangular duct of cross-section 40 cm × 50 cm, with a velocity of [latex]500 m s ^{-1}[/latex]. Determine the Mach number, stagnation temperature, stagnation pressure, and the mass flow rate. Assume helium to be an ideal gas.

Accepted Answer

Given, [latex]p=250 kPa , T=100^{\circ} C =373.15 K , ext { Area }=0.40 imes 0.50=0.2 m ^{2}, V=500 m s ^{-1}[/latex] The gas constant for helium (of molecular weight 4.003) is

[latex]R=\frac{8314}{4.003}=2077 J ( kg K )^{-1}[/latex]

The speed of sound is

[latex]a=\sqrt{\gamma R T}[/latex]

[latex]=\sqrt{1.66 imes 2077 imes 373.15}=1134.3 m s ^{-1}[/latex]

since [latex]\gamma=1.66[/latex] for helium gas. Thus,

[latex]M=\frac{V}{a}=\frac{500}{1134.3}=0.44[/latex]

By isentropic relation, we have the temperature ratio as

[latex]\frac{T_{0}}{T}=1+\frac{\gamma-1}{2} M^{2}[/latex]

Therefore,

[latex]T_{0}=T\left(1+\frac{\gamma-1}{2} M^{2} ight)[/latex]

[latex]=373.15\left(1+0.33 imes 0.44^{2} ight)=397 K[/latex]

By isentropic relation, the pressure ratio is given by

[latex]\frac{p_{0}}{p}=\left(1+\frac{\gamma-1}{2} M^{2} ight)^{\frac{\gamma}{\gamma-1}}[/latex]

Thus,

[latex]p_{0}=250\left(1+0.33 imes 0.44^{2} ight)^{2.515}[/latex]

= 292.13 kPa

The mass flow rate through the duct is

[latex]\dot{m}= ho A V=\frac{p}{R T} A V[/latex]

[latex]=\frac{250 imes 10^{3}}{2077 imes 373.15} imes 0.2 imes 500=32.26 kg s ^{-1}[/latex]

Question 2.6: Helium gas at 250 kPa and 100 °C flows through a rectangular...

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