An aluminum pitot probe is placed in a supersonic air stream of temperature 30°C. If the melting temperature of aluminum is about 660°C, determine the flow velocity at which the probe will begin to melt. Assume air to be an ideal gas.

Question

An aluminum pitot probe is placed in a supersonic air stream of temperature [latex]30^{\circ} C[/latex]. If the melting temperature of aluminum is about [latex]660^{\circ} C[/latex], determine the flow velocity at which the probe will begin to melt. Assume air to be an ideal gas.

Accepted Answer

Given that aluminum will melt at [latex]660^{\circ} C,[/latex] the probe will start melting at a stream stagnation temperature [latex]T_{0} ext { of } 660^{\circ} C.[/latex]

[latex]T_{0}=660^{\circ} C =660+273.15=933.15 K[/latex]

The static temperature of the stream is [latex]T=30^{\circ} C =303.15 K.[/latex] From isentropic relation, we have

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

where M is the stream Mach number. For the present problem, we have

[latex]\frac{933.15}{303.15}=1+0.2 M^{2}[/latex]

since [latex]\gamma=1.4.[/latex] Thus, we get

[latex]M = 3.22 [/latex]

The speed of sound in the air stream is

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

[latex]=\sqrt{1.4 imes 287 imes 303.15}=349 m s ^{-1}[/latex]

Thus, the velocity at which the probe will just begin to melt is

[latex]V=M a=3.22 imes 349[/latex]

[latex]=1123.78 m s ^{-1}[/latex]

Alternatively

This problem can also be solved using the energy equation, as follows.

[latex]h_{0}=h+\frac{V_{2}}{2}[/latex]

where [latex]h_{0}[/latex] and h are the stagnation and static enthalpies of the air stream. For a perfect gas,

[latex]h_{0}=c_{p} T_{0}, \quad h=c_{p} T[/latex]

where [latex]c_{p}=1004.5 J ( kg K )^{-1},[/latex] for air. Therefore,

[latex]T_{0}-T=\frac{V^{2}}{2 c_{p}}[/latex]

[latex]V=\sqrt{2 c_{p}\left(T_{0}-T ight)}[/latex]

[latex]=\sqrt{2 imes 1004.5(933.15-303.15)}[/latex]

[latex]1125.02 m s ^{-1}[/latex]

This is the stream velocity at which the probe will start melting.

Note: The difference between these two answers is due to the fact that the value of Mach number M was terminated at the second decimal place. For this problem, the accurate answer is

[latex]V=1125.02 m s ^{-1}.[/latex]

Question 2.5: An aluminum pitot probe is placed in a supersonic air stream...

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