Discussion
Consider 1-D, steady, inviscid, compressible flow through a convergent nozzle. The total temperature and total pressure are \(T_{o}\), \(P_{o}\) respectively. The flow through the nozzle is choked with a mass flow rate of \(\dot{m}_{o}\). If the total temperature is increased to \(4T_{o}\), with total pressure remaining unchanged, then the mass flow rate through the nozzle
Answer: B
The mass flow rate becomes half of \(\dot{m}_{o}\).
For choked flow through a nozzle of throat area \(A^{*}\), the mass flow rate is
\[ \begin{aligned} \dot{m} &= \frac{A^{*}P_{o}}{\sqrt{T_{o}}} \cdot K \\ K &= \sqrt{\frac{\gamma}{R}}\left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma + 1}{2(\gamma - 1)}} \end{aligned} \]
where \(K\) depends only on the gas. With \(P_{o}\) and the geometry unchanged, \(\dot{m} \propto 1/\sqrt{T_{o}}\). Raising the total temperature to \(4T_{o}\) multiplies the flow by \(1/\sqrt{4} = 1/2\).
Hotter gas is less dense at the same pressure, and although the sonic speed at the throat rises by \(\sqrt{4} = 2\), the density falls by 4, so the net effect is a halving. 'Unchanged', 'twice' and 'four times' ignore or invert this.
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