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Question

# Nitrogen at $900\ \mathrm{kPa}$ and $300^{\circ} \mathrm{C}$ is expanded adiabatically in a closed system to $100\ \mathrm{kPa}$. Find the minimum nitrogen temperature after the expansion.

Solution

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Nitrogen is an ideal gas. To obtain the final temperature when nitrogen is expanded isentropically we will use the isentropic relation of an ideal gas under constant specific heat assumption.

$\frac{T_{2}}{T_1}=\left(\frac{P_{2}}{P_{1}}\right)^{(k-1) / k}$

Since we don't know the final temperature yet, we will assume that the average temperature is $450$ K. The specific heat ratio of nitrogen can be found in Table A-2b to be $k=1.391$. Thus the final temperature is:

\begin{aligned} T_{2} &=& T_{1}\left(\frac{P_{2}}{P_{1}}\right)^{(k-1) / k}\\ &=& (300+273 \mathrm{~K})\left(\frac{100 \mathrm{kPa}}{900 \mathrm{kPa}}\right)^{0.391 / 1.391}\\ &=& \boxed{309 \mathrm{~K}} \end{aligned}

The actual average air temperature is $441$ K, therefore our assumed average temperature of 450 K is good enough.

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