Find the enthalpy change, internal energy change and entropy change across a shock wave.

Find the enthalpy change, internal energy change and entropy change across a shock wave whose upstream air temperature and pressure are \(300\,K\) and \(1\,atm\), and downstream air temperature and pressure are \(700\,K\) and \(9\,atm\).

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Flow passing through a shock waveFlow passing through a shock wave

Specific heat at constant pressure,
\[c_{p}=\frac{\gamma R}{\gamma -1}=\frac{\left ( 1.4 \right )\left ( 287 \right )}{\left ( 1.4-1 \right )}=1004.5\,J/\left  (kg.K \right )\]
Specific heat at condtant volume,
\[c_{v}=\frac{R}{\gamma -1}=\frac{287}{0.4}=717.5\,J/kg.K\]
Enthalpy change across the shock wave,
\[h_{2}-h_{1}=c_{p}\left ( T_{2}-T_{1} \right )=1004.5\left ( 700-300 \right )=4.018\times 10^{5}\,J/kg\]
Internal energy change across the shock wave,
\[e_{2}-e_{1}=c_{v}\left ( T_{2}-T_{1} \right )=717.5\left ( 700-300 \right )=2.87\times 10^{5}\,J/kg\]
Entopy change across the shock wave,
\[s_{2}-s_{1}=c_{p}ln\left ( \frac{T_{2}}{T_{1}} \right )
-Rln\left ( \frac{p_{2}}{p_{1}} \right )
=1004.5ln\left ( \frac{700}{300} \right )-287ln\left ( \frac{9}{1} \right )=220.51\,J/\left (kg.K \right )\]

Answered on 19th April 2021.
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