Problem 2.2:
Air at a temperature of
oC
is flowing at a velocity of
m/s.
A shock wave (covered in details in Chapters 5 and 6) occurs in the flow reducing the velocity to
m/s.
Assuming the flow through the shock wave to be adiabatic, find the temperature of the air behind the shock
wave.
Solution:
Given:
T1 = 25oC
= 298 K,
V1 = 500 m/s,
V2 = 300 m/s,
adiabatic flow \( \implies\dot{q} = 0 \).
To calculate: \( T_{2}=? \)
The schematic diagram of the problem description is shown in Fig. 1.
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Using the conservation of energy, \[ c_{p}T_{1}+\frac{V_{1}^{2}}{2}+\frac{\dot{q}}{\dot{m}}=c_{p}T_{2}+\frac{V_{2}^{2}}{2} \]
\[
1005\times298+\frac{500^{2}}{2}+\cancelto{\text{adiabatic}}{\frac{\dot{q}}{\dot{m}}}=1005\times
T_{2}+\frac{300^{2}}{2}
\]
The solution for \( T_{2} \) can be obtained as,
\[
T_{2}=298+\frac{500^{2}-300^{2}}{2\times1005}
\]
\[
\boxed{T_{2}=377.602\, \text{K}}\ .
\]