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Jet aircrafts fly at altitudes above $30000 \,ft$, where the air is very cold at $-40^{\circ} C$ and the pressure is $0.28 \,atm$. The cabin is maintained at $1 \,atm$ pressure by means of a compressor which exchanges air from outside adiabatically. In order to have a comfortable cabin temperature of $25^{\circ} C$, we will require in addition
a heater to warm the air injected into the cabin
an air-conditioner to cool the air injected into the cabin
neither a heater nor an air-conditioner, the compressor is sufficient
alternatively heating and cooling in the two halves of the compressor cycle
Solution
(b)
Compression of a gas in a compressor is nearly an adiabatic process.
So, by using $p_{\text {in }}^{1-\gamma} \cdot T_{\text {in }}^\gamma=p_{\text {out }}^{1-\gamma} \cdot T_{\text {out }}^\gamma$
We get, $(0.28)^{1-\gamma}(233)^\gamma=(1)^{1-\gamma}(T)^\gamma$
Here, for air, $\gamma=14=\frac{7}{5}$
Hence, $T=(233)(0.28)^{-2 / 7}=\frac{233}{(0.28)^{2 / 7}}$
This temperature is much higher than $298 \,K$ or $25^{\circ} C$.
So, an air-conditioner is needed to cool the air coming out of compressor.
