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## Q. 2.27

A system containing 0.25 m³ of air at a pressure of 4 bar and 150°C expands isentropically to a pressure of 1bar and after this the gas is heated at constant pressure till the enthalpy increases by 60 kJ. Calculate the work done.

If these processes are replaced by a single reversible polytropic process producing the same work between initial and final states, find the index of this expansion. Take $c_p$ = 1.005 kJ/kg. K for air.

## Verified Solution

The p – V diagram is shown in Fig. 2.41.
For isentropic process 1-2,

\begin{aligned}V_2 &=V_1\left\lgroup\frac{p_1}{p_2}\right\rgroup ^{\frac{1}{\gamma}}=0.25\left\lgroup\frac{4}{1}\right\rgroup ^{\frac{1}{1.4}}=0.673 \ m ^3 \\T_2 &=T_1\left\lgroup\frac{p_2}{p_1}\right\rgroup ^{\frac{\gamma-1}{\gamma}}=423\left\lgroup\frac{1}{4}\right\rgroup ^{\frac{1.4-1}{1.4}}=284.66 \ K \\m &=\frac{p_1 V_1}{R T_1}=\frac{4 \times 10^2 \times 0.25}{0.287 \times 423}=0.8237 \ kg\end{aligned}

\begin{aligned}Q_{2-3} &=m c_p\left(T_3-T_2\right) \\ \\60 &=0.8237 \times 1.005\left(T_3-284.66\right) \\ \\T_3 &=357.14 \ K \\ \\V_3 &=\frac{V_2 T_3}{T_2}=\frac{0.673 \times 357.14}{284.66}=0.844 \ m ^3\end{aligned}

\begin{aligned}W_{1-2-3}=W_{1-2} &+W_{2-3} \\&=\left\lgroup\frac{p_1 V_1-p_2 V_2}{\gamma-1}\right\rgroup +p_2\left(V_3-V_2\right) \\&=10^2\left\lgroup\frac{4 \times 0.25-1 \times 0.673}{1.4-1}\right\rgroup +1 \times 10^2(0.844-0.673) \\&=81.75+17.1=98.85 \ kJ\end{aligned}

\begin{aligned}W_{1-3} &=\frac{p_1 V_1-p_2 V_3}{n-1} \\98.85 &=10^2\left\lgroup\frac{4 \times 0.25-1 \times 0.844}{n-1}\right\rgroup \\n &=1.158\end{aligned} 