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## Q. 2.32

A cylinder of 10 cm diameter shown in Fig. 2.44 has a piston loaded with a spring of stiffness 15 kN/m. The initial state of air in the cylinder is 3 bar, 50 × $10^{-6}$ m³, and 298 K. Determine the amount of heat added to the system so that piston moves by 5 cm. Assume $c_ν$ = 0.718 kJ/kg. K, R = 0.287 kJ/kg. K and $\gamma$ =1.4.

## Verified Solution

Piston area,  $\quad \mathrm{A}=\left(\frac{\pi}{4}\right) \times(0.1)^2=7.859 \times 10^{-3} \mathrm{~m}^2$

Compression of spring, δ = 5 cm
Work done by air in compressing the spring,

$W_{1-2}=1 / 2 k \delta^2=\frac{1}{2} \times 15(0.05)^2=0.01875 \mathrm{~kJ}$

Final volume of air in the cylinder,

$V_2=V_1+A \delta=50 \times 10^{-6}+7.859 \times 10^{-3} \times 0.05=0.4427 \times 10^3 \mathrm{~m}^3$

Final Pressure of air in the cylinder,

$p_2=\frac{P_1+k S}{A}=\frac{3+15 \times 0.05}{\left(7.859 \times 10^{-3} \times 10^2\right)}=3.955 \mathrm{~bar}$

Now,

\begin{aligned}W_{1-2} &=\frac{p_1 V_1-p_2 V_2}{n-1} \\0.01875 &=\frac{\left[3 \times 10^2 \times 50 \times 10^{-6}-3.955 \times 10^2 \times 0.4427 \times 10^{-3}\right]}{(n-1)} \\n &=-7.538\end{aligned}

Mass of air,

\begin{aligned}m &=P_1\left\lgroup\frac{V_1}{R T_1}\right\rgroup =\frac{3 \times 10^5 \times 50 \times 10^{-6}}{(287 \times 298)}=5.838 \times 10^5 \mathrm{~kg} \\T_2 &=T_1\left\lgroup \frac{p_2}{p_1}\right\rgroup^{\frac{n-1}{n}}=298\left(\frac{3.955}{3}\right)^{\frac{-7.538-1}{-7.538}} \\&=298(1.318)^{1.1327}=407.5 \mathrm{~K} \\ \\Q_{1-2} &=m\left[\frac{\gamma-1}{n-1}\right] c_v\left(T_2-T_1\right) \\&=5.838 \times 10^{-5}\left[\frac{1.4-1}{-7.538-1}\right] \times 0.718(407.5-298) \\&=0.0215 \mathrm{~kJ}\end{aligned}