Question 2.51: A vessel of capacity 3 m³ contains air at a pressure of 2 ba...
A vessel of capacity 3 m³ contains air at a pressure of 2 bar and temperature 300 K. Additional air is pumped into the system until the pressure rises to 30 bar and temperature to 330 K. Determine (a) the mass of air pumped (b) heat transferred, and (c) the change in entropy if the air is cooled down to 300 K. Take c_ν = 0.718 kJ/kg.K.
(a)
\begin{aligned}&m_1=\frac{p_1 V_1}{R T_1}=\frac{2 \times 10^5 \times 3}{287 \times 300}=6.97 \mathrm{~kg} \\&m_2=\frac{p_2 V_2}{R T_2}=\frac{30 \times 10^5 \times 3}{287 \times 330}=95.027 \mathrm{~kg}\end{aligned}
Mass of air pumped in, m=m_2-m_1=88.057 \mathrm{~kg}
(b) During cooling,
\begin{aligned}d U &=m_2 C_ν\left(T_3-T_2\right)=95.0 \times 0.718(300-330)=-2047.5 \mathrm{~kJ} \\ \\d Q &=d U+d W=-2047.5+0=-2047.5 \mathrm{~kJ}\end{aligned}
(c)
\begin{aligned}\Delta S=& S_3-S_2=m c_ν \ln \frac{T_3}{T_2}=95.027 \times 0.718 \ln \frac{300}{330} \\&=-6.503 \mathrm{~kJ} / \mathrm{K}\end{aligned}