Answer
b) $\mathrm{COP}_{\mathrm{R}} =1.12$
c) $\dot{m}=0.237\text{ kg/s}$
Work Step by Step
(a) The lowest temperature in the cycle occurs at the turbine exit. From the isentropic relations,
$$
\begin{aligned}
& T_2=T_1\left(\frac{P_2}{P_1}\right)^{(\mathrm{k}-1) / \mathrm{k}}=(266 \mathrm{~K})(4)^{0.4 / 1.4}=395.3 \mathrm{~K}=122.3^{\circ} \mathrm{C} \\
& T_5=T_4\left(\frac{P_5}{P_4}\right)^{(\mathrm{k}-1) / \mathrm{k}}=(258 \mathrm{~K})\left(\frac{1}{4}\right)^{0.4 / 1.4}=173.6 \mathrm{~K}=-99.4^{\circ} \mathrm{C}=T_{\min }
\end{aligned}
$$ (b) From an energy balance on the regenerator, $$
\begin{aligned}
\dot{E}_{\text {in }}-\dot{E}_{\text {out }} & =\Delta \dot{E}_{\text {system }}^ {\varphi 0 \text { (steady) }}=0 \\
\dot{E}_{\text {in }} & =\dot{E}_{\text {out }} \\
\sum \dot{m}_e h_e & =\sum \dot{m}_i h_i \longrightarrow \dot{m}\left(h_3-h_4\right)=\dot{m}\left(h_1-h_6\right)
\end{aligned}
$$ or, $$
\dot{m} c_p\left(T_3-T_4\right)=\dot{m} c_p\left(T_1-T_6\right) \longrightarrow T_3-T_4=T_1-T_6
$$ or, $$
T_6=T_1-T_3+T_4=\left(-7^{\circ} \mathrm{C}\right)-27^{\circ} \mathrm{C}+\left(-15^{\circ} \mathrm{C}\right)=-49^{\circ} \mathrm{C}
$$ Then the COP of this ideal gas refrigeration cycle is determined from
$$ \begin{aligned}
\mathrm{COP}_{\mathrm{R}} & =\frac{q_L}{w_{\text {net, in }}}=\frac{q_L}{w_{\text {comp, in }}-w_{\text {turb, out }}} \\
& =\frac{h_6-h_5}{\left(h_2-h_1\right)-\left(h_4-h_5\right)} \\
& =\frac{T_6-T_5}{\left(T_2-T_1\right)-\left(T_4-T_5\right)} \\
& =\frac{-49^{\circ} \mathrm{C}-\left(-99.4^{\circ} \mathrm{C}\right)}{[122.3-(-7)]^{\circ} \mathrm{C}-[-15-(-99.4)]^{\circ} \mathrm{C}}=\mathbf{1 . 1 2}
\end{aligned}
$$ (c) The mass flow rate is determined from $$
\dot{m}=\frac{\dot{Q}_{\text {refrig }}}{q_L}=\frac{\dot{Q}_{\text {refrig }}}{h_6-h_5}=\frac{\dot{Q}_{\text {refrig }}}{c_p\left(T_6-T_5\right)}=\frac{12 \mathrm{~kJ} / \mathrm{s}}{\left(1.005 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right)[-49-(-99.4)]^{\circ} \mathrm{C}}=\mathbf{0 . 2 3 7} \mathrm{kg} / \mathrm{s}
$$
