SSC JE Mechanical Engineering 27 Sep 2019 Shift 1 Paper

A Bell-Coleman air refrigeration cycle works on the reverse Brayton cycle.
he relationship between the various temperature of the cycle is given by the isentropic relationship applied to both compression and expansion processes:
${\mathit{r}}_{\mathit{p}}=\frac{{\mathit{P}}_2}{{\mathit{P}}_1}={\left(\frac{{\mathit{T}}_2}{{\mathit{T}}_1}\right)}^{\frac{\mathrm{\gamma }}{\mathrm{\gamma }-1}}={\left(\frac{{\mathit{T}}_3}{{\mathit{T}}_4}\right)}^{\frac{\mathrm{\gamma }}{\mathrm{\gamma }-1}}$
$\frac{{\mathit{T}}_2}{{\mathit{T}}_1}=\frac{{\mathit{T}}_3}{{\mathit{T}}_4}$ or $\frac{{\mathit{T}}_2}{{\mathit{T}}_3}=\frac{{\mathit{T}}_1}{{\mathit{T}}_4}$

Refrigeration effect: ${\mathit{Q}}_1={\mathit{C}}_{\mathit{p}}\left({\mathit{T}}_1-{\mathit{T}}_4\right)$
Compressor work, ${\mathit{W}}_1={\mathit{C}}_{\mathit{p}}\left({\mathit{T}}_2-{\mathit{T}}_1\right)$
Turbine work, ${\mathit{W}}_2={\mathit{C}}_{\mathit{p}}\left({\mathit{T}}_3-{\mathit{T}}_4\right)$
$\mathit{C}\mathit{O}\mathit{P}=\frac{{\mathit{Q}}_1}{{\mathit{W}}_1-{\mathit{W}}_2}=\frac{\left({\mathit{T}}_1-{\mathit{T}}_4\right)}{\left({\mathit{T}}_2-{\mathit{T}}_1\right)-\left({\mathit{T}}_3-{\mathit{T}}_4\right)}$ $=\frac{\left({\mathit{T}}_1-{\mathit{T}}_4\right)}{\left({\mathit{T}}_2-{\mathit{T}}_3\right)-\left({\mathit{T}}_1-{\mathit{T}}_4\right)}$ $=\frac{1}{\frac{{\mathit{T}}_2-{\mathit{T}}_3}{{\mathit{T}}_1-{\mathit{T}}_4}-1}$
$\mathit{C}\mathit{O}\mathit{P}=\frac{1}{\frac{{\mathit{T}}_2-{\mathit{T}}_3}{{\mathit{T}}_1-{\mathit{T}}_4}-1}$ $=\frac{1}{\frac{{\mathit{T}}_3}{{\mathit{T}}_4}\left(\frac{\frac{{\mathit{T}}_2}{{\mathit{T}}_3}-1}{\frac{{\mathit{T}}_1}{{\mathit{T}}_4}-1}\right)-1}$
$=\frac{1}{\frac{{\mathit{T}}_3}{{\mathit{T}}_4}-1}=\frac{1}{\left({\mathit{r}}_{\mathit{p}}\right)\frac{\mathrm{\gamma }-1}{\mathrm{\gamma }}-1}$
$\mathit{C}\mathit{O}\mathit{P}=\frac{1}{{\mathit{r}}_{\mathit{p}}^{\left(\mathrm{\gamma }-1\right)∕\mathit{\gamma }}-1}$