Chapter 7, Problem 104P

(a)

To determine

The actual coefficient of performance of the refrigerator.

Explanation of Solution

Given:

The volume flow rate at inlet 1$\left({\dot{v}}_1\right)$ is $80\text{L}/\min$.

The heating load rate $\left({\dot{Q}}_{\text{heat}}\right)$ is $250\text{kJ}/\min$.

The other equipment load rate $\left({\dot{Q}}_{\text{equipment}}\right)$ is 0.9 kW.

Calculation:

Refer Table A-12, “Saturated refrigerant-134a: Pressure table”, obtain the properties of refrigerant R-134a at initial pressure $\left(P_1\right)$ of $\text{400 kPa}$ and quality 1.

  $\begin{array}{l}{h}_1=255.61\text{kJ}/\text{kg}\\ v_1=0.05127{\text{m}}^{\text{3}}/\text{kg}\end{array}$

Refer Table A-13, “Superheated refrigerant-134a”, obtain the properties of refrigerant R-134a at exit pressure $\left(P_2\right)$ of $\text{1}\text{.2 MPa}$ and temperature $\left(T_2\right)$ of $70°\text{C}$.

  $h_2=300.63\text{kJ}/\text{kg}$

Calculate the mass flow rate of a refrigerant $\left({\dot{m}}_R\right)$.

  ${\dot{m}}_R=\frac{{\dot{v}}_1}{v_1}$

  $\begin{array}{c}{\dot{m}}_R=\frac{80\text{L}/\min }{0.05127{\text{m}}^{\text{3}}/\text{kg}}\\ =\left(\frac{80\text{L}/\min \left(\frac{1{\text{m}}^{\text{3}}}{1000\text{L}}\right)\left(\frac{1\min }{60\text{s}}\right)}{0.05127{\text{m}}^{\text{3}}/\text{kg}}\right)\\ =0.02601\text{kg}/\text{s}\end{array}$

Calculate the power consumption of the compressor $\left({\dot{W}}_{\text{in}}\right)$.

  ${\dot{W}}_{\text{in}}={\dot{m}}_R\left(h_2-h_1\right)$

  $\begin{array}{c}{\dot{W}}_{\text{in}}=0.02601\text{kg}/\text{s}\left(300.63\text{kJ}/\text{kg}-255.61\text{kJ}/\text{kg}\right)\\ =1.171\text{kW}\end{array}$

Calculate the refrigeration load $\left({\dot{Q}}_L\right)$.

  ${\dot{Q}}_L={\dot{Q}}_{\text{heat}}+{\dot{Q}}_{\text{equipment}}$

  $\begin{array}{c}{\dot{Q}}_L=250\text{kJ}/\min +0.9\text{kW}\\ =250\text{kJ}/\min \left(\frac{1\min }{60\text{}\text{s}}\right)+0.9\text{kW}\\ =\text{5}\text{.067}\text{kW}\end{array}$

Write the formula to calculate the actual coefficient of performance.

  ${\text{COP}}_{\text{actual}}=\frac{{\dot{Q}}_L}{{\dot{W}}_{\text{in}}}$

  $\begin{array}{c}{\text{COP}}_{\text{actual}}=\frac{5.067\text{kW}}{1.171\text{kW}}\\ =4.33\end{array}$

Thus, the actual coefficient of performance of the refrigerator is $4.33$.

(b)

To determine

The maximum coefficient of performance of a reversible refrigerator.

Explanation of Solution

Write the formula to calculate the maximum coefficient of performance of a reversible refrigerator.

  $\begin{array}{c}{\text{COP}}_{\text{max}}=\frac{1}{\frac{T_H}{T_L}-1}\\ {\text{COP}}_{\text{max}}=\frac{1}{\frac{34°\text{C}}{23°\text{C}}-1}\\ {\text{COP}}_{\text{max}}=\frac{1}{\frac{307\text{K}}{296\text{K}}-1}\\ =26.91\end{array}$

Thus, the maximum coefficient of performance of a reversible refrigerator is $26.91$.

(c)

To determine

The minimum volume flow rate at the compressor inlet.

Explanation of Solution

Calculate the minimum power input to the condenser for the same refrigeration load $\left({\dot{W}}_{\text{in,min}}\right)$.

  ${\dot{W}}_{\text{in,min}}=\frac{{\dot{Q}}_L}{{\text{COP}}_{\text{max}}}$

  $\begin{array}{c}{\dot{W}}_{\text{in,min}}=\frac{5.067\text{kW}}{26.91}\\ =0.1883\text{kW}\end{array}$

Calculate the minimum mass flow rate $\left({\dot{m}}_{R,\min }\right)$.

  ${\dot{m}}_{R,\min }=\frac{{\dot{W}}_{\text{in,min}}}{h_2-h_1}$

  $\begin{array}{c}{\dot{m}}_{R,\min }=\frac{0.1883\text{kW}}{300.63\text{kJ}/\text{kg}-255.61\text{kJ}/\text{kg}}\\ =0.004182\text{kg}/\text{s}\end{array}$

Calculate the minimum volume flow rate at the compressor inlet. $\left({\dot{v}}_{\min ,1}\right)$.

  ${\dot{v}}_{\min ,1}={\dot{m}}_{R,\min }{v}_1$

  $\begin{array}{c}{\dot{v}}_{\min ,1}=\left(0.004182\text{kg}/\text{s}\right)\left(0.05127{\text{m}}^{\text{3}}/\text{kg}\right)\\ =0.0002144{\text{m}}^{\text{3}}/\text{s}\\ =12.9\text{L}/\min \end{array}$

Thus, the minimum volume flow rate at the compressor inlet is $12.9\text{L}/\min$.