Question

A refrigerator uses refrigerant-134a as the working fluid and operates on the ideal vapor-compression refrigeration cycle except for the compression process. The refrigerant enters the evaporator at \(120 \mathrm{kPa}\) with a quality of 34 percent and leaves the compressor at \(70^{\circ} \mathrm{C}\). If the compressor consumes \(450 \mathrm{W}\) of power, determine \((a)\) the mass flow rate of the refrigerant, ( \(b\) ) the condenser pressure, and ( \(c\) ) the COP of the refrigerator.

Short answer

Question: Determine the mass flow rate of the refrigerant, condenser pressure, and the coefficient of performance (COP) of the refrigerator using the provided data for an ideal vapor-compression refrigeration cycle with R-134a refrigerant, evaporator inlet state P_1 = 120 kPa, x = 0.34, compressor outlet state T_3 = 70ºC, and compressor power input W_in = 450 W. Answer: Follow the given steps in the solution: 1. Determine the refrigerant properties at each state using the R-134a property tables. 2. Use the compressor power input to find the mass flow rate of the refrigerant using the formula: \(\dot{m} = \frac{W_{\text{in}}}{h_2 - h_1}\). 3. Find the condenser pressure (State 2) from the property tables, which was determined in Step 1. 4. Calculate the COP of the refrigerator using the formula: \(COP = \frac{h_1 - h_{f1}}{h_2 - h_1}\). Using these steps, you can determine the mass flow rate of the refrigerant, condenser pressure, and the COP of the refrigerator.

Step-by-step solution

Determine properties of the refrigerant at each state

Using the refrigerant properties table for R-134a, we can find the properties of the refrigerant at each state. State 1 (Evaporator inlet): - \(P_1 = 120\,\text{kPa, x = 0.34}\) - We can find \(h_1\) and \(s_1\) using the quality and property tables: - \(h_1 = h_{f1} + x(h_{g1}-h_{f1})\) - \(s_1 = s_{f1} + x(s_{g1}-s_{f1})\) State 3 (Compressor outlet): - \(T_3 = 70^\circ C\) - Assuming isentropic compression: \(s_2 = s_1\) - From this value of entropy, we can find \(P_2\) and \(h_2\) from the property tables of R-134a.

Use compressor power input to find the mass flow rate of the refrigerant

Now we know the power input to the compressor, we can use it to find the mass flow rate of the refrigerant: \(\dot{m} = \frac{W_{\text{in}}}{h_2 - h_1}\)

Find the condenser pressure (State 2)

We have already found the pressure at state 2 (\(P_2\)) in Step 1 using the property tables of R-134a.

Calculate the COP of the refrigerator

Now, we will find the COP (Coefficient of Performance) of the refrigerator. The formula to find COP for refrigeration cycles is given by: \(COP = \frac{\text{Desired Effect (cooling)}}{\text{Work Input}}\) In this case, the desired effect is the heat absorbed by the refrigerant in the evaporator, and the work input is the power input to the compressor. So, we can write: \(COP = \frac{\dot{m}(h_1 - h_4)}{\dot{m}(h_2 - h_1)}\) Note that \(h_4 = h_{f1}\) for the ideal vapor-compression refrigeration cycle. Therefore, we can now calculate the COP as: \(COP = \frac{h_1 - h_{f1}}{h_2 - h_1}\) By following these steps, we can find the mass flow rate of the refrigerant, the condenser pressure, and the COP of the refrigerator.