Question

Refrigerant- 134 a enters the condenser of a residential heat pump at \(800 \mathrm{kPa}\) and \(50^{\circ} \mathrm{C}\) at a rate of \(0.022 \mathrm{kg} / \mathrm{s}\) and leaves at \(750 \mathrm{kPa}\) subcooled by \(3^{\circ} \mathrm{C}\). The refrigerant enters the compressor at \(200 \mathrm{kPa}\) superheated by \(4^{\circ} \mathrm{C}\). Determine (a) the isentropic efficiency of the compressor, ( \(b\) ) the rate of heat supplied to the heated room, and ( \(c\) ) the COP of the heat pump. Also, determine \((d)\) the \(\mathrm{COP}\) and the rate of heat supplied to the heated room if this heat pump operated on the ideal vapor-compression cycle between the pressure limits of 200 and 800 kPa.

Short answer

Based on the given information, the states of the heat pump cycle are as follows: State 1: Refrigerant enters the compressor at 200 kPa and 4°C superheat. State 2: Refrigerant enters the condenser at 800 kPa and 50°C. State 3: Refrigerant leaves the condenser at 750 kPa and 3°C subcooling. Calculate the isentropic efficiency of the compressor, the rate of heat supplied to the heated room, the COP of the heat pump, and compare these values with those of an ideal vapor-compression cycle with the same pressure limits.

Step-by-step solution

Identify States

We have the following states in the cycle: State 1: Refrigerant enters the compressor at 200 kPa and superheated by 4°C. State 2: Refrigerant enters the condenser at 800 kPa and 50°C. State 3: Refrigerant leaves the condenser at 750 kPa and subcooled by 3°C. We can find the necessary properties of the refrigerant at each state using the pressure and temperature information provided and consulting a refrigerant-134a property table.

Calculate the specific enthalpy and entropy values for each state

Using the pressure and temperature information from each state, we can find the specific enthalpy (h) and specific entropy (s) values for each point: State 1: \(h_{1} = hg_{200kPa,4°C}\) and \(s_{1} = sg_{200kPa,4°C}\) State 2: \(h_{2} = h_{2s}\) (isentropic) and \(s_{2s} = s_1\) (Since the process between state 1 and state 2 is isentropic) State 3: \(h_{3} = hf_{750kPa,3°C}\) Once we have these specific enthalpy and entropy values, we can proceed further to calculate the required parameters.

Calculate the isentropic efficiency of the compressor

The isentropic efficiency of the compressor can be calculated using the formula: Isentropic efficiency = \(\frac{h_{2s} - h_{1}}{h_{2} - h_{1}}\) We have the specific enthalpy values for state 1 and state 2, and we can plug these values into the formula to find the isentropic efficiency of the compressor.

Calculate the rate of heat supplied to the heated room

Rate of heat supplied to the heated room (Q) can be calculated using the formula: \(Q = \dot{m} \cdot (h_{2} - h_{3})\) Where \(\dot{m}\) is the mass flow rate of the refrigerant, which is given as 0.022 kg/s. We have the specific enthalpy values for state 2 and state 3 that we can plug into this formula to find the rate of heat supplied to the heated room.

Calculate the COP of the heat pump

The coefficient of performance (COP) of the heat pump can be calculated using the formula: COP = \(\frac{Q}{\dot{W}}\) Where \(\dot{W}\) is the rate of work done by the compressor which can be calculated as: \(\dot{W} = \dot{m} \cdot (h_{2} - h_{1})\) We have the heat supplied (Q) and the rate of work done (\(\dot{W}\)). We can plug these values into the formula to find the COP of the heat pump.

Determine the \(\mathrm{COP}\) and the rate of heat supplied to the heated room for the ideal vapor-compression cycle between the pressure limits of 200 and 800 kPa

For the ideal vapor-compression cycle, we can assume that the compressor and expansion device are both isentropic. We can use the same information as above, but this time we will use the isentropic values for state 2s and state 3s. We can then follow the same formulas as in steps 4 and 5 to calculate the COP and the rate of heat supplied to the heated room for the ideal cycle. By calculating the isentropic efficiency of the compressor, the rate of heat supplied to the heated room, and the COP of the heat pump for both the actual and ideal vapor-compression cycles, we can better understand the performance of the heat pump and the effects of inefficiencies on the overall process.