Question

A heat pump with refrigerant-134a as the working fluid is used to keep a space at \(25^{\circ} \mathrm{C}\) by absorbing heat from geothermal water that enters the evaporator at \(50^{\circ} \mathrm{C}\) at a rate of \(0.065 \mathrm{kg} / \mathrm{s}\) and leaves at \(40^{\circ} \mathrm{C}\). The refrigerant enters the evaporator at \(20^{\circ} \mathrm{C}\) with a quality of 23 percent and leaves at the inlet pressure as saturated vapor. The refrigerant loses \(300 \mathrm{W}\) of heat to the surroundings as it flows through the compressor and the refrigerant leaves the compressor at \(1.4 \mathrm{MPa}\) at the same entropy as the inlet. Determine ( \(a\) ) the degrees of subcooling of the refrigerant in the condenser, (b) the mass flow rate of the refrigerant, \((c)\) the heating load and the COP of the heat pump, and (d) the theoretical minimum power input to the compressor for the same heating load.

Short answer

To find the degrees of subcooling of the refrigerant, the mass flow rate of the refrigerant, the heating load and the COP of the heat pump, and the theoretical minimum power input to the compressor for the same heating load, follow these steps: 1. Calculate the heat transferred in the evaporator by finding the change in enthalpy of the geothermal water and using the energy balance equation. 2. Determine the mass flow rate of the refrigerant using the energy balance equation for the evaporator and the enthalpy values of the refrigerant. 3. Calculate the heating load and the coefficient of performance (COP) of the heat pump using the heat absorbed by the refrigerant and the net power input to the compressor. 4. Find the degrees of subcooling in the condenser by comparing the saturation temperature of the refrigerant at the condenser pressure and the actual temperature of the refrigerant leaving the condenser. 5. Calculate the theoretical minimum power input to the compressor using the isentropic efficiency of the compressor and the Second Law of Thermodynamics.

Step-by-step solution

Find the heat transferred in the evaporator

From the given data, we know that geothermal water enters the evaporator at \(50^{\circ}\mathrm{C}\) and leaves it at \(40^{\circ}\mathrm{C}\). Also, the mass flow rate of the geothermal water is \(0.065\,\mathrm{kg/s}\). The heat absorbed from geothermal water in the evaporator of the heat pump can be calculated using the following formula: \(q_{in} = \dot{m}_{w} \cdot \Delta h_{w}\) Where \(\dot{m}_{w}\) is the mass flow rate of the geothermal water, and \(\Delta h_{w}\) is the change in enthalpy of the geothermal water. We can first find the enthalpy changes of water entering and leaving the evaporator using steam tables and then calculate the heat absorbed.

Calculate the mass flow rate of the refrigerant

To calculate the mass flow rate of the refrigerant we can use the energy balance equation for the evaporator: \(\dot{m}_{R} \cdot h_{1} = \dot{m}_{R} \cdot h_{2} - q_{in}\) Where \(\dot{m}_{R}\) is the mass flow rate of the refrigerant, \(h_{1}\) and \(h_{2}\) are the enthalpy of the refrigerant entering and leaving the evaporator, respectively. From given data, we know the refrigerant enters the evaporator at \(20^\circ\mathrm{C}\) with a quality of \(0.23\). Using the refrigerant-134a tables, we can find the enthalpies of refrigerant entering and leaving evaporator, and then calculating the mass flow rate of the refrigerant.

Calculate the heating load and the COP of the heat pump

The heating load is the amount of heat absorbed by the refrigerant in the evaporator and can be calculated as the difference in the heat flow rates of the geothermal water and refrigerant in the evaporator. The coefficient of performance (COP) of the heat pump can be calculated using the following formula: COP = \(\frac{Q_{H}}{W_{net,in}}\) Where \(Q_{H}\) is the heating load, and \(W_{net,in}\) is the net power input to the compressor. The heating load is the same as \(q_{in}\) in step 1 and the power input can be found by multiplying mass flow rate of the refrigerant with the work done per kg of the refrigerant.

Determine the degrees of subcooling in the condenser

The subcooling in the condenser can be determined by finding the difference between the saturation temperature of the refrigerant at the condenser pressure and the actual temperature of the refrigerant leaving the condenser. We can calculate the saturation temperature using the refrigerant-134a tables and the given pressure data, and then finding degree of subcooling of the refrigerant.

Calculate the theoretical minimum power input to the compressor

The theoretical minimum power input to the compressor can be calculated using the isentropic efficiency of the compressor and the Second Law of Thermodynamics. The isentropic efficiency is defined as: \(\eta_c = \frac{W_{in,s}}{W_{in}}\) Where \(W_{in,s}\) is the isentropic work input to the compressor and \(W_{in}\) is the actual work input to the compressor. Using the given values of entropy for the refrigerant at the inlet and outlet of the compressor, we can find the isentropic work input and subsequently, the theoretical minimum power input to the compressor. In conclusion, follow these steps to determine the degrees of subcooling of the refrigerant, the mass flow rate of the refrigerant, the heating load and the COP of the heat pump, and the theoretical minimum power input to the compressor for the same heating load.

Key concepts

Refrigerant-134a

Refrigerant-134a, also known as 1,1,1,2-Tetrafluoroethane, is a haloalkane refrigerant with thermodynamic properties that make it popular for heat pump and refrigeration applications. It operates at a relatively low pressure compared to refrigerants such as R-22, making it safer and more energy-efficient.

When used in heat pumps, Refrigerant-134a undergoes phase changes and energy exchanges through compression and expansion processes. These changes in state are essential for the heat pump cycle, enabling it to absorb heat from one place and release it to another, ensuring a consistent interior temperature.

Geothermal Heat Exchange

Geothermal heat exchange is about harnessing the consistent temperatures found below the Earth's surface to heat or cool a space. In heat pumps, geothermal water serves as a heat source or sink. The exercise mentioned involves a heat pump absorbing heat from geothermal water.

Geothermal heat exchange systems offer high efficiency as the temperature below ground remains relatively stable, making for an effective and sustainable method of temperature control. The key to their operation lies in their capability to transfer heat to or from the ground effectively.

Enthalpy Change

Enthalpy change represents the total heat content change of a system at constant pressure. It's crucial in thermodynamics as it helps to quantify the heat transfer into or out of a system during a process.

In the context of the given exercise, the change in enthalpy of the geothermal water and the refrigerant within the evaporator are used to calculate the heat absorbed by the refrigerant, which is vital for determining the heating load of the heat pump.

Coefficient of Performance (COP)

The coefficient of performance (COP) is a measure of a heat pump's efficiency. It is the ratio of heating or cooling provided to the work input required to produce that heating or cooling.

COP = \(\frac{Q_{H}}{W_{net,in}}\)
In terms of the exercise, calculating the COP can show how well the heat pump uses electrical energy to move heat. A higher COP indicates a more efficient system, which is critical for assessing the performance and cost-effectiveness of the heat pump.

Compressor Work

Compressor work in a heat pump involves compressing the refrigerant, which increases its pressure and temperature, preparing it to release heat in the condenser. Calculating the work done by the compressor is fundamental for determining the power input required by the heat pump system.

Considering the exercise, understanding the work performed by the compressor helps to evaluate the compressor's demand on electrical energy and its impact on the overall enery efficiency of the heat pump.

Subcooling in Condenser

Subcooling in a condenser refers to the process of cooling a refrigerant below its saturation temperature at a given pressure, after it has been condensed from a gas to a liquid. Subcooling ensures the refrigerant leaves the condenser as a cool liquid, which enhances the efficiency of the heat pump.

In the provided problem, determining the degree of subcooling helps to evaluate the performance of the condenser and the condition of the refrigerant before it enters the expansion valve.

Isentropic Efficiency

Isentropic efficiency is a performance metric that compares the actual work of a compressor to the work if the process were isentropic, meaning no entropy is generated and the process is reversible. Isentropic processes are ideal, and achieving high isentropic efficiency implies that the real compressor's performance is close to this theoretical optimum.

For the calculation in the exercise, the isentropic efficiency would be used to find the theoretical minimum power input, which serves as an important benchmark for assessing the real compressor's energy use.