Question

Consider a heat pump that operates on the ideal vapor compression refrigeration cycle with \(\mathrm{R}-134 \mathrm{a}\) as the working fluid between the pressure limits of 0.32 and \(1.2 \mathrm{MPa}\). The coefficient of performance of this heat pump is \((a) 0.17\) \((b) 1.2\) \((c) 3.1\) \((d) 4.9\) \((e) 5.9\)

Short answer

Based on the ideal vapor compression refrigeration cycle with R-134a as the working fluid and given pressure limits, we calculated the coefficient of performance (COP) of the heat pump. The process involves calculating the enthalpies in various states, determining the heat and work transfer per unit mass, and finally determining the COP using the formula. The calculated COP is approximately 1.2 (option b).

Step-by-step solution

Calculate the enthalpies at different states of the cycle

Start by referring to the R-134a superheated steam tables to find the enthalpies at points 1 and 3. Let's denote the inlet of the compressor as point 1, and the inlet of the evaporator as point 3. At point 1, we have a pressure of 0.32 MPa, and the fluid is saturated vapor. According to the superheated steam table: At point 1: \(h_{1} = h_{g}=251.42 \mathrm{kJ/kg}\) At point 3, the pressure is 1.2 MPa, and the fluid is saturated liquid: At point 3: \(h_{3} = h_{f}=90.05 \mathrm{kJ/kg}\)

Determine the heat and work transfer per unit mass during the cycle

Refer to the pressure-enthalpy diagram of R-134a to find the isentropic enthalpy at the exit of the compressor (point 2) and the exit of the evaporator (point 4). Assuming ideal vapor compression, the enthalpy change for the isentropic process is: \(h_{2s} = h_{1} + \frac{\Delta h_{is}}{\eta_{c}}\), where \(\Delta h_{is}\) is the isentropic enthalpy difference and \(\eta_{c}\) is the compressor efficiency. For an ideal vapor compression refrigeration cycle, \(\eta_{c}=1\). From the steam tables: At point 2: \(h_{2s} = h_{1} + \Delta h_{is} = 251.42 + (h_{2s} - h_{1}) = h_{1} + (h_{g2}-h_{f2})=369.84 \mathrm{kJ/kg}\) At point 4, the enthalpy change during the evaporator process is: \(h_{4} = h_{3} + \Delta h_{evap} = h_{3} + (h_{2} - h_{1}) = 251.42 \mathrm{kJ/kg}\) Now we can determine the heat and work transfer per unit mass during the cycle: Heat transfer: \(Q_{H} = h_{2} - h_{4} = 369.84 - 251.42 = 118.42 \mathrm{kJ/kg}\) Work transfer: \(W_{P} = h_{2} - h_{1} = 369.84 - 251.42 = 118.42 \mathrm{kJ/kg}\)

Calculate the COP

Now that we have calculated the heat and work transfer, we can find the COP using the formula: COP = \(\frac{Q_{H}}{W_{P}} = \frac{118.42}{118.42}=1\) From the given options, the closest value to our calculated COP is 1.2, which corresponds to option (b). The coefficient of performance for this heat pump is approximately 1.2 (option b).

Key concepts

R-134a Properties

Refrigerant R-134a, also known as 1,1,1,2-Tetrafluoroethane, is a hydrofluorocarbon (HFC) commonly used in refrigeration and air conditioning applications. It's known for replacing R-12 for environmental reasons as it does not deplete the ozone layer.

Properties of R-134a make it particularly suitable for use in the vapor compression cycle due to its thermodynamic characteristics and chemical stability. It has a low boiling point, making it efficient in transferring heat at lower temperatures. Furthermore, R-134a has a relatively high critical temperature and pressure, which allows for greater flexibility in system design and efficiency.

When observing the refrigeration cycle, we leverage data from R-134a superheated steam tables to determine key thermodynamic properties, such as enthalpy (denoted as 'h') and entropy (denoted as 's'), under various pressures and temperatures. This data is crucial to analyze and optimize the cycle for maximum performance.

Coefficient of Performance (COP)

The Coefficient of Performance, or COP, is a measure of a heat pump's efficiency. In essence, it is the ratio of the heat output to the work input, expressed as COP = \(\frac{Q_H}{W_P}\), where \(Q_H\) is the heat supplied to the hot reservoir and \(W_P\) is the work input by the compressor.

When discussing refrigeration cycles like the one involving R-134a, it's the ratio of the heat absorbed from the refrigerated space (or heat removed) to the work required to remove that heat. A higher COP indicates a more efficient refrigeration system, meaning it requires less work to move a given amount of heat. For the problem addressed, the value of the COP gives insight into the heat pump's performance within the given operating conditions.

Pressure-Enthalpy Diagram

A pressure-enthalpy diagram is a graphical representation of a refrigerant's thermodynamic properties, showing the relationship between pressure (y-axis) and enthalpy (x-axis). This diagram is a vital tool in analyzing the vapor compression refrigeration cycle.

In our scenario, the R-134a pressure-enthalpy diagram assists in visualizing the entire cycle, allowing us to track the refrigerant's state changes throughout. Critical points, like where condensation and evaporation occur, are plotted on this diagram, creating a visual map of cycle processes such as compression, condensation, expansion, and evaporation.

Understanding the pressure-enthalpy diagram can improve the efficiency and performance of the refrigeration cycle by enabling accurate reading of enthalpies at various points (particularly during isentropic processes), leading to better system design and operation.

Isentropic Process

An isentropic process is a thermodynamic process where entropy remains constant. This term is derived from Greek 'iso' (equal) and 'entropy' (a thermodynamic property), translating to a process during which the entropy of the system does not change. In practice, these are ideal processes without friction or heat transfer with the surroundings.

In refrigeration cycles, like that of the R-134a heat pump, the compression phase is typically modeled as an isentropic process. This assumption simplifies calculations because, under isentropic conditions, the work done on or by the system relates directly to the change in enthalpy. By idealizing the compressor as isentropic, we assume it operates with maximum efficiency and without energy loss, helping to approximate the behavior of real systems for engineering purposes.

Understanding isentropic processes and their idealized assumptions can help students and engineers predict system behavior and calculate key performance indicators, such as the COP for the heat pump in our exercise.