Question

A vapor-compression refrigeration system absorbs heat from a space at \(0^{\circ} \mathrm{C}\) at a rate of \(24,000 \mathrm{Btu} / \mathrm{h}\) and rejects heat to water in the condenser. The water experiences a temperature rise of \(12^{\circ} \mathrm{C}\) in the condenser. The COP of the system is estimated to be \(2.05 .\) Determine \((a)\) the power input to the system, in \(\mathrm{kW},(b)\) the mass flow rate of water through the condenser, and \((c)\) the second-law efficiency and the exergy destruction for the refrigerator. Take \(T_{0}=20^{\circ} \mathrm{C}\) and \(c_{p, \text { water }}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\).

Short answer

Based on the provided information and calculations: (a) The power input to the vapor-compression refrigeration system is 3.4312 kW. (b) The mass flow rate of water through the condenser is 0.2064 kg/s. (c) The second-law efficiency of the system is 58.53%, and the exergy destruction rate is 4.9829 kW.

Step-by-step solution

Calculate power input

To find the power input to the system, we need to use the provided COP value which is expressed as: \(\text{COP}_{\text{R}} = \frac{Q_{\text{abs}}}{W_{\text{in}}}\), where \(Q_{\text{abs}}\) is the heat absorbed and \(W_{\text{in}}\) is the power input. First, we need to convert the given heat rate from Btu/h to kW: \(Q_{\text{abs}} = 24000 \frac{\text{Btu}}{\text{h}} \times \frac{0.2931 \text{kW}}{1 \text{Btu/h}} = 7.0344 \text{kW}\) Now, we will use the COP to find the power input: \(W_{\text{in}} = \frac{Q_{\text{abs}}}{\text{COP}_{\text{R}}} = \frac{7.0344 \text{kW}}{2.05} = 3.4312 \text{kW}\) The power input to the system is \(3.4312 \text{kW}\).

Calculate the mass flow rate of water

The heat rejected to the water can be calculated as the sum of heat absorbed and the power input: \(Q_{\text{reject}} = Q_{\text{abs}} + W_{\text{in}} = 7.0344 \text{kW} + 3.4312 \text{kW} = 10.4656 \text{kW}\) We can use the provided temperature rise of water and its specific heat to find the mass flow rate of water in the condenser: \(\dot{m}_{\text{water}} = \frac{Q_{\text{reject}}}{c_{p, \text{water}} \cdot \Delta T_{\text{water}}} = \frac{10.4656 \text{kW}}{4.18 \frac{\text{kJ}}{\text{kg}\cdot{}^{\circ}\text{C}} \times 12^{\circ}\text{C}} = 0.2064 \frac{\text{kg}}{\text{s}}\) The mass flow rate of water is 0.2064 kg/s.

Calculate second-law efficiency and exergy destruction

To obtain the second-law efficiency, we need to use the following formula: \(\eta_{\text{II}} = \frac{\text{COP}_{\text{R}}}{\text{COP}_{\text{Carnot}}} = \frac{\text{COP}_{\text{R}}}{\frac{T_{\text{low}}}{T_{\text{high}} - T_{\text{low}}}}\) Here, we need to convert the temperatures in Kelvin: \(T_{\text{low}} = 0^{\circ}\text{C} + 273.15 = 273.15 \text{K}\) \(T_{\text{high}} = 20^{\circ}\text{C} + 273.15 = 293.15 \text{K}\) Now, we can calculate the second-law efficiency: \(\eta_{\text{II}} = \frac{2.05}{\frac{273.15 \text{K}}{293.15 \text{K} - 273.15 \text{K}}}} = 0.5853\) The second-law efficiency is 0.5853 or 58.53%. To calculate exergy destruction, we can use the following formula: \(X_{\text{destroyed}} = \frac{Q_{\text{abs}}}{\eta_{\text{II}}} - Q_{\text{abs}}\) Substituting the values, \(X_{\text{destroyed}} = \frac{7.0344 \text{kW}}{0.5853} - 7.0344 \text{kW} = 4.9829 \text{kW}\) The exergy destruction rate is \(4.9829 \text{kW}\). In summary: \((a)\) The power input to the system is \(3.4312 \text{kW}\). \((b)\) The mass flow rate of water is \(0.2064 \frac{\text{kg}}{\text{s}}\). \((c)\) The second-law efficiency is \(0.5853\) or \(58.53\%\), and the exergy destruction is \(4.9829 \text{kW}\).

Key concepts

COP (Coefficient of Performance)

The Coefficient of Performance (COP) is a crucial measurement in vapor-compression refrigeration systems. It's defined as the ratio of the amount of cooling provided (or heat absorbed, represented as Q_abs), to the work input (or power input, represented as W_in). This value indicates the efficiency with which the refrigeration system operates. A higher COP means that the system is more efficient, requiring less energy for a given amount of cooling.

In the example provided, the COP is given as 2.05, which implies that for every unit of energy invested in the system, a little over two units of energy are moved from the cooled space. This ratio is fundamental when evaluating the performance and energy consumption of refrigeration systems.

Mass Flow Rate

Understanding the mass flow rate is essential in evaluating the effectiveness of the heat rejection process in the condenser of a refrigeration system. It's a measure of the amount of mass passing through a given point per unit time, typically expressed in kilograms per second (kg/s).

In practice, the mass flow rate in a condenser is calculated based on the amount of heat to be rejected and the specific properties of the coolant - in this case, water. It's a clear indicator of the capacity of the cooling water to absorb heat. The calculation uses the specific heat capacity of water and the temperature rise experienced as it absorbs heat from the refrigerant. A precisely calculated mass flow rate ensures that the condenser operates efficiently and effectively.

Second-Law Efficiency

Second-law efficiency, in the context of refrigeration systems, is an expression of how well the system performs compared to an ideal, reversible system operating between the same temperatures. It is a more insightful measure than first-law (or energy) efficiency because it accounts for the quality of energy conversions as well as the quantity.

To determine the second-law efficiency, one would compare the actual COP of the refrigeration system to the COP of a Carnot refrigerator operating between the same temperatures. The Carnot COP represents the maximum possible efficiency for such a temperature differential. Thus, the real system's second-law efficiency gives us insight into how close the system comes to theoretical perfection, with 100% efficiency being completely reversible operation.

Exergy Destruction

Exergy destruction refers to the irreversible loss of potential work due to inefficiencies in a thermodynamic process, such as those found in vapor-compression refrigeration systems. It's a concept rooted in the second law of thermodynamics, which states that any real process is somewhat irreversible and, therefore, destroys some amount of exergy.

In our example, calculating the exergy destruction allows us to quantify how much work potential is lost due to irreversibilities within the refrigerator. This can include friction, heat loss, and other non-idealities. Understanding exergy destruction is fundamental for engineers aiming to optimize systems, as it shines light on where and how improvements can be made to reduce losses and enhance overall system performance.

Thermodynamic Properties

The understanding of thermodynamic properties is vital for determining the operation of any refrigeration system. These properties include temperature, pressure, specific volume, specific enthalpy, specific entropy, and specific heat capacities. All these properties help in analyzing and designing thermodynamic cycles.

In the context of our exercise, specific properties such as the specific heat capacity of water (c_{p, water}) and the temperature rise in the condenser are used to calculate the mass flow rate of the cooling water. These properties help us to understand how the refrigerant and water interact within the system, ensuring that the system can operate effectively while transferring the necessary heat for maintaining desired temperatures.