Question

A room is kept at \(-5^{\circ} \mathrm{C}\) by a vapor-compression refrigeration cycle with \(\mathrm{R}-134 \mathrm{a}\) as the refrigerant. Heat is rejected to cooling water that enters the condenser at \(20^{\circ} \mathrm{C}\) at a rate of \(0.13 \mathrm{kg} / \mathrm{s}\) and leaves at \(28^{\circ} \mathrm{C}\). The refrigerant enters the condenser at \(1.2 \mathrm{MPa}\) and \(50^{\circ} \mathrm{C}\) and leave as a saturated liquid. If the compressor consumes \(1.9 \mathrm{kW}\) of power, determine (a) the refrigeration load, in \(\mathrm{Btu} / \mathrm{h}\) and the \(\mathrm{COP}\), (b) the second-law efficiency of the refrigerator and the total exergy destruction in the cycle, and \((c)\) the exergy destruction in the condenser. Take \(T_{0}=20^{\circ} \mathrm{C}\) and \(c_{p \text { ,water }}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C} .\)

Short answer

Answer: In this vapor-compression refrigeration cycle, the refrigeration load is 14929.8 Btu/hr, the Coefficient of Performance (COP) is 2.302, the second-law efficiency is 47.2%, and the total exergy destruction in the cycle is 1.003 kJ/s. Additionally, the exergy destruction in the condenser is 0.440 kJ/s.

Step-by-step solution

Calculate the mass flow rate of the refrigerant

From the given data, the cooling water enters the condenser at 20°C and leaves at 28°C. The mass flow rate of the cooling water is 0.13 kg/s. We can calculate the heat transfer rate from the condenser to the cooling water using the formula: Q = m*c_p*(T2 - T1). Here, m is the mass flow rate, T2 and T1 are the final and initial temperatures, and c_p is the specific heat capacity of water. Q = (0.13 kg/s) * (4.18 kJ/(kg * °C))(28°C - 20°C) = 4.3736 kJ/s = 4373.6 W.

Calculate the heat rejected by the refrigerant

The compressor consumes 1.9 kW of power, and the heat transfer rate from the condenser to the cooling water is 4373.6 W. Thus, the heat rejected by the refrigerant is equal to the sum of these two heat transfers. Q_rejected = (1.9 * 1000 W) + 4373.6 W = 6273.6 W.

Calculate the heat absorbed by the refrigerant

Since the refrigerant leaves the condenser as a saturated liquid, the heat absorbed by the refrigerant in the evaporator is equal to the heat rejected minus the work done by the compressor. Q_absorbed = Q_rejected - (1.9 * 1000 W) = 4373.6 W

Determine the refrigeration load in Btu/hr

To convert the heat absorbed by the refrigerant to Btu/hr, we multiply by the conversion factor of 3.412142 Btu/hr-W. Refrigeration_load = 4373.6 W * 3.412142 Btu/(hr-W) = 14929.8 Btu/hr.

Calculate the Coefficient of Performance (COP)

The COP is defined as the ratio of the refrigeration effect (heat absorbed) to the work input (compressor power consumption). COP = Q_absorbed / (1.9 * 1000 W) = 4373.6 W / 1900 W = 2.302 For part (a), the refrigeration load is 14929.8 Btu/hr, and the COP is 2.302. Now, let's calculate the second-law efficiency for part (b) and exergy destruction for part (c).

Calculate the second-law efficiency

The second-law efficiency is defined as the ideal COP divided by the actual COP. For a Carnot cycle, the ideal COP is given by the formula: Ideal COP = T_low / (T_high - T_low). We must convert the temperatures to Kelvin first. T_low (K) = -5°C + 273.15 K = 268.15 K, T_high (K) = 50°C + 273.15 K = 323.15 K. Ideal_COP = 268.15 K / (323.15 K - 268.15 K) = 268.15 K / 55 K = 4.875 The second-law efficiency is calculated as: Second-law_efficiency = Actual_COP / Ideal_COP = 2.302 / 4.875 = 0.472 or 47.2%.

Calculate the total exergy destruction in the cycle

Exergy destruction is related to the second-law efficiency and the work input of the system. The formula for total exergy destruction in the cycle is: Total_Exergy_Destruction = Work_input * (1 - Second-law_efficiency). Total_Exergy_Destruction = (1.9 * 1000 W) * (1 - 0.472) = 1003 W or 1.003 kJ/s. For part (b), the second-law efficiency is 47.2%, and the total exergy destruction in the cycle is 1.003 kJ/s. For part (c), we must calculate the exergy destruction in the condenser. The COP of the refrigerator is given by the formula: COP = 1 - T_low / T_high. From this formula, we can derive the exergy destruction for the condenser as:

Calculate the exergy destruction in the condenser

Exergy_Destruction_Condenser = Work_input * ((1 / Actual_COP) - (1 / Ideal_COP)). Exergy_Destruction_Condenser = (1.9 * 1000 W) * ((1 / 2.302) - (1 / 4.875)) = 440 W or 0.440 kJ/s For part (c), the exergy destruction in the condenser is 0.440 kJ/s.

Key concepts

Vapor-Compression Refrigeration Cycle

The vapor-compression refrigeration cycle is a cornerstone concept in thermodynamics and a common method used in refrigeration and air-conditioning systems. The cycle involves four key processes: evaporation, compression, condensation, and expansion. During evaporation, the refrigerant absorbs heat from the space to be cooled, effectively lowering the temperature. It then gets compressed to a high pressure and temperature, which leads to the condensation process where the refrigerant releases the absorbed heat. Lastly, through expansion, the refrigerant pressure and temperature are reduced before it reenters the evaporator, continuing the cycle.

Understanding this cycle is crucial because it sets the stage for evaluating system performance, efficiency, and environmental impact. For engineers, grasping the intricacies of each process and the transitions between them is essential for optimizing system design and troubleshooting.

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a dimensionless measure that enables engineers and technicians to assess the effectiveness of a refrigeration system. It represents the ratio of useful heating or cooling provided to the work required by the system. A higher COP indicates a more efficient system since more refrigeration effect is produced per unit of work. In mathematical terms, the COP is calculated as COP = \frac{Q_{absorbed}}{W_{input}}where \(Q_{absorbed}\) is the heat absorbed by the refrigerant, typically at the evaporator, and \(W_{input}\) is the work consumed by the compressor. In educational terms, this metric can be akin to reviewing 'bang for your buck'—getting the maximum cooling effect with minimal power consumption.

When analyzing textbook exercises on refrigeration systems, it's essential to convert energy units accordingly and accurately determine the COP. This not only helps with immediate homework problems but also builds foundational knowledge for real-world energy management.

Exergy Destruction

Exergy destruction is a crucial part of the second law of thermodynamics and represents the quality of energy. It is the energy that cannot be used for work due to irreversibilities in a thermodynamic process. Exergy destruction is directly related to entropy production, and it's a measure of the irreversibilities within the system. Understanding and calculating the exergy destruction within components of the refrigeration cycle, such as the compressor and the condenser, gives insight into where inefficiencies lie and how they can be minimized.

For enhanced learning, it is overall significant to comprehend the interplay between energy efficiency and entropy generation. When addressing thermodynamics problems in engineering education, highlighting the importance of reducing exergy destruction leads to improved system designs and, consequently, a more sustainable use of energy resources.