Question

An ideal gas refrigeration system with three stages of compression with intercooling operates with air entering the first compressor at \(50 \mathrm{kPa}\) and \(-30^{\circ} \mathrm{C}\). Each compressor in this system has a pressure ratio of \(7,\) and the air temperature at the outlet of all intercoolers is \(15^{\circ} \mathrm{C}\) Calculate the COP of this system. Use constant specific heats at room temperature.

Short answer

Question: Calculate the coefficient of performance (COP) for a three-stage ideal gas refrigeration system operating under the given conditions: pressure ratio in each stage is 7, ambient temperature at the start of the first stage is -30°C, ambient temperature at the start of the subsequent stages is 15°C, specific heat ratio (k) is 1.4, and specific heat at constant pressure (C_p) is 1005 J/kg/K. Answer: To calculate the COP for the given three-stage refrigeration system, follow these steps: 1. Calculate the total work input by determining the temperature ratio, outlet temperatures, and work done for each stage using the given pressure ratios, ambient temperatures, specific heat ratio, and specific heat. 2. Calculate the heat absorbed (Q_in) in the last stage using the outlet temperature of the final stage and the specific heat. 3. Finally, calculate the COP using the formula COP = Q_in / W_total, where Q_in is the heat absorbed, and W_total is the total work input calculated in previous steps.

Step-by-step solution

Calculate the total work input

For an isentropic process involving an ideal gas, the temperature ratio (T_out/T_in) is related to the pressure ratio (P_out/P_in) by the equation T_out/T_in = (P_out/P_in)^(k-1), where k is the specific heat ratio (C_p/C_v). Given that the pressure ratio in each stage is 7, and the ambient temperature, T_in, at the start of each stage is -30°C and 15°C for the 1st and the subsequent stages respectively, let us first convert these to Kelvin by adding 273. We then use the above relation to find the temperatures at the outlets of each stage. This can be done using the formula T_out = [(P_out/P_in)^(k-1)]*T_in for each stage, where k can be assumed as 1.4 for air at room temperature (as we're using specific heat at room temperature). After finding T_out for each stage, the work done, W, on the air in each stage can be calculated using the formula W = C_p*(T_out - T_in), where C_p is the specific heat of air at constant pressure (assumed 1005 J/kg/K for air at room temperature). Adding up the works done for each stage gives the total work input.

Calculate the heat absorbed

The heat absorbed in the last stage, Q_in, can be calculated by the formula Q_in = C_p*(T_out_final_stage - T_in_final_stage), where T_out_final_stage is the outlet temperature of the air in the final stage.

Calculate the Coefficient of Performance

The Coefficient of Performance (COP) of the refrigeration system can now be calculated by the formula COP = Q_in / W_total, where Q_in is the heat absorbed and W_total is the total work input calculated in the previous steps. This gives the COP of the system.