Question

A gas refrigeration system using air as the working fluid has a pressure ratio of \(5 .\) Air enters the compressor at \(0^{\circ} \mathrm{C} .\) The high- pressure air is cooled to \(35^{\circ} \mathrm{C}\) by rejecting heat to the surroundings. The refrigerant leaves the turbine at \(-80^{\circ} \mathrm{C}\) and enters the refrigerated space where it absorbs heat before entering the regenerator. The mass flow rate of air is \(0.4 \mathrm{kg} / \mathrm{s}\). Assuming isentropic efficiencies of 80 percent for the compressor and 85 percent for the turbine and using variable specific heats, determine ( \(a\) ) the effectiveness of the regenerator, \((b)\) the rate of heat removal from the refrigerated space, and \((c)\) the \(\mathrm{COP}\) of the cycle. Also, determine \((d)\) the refrigeration load and the COP if this system operated on the simple gas refrigeration cycle. Use the same compressor inlet temperature as given, the same turbine inlet temperature as calculated, and the same compressor and turbine efficiencies.

Short answer

To summarize, in order to solve the given gas refrigeration problem, we need to follow these steps: 1. Calculate the temperature and enthalpy at each state point (1, 2, 3, and 4) using the isentropic temperature relationships and isentropic efficiencies for the compressor and the turbine. 2. Calculate the regenerator's effectiveness as the ratio of its actual heat transfer to the maximum possible heat transfer. 3. Calculate the rate of heat removal from the refrigerated space using the mass flow rate of air and the change in enthalpy across the refrigeration space. 4. Calculate the coefficient of performance (COP) as a ratio of the heat removed from the refrigerated space to the work input to the cycle, taking into account the work input for the compressor and the work output from the turbine. 5. Calculate the refrigeration load and COP for the simple gas refrigeration cycle assuming the same temperature as the non-simple cycle. By following these steps, we can determine the effectiveness of the regenerator, the heat removal rate from the refrigerated space, the coefficient of performance for the given gas refrigeration cycle, and the refrigeration load and COP for the simple gas refrigeration cycle.

Step-by-step solution

Calculate the temperature and enthalpy at state points (1, 2, 3, and 4)

To determine the temperature and enthalpy at each state point, we'll be needing the temperatures T1, T3, and T4, a pressure ratio of 5, isentropic efficiencies for the compressor, and the turbine as well as the air mass flow rate. The temperatures are given: T1 = 0°C = 273 K T3 = 35°C = 308 K T4 = -80°C = 193 K The isentropic temperature relationships can be expressed as: Relative pressure (rp) = \(P_2/P_1 = P_3/P_4\) Using the isentropic temperature relationship and the given pressure ratio for the compressor: \(T_{2s}/T_1 = (P_2/P_1)^{(k-1)/k}\) \(T_{2s} = T_1(rp)^{(k-1)/k}\) For the turbine, similarly: \(T_{3}/T_{4s} = (P_3/P_4)^{(k-1)/k}\) \(T_{4s} = T_3(rp)^{-(k-1)/k}\) Now, taking into account the isentropic efficiencies of the compressor and turbine, we have: \(T_{2a} = T_1 + (T_{2s} - T_1)/\eta_c\) \(T_{4a} = T_3 - (T_3 - T_{4s})\eta_t\) Now, we can obtain the enthalpy at each state point using the specific heats for variable temperatures: \(h_1 = C_p T_1\) \(h_2 = C_p T_{2a}\) \(h_3 = C_p T_3\) \(h_4 = C_p T_{4a}\)

Calculate the effectiveness of the regenerator

We can express the regenerator's effectiveness as the ratio of its actual heat transfer to the maximum possible heat transfer: \(\epsilon_r = \frac{h_4 - h_1}{h_3 - h_2}\)

Calculate the rate of heat removal from the refrigerated space

The rate of heat removal (Q_load) can be calculated using the mass flow rate of air and the change in enthalpy across the refrigeration space: \(Q_{load} = \dot{m}(h_1 - h_4)\)

Calculate the COP of the cycle

The coefficient of performance (COP) is the ratio of the heat removed from the refrigerated space to the work input to the cycle. Therefore, we can find the work input for the compressor (W_comp) and the work output from the turbine (W_turb) as follows: \(W_{comp} = \dot{m}(h_2 - h_1)\) \(W_{turb} = \dot{m}(h_3 - h_4)\) The net work input (W_net) can be obtained by subtracting the work output from the work input: \(W_{net} = W_{comp} - W_{turb}\) Now, we can calculate the COP of the gas refrigeration cycle: \(COP = \frac{Q_{load}}{W_{net}}\)

Calculate the refrigeration load and COP for the simple gas refrigeration cycle

For the simple gas refrigeration cycle, we will assume the same temperature as the non-simple cycle. Therefore, the enthalpy and work values will remain the same. We will calculate the values for \(Q_{load}\) and COP based on these values: \(Q_{load(simple)} = \dot{m}(h_1 - h_{4a})\) \(COP_{(simple)} = \frac{Q_{load(simple)}}{W_{net}}\) Now we have determined all the variables required for the given problem statement; including the effectiveness of the regenerator, the heat removal rate from the refrigerated space, the coefficient of performance for the given gas refrigeration cycle, and the refrigeration load and COP for the simple gas refrigeration cycle.