Question

A gas refrigeration cycle with a pressure ratio of 4 uses helium as the working fluid. The temperature of the helium is \(-6^{\circ} \mathrm{C}\) at the compressor inlet and \(50^{\circ} \mathrm{C}\) at the turbine inlet. Assuming isentropic efficiencies of 85 percent for both the turbine and the compressor, determine ( \(a\) ) the minimum temperature in the cycle, \((b)\) the coefficient of performance, and ( \(c\) ) the mass flow rate of the helium for a refrigeration rate of \(25 \mathrm{kW}\).

Short answer

In a refrigeration cycle using helium as the working fluid, with a given pressure ratio, temperatures at compressor and turbine inlets, isentropic efficiencies, and a refrigeration capacity, the minimum temperature in the cycle is the same as the temperature at the compressor inlet, which is 267.15 K. To calculate the coefficient of performance (COP) and mass flow rate, we will first determine the actual temperatures at the compressor and turbine outlets using the isentropic efficiencies. The COP can then be found using the temperature differences in the cycle, and the mass flow rate can be calculated using the refrigeration capacity and the enthalpy difference at the refrigerator.

Step-by-step solution

Making a schematic diagram and identifying known data

We start by drawing a schematic diagram of the gas refrigeration cycle consisting of a compressor, a heat exchanger, a turbine, and a refrigerator. We are given the following data: - Pressure ratio (\(r_{p}\)) = 4 - Compressor inlet temperature (\(T_{1}\)) = \(-6^{\circ} \mathrm{C}\) = \(267.15 \mathrm{K}\) - Turbine inlet temperature (\(T_{3}\)) = \(50^{\circ} \mathrm{C}\) = \(323.15 \mathrm{K}\) - Isentropic efficiencies for the compressor (\(\eta_{c}\)) and turbine (\(\eta_{t}\)) = 85% - Refrigeration capacity = 25 kW

Find the minimum temperature

To find the minimum temperature in the cycle, we need to first determine the temperature at the compressor exit (\(T_{2s}\)) if the compressor is isentropic. We will utilize the isentropic relation for an ideal gas: \(T_{2s} = T_{1} (r_{p})^{(\gamma - 1)/\gamma}\) Assuming helium behaves as an ideal gas with a heat capacity ratio of \(\gamma = \frac{5}{3}\), we can calculate \(T_{2s}\): \(T_{2s} = 267.15 (4)^{(\frac{5}{3}-1)/(\frac{5}{3})}\) Next, we will use the isentropic efficiency of the compressor to find the actual temperature at the compressor exit (\(T_{2}\)): \(\eta_{c} = \frac{T_{2s} - T_{1}}{T_{2} - T_{1}}\) Solving for \(T_{2}\), we get: \(T_{2} = T_{1} + \frac{T_{2s} - T_{1}}{\eta_{c}}\) The minimum temperature in the cycle is the one at the refrigerator exit, which is the same as the temperature at the compressor inlet, \(T_{1}\) = \(267.15 \mathrm{K}\).

Coefficient of performance calculation

The coefficient of performance (COP) is given by the ratio of the desired output (refrigeration capacity) to the net work input. For the gas refrigeration cycle, the COP is expressed as: \(COP = \frac{Q_{L}}{W_{net}} = \frac{m\cdot (h_{4} - h_{1})}{m\cdot (h_{2} - h_{3})} = \frac{h_{4} - h_{1}}{h_{2} - h_{3}}\) Since we need the enthalpy difference values, we can approximately use the specific heats (\(c_{v}\) and \(c_{p}\)) for helium and find the values of enthalpy changes. \(h_{4} - h_{1} = c_{p} (T_{4} - T_{1})\) and \(h_{2} - h_{3} = c_{p} (T_{2} - T_{3})\) Substitute in the COP formula, we get: \(COP = \frac{T_{4} - T_{1}}{T_{2} - T_{3}}\) To find \(T_{4}\) (temperature at the turbine outlet), we need to first calculate the isentropic temperature at the turbine exit (\(T_{4s}\)) using the pressure ratio 4, we will use this relation: \(T_{4s} = T_{3} (1/r_p)^{(\gamma - 1)/\gamma}\) Now, use the isentropic efficiency of the turbine to find the actual temperature at the turbine exit (\(T_{4}\)): \(\eta_{t} = \frac{ T_{3} - T_{4}}{T_{3} - T_{4s}}\) We can solve for \(T_{4}\) and substitute the temperatures in the COP formula to find the coefficient of performance.

Mass flow rate calculation

To find the mass flow rate of the helium, use the refrigeration capacity and the enthalpy difference at the refrigerator: \(25 \mathrm{kW} = m \cdot c_{p} (T_{4} - T_{1})\) Solve for the mass flow rate of helium: \(m = \frac{25 \mathrm{kW}}{c_{p} (T_{4} - T_{1})}\)