Question

An ideal Stirling engine using helium as the working fluid operates between temperature limits of 300 and 2000 K and pressure limits of \(150 \mathrm{kPa}\) and 3 MPa. Assuming the mass of the helium used in the cycle is \(0.12 \mathrm{kg}\), determine \((a)\) the thermal efficiency of the cycle, \((b)\) the amount of heat transfer in the regenerator, and \((c)\) the work output per cycle.

Short answer

Question: Calculate the thermal efficiency, heat transfer in the regenerator, and work output per cycle for an ideal Stirling engine using helium as the working fluid with temperature limits of 2000 K (high) and 300 K (low) and pressure limits of 10 MPa and 1 MPa, respectively. The mass of helium is 0.12 kg. Answer: To calculate the thermal efficiency, heat transfer in the regenerator, and work output per cycle, first determine the heat transfer for the isothermal processes at the high and low temperatures using the formula \(Q = m \times R \times T \times \ln{(V_f / V_i)}\). Next, find the volume ratios for each process, then calculate the net work done and thermal efficiency using the formulas \(W_{net} = Q_h - Q_c\) and \(\eta = \frac{W_{net}}{Q_h}\). Finally, determine the heat transfer in the regenerator and work output per cycle.

Step-by-step solution

Determine the isothermal processes' heat transfer

The Stirling cycle has two isothermal processes, one at the high temperature (T_h) and the other at the low temperature (T_c). We need to find the heat transfer (Q_h and Q_c) for these processes. To do this, we will use the following equation: \(Q = m \times R \times T \times \ln{(V_f / V_i)}\) For the high and low temperature processes: \(Q_h = m \times R \times T_h \times \ln{(V_2 / V_1)} \quad\) and \(\quad Q_c = m \times R \times T_c \times \ln{(V_3 / V_4)}\) where: \(m\) = mass of helium (\(0.12 kg\)) \(R\) = specific gas constant for helium (\(2.077 kJ/(kg \cdot K)\)) \(T_h\) = high temperature (\(2000 K\)) \(T_c\) = low temperature (\(300 K\)) \(V_1\), \(V_2\), \(V_3\), \(V_4\) = initial and final volumes of the isothermal processes

Find volumes of isothermal processes

To find the initial and final volumes, we can use the isochoric processes connecting the isothermal processes. In this case, we are given the pressure limits for these processes. We can use the ideal gas law equation to find the volumes: \(P_1 V_1 = m R T_1\) \(P_2 V_2 = m R T_2\) We can solve for \(V_2/V_1\) and then relate it to \(P_1/P_2\) using the volume ratios: \(V_2/V_1 = (P_1/P_2)(T_2/T_1)\) We can similarly find the volume ratios \(V_3/V_4\).

Calculate heat transfer for isothermal processes

Now that we have the volumes ratio, we can go back to Step 1 and calculate \(Q_h\) and \(Q_c\) for the high and low temperature processes.

Determine thermal efficiency

Thermal efficiency (\(\eta\)) can be calculated using the following equation: \(\eta = \frac{W_{net}}{Q_h}\) where \(W_{net}\) is the net work done during the cycle. It can be calculated as: \(W_{net} = Q_h - Q_c\) Now, we can find the thermal efficiency of the cycle.

Determine the heat transfer in the regenerator

For an ideal Stirling engine, the heat transfer in the regenerator, \(Q_{regen}\), can be calculated as the difference between the high and low temperature heat transfer: \(Q_{regen} = Q_h - Q_c\)

Calculate work output per cycle

Since we already found the net work done in Step 4, we can use that value as the work output per cycle. These steps will help us determine the thermal efficiency, the amount of heat transfer in the regenerator, and the work output per cycle for the given Stirling engine.