Question

Calculate the exergy destruction for each process of Stirling cycle of Prob. \(9-74,\) in \(\mathrm{kJ} / \mathrm{kg}\)

Short answer

Answer: The five steps to find the exergy destruction for each process in a Stirling cycle are: 1. Identify the processes in the Stirling cycle, including isothermal compression, constant-volume heat transfer, isothermal expansion, and constant-volume heat transfer. 2. Calculate the exergy transfer due to heat for each process using the formula \(Q_{X} = Q_{in} \cdot (1 - \frac{T_0}{T})\). 3. Calculate the exergy transfer due to work for the isothermal processes using the formula \(W_{X} = W_{in} = -mRT\ln\frac{V_2}{V_1}\). 4. Calculate exergy changes for each process using the exergy balance equation, \(\Delta X = Q_{X} + W_{X}\). 5. Calculate the exergy destruction for each process as the difference between the exergy change and the reversible exergy transfer, \(X_{destruction} = \Delta X - W_{X}\).

Step-by-step solution

Identify the processes in the Stirling cycle

Isothermal compression, constant-volume heat transfer, isothermal expansion, and constant-volume heat transfer are the four processes in the Stirling cycle.

Calculate the exergy transfer due to heat

Recall that exergy transfer due to heat (\(Q_{X}\)) for each process can be calculated using the formula \(Q_{X} = Q_{in} \cdot (1 - \frac{T_0}{T})\), where \(Q_{in}\) is the heat transfer during the process, \(T_0\) is the dead state temperature, and \(T\) is the temperature at which the process takes place. We will use this formula for calculating exergy transfer for the isothermal processes and the constant-volume heat transfer processes.

Calculate the exergy transfer due to work

Since work is a reversible process, exergy transfer due to work (\(W_{X}\)) is equal to the work transfer. For the isothermal processes (compression and expansion), we can use the formula \(W_{X} = W_{in} = -mRT\ln\frac{V_2}{V_1}\), where \(m\) is the mass, \(R\) is the gas constant, and \(V_2\) and \(V_1\) are the final and initial volumes, respectively.

Calculate exergy changes for each process

According to the exergy balance equation, the exergy change for each process can be calculated as \(\Delta X = Q_{X} + W_{X}\). For each process, calculate the exergy change using the formula above, considering the heat and work interactions for each process.

Calculate the exergy destruction for each process

The exergy destruction for each process can be calculated as the difference between the exergy change and the reversible exergy transfer, which is the work transfer for isothermal processes and zero for constant-volume heat transfer processes: \(X_{destruction} = \Delta X - W_{X}\). For each process, calculate the exergy destruction using the formula above. By following these steps, we can calculate the exergy destruction for each process in the Stirling cycle. Note that we need numerical values for heat and work interactions and initial and final states of each process to apply these calculations.