Question

Consider a simple Brayton cycle using air as the working fluid; has a pressure ratio of \(12 ;\) has a maximum cycle temperature of \(600^{\circ} \mathrm{C} ;\) and operates the compressor inlet at \(100 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C} .\) Which will have the greatest impact on the back-work ratio: a compressor isentropic efficiency of 80 percent or a turbine isentropic efficiency of 80 percent? Use constant specific heats at room temperature.

Short answer

Solution: Perform the steps 5 from Case 1 and Case 2, calculate the back-work ratios, and compare the results to determine which case has the higher back-work ratio. The case with the higher back-work ratio will have a greater impact on the Brayton cycle's performance.

Step-by-step solution

1. Given Parameters

The given parameters for the simple Brayton cycle are: - Pressure ratio: \(r_p=12\) - Maximum cycle temperature: \(T_3 = 600^{\circ} \mathrm{C}\) - Compressor inlet pressure & temperature: \(P_1 =100 \mathrm{kPa}\), and \(T_1 = 15^{\circ} \mathrm{C}\) - Compressor isentropic efficiency (Case 1): \(\eta_c = 0.8\) - Turbine isentropic efficiency (Case 2): \(\eta_t = 0.8\)

2. Brayton Cycle Analysis for Compressor Isentropic Efficiency of 80% (Case 1)

In this case, we will analyze the Brayton cycle considering a compressor isentropic efficiency of 80%. Step 1 - Isentropic compression (1-2s): Using the pressure ratio, find the isentropic exit temperature, \(T_{2s}\), from the compressor using the relation \( \frac{T_{2s}}{T_1} = \left(\frac{P_2}{P_1}\right)^{(\gamma -1)/\gamma}\). Step 2 - Actual compression (1-2): Use the compressor isentropic efficiency to calculate the actual exit temperature, \(T_2\), from the compressor using the relation \( \eta_c = \frac{T_{2s}-T_1}{T_2-T_1}\). Step 3 - Isentropic expansion in the turbine (3-4s): Use the pressure ratio to find the isentropic exit temperature, \(T_{4s}\), from the turbine using the relation \( \frac{T_{4s}}{T_3} = \left(\frac{P_1}{P_2}\right)^{(\gamma - 1) / \gamma }\). Step 4 - Actual expansion in the turbine (3-4): Since the turbine has an isentropic efficiency of 100%, the actual exit temperature from the turbine is \( T_4 = T_{4s}\). Step 5 - Find the back-work ratio: Calculate the back-work ratio, \(BWR\), using the relation \(BWR = \frac{T_1(T_{2s} - T_1)}{ T_3 (T_3 - T_{4s})}\).

3. Brayton Cycle Analysis for Turbine Isentropic Efficiency of 80% (Case 2)

In this case, we will analyze the Brayton cycle considering a turbine isentropic efficiency of 80%. Step 1 - Isentropic compression (1-2s): This process remains the same as in Case 1 and we can use the isentropic exit temperature, \(T_{2s}\). Step 2 - Actual compression (1-2): Since the compressor has an isentropic efficiency of 100%, the actual exit temperature from the compressor is \(T_2 = T_{2s}\). Step 3 - Isentropic expansion in the turbine (3-4s): This process remains the same as in Case 1 and we can use the isentropic exit temperature, \(T_{4s}\). Step 4 - Actual expansion in the turbine (3-4): Use the turbine isentropic efficiency to calculate the actual exit temperature, \(T_4\), from the turbine using the relation \(\eta_t = \frac{T_3 - T_{4s}}{T_3 - T_4}\). Step 5 - Find the back-work ratio: Calculate the back-work ratio, \(BWR\), using the relation \(BWR = \frac{T_1(T_{2s} - T_1)}{ T_3 (T_3 - T_4)}\).

4. Compare the Back-Work Ratios for Both Cases

To determine which case will have the greatest impact on the back-work ratio, we will compare the back-work ratios calculated in steps 5 of both cases (Case 1 and Case 2). The case with the higher back-work ratio will have a greater impact on the Brayton cycle's performance.