Question

Consider an ideal gas-turbine cycle with one stage of compression and two stages of expansion and regeneration. The pressure ratio across each turbine stage is the same. The highpressure turbine exhaust gas enters the regenerator and then enters the low-pressure turbine for expansion to the compressor inlet pressure. Determine the thermal efficiency of this cycle as a function of the compressor pressure ratio and the high-pressure turbine to compressor inlet temperature ratio. Compare your result with the efficiency of the standard regenerative cycle.

Short answer

Answer: The given gas-turbine cycle will be more efficient than the standard regenerative cycle when the following inequality holds true: \(\frac{T_1(\frac{T_{r3} - T_{r1}}{2})}{T_{r3}(\frac{T_{r3} - T_{r1}}{r_c^{\gamma - 1} - 1}(r_c^{\gamma - 1}) + T_1 - T_1)} < \frac{T_1}{T_2}(r_c^{\gamma - 1} - 1)\). The efficiency comparison depends on the values of the compressor pressure ratio (\(r_c\)), high-pressure turbine to compressor inlet temperature ratio (\(\frac{T_{r3}}{T_1}\)), and specific heat ratio (\(\gamma\)).

Step-by-step solution

Recall the efficiency expressions of ideal gas-turbine cycle and standard regenerative cycle

In order to determine the efficiency of the given cycle, we will first recall the formula for the efficiency of an ideal gas-turbine cycle, which is: \(\eta_{gas-turbine} = 1 - \frac{T_1(T_{r2} - T_{r1})}{T_{r3}(T_{2} - T_{1})}\) Also, the efficiency of the standard regenerative cycle is given by: \(\eta_{regenerative} = 1 - \frac{T_1}{T_2}(r_c^{\gamma - 1} - 1)\) where: - \(T_1\) and \(T_2\) are the compressor inlet and outlet temperatures, respectively - \(T_{r1}\), \(T_{r2}\), and \(T_{r3}\) are the regenerator inlet, intermediate, and outlet temperatures, respectively - \(r_c\) is the compressor pressure ratio - \(\gamma\) is the specific heat ratio

Express cycle efficiency as a function of given parameters

Now, we'll express the cycle efficiency as a function of the compressor pressure ratio (\(r_c\)) and the high-pressure turbine to compressor inlet temperature ratio (\(\frac{T_{r3}}{T_1}\)). We need to express \(T_{r2}\) and \(T_2\) in terms of the given parameters. To do that, we use the equations: \(T_2 = \frac{T_{r3} - T_{r1}}{r_c^{\gamma - 1} - 1} (r_c^{\gamma - 1}) + T_1\) \(T_{r2} = \frac{T_{r3} - T_{r1}}{2} + T_{r1}\) Substituting these equations into the efficiency equation for the ideal gas-turbine cycle, we get: \(\eta_{gas-turbine}(\frac{T_{r3}}{T_1}, r_c) = 1 - \frac{T_1(\frac{T_{r3} - T_{r1}}{2})}{T_{r3}(\frac{T_{r3} - T_{r1}}{r_c^{\gamma - 1} - 1}(r_c^{\gamma - 1}) + T_1 - T_1)}\)

Compare the efficiency of the given cycle with the efficiency of the standard regenerative cycle

Now that we have an expression for the efficiency of the gas-turbine cycle as a function of the given parameters, we can compare it with the efficiency of the standard regenerative cycle. We can note that the given cycle will have a higher efficiency than the standard regenerative cycle when: \(\eta_{gas-turbine}(\frac{T_{r3}}{T_1}, r_c) > \eta_{regenerative}\) This inequality holds when: \(\frac{T_1(\frac{T_{r3} - T_{r1}}{2})}{T_{r3}(\frac{T_{r3} - T_{r1}}{r_c^{\gamma - 1} - 1}(r_c^{\gamma - 1}) + T_1 - T_1)} < \frac{T_1}{T_2}(r_c^{\gamma - 1} - 1)\) By comparing the efficiencies of the given cycle and the standard regenerative cycle, we can observe that the given gas-turbine cycle will have higher efficiency than the standard regenerative cycle when the inequality above holds true. This will depend on the values of the compressor pressure ratio (\(r_c\)), high-pressure turbine to compressor inlet temperature ratio (\(\frac{T_{r3}}{T_1}\)), and specific heat ratio (\(\gamma\)).