Question

Consider a steam power plant that operates on the ideal reheat Rankine cycle. The plant maintains the boiler at \(5000 \mathrm{kPa},\) the reheat section at \(1200 \mathrm{kPa}\), and the condenser at 20 kPa. The mixture quality at the exit of both turbines is 96 percent. Determine the temperature at the inlet of each turbine and the cycle's thermal efficiency.

Short answer

Answer: The temperature at the inlet of the high-pressure turbine is approximately 486.7°C, and at the inlet of the low-pressure turbine is approximately 311.7°C. The thermal efficiency of the ideal reheat Rankine cycle is approximately 34.93%.

Step-by-step solution

Find the enthalpy and entropy values for different stages of the cycle

To find the enthalpy and entropy values for each stage of the cycle, we can use the steam tables at the given pressures and mixture quality. - State 1: At the exit of the boiler (inlet of the high-pressure turbine) Boiler pressure: \(P_1 = 5000\,\mathrm{kPa}\) Since it's the exit of the boiler, the steam will be in the saturated vapor state. From the steam tables, we can find the enthalpy (\(h_1\)) and entropy (\(s_1\)) values for this point: \(h_1 = h_\mathrm{g}(P_1) \approx 3380.4\, \mathrm{kJ/kg}\) \(s_1 = s_\mathrm{g}(P_1) \approx 6.048\,\mathrm{kJ/kg\cdot K}\) - State 2: At the exit of the high-pressure turbine (inlet of the reheat section) Reheat section pressure: \(P_2 = 1200\,\mathrm{kPa}\) Mixture quality at the exit of the high-pressure turbine: \(x_2 = 0.96\) To find the enthalpy (\(h_2\)) and entropy (\(s_2\)) values, we'll use the steam tables again: \(h_2 = h_\mathrm{f}(P_2) + x_2[h_\mathrm{g}(P_2) - h_\mathrm{f}(P_2)] \approx 2818.55\,\mathrm{kJ/kg}\) \(s_2 = s_\mathrm{f}(P_2) + x_2 [s_\mathrm{g}(P_2) - s_\mathrm{f}(P_2)] \approx 6.048\,\mathrm{kJ/kg\cdot K}\) - State 3: At the reheat section (inlet of the low-pressure turbine) Reheat section pressure: \(P_3 = 1200\,\mathrm{kPa}\) Since it's at the reheat point, the steam will be in the saturated vapor state. From the steam tables: \(h_3 = h_\mathrm{g}(P_3) \approx 3219\,\mathrm{kJ/kg}\) \(s_3 = s_\mathrm{g}(P_3) \approx 7.057\,\mathrm{kJ/kg\cdot K}\) - State 4: At the exit of the low-pressure turbine (inlet of the condenser) Condenser pressure: \(P_4 = 20\,\mathrm{kPa}\) Mixture quality at the exit of the low-pressure turbine: \(x_4 = 0.96\) To find the enthalpy (\(h_4\)) and entropy (\(s_4\)) values, we'll use the steam tables again: \(h_4 = h_\mathrm{f}(P_4) + x_4[h_\mathrm{g}(P_4) - h_\mathrm{f}(P_4)] \approx 2433.8\,\mathrm{kJ/kg}\) \(s_4 = s_\mathrm{f}(P_4) + x_4 [s_\mathrm{g}(P_4) - s_\mathrm{f}(P_4)] \approx 7.057\,\mathrm{kJ/kg\cdot K}\) Now we have the enthalpy and entropy values for each stage of the cycle.

Calculate the temperature at the inlet of each turbine

Using the enthalpy and entropy values, we can now find the temperature at the inlet of each turbine. - Temperature at the inlet of the high-pressure turbine (\(T_1\)): \(T_1 = T(P_1, s_1) \approx 486.7\,\mathrm{^\circ C}\) - Temperature at the inlet of the low-pressure turbine (\(T_3\)): \(T_3 = T(P_3, s_3) \approx 311.7\,\mathrm{^\circ C}\)

Compute the cycle's thermal efficiency

To calculate the thermal efficiency of the cycle, we'll use the enthalpy values obtained in step 1. The thermal efficiency (\(\eta_\mathrm{thermal}\)) of the cycle can be calculated as follows: $$\eta_\mathrm{thermal} = 1 - \frac{h_4 - h_1 - h_3 + h_2}{h_1 - h_3 + h_2} \approx 34.93\%$$ The thermal efficiency of the ideal reheat Rankine cycle is approximately 34.93%.