Question

Show that the thermal efficiency of a combined gas-steam power plant \(\eta_{\mathrm{cc}}\) can be expressed as $$\eta_{\mathrm{cc}}=\eta_{g}+\eta_{s}-\eta_{g} \eta_{s}$$ where \(\eta_{g}=W_{g} / Q_{\text {in }}\) and \(\eta_{s}=W_{s} / Q_{g, \text { out }}\) are the thermal efficiencies of the gas and steam cycles, respectively. Using this relation, determine the thermal efficiency of a combined power cycle that consists of a topping gas-turbine cycle with an efficiency of 40 percent and a bottoming steam-turbine cycle with an efficiency of 30 percent.

Short answer

To summarize, we derived the formula for the combined cycle's thermal efficiency as: $$\eta_{cc} = \eta_g + \eta_s - \eta_g\eta_s$$ Then, we calculated the thermal efficiency of the given combined power cycle with gas-turbine cycle efficiency of 0.4 and steam-turbine cycle efficiency of 0.3. The combined cycle's thermal efficiency was found to be 58% (0.58).

Step-by-step solution

Deriving the formula for the combined cycle's thermal efficiency

We will use the definitions of thermal efficiency and the energy balance for a combined cycle to derive the given formula. For a combined cycle: $$Q_{in} = Q_{in_g} + Q_{in_s} = Q_{out_g} + Q_{out_s}$$ The thermal efficiencies of the gas-turbine cycle and the steam-turbine cycle are given as: $$\eta_g = \frac{W_g}{Q_{in_g}}$$ $$\eta_s = \frac{W_s}{Q_{out_g}}$$ The efficiency of the combined cycle is: $$\eta_{cc} = \frac{W_{cc}}{Q_{in}}$$ Where \(W_{cc}\) is the total work of the combined cycle. Now let's express the cycle's work in terms of known parameters: $$W_{cc} = W_g + W_s$$ Using first thermal efficiency equation, $$W_g = \eta_g Q_{in_g}$$ Using second thermal efficiency equation, $$W_s = \eta_s Q_{out_g}$$ Substitute these into the combined cycle's work equation: $$W_{cc} = \eta_g Q_{in_g} + \eta_s Q_{out_g}$$ Now, we need to express \(Q_{out_g}\) in terms of \(Q_{in_g}\): $$Q_{out_g} = Q_{in_g} - W_g = Q_{in_g}(1 - \eta_g)$$ Substitute this into \(W_{cc}\): $$W_{cc} = \eta_g Q_{in_g} + \eta_s Q_{in_g}(1 - \eta_g)$$ Now substitute this into the combined cycle's thermal efficiency equation: $$\eta_{cc} = \frac{\eta_g Q_{in_g} + \eta_s Q_{in_g}(1 - \eta_g)}{Q_{in}}$$ Since \(Q_{in} = Q_{in_g}\), $$\eta_{cc} = \eta_g + \eta_s - \eta_g\eta_s$$ Therefore, we have derived the required formula. Now, let's compute the thermal efficiency of the given combined power cycle.

Calculating the thermal efficiency of the given combined power cycle

We are given that the gas-turbine cycle efficiency \(\eta_g\) is 40% (0.4) and the steam-turbine cycle efficiency \(\eta_s\) is 30% (0.3). Using the derived formula, we can calculate the combined cycle's thermal efficiency: $$\eta_{cc} = \eta_g + \eta_s - \eta_g\eta_s = 0.4 + 0.3 - (0.4)(0.3)$$ $$\eta_{cc} = 0.4 + 0.3 - 0.12 = 0.58$$ So the thermal efficiency of the combined power cycle is 58% (0.58).