Question

A simple Rankine cycle uses water as the working fluid. The boiler operates at \(6000 \mathrm{kPa}\) and the condenser at \(50 \mathrm{kPa} .\) At the entrance to the turbine, the temperature is \(450^{\circ} \mathrm{C} .\) The isentropic efficiency of the turbine is 94 percent, pressure and pump losses are negligible, and the water leaving the condenser is subcooled by \(6.3^{\circ} \mathrm{C}\). The boiler is sized for a mass flow rate of \(20 \mathrm{kg} / \mathrm{s}\). Determine the rate at which heat is added in the boiler, the power required to operate the pumps, the net power produced by the cycle, and the thermal efficiency.

Short answer

Also, calculate the rate of heat addition in the boiler, the power required to operate the pumps, and the thermal efficiency of the cycle.

Step-by-step solution

State Points and Known Parameters

Determine the state points and parameters. 1. Turbine inlet (State 1): \(P_1 = 6000 \ \mathrm{kPa}, T_1 = 450^{\circ}\mathrm{C}\) 2. Turbine outlet (State 2): \(P_2 = 50 \ \mathrm{kPa}\) 3. Condenser outlet (State 3): \(P_3 = 50 \ \mathrm{kPa}, T_3 = T_\mathrm{sat} - 6.3^{\circ} \mathrm{C}\) 4. Pump outlet (State 4): \(P_4 = 6000 \ \mathrm{kPa}\) Other given parameters: - Isentropic efficiency of the turbine: \(\eta_\mathrm{turbine} = 94\%\) - Mass flow rate: \(m = 20 \ \mathrm{kg/s}\)

Calculate Turbine Work

Using the isentropic efficiency of the turbine, calculate the actual turbine work. 1. Find the enthalpy at state 1: \(h_1 = h(T_1, P_1)\) 2. Calculate the isentropic enthalpy at state 2: \(h_{2s} = h(T_{2s}, P_2)\) 3. Calculate the actual enthalpy at state 2: \(h_2 = h_1 - \eta_\mathrm{turbine}(h_1 - h_{2s})\) 4. Calculate the turbine work: \(W_\mathrm{turbine} = m(h_1 - h_2)\)

Calculate Pump Work

Neglecting pressure and pump losses, calculate the pump work. 1. Find the specific volume at state 3: \(v_3 = v(T_3, P_3)\) 2. Calculate the pump work: \(W_\mathrm{pump} = m \cdot v_3 (P_4 - P_3)\)

Calculate Heat Added in Boiler

Calculate the rate of heat addition in the boiler. 1. Calculate the heat added in the boiler: \(Q_\mathrm{in} = m(h_1 - h_4)\)

Calculate Net Power and Thermal Efficiency

Calculate the net power produced by the cycle and the thermal efficiency. 1. Calculate the net power: \(W_\mathrm{net} = W_\mathrm{turbine} - W_\mathrm{pump}\) 2. Calculate the thermal efficiency: \(\eta_\mathrm{cycle} = \frac{W_\mathrm{net}}{Q_\mathrm{in}}\) With the above steps, we can find the rate of heat addition in the boiler, the power required to operate the pumps, the net power produced by the cycle, and the thermal efficiency of the given Rankine cycle.