1. **Problem Statement:** Calculate the turbine work, pump work, heat added in the boiler, and thermal efficiency for an ideal Rankine cycle with steam entering the turbine at 16 MPa and 540°C and condensing at 10 kPa. 2. **Given Data:** - Turbine inlet pressure $P_1 = 16$ MPa - Turbine inlet temperature $T_1 = 540^\circ C$ - Condenser pressure $P_2 = 10$ kPa 3. **Step 1: Find properties at key points using steam tables:** - At turbine inlet (state 1): superheated steam at $16$ MPa and $540^\circ C$. From steam tables, find enthalpy $h_1$ and entropy $s_1$. - At turbine outlet (state 2): pressure $10$ kPa, isentropic expansion so $s_2 = s_1$. Find enthalpy $h_2$ at $P_2$ and $s_2$. - At condenser outlet (state 3): saturated liquid at $10$ kPa. Find enthalpy $h_3$ and specific volume $v_3$. - At pump outlet (state 4): pressure $16$ MPa, isentropic compression. Calculate enthalpy $h_4$ using $h_4 = h_3 + v_3 (P_4 - P_3)$. 4. **Step 2: Calculate turbine work per kg steam:** $$W_{turbine} = h_1 - h_2$$ 5. **Step 3: Calculate pump work per kg steam:** $$W_{pump} = h_4 - h_3$$ 6. **Step 4: Calculate heat added in boiler per kg steam:** $$Q_{in} = h_1 - h_4$$ 7. **Step 5: Calculate thermal efficiency:** $$\eta = \frac{W_{net}}{Q_{in}} = \frac{W_{turbine} - W_{pump}}{Q_{in}}$$ --- **Using steam tables approximate values:** - At $16$ MPa, $540^\circ C$: $h_1 \approx 3445.5$ kJ/kg, $s_1 \approx 6.6$ kJ/kg.K - At $10$ kPa, saturated liquid: $h_3 \approx 191.8$ kJ/kg, $v_3 \approx 0.00101$ m³/kg - At $10$ kPa, $s_2 = s_1 = 6.6$ kJ/kg.K, find $h_2$ from steam tables for quality $x$: Using $s_2 = s_f + x s_{fg}$, solve for $x$ and then $h_2 = h_f + x h_{fg}$. At $10$ kPa: $s_f = 0.6492$, $s_{fg} = 8.1489$, $h_f = 191.8$, $h_{fg} = 2392.8$. $$x = \frac{6.6 - 0.6492}{8.1489} = 0.74$$ $$h_2 = 191.8 + 0.74 \times 2392.8 = 191.8 + 1770.7 = 1962.5 \text{ kJ/kg}$$ - Pump work: $$W_{pump} = v_3 (P_4 - P_3) = 0.00101 \times (16 \times 10^3 - 10) = 0.00101 \times 15990 = 16.15 \text{ kJ/kg}$$ $$h_4 = h_3 + W_{pump} = 191.8 + 16.15 = 207.95 \text{ kJ/kg}$$ 8. **Calculate turbine work:** $$W_{turbine} = 3445.5 - 1962.5 = 1483 \text{ kJ/kg}$$ 9. **Calculate heat added:** $$Q_{in} = 3445.5 - 207.95 = 3237.55 \text{ kJ/kg}$$ 10. **Calculate thermal efficiency:** $$\eta = \frac{1483 - 16.15}{3237.55} = \frac{1466.85}{3237.55} = 0.453 = 45.3\%$$ **Final answers:** - (a) Turbine work per kg steam = 1483 kJ/kg - (b) Pump work per kg steam = 16.15 kJ/kg - (c) Heat added in boiler per kg steam = 3237.55 kJ/kg - (d) Thermal efficiency = 45.3%