Question

A simple ideal Rankine cycle with water as the working fluid operates between the pressure limits of 2500 psia in the boiler and 5 psia in the condenser. What is the minimum temperature required at the turbine inlet such that the quality of the steam leaving the turbine is not below 80 percent. When operated at this temperature, what is the thermal efficiency of this cycle?

Short answer

Answer: To find the minimum temperature at the turbine inlet, we need to analyze the Rankine cycle and use the given information on boiler and condenser pressures. The temperature at the turbine inlet can be calculated using the entropy at state 3 and the saturated steam tables. Once the temperature is determined, the thermal efficiency of the cycle can be calculated using the work done by the turbine and pump and the heat input to the cycle.

Step-by-step solution

Initial Information and Calculations

First, we need to convert the given pressures to SI units (pascals): Boiler pressure (P_2) = 2500 psia = 2500 * 6894.76 Pa = 17,236,900 Pa Condenser pressure (P_1) = 5 psia = 5 * 6894.76 Pa = 34,473.8 Pa Now, let's calculate the enthalpy and entropy of water vapor and liquid at both pressures: State 2 (P_2): Enthalpy of vapor (h_2v) = h_fg (P_2) Entropy of vapor (s_2v) = s_fg (P_2) State 1 (P_1): Enthalpy of liquid (h_1f) = h_f (P_1) Enthalpy of vapor (h_1g) = h_g (P_1) Entropy of vapor (s_1g) = s_g (P_1) We can use steam tables to find these values.

Turbine Inlet Temperature

We are given that the quality of steam leaving the turbine should not be below 80%. So, we have: X = 0.8 where X is the quality of steam. We can now calculate the entropy of the steam at state 3 using the isentropic expansion process (s_3 = s_2v). The quality of steam at the outlet can be determined as follows: s_3 = s_1f + X * (s_1g - s_1f) Now, we can calculate the temperature at the turbine inlet (T_3) using the entropy at state 3: T_3 = T_{sat}(s_3)

Turbine Work, Pump Work, and Heat Input

Next, we can calculate the work done by the turbine (W_turbine), the work done by the pump (W_pump), and the heat input to the cycle (Q_in): W_turbine = h_3 - h_4 = h_3 - (h_1f + X * (h_1g-h_1f)) W_pump = v_1 * (P_2 - P_1) ≈ h_2f - h_1f Q_in = h_3 - h_2v

Thermal Efficiency

Finally, we can find the thermal efficiency (η) of the Rankine cycle using the following formula: η = (W_turbine - W_pump) / Q_in Now, we have calculated the minimum required temperature at the turbine inlet and the thermal efficiency of the cycle.