How to calculate Steam Generator Pressure in this cycle if you only have one turbine inlet condition?

Question

A simple, superheated Rankine cycle has turbine inlet temperature of 900 K and condenser pressure of 20 kPa. The turbine has efficiency of 92% and the pump has efficiency of 85%. If the overall cycle thermal efficiency is 36%, what is the steam generator pressure (kPa)?

Answer 1

to solve this problem, you can use the first law of thermodynamics to balance the energy inputs and outputs of the Rankine cycle.

First, assume that the mass flow rate of the steam is 1 kg/s. The thermal efficiency of the cycle is defined as the ratio of the net work output to the heat input, so the heat input to the steam generator can be calculated as:

Q_in = W_net / η_th = (m * (h_2 - h_1)) / η_th

where Q_in is the heat input, W_net is the net work output, η_th is the thermal efficiency, m is the mass flow rate, h_2 is the enthalpy at the turbine inlet, and h_1 is the enthalpy at the steam generator outlet.

The enthalpies at the turbine inlet and the steam generator outlet can be calculated using the steam tables, given the temperature and pressure conditions. The enthalpy at the turbine inlet can be calculated as:

h_2 = h(900 K, P)

where P is the pressure at the steam generator outlet. The enthalpy at the steam generator outlet can be calculated as:

h_1 = h(T_c, 20 kPa)

where T_c is the condenser temperature.

The net work output of the cycle can be calculated as the work output of the turbine minus the work input to the pump:

W_net = W_turbine - W_pump = m * (h_2 - h_1) * η_turbine - m * (h_3 - h_4) * η_pump

where W_turbine is the work output of the turbine, W_pump is the work input to the pump, η_turbine is the turbine efficiency, h_3 is the enthalpy at the pump inlet, and h_4 is the enthalpy at the pump outlet.

The enthalpies at the pump inlet and outlet can be calculated using the steam tables, given the temperature and pressure conditions. The enthalpy at the pump inlet can be calculated as:

h_3 = h(T_c, 20 kPa)

where T_c is the condenser temperature. The enthalpy at the pump outlet can be calculated as:

h_4 = h(T_s, P)

where T_s is the saturation temperature corresponding to the pressure P at the steam generator outlet.

Substituting these equations into the expression for the heat input, we can solve for the pressure P at the steam generator outlet:

Q_in = (m * (h_2 - h_1)) / η_th = (m * (h(900 K, P) - h(T_c, 20 kPa))) / η_th

Plugging in the given values for the turbine inlet temperature, condenser pressure, thermal efficiency, turbine efficiency, and pump efficiency, we can solve for the pressure P at the steam generator outlet:

P = ?

Solving this equation will give you the pressure at the steam generator outlet.