Question

A steam power plant operates on a simple ideal Rankine cycle between the pressure limits of 1250 and 2 psia. The mass flow rate of steam through the cycle is \(75 \mathrm{lbm} / \mathrm{s}\). The moisture content of the steam at the turbine exit is not to exceed 10 percent. Show the cycle on a \(T-s\) diagram with respect to saturation lines, and determine \((a)\) the minimum turbine inlet temperature, \((b)\) the rate of heat input in the boiler, and \((c)\) the thermal efficiency of the cycle.

Short answer

Question: Determine the minimum allowable temperature at the turbine inlet, the rate of heat input in the boiler, and the thermal efficiency of the Rankine cycle with the given high and low pressure limits and mass flow rate.

Step-by-step solution

Label the Rankine cycle processes

on a T-s diagram. We will have four processes, and they are: 1‐2: Isentropic process in the turbine 2‐3: Constant pressure process in the condenser 3‐4: Isentropic process in the pump 4‐1: Constant pressure process in the boiler

Determine the minimum turbine inlet temperature

We know the moisture content of the steam at the turbine exit should not exceed 10%, so we use the quality of the vapor x to find the isentropic entropy (\(s_2\)) at the known pressure (\(P_2\)=2 psia). \(s_2 = s_f + x \times (s_g−s_f)\) where \(s_f\) and \(s_g\) are the entropies of saturated liquid and saturated vapor, respectively. Assuming 10% moisture, we plug in x = 0.9. Now, use the isentropic relation \(s_2 = s_1\), find the minimum turbine inlet temperature, which corresponds to the entropy \(s_1=s_2\) and pressure \(P_1=1250 \, psia\).

Calculate the enthalpies at each stage of the cycle

Using the determined minimum turbine inlet temperature and entropy, find the specific enthalpy at the turbine inlet (\(h_1\)) from the steam tables. For process 1-2, we'll use the isentropic relation and find the specific enthalpy at the exit of the turbine (\(h_2\)). For process 2-3, we can easily find enthalpy \(h_3\) in the steam tables because it is a saturated liquid at the known pressure (\(P_2=2 \, psia\)). For process 3-4, use the isentropic relation or assume the ideal pump (no heat and friction). In an ideal pump, the change in enthalpy is equal to the specific volume of the fluid multiplied by the difference in pressure: \(\Delta h = v_f(P_1 - P_2)\). Then, determine \(h_4 = h_3 + \Delta h\). Finally, for process 4-1, find \(h_1\) from the steam tables using the known pressure and temperature.

Calculate the work outputs and heat inputs

The work output of the turbine (\(W_{t}\)) and the work needed for the pump (\(W_{p}\)) can be calculated as follows: \(W_{t} = m(h_1 - h_2)\) \(W_{p} = m(h_4 - h_3)\) Where \(m=75 \,\mathrm{lbm} / \mathrm{s}\) is the mass flow rate of steam through the cycle. Now, the net work output of the cycle is the difference between the turbine work output and the pump work input: \(W_{net} = W_{t} - W_{p}\) Rate of heat input in the boiler is calculated by the difference in enthalpies during process 4‐1. \(Q_{in} = m(h_1 - h_4)\)

Find the thermal efficiency of the cycle

The thermal efficiency of the cycle can be determined by the ratio of the net work output to heat input: \(\eta_{thermal} = \frac{W_{net}}{Q_{in}}\) Now we have the requested information: (a) Minimum turbine inlet temperature (b) Rate of heat input in the boiler: \(Q_{in}\) (c) Thermal efficiency of the cycle: \(\eta_{thermal}\)