Question

A steam turbine is equipped to bleed 6 percent of the inlet steam for feedwater heating. It is operated with 500 psia and \(600^{\circ} \mathrm{F}\) steam at the inlet, a bleed pressure of 100 psia, and an exhaust pressure of 5 psia. The turbine efficiency between the inlet and bleed point is 97 percent, and the efficiency between the bleed point and exhaust is 95 percent. Calculate this turbine's second-law efficiency. Take \(T_{0}=77^{\circ} \mathrm{F}\).

Short answer

Question: Calculate the second-law efficiency of a steam turbine with given conditions: Inlet pressure - 500 psia, Inlet temperature - \(600^{\circ} \mathrm{F}\), Bleed pressure - 100 psia, Exhaust pressure - 5 psia, Bleed percentage - 6%, Turbine efficiency between inlet and bleed point - 97%, Turbine efficiency between bleed point and exhaust - 95%, Ambient temperature - \(77^{\circ} \mathrm{F}\). Answer: Use the step-by-step solution provided to determine the specific entropy and enthalpy values at different points in the process. Use these values along with the turbine efficiencies to calculate the second-law efficiency for the steam turbine.

Step-by-step solution

Define the given variables

Inlet pressure, \(P_1\) = 500 psia Inlet temperature, \(T_1\) = \(600^{\circ} \mathrm{F}\) Bleed pressure, \(P_2\) = 100 psia Exhaust pressure, \(P_3\) = 5 psia Bleed percentage = 6% Turbine efficiency between inlet and bleed point, \(\eta_{1-2}\) = 97% Turbine efficiency between bleed point and exhaust, \(\eta_{2-3}\) = 95% Ambient temperature, \(T_{0}\) = \(77^{\circ} \mathrm{F}\)

Determine entropy and enthalpy at the inlet

Refer to the steam tables to determine the specific entropy (\(s_1\)) and specific enthalpy (\(h_1\)) of the steam at the inlet condition (\(P_1\) and \(T_1\)).

Determine the isentropic enthalpy, \(h_{2s}\), of the steam at the bleed condition

Since the process is isentropic: \(s_{2s} = s_1\) Determine the isentropic enthalpy (\(h_{2s}\)) at the bleed pressure (\(P_2\)) and entropy (\(s_{2s}\)) using the steam tables.

Calculate the actual enthalpy, \(h_2\), of the steam at the bleed condition

Use the turbine efficiency between the inlet and bleed point to determine the actual enthalpy of the steam at the bleed condition: \(\eta_{1-2} = \frac{h_{1}-h_{2}}{h_{1}-h_{2s}}\) \(h_{2} = h_{1}-\eta_{1-2}(h_{1}-h_{2s})\)

Determine the isentropic enthalpy, \(h_{3s}\), of the steam at the exhaust condition

Since the process is isentropic between bleed and exhaust: \(s_{3s} = s_2\) Determine the isentropic enthalpy (\(h_{3s}\)) at the exhaust pressure (\(P_3\)) and entropy (\(s_{3s}\)) using the steam tables.

Calculate the actual enthalpy, \(h_3\), of the steam at the exhaust condition

Use the turbine efficiency between the bleed point and exhaust to determine the actual enthalpy of the steam at the exhaust condition: \(\eta_{2-3} = \frac{h_{2}-h_{3}}{h_{2}-h_{3s}}\) \(h_{3} = h_{2}-\eta_{2-3}(h_{2}-h_{3s})\)

Calculate the total entropy change for the process

Calculate the total entropy change, \(\Delta S\), for the process using the actual entropies at each condition: \(\Delta S = \Delta S_1 - \Delta S_2\) \(\Delta S_1 = \left(1 - \frac{Bleed\, Percentage}{100}\right)(s_2 - s_1)\) \(\Delta S_2 = \frac{Bleed\, Percentage}{100}(s_3 - s_2)\) Use the values of \(s_1\), \(s_2\), and \(s_3\) from previous steps to calculate the entropy change at each stage.

Calculate the second-law efficiency

Calculate the second-law efficiency of the turbine, \(\eta_{II}\): \(\eta_{II} = 1 - \frac{T_0 \Delta S}{h_1 - h_3}\) Use the enthalpy values from previous steps and the ambient temperature to obtain the second-law efficiency. After step by step calculation, we will get the second-law efficiency value for this steam turbine.