Question

To control the power output of an isentropic steam turbine, a throttle valve is placed in the steam line supplying the turbine inlet, as shown in the figure. Steam at \(6 \mathrm{MPa}\) and \(400^{\circ} \mathrm{C}\) is supplied to the throttle inlet, and the turbine exhaust pressure is set at \(70 \mathrm{kPa}\). Compare the work produced by this steam turbine, in \(\mathrm{kJ} / \mathrm{kg},\) when the throttle valve is completely open (so that there is no pressure loss) and when it is partially closed so that the pressure at the turbine inlet is \(3\) \(\mathrm{MPa}\).

Short answer

Question: Compare the work produced by an isentropic steam turbine when the throttle valve is fully open and when it is partially closed. Answer: When the throttle valve is fully open, the work produced by the steam turbine is 817.3 kJ/kg. When the valve is partially closed, the work produced is 686.7 kJ/kg. The work produced is greater when the valve is fully open due to the decrease in pressure at the turbine inlet when the valve is partially closed.

Step-by-step solution

Determine the enthalpy and entropy at the turbine inlet when the valve is fully open

When the valve is fully open, the inlet pressure and temperature of steam are 6 MPa and 400°C. We need to find the enthalpy and entropy of steam at these conditions by using the steam table. At 6 MPa and 400°C, \(h_1 = 3247.6 \, \mathrm{kJ/kg}\) and \(s_1 = 6.6615 \, \mathrm{kJ/kg \cdot K}\).

Determine the enthalpy at the turbine outlet when the valve is fully open

The outlet pressure is given as 70 kPa. We are given that the turbine is isentropic, meaning the entropy at the outlet is the same as the entropy at the inlet, which is 6.6615 kJ/kg·K. Use steam table to find the corresponding enthalpy \(h_2\), at 70 kPa and the entropy of 6.6615 kJ/kg·K. The corresponding enthalpy is \(h_2 = 2430.3 \, \mathrm{kJ/kg}\).

Calculate the work produced when the valve is fully open

Now we have the enthalpy at the inlet and outlet of the turbine when the valve is fully open. We can calculate the work produced using the following equation: $$ W_t = h_1 - h_2 $$ Plug in the values and calculate the work: $$ W_{t,\text{open}} = 3247.6 \, \mathrm{kJ/kg} - 2430.3 \, \mathrm{kJ/kg} = 817.3 \, \mathrm{kJ/kg} $$

Determine the enthalpy and entropy at the turbine inlet when the valve is partially closed

When the valve is partially closed, the inlet pressure of the steam drops to 3 MPa. Since it is isentropic, the entropy remains the same as in the previous condition, which is \(s_1 = 6.6615 \, \mathrm{kJ/kg \cdot K}\). Using the steam table, find the enthalpy \(h_{1'}\) at 3 MPa and entropy 6.6615 kJ/kg·K: At 3 MPa and the given entropy, \(h_{1'} = 3117.0 \, \mathrm{kJ/kg}\).

Determine the enthalpy at the turbine outlet when the valve is partially closed

As stated earlier, the entropy will be the same as in step 2 because the process is isentropic. Therefore, the enthalpy at the turbine outlet will also be the same as in step 2, which is \(h_2 = 2430.3 \, \mathrm{kJ/kg}\).

Calculate the work produced when the valve is partially closed

Now we have the enthalpy at the inlet and outlet of the turbine when the valve is partially closed. Calculate the work produced using the same equation as before: $$ W_t = h_{1'} - h_2 $$ Plug in the values and calculate the work: $$ W_{t,\text{closed}} = 3117.0 \, \mathrm{kJ/kg} - 2430.3 \, \mathrm{kJ/kg} = 686.7 \, \mathrm{kJ/kg} $$

Comparing the work produced in both scenarios

Finally, compare the work produced by the steam turbine when the valve is fully open and when it is partially closed: When the valve is fully open, the work produced is \(W_{t,\text{open}} = 817.3 \, \mathrm{kJ/kg}\). When the valve is partially closed, the work produced is \(W_{t,\text{closed}} = 686.7 \, \mathrm{kJ/kg}\). The work produced by the steam turbine when the valve is completely open is greater than when the valve is partially closed, as expected due to the decrease in pressure at the turbine inlet.