Question

Superheated steam at \(8 \mathrm{MPa}\) and \(480^{\circ} \mathrm{C}\) leaves the steam generator of a vapor power plant. Heat transfer and frictional effects in the line connecting the steam generator and the turbine reduce the pressure and temperature at the turbine inlet to \(7.6 \mathrm{MPa}\) and \(440^{\circ} \mathrm{C}\), respectively. The pressure at the exit of the turbine is \(10 \mathrm{kPa}\), and the turbine operates adiabatically. Liquid leaves the condenser at \(8 \mathrm{kPa}, 36^{\circ} \mathrm{C}\). The pressure is increased to \(8.6 \mathrm{MPa}\) across the pump. The turbine and pump isentropic efficiencies are \(88 \%\). The mass flow rate of steam is \(79.53 \mathrm{~kg} / \mathrm{s}\). Determine (a) the net power output, in \(\mathrm{kW}\). (b) the thermal efficiency. (c) the rate of heat transfer from the line connecting the steam generator and the turbine, in \(\mathrm{kW}\). (d) the mass flow rate of condenser cooling water, in \(\mathrm{kg} / \mathrm{s}\), if the cooling water enters at \(15^{\circ} \mathrm{C}\) and exits at \(35^{\circ} \mathrm{C}\) with negligible pressure change.

Short answer

The net power output is found using mass flow rate and enthalpy changes. Thermal efficiency follows from net power over heat added. The heat transfer rate in the line is the product of mass flow rate and enthalpy difference. Cooling water flow rate uses heat balance in the condenser section.

Step-by-step solution

- Determine state properties for steam entering and exiting the turbine

First, identify the initial conditions of the steam. At the steam generator, the steam is at 8 MPa and 480°C. At the turbine inlet, the pressure and temperature drop to 7.6 MPa and 440°C, respectively. At the turbine exit, the steam is at 10 kPa.

- Calculate the Isentropic Efficiencies

Given the turbine and pump isentropic efficiencies are each 88%, apply these to calculate the actual changes in specific enthalpy for both components. For instance,

- Determine Pump Work

Determine the work done by the pump using the isentropic efficiency. The pump increases the pressure from 8 kPa to 8.6 MPa. Use the specific volume of the liquid at the inlet conditions and the isentropic efficiency to compute this.

- Calculate Power Output

Using the mass flow rate of steam (79.53 kg/s) and the enthalpy changes for both the turbine and the pump, calculate the net power output. Apply the formula

- Calculate Thermal Efficiency

Using the net power output and the heat added in the boiler, calculate the thermal efficiency of the power plant using the formula

- Determine Heat Transfer Rate in Line Connection

To find the rate of heat loss from the connecting line, use the enthalpy difference between the steam generator exit and the turbine inlet multiplied by the mass flow rate of the steam.

- Calculate Mass Flow Rate of Cooling Water

Finally, use the heat balance in the condenser to find the mass flow rate of cooling water. The cooling water enters at 15°C and exits at 35°C, so use the temperature rise and the specific heat capacity to find this.

Key concepts

Isentropic Efficiency

Isentropic efficiency is a crucial parameter in evaluating the performance of turbines and pumps in a vapor power plant. It compares the actual performance of these components to the ideal, isentropic process where no entropy is generated.
For a turbine, isentropic efficiency (\(\eta_{t}\)) is the ratio of the actual work output to the work output if the process were isentropic:

\[\eta_{t} = \frac{h_{1} - h_{2a}}{h_{1} - h_{2s}} \]

Here, \(h_{1}\) is the specific enthalpy at the turbine inlet, \(h_{2a}\) is the actual specific enthalpy at the turbine outlet, and \(h_{2s}\) is the specific enthalpy at the outlet if the process were isentropic. For pumps, isentropic efficiency is defined similarly but in reverse, considering the work input rather than the output.

Thermal Efficiency

Thermal efficiency is a measure of how well a power plant converts heat from fuel into useful work. It is the ratio of the net power output to the heat input.
Thermal efficiency (\(\eta_{th}\)) can be computed using the formula:

\[\eta_{th} = \frac{W_{net}}{Q_{in}} \]

\(W_{net}\) is the net work output of the system, which is the difference between the turbine work output and the pump work input. \(Q_{in}\) is the rate of heat added in the boiler. High thermal efficiency means more energy from the fuel is effectively converted into work.

Turbine Work Output

The work done by the turbine (\(W_{t}\)) is a critical component of the overall power plant performance. It represents the energy converted from steam to mechanical work. The turbine work output is calculated using:

\[W_{t} = \dot{m} (h_{1} - h_{2a}) \]

where \(\dot{m}\) is the mass flow rate of the steam, \(h_{1}\) is the specific enthalpy at the turbine inlet, and \(h_{2a}\) is the specific enthalpy at the actual turbine outlet. The turbine should maximize output with minimal energy loss to achieve higher efficiencies. The turbine efficiency, therefore, directly affects the net power output.

Condensers in Power Plants

The condenser's role in a vapor power plant is to condense the exhaust steam from the turbine back into liquid water. This process releases a significant amount of heat. The condensed liquid is then pumped back into the boiler.
Effective condensation ensures that the next cycle starts with water at a suitable pressure and temperature for pumping. Moreover, condensers aid in maintaining low back pressure in the turbine, thus improving the overall efficiency.
The heat rejected \(Q_{out}\) in the condenser is crucial and is calculated by:

\[Q_{out} = \dot{m} (h_{2a} - h_{f}) \]

Here, \(h_{f}\) represents the specific enthalpy of the liquid leaving the condenser.

Pump Work Input

In a vapor power plant, the pump work input (\(W_{p}\)) is essential in ensuring that the liquid water is brought back to the high pressure required in the boiler. The pump's task is to increase the pressure from the condenser's outlet conditions to the boiler's inlet conditions.
The pump work input is determined by:

\[W_{p} = \dot{m} \frac{v (P_{2} - P_{1})}{\eta_{p}} \]

where \(v\) is the specific volume of the liquid water, \(P_{2}\) and \(P_{1}\) are the outlet and inlet pressures respectively, and \(\eta_{p}\) is the isentropic efficiency of the pump. Minimizing pump work input without compromising performance enhances the net work output and the overall efficiency of the power plant.