Question

A steam power plant operates on an ideal regenerative Rankine cycle with two open feedwater heaters. Steam enters the turbine at \(8 \mathrm{MPa}\) and \(550^{\circ} \mathrm{C}\) and exhausts to the condenser at \(10 \mathrm{kPa}\). Steam is extracted from the turbine at 0.6 and 0.2 MPa. Water leaves both feedwater heaters as a saturated liquid. The mass flow rate of steam through the boiler is \(16 \mathrm{kg} / \mathrm{s}\). Show the cycle on a \(T\) -s diagram, and determine (a) the net power output of the power plant and ( \(b\) ) the thermal efficiency of the cycle.

Short answer

Question: Determine the net power output and thermal efficiency of the power plant operating on an ideal regenerative Rankine cycle with two open feedwater heaters, given the following operating conditions: Boiler mass flow rate is 16 kg/s; steam entry point at 8 MPa and 550°C, first extraction point at 0.6 MPa, second extraction point at 0.2 MPa, and condenser outlet at 10 kPa saturation liquid. Answer: To calculate the net power output (W_net) and thermal efficiency (η) of the power plant, follow these steps: 1. Draw the T-s Diagram and identify key points (1, 2a, 3a, and 4a). 2. Determine the states of the working fluid at all key points using steam tables. 3. Calculate the mass flow rates (m_2 and m_3) at different extraction points by solving mass balance equations. 4. Calculate the power output of the turbine (W_t) and work input to the pump (W_p) using the enthalpy values and mass flow rates. 5. Calculate the net power output (W_net) of the power plant as the difference between the work output of the turbine and the work input to the pumps. 6. Calculate the heat input (Q_in) to the boiler using the enthalpy values and mass flow rate. 7. Calculate the thermal efficiency (η) of the cycle as the ratio of the net work output to the heat input. By following these steps and performing calculations, the net power output and thermal efficiency of the power plant can be determined.

Step-by-step solution

Draw the T-s Diagram and Identify Key Points

Begin by drawing the T-s diagram for the given cycle. Identify the key points in the cycle, which include the steam entry point, extraction points, and the condenser outlet. These points are: 1. Steam entry point at 8 MPa and 550°C 2. First extraction point at 0.6 MPa 3. Second extraction point at 0.2 MPa 4. Condenser outlet at 10 kPa and saturation liquid We will label these points as 1, 2a, 3a, and 4a.

Determine the states of working fluid at all key points

By using the given information and the saturated steam tables, we can calculate the states of working fluid at all key points. These include temperature, pressure, enthalpy, and entropy. We also need to find the enthalpy values at points 2, 3, and 4 (labeled as 2b, 3b, and 4b). 1. At point 1, the pressure is 8 MPa, and the temperature is 550°C. 2. At points 2a and 2b, the pressure is 0.6 MPa. 3. At points 3a and 3b, the pressure is 0.2 MPa. 4. At points 4a and 4b, the pressure is 10 kPa. We can find the enthalpy values using steam tables.

Calculate the mass flow rates at different extraction points

The mass flow rate (m) of steam through the boiler is given as \(16 \mathrm{kg} / \mathrm{s}\). Let us denote the mass flow rates at extraction points 2a and 3a as \(m_{2}\) and \(m_{3}\), respectively. For a regenerative Rankine cycle with open feedwater heaters, the mass balance equation for the feedwater heaters can be written as: For Heater 1 (at 0.6 MPa): \(m = m_{2} + m_{3}\) For Heater 2 (at 0.2 MPa): \(m_{3} = m - m_{2}\) Solve these equations simultaneously to find \(m_{2}\) and \(m_{3}\).

Calculate the power output of the turbine and work input to the pump

Using the enthalpy values from Step 2 and the mass flow rates from Step 3, we can now determine the power output of the turbine (W_t) and the work input to the pump (W_p). The work output of the turbine is given by: \(W_{t} = m[(h_{1} - h_{2a}) + (1 - m_{2})(h_{2a} - h_{3a})]\) The work input to the pump is the sum of the work inputs to both feedwater pumps: \(W_{p} = m[(1 - m_{2})(v_{4b})(P_{3b} - P_{4b}) + (1 - m_{2} - m_{3})(v_{3b})(P_{2b} - P_{3b})]\)

Calculate the net power output of the power plant

The net power output (W_net) of the power plant is the difference between the work output of the turbine and the work input to the pumps: \(W_{net} = W_{t} - W_{p}\)

Calculate the heat input to the boiler

The heat input (Q_in) to the boiler can be calculated as follows: \(Q_{in} = m(h_{1} - h_{4b})\)

Calculate the thermal efficiency of the cycle

Finally, the thermal efficiency (η) of the cycle can be calculated as the ratio of the net work output to the heat input: \(η = \frac{W_{net}}{Q_{in}}\) By following these steps, we can calculate the net power output (W_net) and the thermal efficiency (η) of the power plant operating on an ideal regenerative Rankine cycle with two open feedwater heaters.

Key concepts

Feedwater Heaters

Feedwater heaters play a crucial role in improving the efficiency of a steam power plant. They are essentially heat exchangers that preheat the boiler's feedwater by using steam extracted from different stages of the turbine. This process increases the temperature of the water before it enters the boiler, reducing the amount of heat required to produce steam. In our ideal regenerative Rankine cycle example, there are two open feedwater heaters that receive steam at pressures 0.6 and 0.2 MPa.

By transferring heat from the extracted steam to the feedwater, the overall thermal efficiency of the cycle is enhanced. The extracted steam, which provides the heat, condenses and becomes a saturated liquid, and is then mixed with the feedwater. This process contributes to a more efficient cycle as it lowers the required energy input and optimizes the steam’s thermal usage.

Steam Power Plant

A steam power plant is a facility designed to generate electricity by utilizing the thermal energy contained in steam. Composed of a boiler, turbine, condenser, and feedwater pumps, the plant operates on a thermodynamic cycle known as the Rankine cycle.

In our specific scenario, the plant operates on a regenerative Rankine cycle with the goal of improving efficiency. Regeneration takes advantage of the turbine's extracted steam to preheat the feedwater, which reduces the fuel consumption and increases the output of the power plant. Steam enters the turbine at a high pressure and temperature, expands through the turbine producing work, and finally condenses back into water to be reused in the boiler. The regenerative cycle introduces an intermediate step with feedwater heaters to enhance the cycle's overall performance.

Thermal Efficiency

Thermal efficiency is a measure of how well a power plant converts heat from fuel into work or electricity. It is defined as the ratio of the net work output of the plant to the heat input required to generate the steam. In mathematical terms, it is expressed as:
\[\begin{equation}η = \frac{W_{net}}{Q_{in}}\end{equation}\]

For a regenerative Rankine cycle, the use of feedwater heaters improves thermal efficiency by reducing the energy needed to convert water into steam. This is because the feedwater is preheated, thus requiring less heat from the boiler. The example problem showcases how the efficiency is impacted by the various parameters like steam extraction for the heaters, and it also demonstrates the calculation of net work output and heat input.

T-s Diagram

A T-s diagram, short for temperature-entropy diagram, is a graphical representation of a thermodynamic cycle. It plots the temperature (T) against entropy (s) and visually displays the heat and work transfers during the process. In the T-s diagram for the regenerative Rankine cycle with feedwater heaters, we can identify and label key points corresponding to the specific states of the working fluid, like entering and exiting steam conditions at the turbine and condenser.

The diagram provides valuable insights into the cycle, such as identifying where heat is added or removed, and understanding entropy changes throughout the process. It’s an indispensable tool for analyzing the cycle’s performance and for calculating the thermal efficiency. The problem provided demonstrates the significance of the T-s diagram in illustrating the cycle and aiding in the calculation of different key thermodynamic properties.

Saturated Steam Tables

Saturated steam tables are indispensable resources that provide information about water and steam properties at the saturation point, the state where water and steam coexist at a given pressure and temperature. The tables list values such as temperature, pressure, specific enthalpy (heat content), specific entropy, and specific volume.

In our regenerative Rankine cycle, we use the saturated steam tables to determine the enthalpies at various points where the working fluid changes phase or when steam is extracted for feedwater heating. Accurate enthalpy values are essential for calculating work and heat transfers within the cycle. For instance, one would look up the enthalpy of water and steam at the pressure corresponding to the condenser to find the required inputs for calculating efficiency. The correct utilization of these tables is vital for precise thermodynamic calculations in the power plant operation.