Question

Using EES (or other) software, investigate the effect of extraction pressure on the performance of an ideal regenerative Rankine cycle with one open feedwater heater. Steam enters the turbine at \(15 \mathrm{MPa}\) and \(600^{\circ} \mathrm{C}\) and the condenser at 10 kPa. Determine the thermal efficiency of the cycle, and plot it against extraction pressures of 12.5,10,7,5 \(2,1,0.5,0.1,\) and \(0.05 \mathrm{MPa}\), and discuss the results.

Short answer

Answer: By calculating the thermal efficiency of the cycle for various extraction pressures (12.5, 10, 7, 5, 2, 1, 0.5, 0.1, and 0.05 MPa) and plotting the results, we can identify the relationship between extraction pressure and thermal efficiency. The optimal extraction pressure for maximum thermal efficiency can also be determined, which helps understanding the performance implications for an ideal regenerative Rankine cycle.

Step-by-step solution

Determine the states in the Rankine cycle

To begin, we need to determine the states (pressure, temperature, and enthalpy) at each point in the Rankine cycle: points 1, 2, 3, 4, 5, and 6. We can do this using the steam tables and the input parameters (15 MPa, 600°C, and condenser at 10 kPa). We already know the conditions for points 1, 2, and 6, so we only need to determine the remaining points.

Perform an energy balance on the feedwater heater

Perform an energy balance on the open feedwater heater to determine the mass fraction of steam (y) extracted at the various extraction pressures. We can do this with the following equation: \[ y = \frac{h_4 - h_3}{h_5 - h_3} \]

Calculate the work and heat input/output

Calculate the work of the turbine, the work of the pump, and the heat input/output at the boiler and condenser using the enthalpies determined in Step 1. The work for turbine and pump can be calculated using these equations: \[ W_\mathrm{turbine} = (1-y)(h_1 - h_2) \] \[ W_\mathrm{pump} = h_4 - h_6 \] The heat for boiler and condenser can be calculated as: \[ Q_\mathrm{in} = h_1 - h_6 \] \[ Q_\mathrm{out} = (1-y)(h_2 - h_5) + y(h_2 - h_3) \]

Determine the thermal efficiency

With the work and heat input/output calculated, we can now determine the thermal efficiency using the following equation: \[ \eta_\mathrm{thermal} = \frac{W_\mathrm{turbine} - W_\mathrm{pump}}{Q_\mathrm{in}} \] The thermal efficiency should be calculated for each of the specified extraction pressures: 12.5, 10, 7, 5, 2, 1, 0.5, 0.1, and 0.05 MPa.

Plot thermal efficiency against extraction pressure

After calculating the thermal efficiency of the cycle for all specified extraction pressures, you should plot the results. Create a graph with the extraction pressures on the x-axis and the thermal efficiency on the y-axis.

Discuss the results

By examining the plotted graph, you can identify trends and discuss the relationship between extraction pressure and thermal efficiency. This may include identifying the optimal extraction pressure for maximum thermal efficiency and discussing the implications of this on the performance of an ideal regenerative Rankine cycle.

Key concepts

Thermal Efficiency Calculation

Understanding the thermal efficiency of power cycles, such as the regenerative Rankine cycle, is key to analyzing system performance. The calculation of thermal efficiency is a measure of how well a cycle converts heat into work. In a simple Rankine cycle, where no regeneration takes place, the efficiency is determined using the equation:
\[ \eta_{\text{thermal}} = \frac{W_{\text{net}}}{Q_{\text{in}}} \]
where \( W_{\text{net}} = W_{\text{turbine}} - W_{\text{pump}} \) is the net work output of the cycle, and \( Q_{\text{in}} \) is the heat added in the boiler.
Regeneration improves efficiency by preheating the feedwater, which reduces the need for excessive heating in the boiler.
When analyzing the effect of extraction pressure, as in the given exercise, the thermal efficiency for each pressure setting is calculated and compared. A graph with extraction pressure versus efficiency is beneficial to visualize the results and draw conclusions about the cycle's performance in relation to changes in extraction pressure.

Energy Balance in Thermodynamics

Energy balance is a fundamental concept in thermodynamics, ensuring the conservation of energy within a system. For any device within a Rankine cycle, like a feedwater heater, boiler, or condenser, an energy balance must be applied. This means the energy entering the device must equal the energy leaving it.
In the context of the regenerative Rankine cycle, the energy balance for the open feedwater heater would involve the incoming water stream from the pump and the extraction steam from the turbine. The balance equation is:
\[ y(h_5 - h_3) + (1 - y)(h_4 - h_3) = 0 \]
where \( h_x \) is the specific enthalpy at state x and y is the fraction of steam extracted. Solving this equation helps in determining the mixing ratio critical for analyzing the cycle's performance.

Steam Turbine Performance

Steam turbines are at the heart of Rankine cycles, and their performance is a significant determinant of the overall efficiency. In this exercise, the work output of the turbine is calculated using the enthalpies at different points in the cycle:
\[ W_{\text{turbine}} = (1-y)(h_1 - h_2) \]
This equation exemplifies that the turbine's work is dependent on the enthalpy difference between the steam inlet and outlet conditions, as well as the mass fraction of steam that is extracted (1 - y). The performance is influenced by the extraction pressure; as it varies, so does the amount of work generated by the turbine. Therefore, optimizing the extraction pressure is crucial for maximizing work output and cycle efficiency.

Feedwater Heater Analysis

In a regenerative Rankine cycle, the feedwater heater plays an important role by utilizing the extracted steam to preheat the feedwater heading back into the boiler. Analyzing its performance includes using energy balance to determine the optimal extraction pressure, which maximizes cycle efficiency.
The effectiveness of the feedwater heater can be gauged by considering the amount of heat transferred and the impact of different extraction pressures. The energy balance approach, as shown earlier, allows one to calculate the mass flow fraction of the extracted steam necessary for heating the feedwater.
Furthermore, this analysis helps in determining the right balance between maximizing the heater's benefit and minimizing the drawbacks of over-extraction, such as a reduction in the available energy to produce work in the turbine.