Question

A steam power plant operates on an ideal reheat regenerative Rankine cycle with one reheater and two feedwater heaters, one open and one closed. Steam enters the high-pressure turbine at \(15 \mathrm{MPa}\) and \(600^{\circ} \mathrm{C}\) and the low-pressure turbine at 1 MPa and \(500^{\circ} \mathrm{C}\). The condenser pressure is 5 kPa. Steam is extracted from the turbine at \(0.6 \mathrm{MPa}\) for the closed feedwater heater and at 0.2 MPa for the open feedwater heater. In the closed feedwater heater, the feedwater is heated to the condensation temperature of the extracted steam. The extracted steam leaves the closed feedwater heater as a saturated liquid, which is subsequently throttled to the open feedwater heater. Show the cycle on a \(T-s\) diagram with respect to saturation lines. Determine \((a)\) the fraction of steam extracted from the turbine for the open feedwater heater, \((b)\) the thermal efficiency of the cycle, and \((c)\) the net power output for a mass flow rate of \(42 \mathrm{kg} / \mathrm{s}\) through the boiler

Short answer

#tag_title# Answer #tag_content# 1. To sketch the T-s diagram, we would plot the temperature (T) on the vertical axis and entropy (s) on the horizontal axis. Locate the given enthalpies, pressures, and extracted steam, and connect the points to represent the different components of the Rankine cycle. 2. To determine the fraction of steam extracted for the open feedwater heater (\(x_6\)), we should first find the enthalpies at states 6, 9, and 10 using the given pressures and temperatures, and then solve the equation \(h_{6}x_{6}+h_{10}(1-x_{6})=h_{9}\) for \(x_6\). 3. The thermal efficiency (\(\eta\)) of the cycle can be calculated by finding the specific enthalpies at each state, using the equations for specific work, and dividing the specific net work output (\(W_{net}/s\)) by the specific heat input (\(Q_{in}/s\)). 4. The net power output can be calculated by multiplying the specific net work output (\(W_{net}/s\)) obtained in Step 3 by the mass flow rate (\(\dot{m}\)): \(\dot{W}_{net}=\dot{m} \cdot (W_{net}/s)\). Use the given mass flow rate (42 kg/s) and values found in the previous steps to determine the net power output of the system.

Step-by-step solution

Sketch the T-s diagram with respect to saturation lines

First, draw a T-s diagram. Place state points for the given steam enthalpies, reheater and condenser pressure, and the extracted steam pressure. Connect the points representing parts of the cycle. Make sure the cycle is shown clearly and completely.

Determine the fraction of steam extracted for the open feedwater heater

To determine the fraction of steam extracted, we'll need to calculate the mass flow rate of the extracted steam. We can do this by applying energy balance equations in the open feedwater heater, which is a mixing chamber: \(h_{6}x_{6}+h_{10}(1-x_{6})=h_{9},\) where \(x_6\) is the fraction of steam extracted from the turbine, and \(h_6\), \(h_9\), \(h_{10}\) are the enthalpies of the extracted steam, mixed fluid, and liquid feedwater, respectively. We have to find the respective enthalpies using the given pressures and temperatures, and then solve the equation for \(x_6\)

Calculate the thermal efficiency of the cycle

The thermal efficiency (\(\eta\)) of the cycle can be calculated using the following equation: \(\eta=\frac{W_{net}}{Q_{in}},\) where \(W_{net}\) is the net work output of the cycle and \(Q_{in}\) is the heat input to the cycle. To calculate the work output and heat input, we need to use the First Law of Thermodynamics applied to each component of the cycle. The net work output is given by the sum of the work output of the high-pressure turbine (\(W_{HP}\)), the work output of the low-pressure turbine (\(W_{LP}\)), and the work input to the pump (\(W_P\)). \(W_{net}=W_{HP}+W_{LP}-W_P,\) we will use the following equations for calculating specific work: \(W_{HP}=(h_1-h_2)\), \(W_{LP}=(h_3-h_4)\), \(W_P=(h_{11}-h_{10})\) The heat input to the cycle is given by heat addition in both the boiler and reheater: \(Q_{in}=Q_{boiler}+Q_{reheater}=(h_1-h_{11})+(h_3-h_2)\) Calculate the specific enthalpies of each state using steam tables or an online steam properties calculator. Plug the specific enthalpies into the equations above to find \(W_{net}/s\) and \(Q_{in}/s\) and divide \(W_{net}/s\) by \(Q_{in}/s\) to obtain the cycle's thermal efficiency.

Calculate the net power output

We're given a mass flow rate (\(\dot{m}\)) of 42 kg/s through the boiler. Now, to determine the net power output of the system, simply multiply the specific net work output (\(W_{net}/s\)) obtained in Step 3 by the mass flow rate: \(\dot{W}_{net}=\dot{m} \cdot (W_{net}/s)\) Using the values found in the previous steps, calculate the net power output of the system.

Key concepts

Steam Power Plant

A steam power plant is a thermal system that converts the energy stored in fuel into electrical energy. In traditional steam power plants, water is heated in a boiler to create steam, which then expands through a turbine, producing mechanical energy that is converted into electricity with a generator. The key components involve a boiler, turbine, condenser, and pump, circulating water through a closed loop. The efficiency of such plants is a critical measure, as it signifies the ratio of the useful energy output to the energy input, often facing losses due to friction, heat transfer, and other thermodynamic inefficiencies.

The reheat regenerative Rankine cycle, a variation of the simple Rankine cycle, introduces additional steps to reclaim waste heat and improve efficiency, which includes reheating the steam after it has partially expanded through the turbine and using feedwater heaters to preheat the water entering the boiler using extracted steam, thereby reducing the fuel needed to reach boiling point.

Thermal Efficiency

Thermal efficiency is fundamentally a measure of the fraction of heat converted into useful work in a thermal system. In a steam power plant operating on the Rankine cycle, we calculate this by taking the ratio of the net work output of the cycle to the heat input. Higher efficiency indicates that more of the input energy is being effectively converted to electricity, while lower efficiency implies more energy is lost to the environment. Improvements in the thermal efficiency can significantly reduce fuel consumption and operational costs and decrease the environmental impact.

For students looking to calculate thermal efficiency in their exercises, understanding how to apply the First Law of Thermodynamics to various components of the power cycle is imperative. Extracting insights from the problem statement can also guide which calculations are necessary, and employing accurate measurements of specific work and heat input derived from enthalpies at different states is crucial.

T-s Diagram

A T-s diagram, or temperature-entropy diagram, is an essential graphical representation in thermodynamics that plots temperature (T) against entropy (s). It's a powerful tool used to visualize the heat transfer and work done during various processes in thermodynamic cycles, such as the Rankine cycle used in steam power plants. On the diagram, isobars indicate constant pressure, while saturation lines distinguish the phases of water and steam.

For students analyzing such cycles, the T-s diagram offers a way to see the relationships between different states and processes — addition of heat, expansion, heat removal, and compression. Key transitions between states can be marked on the diagram, helping to understand the flow of thermal energy and efficiency of the cycle. Knowing how to read and interpret T-s diagrams is crucial for solving problems related to thermal systems.

Mass Flow Rate

The mass flow rate is a quantity measuring how much mass passes through a given surface per unit of time, typically denoted with \( \dot{m} \) and expressed in kilograms per second (kg/s). In a steam power plant, the mass flow rate is the amount of steam flowing through the boiler, turbine, and other components in a given period. It is an integral parameter influencing the plant's power output as changes in the mass flow rate directly affect the energy transfer rates and therefore the net power the plant can generate.

Understanding how to calculate and apply the mass flow rate is essential for students dealing with power plant exercises. By multiplying the specific net work output obtained from a Rankine cycle by the mass flow rate, students can determine the system's net power output, a critical factor in designing and evaluating power plant performance.

Feedwater Heaters

Feedwater heaters are devices designed to improve the efficiency of steam power plants by preheating the water before it enters the boiler. There are typically two types of feedwater heaters: open and closed. Open feedwater heaters mix the feedwater directly with steam extracted from the turbine, while closed feedwater heaters do not allow direct mixing and instead use heat exchangers to transfer heat to the feedwater.

By using the steam extracted from various stages of the turbine, these heaters significantly reduce the fuel required to reach the necessary boiler temperature, thereby increasing the cycle's efficiency. When working on exercises involving feedwater heaters, it is important for students to clearly understand the energy transfer occurring within these components and the application of energy balance equations to find unknowns, such as the fraction of steam extracted for the open feedwater heater. Proper application of thermodynamic principles to these devices is key to grasping the overall efficiency improvements in the Rankine cycle.