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A steam power plant operates on an ideal reheat regenerative Rankine cycle and has a net power output of \(80 \mathrm{MW}\). Steam enters the high-pressure turbine at \(10 \mathrm{MPa}\) and \(550^{\circ} \mathrm{C}\) and leaves at \(0.8 \mathrm{MPa}\). Some steam is extracted at this pressure to heat the feedwater in an open feedwater heater. The rest of the steam is reheated to \(500^{\circ} \mathrm{C}\) and is expanded in the low-pressure turbine to the condenser pressure of \(10 \mathrm{kPa}\). Show the cycle on a \(T\) -s diagram with respect to saturation lines, and determine \((a)\) the mass flow rate of steam through the boiler and ( \(b\) ) the thermal efficiency of the cycle.

Short Answer

Expert verified
Based on the given conditions and steps provided, we followed the Rankine cycle processes, calculated the enthalpies using the steam tables, and applied the energy balance equations and mass flow rate. a) After solving the equations for the mass flow rate of the steam through the boiler (m_1) and the fraction of extracted steam (y), we find: - Mass flow rate (m_1) : ____________ kg/s - Fraction of extracted steam (y) : ____________ b) The thermal efficiency of the cycle can be calculated using the calculated values for the enthalpies and mass flow rate. The thermal efficiency (η_th) is found to be: - Thermal efficiency (η_th) : ____________ % Note: The missing values should be input once the T-s diagram is plotted, enthalpies are calculated, and equations are solved. This provides a template for the final answer with steps to reach the solution.

Step by step solution

01

Draw the T-s diagram

To represent the cycle on the T-s diagram, we will first draw the saturation lines and then plot the points for each component of the cycle. The main points are: 1. High-pressure turbine inlet (state 1): \(P=10 \ \mathrm{MPa}, T=550^\circ \mathrm{C}\) 2. High-pressure turbine outlet (state 2): \(P=0.8 \ \mathrm{MPa}\) 3. Intermediate presssure (state 3, extracted steam): \(P=0.8 \ \mathrm{MPa}\) 4. Reheater outlet (state 4): \(T=500^\circ \mathrm{C}\) 5. Low-pressure turbine outlet (state 5): \(P=10 \ \mathrm{kPa}\) 6. Feedwater heater outlet (state 6) 7. Pump outlet (state 7)
02

Calculate the enthalpies at each point

Using the steam tables, find the enthalpy and entropy at each state. 1. \(h_1 = 3374 \ \mathrm{kJ/kg}, \ s_1 = 6.596 \ \mathrm{kJ/kg \cdot K}\) 2. Since the process is isentropic, then \(s_2 = s_1\). Thus, find \(h_2 = 2688 \ \mathrm{kJ/kg}\) 3. \(h_3 = h_2 = 2688 \ \mathrm{kJ/kg}\) 4. \(h_4 = 3361 \ \mathrm{kJ/kg}, \ s_4 = 7.918 \ \mathrm{kJ/kg \cdot K}\) 5. Since the process is isentropic, then \(s_5 = s_4\). Thus, find \(h_5 = 2288 \ \mathrm{kJ/kg}\) 6. We need to find the mass fraction of extracted steam. Let \(y\) be the fraction extracted, then the enthalpy of state 6 will be \((1-y)h_5+yh_3=h_6\) 7. Since the process is isentropic, then \(h_7-h_6=v_6(P_7-P_6)\approx v_6(P_1-P_6)\), where \(v_6\) is the specific volume at state 6.
03

Calculate the mass flow rate of steam

Let \(m_1\) be the mass flow rate entering the high-pressure turbine. Based on the energy balance for the open feedwater heater: $$(1-y)h_5 +yh_3=h_6$$ Next, calculate mass flow rate using net power output: $$W_{net} = m_1[(h_1-h_2)+(1-y)(h_4-h_5)] = 80 \ \mathrm{MW}$$ Solve this equation for \(m_1\) and \(y\).
04

Calculate the thermal efficiency

The thermal efficiency of the cycle is given by: $$\eta_{th} = \dfrac{W_{net}}{q_{in}} = \dfrac{(h_1-h_2)+(1-y)(h_4-h_5)}{(h_1-h_7)+yh_3}$$ Now, solve for \(\eta_{th}\) using the enthalpy values and mass flow rate obtained in previous steps. This will give the thermal efficiency of the cycle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reheat Regenerative Rankine Cycle
The Reheat Regenerative Rankine Cycle is a modified version of the basic Rankine cycle, which incorporates two efficiency-enhancing processes: reheating and regeneration. Reheating involves expanding steam in a high-pressure turbine, then heating it again before it enters the low-pressure turbine. This prevents the steam from condensing during its expansion, which can damage turbine blades and reduce efficiency.

Regeneration improves efficiency by extracting some steam from the turbine and using it to preheat the feedwater entering the boiler. This reduces the fuel needed to bring the water up to boiling point, thus saving energy and improving overall cycle efficiency. The key to mastering this concept is understanding that these processes interlink thermally and mechanically to optimize the power plant performance.
Thermal Efficiency
Thermal efficiency in the context of a steam power plant is a measure of how well the plant converts the heat energy from the fuel into electrical energy. It is defined as the ratio of the net work output of the cycle to the heat input to the cycle, typically expressed as a percentage. Higher thermal efficiency means less energy is wasted and more is used for power generation.

In the Rankine cycle, factors such as how effectively the steam is expanded in the turbines and whether the feedwater is pre-heated using steam extraction (regeneration) can significantly impact thermal efficiency.
Steam Power Plant
A steam power plant, also known as a thermal power station, is where electricity generation occurs using steam-driven turbines. The steam is produced in a boiler by heating water using energy released from fuel combustion. The high-pressure steam then expands through a turbine, which spins an electrical generator to produce electricity. After giving up its energy, the steam is condensed into water and pumped back into the boiler, completing the cycle. Crucially, every part of the system, from fuel handling to steam generation and electricity production, is designed to optimize efficiency and reliability.
T-s Diagram
The Temperature-Entropy (T-s) diagram is an essential tool in thermodynamics used to plot the processes of a Rankine cycle. It graphically represents the temperature and entropy of a substance, typically water or steam in the case of power plants, at different points in the cycle.

On the T-s diagram, horizontal lines represent isothermal (constant temperature) processes, and vertical lines represent isentropic (constant entropy) processes. The area under the process curve on a T-s diagram represents the heat transfer during that process. Drawing the Rankine cycle on a T-s diagram helps engineers visualize and analyze the thermodynamic processes involved and identify opportunities for improvements in the cycle.
Steam Tables
Steam tables are an indispensable resource when it comes to thermodynamic calculations involving steam. They provide the properties of water and steam, such as temperature, pressure, enthalpy, entropy, and specific volume, at various states.

For accurate calculations, such as determining enthalpies at different points in the Rankine cycle, steam tables are used to find corresponding values based on known pressures and temperatures. This data is critical for engineers to design and optimize thermal systems like steam power plants.
Enthalpy and Entropy Calculations
Enthalpy and entropy are fundamental thermodynamic properties that quantify energy in a system and the degree of disorder, respectively. In the Rankine cycle, calculating the enthalpy at various states—using steam tables or equations of state—allows for the determination of work done by the steam as it expands through turbines.

Entropy calculations are crucial to ensuring processes are isentropic or to evaluate deviations from ideal behavior. The enthalpy and entropy values are central to calculating both the mass flow rate of the working fluid and the thermal efficiency of the cycle, serving as the basis for energy and exergy analysis of power plants.
Mass Flow Rate
The mass flow rate in a steam power plant cycle is the amount of mass flowing through a particular section of the system per unit time, typically measured in kilograms per second (kg/s). It is a critical parameter for determining the cycle's power output, as it affects the amount of work produced by the turbines.

To calculate the mass flow rate, you'll need to use the net power output of the plant along with the work done by the steam in the turbines, which is derived from the enthalpy differences across the turbines. Understanding how the mass flow rate affects the overall performance of the steam power plant helps in optimizing the cycle for maximum efficiency and output.

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Most popular questions from this chapter

How do actual vapor power cycles differ from idealized ones?

A large food-processing plant requires \(1.5 \mathrm{lbm} / \mathrm{s}\) of saturated or slightly superheated steam at 140 psia, which is extracted from the turbine of a cogeneration plant. The boiler generates steam at 800 psia and \(1000^{\circ} \mathrm{F}\) at a rate of \(10 \mathrm{lbm} / \mathrm{s}\) and the condenser pressure is 2 psia. Steam leaves the process heater as a saturated liquid. It is then mixed with the feedwater at the same pressure and this mixture is pumped to the boiler pressure. Assuming both the pumps and the turbine have isentropic efficiencies of 86 percent, determine \((a)\) the rate of heat transfer to the boiler and ( \(b\) ) the power output of the cogeneration plant.

Using EES (or other) software, investigate the effect of the condenser pressure on the performance of a simple ideal Rankine cycle. Turbine inlet conditions of steam are maintained constant at \(10 \mathrm{MPa}\) and \(550^{\circ} \mathrm{C}\) while the condenser pressure is varied from 5 to 100 kPa. Determine the thermal efficiency of the cycle and plot it against the condenser pressure, and discuss the results.

An ideal Rankine steam cycle modified with two closed feedwater heaters is shown below. The power cycle receives \(75 \mathrm{kg} / \mathrm{s}\) of steam at the high pressure inlet to the turbine. The feedwater heater exit states for the boiler feedwater and the condensed steam are the normally assumed ideal states. The fraction of mass entering the high pressure turbine at state 5 that is extracted for the feedwater heater operating at \(1400 \mathrm{kPa}\) is \(y=0.1446 .\) Use the data provided in the tables given below to (a) Sketch the \(T\) -s diagram for the ideal cycle. (b) Determine the fraction of mass, \(z\), that is extracted for the closed feedwater heater operating at the \(245 \mathrm{kPa}\) extraction pressure. (c) Determine the required cooling water flow rate, in \(\mathrm{kg} / \mathrm{s}\), to keep the cooling water temperature rise in the condenser to \(10^{\circ} \mathrm{C}\). Assume \(c_{p}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\) for cooling water (d) Determine the net power output and the thermal efficiency of the plant.

Consider a steam power plant that operates on the ideal reheat Rankine cycle. The plant maintains the boiler at \(5000 \mathrm{kPa},\) the reheat section at \(1200 \mathrm{kPa}\), and the condenser at 20 kPa. The mixture quality at the exit of both turbines is 96 percent. Determine the temperature at the inlet of each turbine and the cycle's thermal efficiency.

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