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$$ \begin{array}{l}{\cdot \text { A coal-fired power plant that operates at an efficiency of }} \\ {38 \% \text { generates } 750 \mathrm{MW} \text { of electric power. How much heat }} \\ {\text { does the plant discharge to the environment in one day? }}\end{array} $$

Short Answer

Expert verified
The plant discharges 2.06 x 10^7 MJ of heat to the environment in one day.

Step by step solution

01

Understand the Efficiency Formula

The efficiency ( \( e \) ) of the power plant is given by the formula: \[ e = \frac{\text{useful energy output}}{\text{total energy input}} \] where useful energy output is the electric power generated and total energy input is the heat energy supplied. For this problem, the efficiency is 38%, or 0.38 in decimal form.
02

Calculate Total Energy Input

We know the plant generates 750 MW of electric power (useful energy output). To find the total energy input, we rearrange the efficiency formula: \[ \text{total energy input} = \frac{\text{useful energy output}}{e} \] Substituting the known values: \[ \text{total energy input} = \frac{750 \text{ MW}}{0.38} \] This calculates the heat energy supplied to the plant.
03

Calculate Heat Discharged to Environment

The heat discharged to the environment is the difference between the total energy input and the useful energy output: \[ \text{heat discharged} = \text{total energy input} - \text{useful energy output} \]
04

Convert Energy to Heat Discharged Per Day

Since power is in MW (or \( \text{MJ/s} \)), we need to find the total heat discharged in one day (24 hours). Thus, \( 1 \text{ day} = 24 \times 3600 \text{ seconds} \). Multiply the heat discharged rate by this time period to get the total heat discharged in one day.

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Key Concepts

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

Energy Efficiency
Energy efficiency in power plants is a critical concept that measures how well the plant converts fuel into useful energy. In our exercise's context, we deal with a coal-fired power plant that runs at 38% efficiency. This means that only 38% of the energy in the coal is transformed into electricity. The remaining 62% is lost as waste heat.

Understanding energy efficiency helps us analyze how much potential is being used effectively. For a more efficient plant, this percentage would be higher, indicating better utilization of the energy supplied by the fuel. Efficient power plants are desirable as they produce more electricity with less fuel, saving resources and reducing environmental impact.
Power Generation
Power generation is the process where power plants convert fuel sources, such as coal, into electricity. In our scenario, the coal-fired plant produces 750 MW of electric power. This output is often referred to as the plant’s capacity.

Generating power involves complex systems, where heat derived from burning coal is used to produce steam. This steam then drives turbines connected to electrical generators, which produce electricity.
  • The effectiveness of this process directly impacts overall energy efficiency.
  • Advancements in technology encourage methods that increase output without increasing the amount of fuel burned.
These innovations can lead to lower emissions and resource conservation, making power generation both cost-effective and environmentally friendly.
Heat Transfer
Heat transfer in power plants is a fundamental principle. It is the process by which heat moves from the fuel source to generate electricity and then is dissipated as waste heat to the environment.

In our coal-fired power plant example, after 38% of the energy is converted to electricity, the remaining energy is not lost entirely. Instead, it is transferred as residual heat, often released into the air or nearby water bodies. Managing this aspect is crucial for environmental protection, preventing issues like thermal pollution.
  • Heat exchangers and cooling towers are typical structures used to control how heat is discharged.
  • Innovative approaches may involve capturing this residual heat for secondary uses, improving overall energy efficiency.
This effective handling of heat transfer affects not only the plant's efficiency but also its environmental footprint.
Energy Conversion
Energy conversion is the transformation of energy from one form to another. At the heart of any thermodynamic system, it allows the conversion of chemical energy in coal into electrical energy in our power plant.

The process follows several stages, each following the principles of conservation of energy. In our context:
  • The chemical energy stored in coal is converted to thermal energy through combustion.
  • This thermal energy is transformed into mechanical energy via steam-driven turbines.
  • Finally, mechanical energy is converted into electrical energy as generators produce electricity.
Each conversion step involves energy losses, primarily as heat, influencing the plant's overall efficiency. The plant's design aims to minimize these losses while maximizing electrical output, drawing a direct link between energy conversion and efficiency.

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Most popular questions from this chapter

An experimental power plant at the Natural Energy Labo- ratory of Hawaii generates electricity from the temperature gradient of the ocean. The surface and deep-water tempera- tures are \(27^{\circ} \mathrm{C}\) and \(6^{\circ} \mathrm{C}\) , respectively. (a) What is the maximum theoretical efficiency of this power plant? (b) If the power plant is to produce 210 \(\mathrm{kW}\) of power, at what rate must heat be extracted from the warm water? At what rate must heat be absorbed by the cold water? Assume the maximum theoretical efficiency. (c) The cold water that enters the plant leaves it at a temperature of \(10^{\circ} \mathrm{C}\) . What must be the flow rate of cold water through the system? Give your answer in \(\mathrm{kg} / \mathrm{h}\) and \(\mathrm{L} / \mathrm{h}\) .

\(\bullet\) A Carnot freezer that runs on electricity removes heat from the freezer compartment, which is at \(-10^{\circ} \mathrm{C}\) , and expels it into the room at \(20^{\circ} \mathrm{C}\) . You put an ice-cube tray con- taining 375 \(\mathrm{g}\) of water at \(18^{\circ} \mathrm{C}\) into the freezer. (a) What is the coefficient of performance of this freezer? (b) How much energy is needed to freeze this water? (c) How much electrical energy must be supplied to the freezer to freeze the water? (d) How much heat does the freezer expel into the room while freezing the ice?

Solar water heater. A solar water heater for domestic hot- water supply uses solar collecting panels with a collection efficiency of 50\(\%\) in a location where the average solar-energy input is 200 \(\mathrm{W} / \mathrm{m}^{2} .\) If the water comes into the house at \(15.0^{\circ} \mathrm{C}\) and is to be heated to \(60.0^{\circ} \mathrm{C},\) what volume of water can be heated per hour if the area of the collector is 30.0 \(\mathrm{m}^{2} ?\)

A certain nuclear power plant has a mechanical power out- put (used to drive an electric generator) of 330 \(\mathrm{MW}\) . Its rate of heat input from the nuclear reactor is 1300 \(\mathrm{MW}\) . (a) What is the thermal efficiency of the system? (b) At what rate is heat discarded by the system?

. An ice-making machine operates in a Carnot cycle. It takes heat from water at \(0.0^{\circ} \mathrm{C}\) and rejects heat to a room at \(24.0^{\circ} \mathrm{C}\) . Suppose that 85.0 \(\mathrm{kg}\) of water at \(0.0^{\circ} \mathrm{C}\) are converted to ice at \(0.0^{\circ} \mathrm{C}\) (a) How much heat is rejected to the room? (b) How much energy must be supplied to the device?

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