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(I) A certain power plant puts out \(580 \mathrm{MW}\) of electric power. Estimate the heat discharged per second, assuming that the plant has an efficiency of \(35 \%\).

Short Answer

Expert verified
Heat discharged per second is approximately 1077.14 MW.

Step by step solution

01

Understanding Efficiency

Efficiency ( extit{η}) is the ratio of useful energy output (electric power) to the total energy input. The formula is given by \[η = \frac{P_{out}}{P_{in}}\]where \(P_{out}\) is the output power, and \(P_{in}\) is the input power. The efficiency \η\ is 35\(\%\), or 0.35.
02

Rearranging the Efficiency Formula

To find the total energy input \(P_{in}\), rearrange the efficiency formula:\[P_{in} = \frac{P_{out}}{η}\]Plug the given values into this formula, where \(P_{out} = 580 \, \text{MW}\), and \(η = 0.35\).
03

Calculating Total Energy Input

Using the rearranged formula:\[P_{in} = \frac{580 \, \text{MW}}{0.35} \, \approx 1657.14\, \text{MW}\]This is the total energy input into the plant per second.
04

Finding the Heat Discharged

The heat discharged, \(Q_{dis}\), is the difference between the total energy input and the useful energy output:\[Q_{dis} = P_{in} - P_{out} = 1657.14 \, \text{MW} - 580 \, \text{MW} = 1077.14 \, \text{MW}\]So, approximately 1077.14 MW of heat is discharged per second.

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Key Concepts

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

Thermal Power Plants
Thermal power plants are facilities that generate electricity by using heat energy. This heat energy is typically produced by burning fossil fuels like coal, oil, or gas, or by harnessing the heat from nuclear reactions. The fundamental principle behind these plants revolves around converting thermal energy into mechanical energy and, subsequently, into electrical energy.
Inside a thermal power plant, fuel is burned in a boiler to produce steam. This steam drives a turbine connected to a generator, which then produces electricity. The steam cycle is often referred to as the Rankine cycle, which includes heating, expanding, cooling, and compressing the working fluid.
  • Fuel is burned to create heat.
  • Heat converts water to steam.
  • Steam spins the turbine.
  • The turbine powers a generator.
  • The generator outputs electricity.
In these plants, one of the key challenges is managing heat and improving efficiency, which often involves handling excess heat that isn't converted into electrical energy.
Energy Input and Output
In the context of an energy system like a thermal power plant, energy input and output are vital considerations for determining efficiency. Energy input refers to the initial energy supplied to the power plant, usually from burning fuels. This is the total energy available for conversion into useful work.
Energy output, on the other hand, is the useful energy that the power plant provides after conversion. Typically, this is the electricity that can be used for homes, businesses, and industries. Efficiency is calculated as the ratio of energy output to energy input, indicating how well the power plant converts the input energy into output energy.
  • Energy input: Energy supplied from fuel combustion.
  • Energy output: Electricity generated by the plant.
  • Efficiency: Ratio of output to input energy.
In many thermal power plants, not all the energy input is converted to output. It is common for a significant portion of the energy to be lost, primarily as heat.
Heat Discharge
Heat discharge in thermal power plants is a by-product of the energy conversion process. After extracting as much useful energy as possible, the remaining heat energy is expelled. This step is crucial because excess heat needs to be dispersed safely and efficiently to prevent damage and maintain system performance.
Much of the heat discharge occurs in the cooling phase. Commonly, there are cooling towers or heat exchangers at play, which help in transferring heat to the air or water medium, thereby reducing the plant's temperature.
  • Heat discharge is unavoidable.
  • Efficient heat management prevents damage.
  • Cooling towers aid in dissipating excess heat.
Managing heat discharge efficiently helps in reducing the environmental impact and improves the overall efficiency of the plant. In the previously given exercise, approximately 1077.14 MW of this energy is lost as heat, which highlights the challenge and necessity of efficient thermal management.

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Most popular questions from this chapter

1\. (III) Consider an ideal gas of \(n\) moles with molar specific heats \(C_{V}\) and \(C_{P}\). (a) Starting with the first law, show that when the temperature and volume of this gas are changed by a reversible process, its change in entropy is given by $$ d S=n C_{V} \frac{d T}{T}+n R \frac{d V}{V} $$ (b) Show that the expression in part \((a)\) can be written as $$ d S=n C_{V} \frac{d P}{P}+n C_{P} \frac{d V}{V} $$ (c) Using the expression from part \((b)\), show that if \(d S=0\) for the reversible process (that is, the process is adiabatic), then \(P V^{\gamma}=\) constant, where \(\gamma=C_{P} / C_{V}\)

(II) If \(0.45 \mathrm{~kg}\) of water at \(100^{\circ} \mathrm{C}\) is changed by a reversible process to steam at \(100^{\circ} \mathrm{C}\), determine the change in entropy of \((a)\) the water, \((b)\) the surroundings, and \((c)\) the universe as a whole. \((d)\) How would your answers differ if the process were irreversible?

A 126.5 -g insulated aluminum cup at \(18.00^{\circ} \mathrm{C}\) is filled with 132.5 \(\mathrm{g}\) of water at \(46.25^{\circ} \mathrm{C}\) . After a few minutes, equilibrium is reached. Determine \((a)\) the final temperature, and \((b)\) the total change in entropy.

(I) A heat engine exhausts \(7800 \mathrm{~J}\) of heat while performing \(2600 \mathrm{~J}\) of useful work. What is the efficiency of this engine?

(II) An ideal heat pump is used to maintain the inside temperature of a house at \(T_{\text {in }}=22^{\circ} \mathrm{C}\) when the outside temperature is \(T_{\text {out }} .\) Assume that when it is operating, the heat pump does work at a rate of \(1500 \mathrm{~W}\). Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by \(\left(650 \mathrm{~W} / \mathrm{C}^{\circ}\right)\left(T_{\mathrm{in}}-T_{\text {out }}\right)\) (a) For what outside temperature would the heat pump have to operate at all times in order to maintain the house at an inside temperature of \(22^{\circ} \mathrm{C} ?(b)\) If the outside temperature is \(8^{\circ} \mathrm{C},\) what percentage of the time does the heat pump have to operate in order to maintain the house at an inside temperature of \(22^{\circ} \mathrm{C} ?\)

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