/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 A power plant taps steam superhe... [FREE SOLUTION] | 91Ó°ÊÓ

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A power plant taps steam superheated by geothermal energy to \(505 \mathrm{~K}\) (the temperature of the hot reservoir) and uses the steam to do work in turning the turbine of an electric generator. The steam is then converted back into water in a condenser at 323 \(\mathrm{K}\) (the temperature of the cold reservoir), after which the water is pumped back down into the earth where it is heated again. The output power (work per unit time) of the plant is 84000 kilowatts. Determine (a) the maximum efficiency at which this plant can operate and (b) the minimum amount of rejected heat that must be removed from the condenser every twenty-four hours.

Short Answer

Expert verified
(a) Maximum efficiency is 36.04%. (b) Rejected heat is 1.288 x 10^{13} J per 24 hours.

Step by step solution

01

Understand Carnot efficiency

The maximum efficiency of a heat engine operating between two temperatures is given by the Carnot efficiency. This is expressed as \( \eta = 1 - \frac{T_c}{T_h} \), where \( T_c \) is the absolute temperature of the cold reservoir and \( T_h \) is the absolute temperature of the hot reservoir. Both temperatures must be in Kelvin (K).
02

Calculate the Carnot efficiency

The temperatures provided are \( T_h = 505 \; \mathrm{K} \) and \( T_c = 323 \; \mathrm{K} \). Using the Carnot efficiency formula: \[ \eta = 1 - \frac{323}{505} \] \[ \eta = 1 - 0.6396 \] \[ \eta = 0.3604 \] The maximum efficiency \( \eta \) is approximately 36.04%.
03

Relate efficiency to work and heat

The efficiency of a heat engine is also defined as \( \eta = \frac{W}{Q_h} \), where \( W \) is the work done, and \( Q_h \) is the heat input. The work done per second is given as 84,000 kilowatts or \( 84,000 \times 10^3 \; \text{W} \).
04

Find heat input required per second

Given \( \eta = 0.3604\) and \( W = 84,000 \times 10^3 \; \mathrm{W} \), rearrange the efficiency formula to find \( Q_h \):\( Q_h = \frac{W}{\eta} \)\[ Q_h = \frac{84,000 \times 10^3}{0.3604} \]\[ Q_h = 2.331 \times 10^8 \; \mathrm{W} \]
05

Determine rejected heat per second

The rejected heat \( Q_c \) is given by \( Q_c = Q_h - W \). Substituting in the values:\[ Q_c = 2.331 \times 10^8 - 84,000 \times 10^3 \]\[ Q_c = 1.491 \times 10^8 \; \mathrm{W} \]
06

Calculate rejected heat per 24 hours

Convert \( Q_c \) from watts to joules over 24 hours. Since 1 watt is 1 joule per second, multiply \( Q_c \) by the number of seconds in 24 hours (24 \( \times \) 60 \( \times \) 60 = 86,400 seconds):\[ Q_c = 1.491 \times 10^8 \times 86,400 \]\[ Q_c = 1.288 \times 10^{13} \; \mathrm{J} \]
07

Conclusion

The maximum efficiency at which the plant can operate is approximately 36.04%. The minimum amount of rejected heat that must be removed from the condenser every twenty-four hours is approximately \( 1.288 \times 10^{13} \; \mathrm{J} \).

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Key Concepts

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

Geothermal Energy
Geothermal energy is a fascinating source of renewable energy that originates from within the Earth. It harnesses the heat produced by natural geological processes. This type of energy is both sustainable and largely unaffected by weather conditions, making it a reliable choice for power generation.
Geothermal power plants typically utilize heated water or steam from the Earth's crust to produce electricity. They extract this steam and channel it to power turbines, which can drive generators to produce electrical power.
The extracted water or steam is often returned to the Earth through injection wells, which allows the geothermal cycle to continue, thereby maintaining sustainability. Geothermal energy is environmentally friendly because it has minimal greenhouse gas emissions compared to fossil fuels.
Heat Engine
A heat engine is a fascinating mechanism that converts heat energy into mechanical work. It operates on a cycle where it absorbs heat from a hot reservoir and partially converts this energy into work.
A common example of a heat engine's application is in electric power plants, like the one utilizing geothermal energy. The operation of a heat engine is driven by the flow of heat from a high-temperature source to a lower temperature sink. It is crucial to understand that not all absorbed heat can be converted into work; some of this heat must be released to a cooler environment.
Heat engines are governed by thermodynamic principles, ensuring that they operate efficiently only under specific conditions, like those described by Carnot's theorem. This concept is central to understanding the theoretical limits of heat engine efficiency.
Thermodynamics
Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work. It primarily deals with the understanding of energy transfer processes.
The principles of thermodynamics are crucial in analyzing the performance of systems like heat engines. One of its key principles is the Second Law of Thermodynamics, which dictates the direction of heat transfer and explains why energy cannot flow spontaneously from a cooler body to a hotter body.
This principle also outlines the concept of efficiency in heat engines, suggesting that no engine can be perfectly efficient. The Carnot cycle, a theoretical model that defines the maximum efficiency achievable by a heat engine, is rooted deeply in thermodynamic laws.
Rejected Heat
Rejected heat is a vital concept when it comes to evaluating the performance of heat engines. It refers to the portion of heat energy that is not converted into work and is instead released into the environment.
Rejected heat is inevitable due to the natural limitations imposed by the laws of thermodynamics. In a geothermal power plant, for instance, even though a significant amount of energy is harnessed for work, much of it ultimately becomes rejected heat, usually dissipated through cooling towers or condensers.
Understanding rejected heat is important to improve the efficiency of existing systems. Engineers aim to minimize this rejected heat to increase the overall efficiency of the engine, though complete elimination is impossible according to the Second Law of Thermodynamics.

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Most popular questions from this chapter

Go Concept Questions A system gains a certain amount of energy in the form of heat at constant pressure, and the internal energy of the system increases by an even greater amount. (a) Is any work done and, if so, is it done on or by the system? (b) If there is work, is it positive or negative, according to our convention? (c) Does the volume of the system increase, decrease, or remain the same? Give your reasoning.

A monatomic ideal gas expands at constant pressure, (a) What percentage of the heat being supplied to the gas is used to increase the internal energy of the gas? (b) What percentage is used for doing the work of expansion?

When a .22 -caliber rifle is fired, the expanding gas from the burning gunpowder creates a pressure behind the bullet. This pressure causes the force that pushes the bullet through the barrel. The barrel has a length of \(0.61 \mathrm{~m}\) and an opening whose radius is \(2.8 \times 10^{-3} \mathrm{~m} .\) A bullet (mass \(=2.6 \times 10^{-3} \mathrm{~kg}\) ) has a speed of \(370 \mathrm{~m} / \mathrm{s}\) after passing through this barrel. Ignore friction and determine the average pressure of the expanding gas.

\(55 \mathrm{~m}\) A monatomic ideal gas \(\left(\gamma=\frac{5}{3}\right)\) is contained within a perfectly insulated cylinder that is fitted with a movable piston. The initial pressure of the gas is \(1.50 \times 10^{5} \mathrm{~Pa}\). The piston is pushed so as to compress the gas, with the result that the Kelvin temperature doubles. What is the final pressure of the gas?

Multiple-Concept Example 6 deals with the same concepts as this problem does. What is the efficiency of a heat engine that uses an input heat of \(5.6 \times 10^{4} \mathrm{~J}\) and rejects \(1.8 \times 10^{4} \mathrm{~J}\) of heat?

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