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College Physics (Urone)
Found in: Page 551

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Short Answer

A coal-fired electrical power station has an efficiency of 38%. The temperature of the steam leaving the boiler is 550 ºC. What percentage of the maximum efficiency does this station obtain? (Assume the temperature of the environment is 20 ºC.)

The maximum efficiency is \(64.4\% \) for the station. This efficiency is \(59\% \) of the maximum efficiency.

See the step by step solution

Step by Step Solution

Step 1: Introduction

Efficiency of the Carnot’s engine is the ratio of work done by the engine to the heat intake from the hot body. The maximum efficiency of the Carnot’s engine is given by the formula

\({\bf{\eta = 1 - }}\frac{{{{\bf{T}}_{\bf{c}}}}}{{{{\bf{T}}_{\bf{h}}}}}\)


\(\eta \) - efficiency of the engine.

\({T_c}\)- the temperature of the cold reservoir.

\({T_h}\)- temperature of the hot reservoir

Step 2:  Given parameters and formula for efficiency of heat engine

Temperature of cold reservoir \({T_c} = 20{\;^{\rm{o}}}{\rm{C}} = 293\;{\rm{K}}\)

Temperature of hot reservoir \({T_h} = 550{\;^{\rm{o}}}{\rm{C}} = 823\;{\rm{K}}\)

The efficiency of the Carnot engine \(\eta = 1 - \frac{{{T_c}}}{{{T_h}}}\)

Step 3:  Calculate the efficiency of the engine

Maximum efficiency can be calculated by calculating Carnot efficiency as

\(\begin{aligned}{}\eta &= 1 - \frac{{{T_c}}}{{{T_h}}}\\ &= 1 - \frac{{293\;}}{{823}}\\ &= 0.644\\ &= 64.4\% \end{aligned}\)

Step 4:  Calculate what percentage real efficiency is of maximum efficiency

Real efficiency \({\eta _1} = 38\% \)

Percentage real efficiency is of maximum efficiency \(\left( {N\% } \right)\)

\(\begin{aligned}{}N\% & = (38\% /64.4\% ) \times 100\\ &= 59\% \end{aligned}\)

Therefore, the maximum efficiency is \(64.4\% \)for the station. The real efficiency is \(59\% \) of the Carnot efficiency.

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