Question

Steam locomotives have an efficiency of \(17\% \) and operate with a hot steam temperature of \(425{\;^o}C\). (a) What would the cold reservoir temperature be if this were a Carnot engine? (b) What would the maximum efficiency of this steam engine be if its cold reservoir temperature were \(150{\;^o}C\)?

Short answer

For a Carnot’s engine, the temperature of the cold reservoir is \(306.34\;^\circ {\rm{C}}\). Maximum efficiency can be achieved, when the temperature of the cold reservoir is \(150\;^\circ {\rm{C}}\), is \(39.4\% \).

Step-by-step solution

Introduction

We calculate the temperature of the hot reservoir by using the formula for efficiency of the Carnot engine when the efficiency and temperature of the cold reservoir are given. We further find the efficiency when the temperature of the cold reservoir is 150^oC.

 Given parameters and formula for efficiency of heat engine

Efficiency of engine\(\eta = 17\% \)

The temperature of the hot reservoir\({T_h} = 425\;^\circ {\rm{C}} = 698\;{\rm{K}}\)

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

Here,

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

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

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

 Calculate the temperature of cold reservoir

The temperature of cold reservoir is calculated as

\(\begin{aligned}{}h &= 1 - \frac{{{T_c}}}{{{T_h}}}\\0.17 &= 1 - \frac{{{T_c}}}{{698\;{\rm{K}}}}\\\frac{{{T_c}}}{{698\;{\rm{K}}}} = 1 - 0.17\end{aligned}\)

\(\begin{aligned}{}{T_c} &= 698\;{\rm{K}} \times 0.83\\ &= 579.34\;{\rm{K}}\\ &= 306.34{\;^o}{\rm{C}}\end{aligned}\)

 Calculate efficiency when temperature of cold reservoir is 150oC

Temperature of cold reservoir\({T_c} = 150\;^\circ {\rm{C}}\)

Efficiency of engine is

\(\begin{aligned}{}h &= 1 - \frac{{{T_c}}}{{{T_h}}}\\ &= 1 - \frac{{423\;{\rm{K}}}}{{698\;{\rm{K}}}}\\ &= 0.394\\ &= 39.4\% \end{aligned}\)

Therefore, the temperature of cold reservoir is \(306.34{\;^o}{\rm{C}}\) for Carnot engine. Maximum efficiency is \(39.4\% \)when temperature of cold reservoir is \(150\;^\circ {\rm{C}}\).