Question

unreasonable Results

(a) Suppose you want to design a steam engine that has heat transfer to the environment at 270ºC and has a Carnot efficiency of 0.800. What temperature of hot steam must you use? (b) What is unreasonable about the temperature? (c) Which premise is unreasonable?

Short answer

The temperature of hot steam is \(2442\;^\circ {\rm{C}}\) which is to be used. The thing that is unreasonable about the temperature is that it is a very high temperature that melts metals like iron, etc. This efficiency and this high temperature is unreasonable.

Step-by-step solution

Introduction

The ratio of the amount of work done by the engine to the amount of heat intake from the source gives the efficiency of the heat engine.

 Given parameters and formula for efficiency of heat engine

Efficiency of engine \(\eta = 0.8\)

The temperature of cold reservoir\({T_c} = 270\;^\circ {\rm{C}} = {\rm{543}}\;{\rm{K}}\)

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

Here,

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

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

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

 Calculate the temperature of hot steam

The temperature of hot steam is calculated as

\(\begin{aligned}{}\eta & = 1 - \frac{{{T_c}}}{{{T_h}}}\\0.8 &= 1 - \frac{{543\;{\rm{K}}}}{{{T_h}}}\\\frac{{543\;{\rm{K}}}}{{{T_h}}} &= 1 - 0.8\end{aligned}\)

\(\begin{aligned}{}{T_h} &= \frac{{543\;{\rm{K}}}}{{0.2}}\\ &= 2715 {\rm{K}}\\ &= 2442{\;^{\rm{o}}}{\rm{C}}\end{aligned}\)

Therefore, the temperature of hot steam is\(2442\;^\circ {\rm{C}}\)which is to be used. The thing that is unreasonable about the temperature is that it is very high temperature,higher than melting point of metals like iron, etc. This would be unreasonable which melts metals like iron, etc. This efficiency and this high temperature is unreasonable.