Chapter 15, Problem 60GP

To determine

The increase in the operating temperature for desired efficiency.

Answer to Problem 60GP

The increase in temperature to obtain desired efficiency is $69.76\text{K}$ .

Explanation of Solution

Given info:

The initial efficiency of the Carnot engine is, ${\eta }_1=30\%=0.3$ .

The final efficiency of the Carnot engine is, ${\eta }_2=40\%=0.4$ .

The lower operating temperature of the Carnot engine is, $T_L=20°\text{C}=\left(20°\text{C}+273\right)\text{K}=293\text{K}$ .

Formula Used:

The expression to calculate the initial higher temperature is,

  $T_{h1}=\frac{T_L}{1-{\eta }_1}$

The expression to calculate the final higher temperature is,

  $T_{h2}=\frac{T_L}{1-{\eta }_2}$

The expression to calculate the increase in temperature to obtain desired efficiency is,

  $\Delta T=T_{h2}-T_{h1}$

Calculation:

Substitute all the values in the above expression.

  $\begin{array}{c}{T}_{h1}=\frac{293\text{K}}{1-0.3}\\ =418.57\text{K}\end{array}$

Substitute all the values in the above expression.

  $\begin{array}{c}{T}_{h2}=\frac{293\text{K}}{1-0.4}\\ =488.33\text{K}\end{array}$

Substitute all the values in the above expression.

  $\begin{array}{c}\Delta T=488.33\text{K}-418.57\text{K}\\ =69.76\text{K}\end{array}$

Conclusion:

Thus, the increase in temperature to obtain desired efficiency is $69.76\text{K}$ .