Question

Question:(I) The exhaust temperature of a heat engine is 230°C. What is the high temperature if the Carnot efficiency is 34%?

Short answer

The high temperature of the Carnot engine is \(489.12^\circ {\rm{C}}\).

Step-by-step solution

Understanding the efficiency of Carnot engine

The efficiency of the Carnot engine is dependent on two temperatures, namely the temperatures of the hot source and the cold sink.

It is the ratio of the difference between both temperatures to the hot reservoir temperature.

Identification of given data

The given data can be listed below as:

• Theefficiency of the Carnot engine is\(e = 34\% \left( {\frac{1}{{100}}} \right) = 0.34\).
• The temperature of the cold reservoir is \({T_{\rm{L}}} = 230^\circ {\rm{C}} = \left( {230^\circ {\rm{C}} + 273} \right){\rm{ K}} = 503{\rm{ K}}\).

Determination of the high temperature of the Carnot engine

The efficiency of the Carnot engine can be expressed as:

\(\begin{aligned}{c}e &= \left( {1 - \frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}}}}} \right)\\\frac{{{T_{\rm{L}}}}}{{{T_{\rm{H}}}}} &= 1 - e\\{T_{\rm{H}}} &= \frac{{{T_{\rm{L}}}}}{{\left( {1 - e} \right)}}\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}{T_{\rm{H}}} &= \frac{{503{\rm{ K}}}}{{\left( {1 - 0.34} \right)}}\\ &= 762.12{\rm{ K}}\\ &= \left( {762.12{\rm{ K}} - {\rm{273}}} \right)^\circ {\rm{C}}\\ &= 489.12^\circ {\rm{C}}\end{aligned}\)

Thus, the high temperature of the Carnot engine is \(489.12^\circ {\rm{C}}\).