Which Temperature Adjustment Improves Carnot Engine Efficiency More?

[kwuk]

Homework Statement

Determine which gives the higher efficiency of a Carnot engine, increasing the temperature of the hot reservoir, or decreasing the temperature of the cold reservoir by the same amount. Justify your answer carefully.

Homework Equations

(T_H-T_C)/T_H)

The Attempt at a Solution

By entering some random numbers into the efficiency equation, I have determined that reducing the temperature of the cold reservoir yeilds the higher efficiency, but I am unable to explain why this is the case, considering that the temperature difference between the reservoirs would be the same if the hot reservoir had been increased instead.

Any advice would be greatly appreciated.

 

[vela]

Try comparing $\partial \eta/\partial T_H$ and $\partial \eta/\partial T_C$ where $\eta$ is the Carnot efficiency.

 

[Andrew Mason]

You are correct that if you change Th by a certain amount or decrease Tc by the same amount, there is the same increase in the numerator. But if you do it by increasing Th what happens to the denominator? Is there any change in the denominator if you decrease Tc?

It is perhaps easier to see the physical explanation if you turn the efficiency upside down and look at inefficiency. Inefficiency is due to Qc - heat flow to the cold reservoir, which is wasted heat (it does not produce work). The only way to improve efficiency is to reduce the proportion of heat flowing to the cold reservoir to the work being done. You want to reduce Qc/(Qh-Qc). In a Carnot cycle, $Q_c/(Q_h-Q_c) = T_c/(T_h-T_c)$. For a given denominator, this ratio reduces only if Tc is reduced.

AM