Carnot Engine Efficiency

In summary: Carnot engine is determined by the temperature difference between the hot and cold reservoirs. If you decrease the temperature of the cold reservoir, the temperature difference decreases, resulting in a higher efficiency. This is because the efficiency is directly proportional to the temperature difference.
  • #1
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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


(TH-TC)/TH)

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.
 
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  • #2
Try comparing [itex]\partial \eta/\partial T_H[/itex] and [itex]\partial \eta/\partial T_C[/itex] where [itex]\eta[/itex] is the Carnot efficiency.
 
  • #3
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, [itex]Q_c/(Q_h-Q_c) = T_c/(T_h-T_c)[/itex]. For a given denominator, this ratio reduces only if Tc is reduced.

AM
 
Last edited:

What is Carnot Engine Efficiency?

Carnot Engine Efficiency is a measure of how well a Carnot engine converts heat energy into mechanical work. It is the maximum possible efficiency for a heat engine operating between two temperatures.

How is Carnot Engine Efficiency calculated?

Carnot Engine Efficiency is calculated using the formula: Efficiency = (Th - Tc) / Th, where Th is the hot temperature and Tc is the cold temperature.

What factors affect Carnot Engine Efficiency?

The two main factors that affect Carnot Engine Efficiency are the hot and cold temperatures. The greater the temperature difference, the higher the efficiency. Additionally, the type of working fluid used in the engine can also impact efficiency.

Why is Carnot Engine Efficiency considered to be the maximum possible efficiency?

Carnot Engine Efficiency is considered to be the maximum possible efficiency because it is based on the Second Law of Thermodynamics, which states that no engine can be more efficient than a Carnot engine. This is because a Carnot engine operates in a reversible cycle, meaning all energy is converted into work without any energy losses.

How is Carnot Engine Efficiency used in real-world applications?

Carnot Engine Efficiency is used as a benchmark for comparing the efficiency of other heat engines. It also serves as a theoretical limit for the maximum efficiency that can be achieved in power plants and other industrial processes that involve heat energy conversion.

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