Carnot engine efficiency * Carnot refrigerator efficiency

In summary, the Carnot engine efficiency is η_work = 1 - T_c/T_h and the Carnot refrigeration efficiency is η_cool = T_c/(T_h-T_c). When combined, the efficiency becomes η_combined = T_c/T_h. This result may seem counter-intuitive, but it balances out when considering the fraction of input energy that gets dumped out as heat. This may also explain how thermal insulators work as a mass assembly of tiny Carnot heat engines and refrigerators. Additionally, the comparison of efficiency and COP shows that the wider the temperature difference, the lower the efficiency.
  • #1
kmarinas86
979
1
Carnot engine efficiency is:
[itex]\eta_{work} = 1 - \frac{T_c}{T_h}[/itex]
Carnot refrigeration efficiency is:
[itex]\eta_{cool} = \frac{ T_c }{T_h-T_c}[/itex]
[itex]\eta_{cool} = \frac{ 1 }{\frac{T_h}{T_c} - 1}[/itex]

Simple multiplication should give me the efficiency where both the engine and the refrigeration share the same hot and cold reservoirs:
[itex]\eta_{combined} = \frac{ 1 - \frac{T_c}{T_h} }{\frac{T_h}{T_c} - 1}[/itex]

Combining this we get:
[itex]\eta_{combined} = \frac{T_c}{T_h}[/itex]

This is a rather strange result. It seems as though we could maximize the efficiency of energy consumption if we simply balanced heat engine work with refrigeration and relied on very small ambient temperature differences. That's very counter-intuitive.

Though, [itex]\frac{T_c}{T_h}[/itex] is also the fraction of input energy into the Carnot engine that gets dumped out as heat. So that balances it out I guess.

Maybe its basically a fancy way to slow down the transfer of heat from hot to cold.
Maybe that's what "thermal insulators" actually are - a mass assembly of very tiny Carnot heat engines and refrigerators :rolleyes:
 
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  • #2
Are you familiar with efficiency and COP ( coeficient of performance )

See
http://en.wikipedia.org/wiki/Carnot_cycle
and
http://en.wikipedia.org/wiki/Coefficient_of_performance

Carnot engine efficiency is:
η work =1−T c T h

That is correct :
η = work ouput / heat input from the hot resevoir
or
η = W / Qhot

Carnot refrigeration efficiency is:
η cool =T c T h −T c

Incorrect. That is COP and not efficiency

For cooling
COP = heat removed from cold resevoir / amount of work input
or
COP = Qcold / Work

So you see, you are not comparing the same resevoir.
 
  • #3
Either way, what was proved here was that if you hook up a carnot heat pump to a carnot heat engine, the best efficiency you can hope for is 100% and the wider the temperature difference, the lower the efficiency gets.
 

1. What is a Carnot engine and how does it work?

A Carnot engine is an idealized heat engine that operates on the Carnot cycle. It consists of a hot reservoir, a cold reservoir, and a working substance (usually a gas) that is alternately heated and cooled to produce work. The engine works by taking in heat from the hot reservoir, converting some of it into work, and exhausting the remaining heat into the cold reservoir. This process is repeated in a continuous cycle to produce mechanical work.

2. What is Carnot engine efficiency and how is it calculated?

Carnot engine efficiency is a measure of the maximum possible efficiency for a heat engine operating between two temperature reservoirs. It is calculated by taking the temperature difference between the hot and cold reservoirs and dividing it by the temperature of the hot reservoir. This can also be expressed as 1 - (temperature of cold reservoir/temperature of hot reservoir).

3. How does Carnot refrigerator efficiency differ from Carnot engine efficiency?

Carnot refrigerator efficiency is a measure of the maximum possible coefficient of performance (COP) for a refrigeration cycle between two temperature reservoirs. It is calculated by taking the temperature difference between the cold and hot reservoirs and dividing it by the temperature of the cold reservoir. This can also be expressed as temperature of cold reservoir/ (temperature of hot reservoir - temperature of cold reservoir).

4. What factors affect the efficiency of a Carnot engine?

The efficiency of a Carnot engine is affected by the temperature difference between the hot and cold reservoirs, the type of working substance used, and the heat transfer processes involved. Additionally, any energy losses due to friction or heat leaks can also decrease the efficiency of the engine.

5. Can the efficiency of a Carnot engine ever be 100%?

No, according to the second law of thermodynamics, it is impossible for any heat engine to have an efficiency of 100%. This is because some energy is always lost in the conversion process, and it is impossible to have a completely reversible cycle in real-world conditions. The maximum theoretical efficiency for a Carnot engine is given by the Carnot efficiency formula discussed in question 2.

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