# Theoretical maximum efficiency of a heat engine without Carnot

• I
In summary: Robert Sterling: So, we don't need to invoke entropy or the second law of thermodynamics to find out the maximum theoretical efficiency of a heat engine?Sadi Carnot: That's right.
TL;DR Summary
It seems possible to find out the theoretical maximum efficiency of a closed-circuit heat engine without invoking the Carnot cycle and its ideal adiabatic processes. Is it?
Through an intriguing fictitious dialog between Sadi Carnot and Robert Sterling, Prof. Israel Urieli of the Ohio University shows that it is not required to invoke entropy, the second law of thermodynamics, and the Carnot cycle with the [ideal] adiabatic processes in order to find out the maximum theoretical efficiency of a heat engine: https://www.ohio.edu/mechanical/thermo/Intro/Chapt.1_6/Carnot_Stirling/index.html

For the sake of completeness, I reproduce it here:

I think the analysis is plausible; the interesting thing is about the heat exchanger (regenerator) which, under ideal conditions, provides a perfect heat transfer between the gas and the regenerator equipment (including the coolant). The only slight confusion is whether ##q_{supplied} = q_{34}## is the only energy transferred into the engine by heat during the isothermal expansion?

Does anyone disagree with this analysis? Please elucidate either way. Note that apparently the professor won a prize for his http://www.centrostirling.com/isec2014/index-isec2014.html.

Last edited:
BvU
Sadi Carnot: Robert, I believe that the maximum efficiency of a heat engine is determined by the temperature difference between its hot and cold reservoirs.Robert Sterling: But Sadi, how do we determine the maximum efficiency without invoking entropy and the second law of thermodynamics?Sadi Carnot: We don't need to invoke entropy or the second law of thermodynamics to find out the maximum theoretical efficiency of a heat engine. We can use an ideal adiabatic process and simply calculate the energy balance between the energy supplied and the energy extracted from the heat engine. The energy supplied is the energy transferred into the engine by heat during the isothermal expansion. The energy extracted is the energy transferred out of the engine by work during the isothermal compression. The maximum efficiency is then the ratio of the energy extracted to the energy supplied.

## 1. What is the theoretical maximum efficiency of a heat engine without Carnot?

The theoretical maximum efficiency of a heat engine without Carnot is known as the Curzon-Ahlborn efficiency, which is given by the formula: η = 1 - √(Tc/Th), where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir. This efficiency is always less than the Carnot efficiency, which is given by the formula: η = 1 - (Tc/Th).

## 2. How does the Curzon-Ahlborn efficiency compare to the Carnot efficiency?

The Curzon-Ahlborn efficiency is always less than the Carnot efficiency, meaning that the Carnot engine is more efficient than any other heat engine without Carnot. This is due to the Carnot cycle being the most efficient thermodynamic cycle possible.

## 3. Can the Curzon-Ahlborn efficiency ever reach 100%?

No, the Curzon-Ahlborn efficiency can never reach 100%. This is because the Carnot efficiency is the maximum possible efficiency for any heat engine, and the Curzon-Ahlborn efficiency is always less than the Carnot efficiency.

## 4. What factors affect the efficiency of a heat engine without Carnot?

The efficiency of a heat engine without Carnot is affected by the temperature difference between the hot and cold reservoirs, as well as the type of thermodynamic cycle being used. Additionally, factors such as friction, heat loss, and the properties of the working fluid can also impact the efficiency.

## 5. How is the Curzon-Ahlborn efficiency used in real-world applications?

The Curzon-Ahlborn efficiency is used as a benchmark for comparing the efficiency of real-world heat engines. While it may not be achievable in practice, it serves as a theoretical limit for the maximum possible efficiency of a heat engine without Carnot. This information can be used to improve the design and efficiency of actual heat engines.

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