I Efficiency of cycles bounded by two isotherms

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The discussion focuses on reversible cycles involving isothermal expansion and compression, with various other processes allowed as long as they remain within specified temperature limits. It seeks to establish whether the Carnot cycle demonstrates superior efficiency compared to other cycles under these constraints. The efficiency of such cycles is influenced by the temperature difference between the hot and cold reservoirs, with greater differences leading to higher efficiency. It is noted that only the Carnot engine and certain engines with regenerators have efficiencies determined solely by these temperatures. The conversation emphasizes the significance of isothermal process temperatures in defining cycle efficiency.
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I want to consider all possible reversible cycles that consist of an isothermal expansion at TH and an isothermal compression at TC.
The other two processes can be isochoric, isobaric, adiabatic or anything else, but they should never leave the temperature range between the two isotherms.
I also want to explicitly exclude heat recycling using a regenerator.
Pressure and volume of the system should always remain finite and the temperature is always finite and larger than 0 K.

Is there a general proof that the Carnot cycle has a higher efficiency than all other cycles considered above?
 
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Let ##dQ_H## be the differential increments of heat received by the engine during the heating part of the cycle, ##dQ_C## be the increments of heat rejected by the engine during the cooling part of the cycle, ##T_{max}## be the maximum temperature during the heating part of the cycle, and ##T_{min}## be the minimum temperature during the cooling part of the cycle. Then, $$\int{\frac{dQ_H}{T}}=\frac{Q_H}{T_{max}}+\delta_H$$where ##\delta_H## is positive (since T < ##T_{max}##). Similarly, $$\int{\frac{dQ_C}{T}}=\frac{Q_C}{T_{min}}-\delta_C$$ where ##\delta_C is positive (## since T > ##T_{min}##). So, $$\Delta S=\frac{Q_H}{T_{max}}+\delta_H-\frac{Q_C}{T_{min}}+\delta_C=0$$Therefore, $$Q_C=\frac{T_{min}}{T_{max}}Q_H+T_{min}(\delta_H+\delta_C)$$So the efficiency is $$e=\frac{Q_H-Q_C}{Q_H}=\frac{T_{max}-T_{min}}{T_{max}}-\frac{T_{min}(\delta_H+\delta_C)}{Q_H}$$
 
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Chestermiller said:
Let ##dQ_H## be the differential increments of heat received by the engine during the heating part of the cycle, ##dQ_C## be the increments of heat rejected by the engine during the cooling part of the cycle, ##T_{max}## be the maximum temperature during the heating part of the cycle, and ##T_{min}## be the minimum temperature during the cooling part of the cycle. Then, $$\int{\frac{dQ_H}{T}}=\frac{Q_H}{T_{max}}+\delta_H$$where ##\delta_H## is positive (since T < ##T_{max}##). Similarly, $$\int{\frac{dQ_C}{T}}=\frac{Q_C}{T_{min}}-\delta_C$$ where ##\delta_C is positive (## since T > ##T_{min}##). So, $$\Delta S=\frac{Q_H}{T_{max}}+\delta_H-\frac{Q_C}{T_{min}}+\delta_C=0$$Therefore, $$Q_C=\frac{T_{min}}{T_{max}}Q_H+T_{min}(\delta_H+\delta_C)$$So the efficiency is $$e=\frac{Q_H-Q_C}{Q_H}=\frac{T_{max}-T_{min}}{T_{max}}-\frac{T_{min}(\delta_H+\delta_C)}{Q_H}$$
Thanks! That's a great proof.
 
The temperatures of the isothermal processes determine the efficiency of cycles bounded by two isotherms, like a Carnot cycle. As the temperature difference between the hot and cold reservoirs grows, efficiency rises.
 
thulsidass said:
The temperatures of the isothermal processes determine the efficiency of cycles bounded by two isotherms, like a Carnot cycle. As the temperature difference between the hot and cold reservoirs grows, efficiency rises.
I would say only the Carnot engine and certain other engines with regenerators have an efficiency that's only determined by the temperatures of the isotherms.
 
thulsidass said:
The temperatures of the isothermal processes determine the efficiency of cycles bounded by two isotherms, like a Carnot cycle. As the temperature difference between the hot and cold reservoirs grows, efficiency rises.
Can you explain more precisely? What do you mean by "cycles bounded by two isotherms, like a Carnot cycle"?
 
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