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We consider the standard Carnot-cycle machine, which can be thought of as having a piston moving within a cylinder, and having the following characteristics:
By definition, the isothermal segments (AB and CD) occur when there is perfect thermal contact between the working fluid and one of the reservoirs, so that whatever heat is needed to maintain constant temperature will flow into or out of the working fluid, from or to the reservoir.
By definition, the adiabatic segments (BC and DA) occur when there is perfect thermal insulation between the working fluid and the rest of the universe, including both reservoirs, thereby preventing the flow of any heat into or out of the working fluid.
The isothermal curves (but not the adiabatic curves) are hyperbolas, according to PV = nRT. The enclosed area (and therefore the mechanical work done) will depend on the two temperatures ("height") and on the amount of heat transferred, which depends in turn on the extent of the isothermal compression or expansion ("width"), during which heat must be transferred to maintain the constant temperature.
We will denote the heat transferred to or from the high-temperature reservoir (during the transition between points A and B) as Qh.
We will denote the heat transferred to or from the low-temperature reservoir (during the transition between points C and D) as Qc.
If a Carnot machine cycles around the path clockwise, a high-temperature isothermal expansion from A to B, an adiabatic expansion cooling down from B to C, a low-temperature isothermal compression from C to D, and finally an adiabatic compression warming up from D to A, it functions as a heat engine, removing energy from the high-temperature reservoir as heat, transforming a portion of that energy to useful mechanical work (the enclosed area) done on the external world, and ejecting the remainder of the energy as waste heat to the low-temperature reservoir.
If a Carnot machine is driven (by an external agency, such as a motor) around the cycle counter clockwise, an adiabatic expansion cooling down from A to D, a low-temperature isothermal expansion from D to C, an adiabatic compression warming up from C to B, and finally a high temperature isothermal compression from B to A, then it functions as either a refrigerator or a heat pump, depending on whether removing heat from the low-temperature reservoir or adding heat to the high-temperature reservoir is of primary interest. The mechanical energy required to force the machine around the cycle is the work done on the machine, the area enclosed.
In general, the "efficiency" or "effectiveness" of a process is the ratio calculated by dividing what you want to accomplish by what you have to pay to get it done. Hence, a larger value is better.
This ratio is the interesting one because you pay for the fuel to obtain Qh, in order to get the benefit of the work done, W. For a Carnot engine, this is entirely determined by the temperatures of the hot and cold reservoirs:
The efficiency depends on the ratio of the temperature difference between the two reservoirs to the absolute temperature of the hot reservoir; alternatively, the efficiency depends on the ratio of the absolute temperatures of the hot and cold reservoirs.
This temperature dependence is a direct consequence of the second law of thermodynamics and the fact that all heat transfers occur during isothermal expansion and contraction, with no temperature difference between the heat reservoir and the working fluid, so that the entropy gained by one exactly matches the entropy lost by the other, with no net change in entropy for the system as a whole. This condition is of course an ideal one, and cannot be met in practice by any real machine. Thus, the Carnot efficiency is the best possible even theoretically; all real machines will be strictly worse than this.
Another consequence of these calcuations is the following: in most practical situations the cold temperature reservoir is essentially the ambient temperature; therefore in order to improve efficiency, the high temperature must be increased. In order to increase the boiling temperature of water, it is necessary to increase the pressure. As a result, large-scale steam-based power installations (whether fueled by coal, oil, natural gas, or nuclear energy) will routinely be designed to operate at high temperatures and presssures, posing significant challenges from mechanical, chemical, and metallurgical perspectives.
The effectiveness of a refrigerator is sometimes called the coefficient of performance ("COP"). The effectiveness will be greater than 1 only if the absolute temperature of the cold reservoir is warmer than half that of the hot reservoir. We can see that refrigeration to extremely cold temperatures is very difficult.
The effectiveness of a heat pump is sometimes called the performance factor ("PF"). For heat pumps, the effectiveness is always greater than 1. Electrically powered heat pumps can make economic sense only if the effectiveness of the heat pump times the efficiency of the electrical generation and transmission process exceeds 1. Otherwise, only part of the fuel burned to produce the electricity would have to be burned to provide the heat needed. (Modern natural gas furnaces can easily transfer more than 95% of the combustion heat to the heated space.) As the temperature of the cold reservoir (the outside temperature) declines, the effectiveness of the heat pump decreases toward 1. Because large electrical generators and high-voltage transmission lines deliver about one-third as much electical energy as the heat value of the fuel they consume, as soon as the PF is less than about 3, it would be cheaper to burn the original fuel directly for the heat, rather than generate electricity to operate a heat pump. This limits the geographical regions where heat pumps make economic sense.
The Carnot heat pump would reach this technical break-even point for heating ordinary living spaces at an outside temperature of about -77 degrees Celsius (-106 degrees Fahrenheit). The real, economic break-even, however, occurs at appreciably warmer outside temperatures, because the real heat pumps do not achieve Carnot effectiveness, and because they must save so much operating fuel cost that they pay for their construction and installation.
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Dick Piccard revised this file (http://www.ohio.edu/people/piccard/phys202/carnot/carnot.html) on January 22, 2013.
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