Carnot formula

These delicate engines provide value as educational tools, but they immediately inspire curiosity into the possibility of generating power from one of the many sources of low temperature waste heat (less than 100°C) that are available. A quick look at the Carnot formula shows that an engine operating with a hot side at 100°C and a cold side at 23°C will have a maximum Carnot efficiency of [((373 K—296 K)/373 K) X 100] about 21 percent. If an engine could be built that achieved 25 percent of the possible 21 percent Carnot efficiency it would have about 5 percent overall Carnot efficiency. 

The maximum possible conversion efficiency is then expressed by what appears to be simply the Carnot formula applied to radiation ... 

Since there is nothing special about the temperature of the cold reservoir, except that d > do, Eqs. (8.18) and (8.19) apply to any reversible heat engine operating between any two thermodynamic temperatures d and do - Equation (8.18) shows that the work produced in a reversible heat engine is directly proportional to the difference in temperatures on the thermodynamic scale, while the efficiency is equal to the ratio of the difference in temperature to the temperature of the hot reservoir. The Carnot formula, Eq. (8.19), which relates the efficiency of a reversible engine to the temperatures of the reservoirs is probably the most celebrated formula in all of thermodynamics. 

Stirling engines also have the maximum theoretical possible efficiency because their power cycle (their theoretical pressure volume diagram) matches the Carnot cycle. The Carnot cycle, first described by the French physicist Sadi Carnot, determines the maximum theoretical efficiency of any heat engine operating between a hot and a cold reservoir. The Carnot efficiency formula is... 

The ideal-gas Carnot cycle is a particular case of the Carnot cycle its efficiency must be given by the general formula of Carnot cycles. Comparison of Equation (4.45) with Equation (4.40) shows... 

We first check that this formula for a thermodynamically maximal temperature yields the Carnot efficiency. As heat transfer is included explicitly in the formulae, we simply remove the sink (E = S =0) and associate a temperature T E E /S with ss PP P... 

Thus, we obtain the same formula for the efficiency of a heat engine as from the Carnot cycle. Along the way, we realized that when heat flows from hot to cold, the environment wUl gain the exhaust heat 1 1 so that the environment gains enfropy. A profound result of this sort of analysis is that entropy tends to increase in the environment unless there is some other condition and the overall entropy in the universe tends to increase. Especially for biology majors and generally for aU of us. 

Show that = 0 for a Carnot cycle derive the efficiency formula. 

Show that ° for a Carnot cycle, derive the Camot efficiency formula. 

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