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previous index next Heat Engines: the Carnot Cycle Flashlet here ! Michael Fowler 7/3/08 The Ultimate in Fuel Efficiency All standard heat engines (steam, gasoline, diesel) work by supplying heat to a gas, the gas then expands in a cylinder and pushes a piston to do its work. The catch is that the heat and/or the gas must somehow then be dumped out of the cylinder to get ready for the next cycle. Our aim in this lecture is to figure out just how efficient such a heat engine can be: what’s the most work we can possibly get for a given amount of fuel? We’ll examine here the simplest possible cyclical model: an ideal gas enclosed in a cylinder, with external connections to supply and take away heat, and a frictionless piston for the gas to perform and absorb mechanical work: Carnot Engine The efficiency question was first posed—and solved—by Sadi Carnot in 1820, not long after steam engines had become efficient enough to begin replacing water wheels, at that time the main power sources for industry. Not surprisingly, perhaps, Carnot visualized the heat engine as a kind of water wheel in which heat (the “fluid”) dropped from a high temperature to a low temperature, losing “potential energy” which the engine turned into work done, just like a water wheel. ( Historical Note : actually, Carnot thought at the time that heat was a fluid—he believed in the Caloric Theory . Remarkably, the naïve “potential energy of a caloric fluid” approach gives exactly the right answer for the efficiency of an ideal engine! Carnot accepted that there was an absolute zero of temperature, from which he figured out that on being cooled to absolute zero, the caloric fluid would give up all its heat energy.
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2 Therefore, if it falls only half way to absolute zero from its beginning temperature, it will give up half its heat, and an engine taking in heat at T and shedding it at ½ T will be utilizing half the possible heat, and be 50% efficient. Picture a water wheel that takes in water at the top of a waterfall, but lets it out halfway down. So, the efficiency of an ideal engine operating between two temperatures will be equal to the fraction of the temperature drop towards absolute zero that the heat undergoes. This turns out to be exactly correct, even though the reasoning is based on a false model.) The water wheel analogy proved to be useful in another way: Carnot knew that the most efficient water wheels were those that operated smoothly, the water went into the buckets at the top from the same level, it didn’t fall into them through any height, and didn’t splash around. In the idealized limit of a frictionless water wheel, with gentle flow on and off the wheel, such a machine would be reversible —if the wheel is run backwards by power supplied from the outside, so it raises water back up, it will take the same power that the wheel was itself delivering in normal operation. This idealized water wheel is clearly perfectly efficient, so the analogs of zero friction and gentle flow are what we need in the perfect heat engine.
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This note was uploaded on 12/07/2011 for the course PHYSICS 152 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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CarnotEngine - previous index next Heat Engines: the Carnot...

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