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Carnot's Thereom states:

The result is the following idea: This equation represents the maximum possible efficiency of a heat engine. You can't do any better than that.

This statement is wrong.

Quite simply, rather than reject the heat during the compression phase, simply run a stirling engine to cool the gas.

In fact, the limit is 100% effiency if we cool our carnot engine with another carnot engine. Then get two more carnot engines for the next cylce of cooling, then 4 more for the next cycle and so on. This limits to 100% efficiency

Remember, we can design any kind of heat engine we want. In fact, lets put our heat engine in a box, so that nobody can see how it is designed. We add heat, through a heat portal in the box, and measure the work that we can get from the heat energy input. We add the energy, use it to perform the work cycle on a carnot engine. Then, during the compressing cycle, we don't throw the heat away into the low temp reservoir. We cool the gas by attaching a stirling engine to it. Hey, look at that, we have just beaten the 1 - Tcold/Thot limit.

So, 1 - Tcold/Thot is not the the maximum possible efficiency of a heat engine.

It seems ridiculous to me that a heat engine design calls for throwing away heat energy. Why would you do that? Simply apply it to a secondary heat engine and so on.

Car engines are about 25% efficient. 75% of the chemical energy used to move your car down the road is wasted heat. They even need heat radiators to get rid of all the energy. How heavy would a secondary heat engine be? Rather than throw all that energy away, could we run a cascade of ever smaller heat engines to increase effiecency?

Anyway, 1 - Tcold/Thot is not the maximum possible efficiency of a heat engine.

To conclude, I can hear the objection "It is the maximum efficiency for a carnot engine". You are right. But remember, we are not attempting to disprove that, we are disproving the statement: This equation represents the maximum possible efficiency of a heat engine. You give me some heat energy, then I do as much work with it as possible, with any heat engine design I choose. I can beat 1 - Tcold/Thot.