# Maximum possible efficiency

An individual walking the roads at night indicates an invention he is working on produces 55KW with 70KW of thermal energy supplying it. The cold side is at 40C, the hot side at 1650C. Is this something to consider further? Explain.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

Let's see if the claim is consistent with the laws of thermodynamics. If it isn't then this invention can't possibly work and wouldn't be worth considering.

Assume that after the engine starts working a steady state situation is reached with the conditions as described in the problem. This means that all thermodynamical variables should remain constant on average. If the engine contains moving parts such as a piston then after one cycle is completed the engine should return to the same state.

The thermodynamical state can be specified by the total internal energy E and the entropy S. E must remain constant. This means that if some heat is absorbed by the engine and some heat is dumped in the environment, the difference is work performed by the engine. Denoting the heat absorbed as Q1, the heat dumped as Q2 and the work performed as W, we can write:

Q1 = Q2 + W (1)

If some heat Q is absorbed at temperature T the entropy increases by an amount Q/T. Work doesn't lead to changes in entropy. Now the inventor wants to make W in equation (1) as large as possible for fixed amount of heat, Q1, absorbed by the engine. If the heat is absorbed at temperature T1 and dumped at temperature T2, then the change in entropy is Q1/T1 - Q2/T2. But because the engine must be in the same state after performing work this must be zero, so we have:

Q1/T1 = Q2/T2 (2).

Making W in (1) as large as possible amounts to making Q2 as small as possible. From (2) you can see that for fixed Q1 this happens when T1 is as large as possible and T2 as small as possible. From the description of the engine T2 can at most be 1650 C = 1923 K and T2 has to be higher than 40 C = 313 K

From (2) and (1) it follows that:

W = (T1-T2)/T1 Q1

Using the bounds on T1 and T2 you get:

W < 0.837 Q1

The inventor says that he can produce 55 KW for every 70 KW of heat absorbed. 55 is indeed less than 0.837*70, so such an engine is not impossible.

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