This derivation is a slight simplification of Curzon & Ahlborn.
Consider a heat engine, with a single working fluid cycling around the engine. On one side, the working fluid has temperature , and is in direct contact with the hot heat bath. On the other side, it has temperature , and is in direct contact with the cold heat bath.
The heat flow into the engine is , where  is the heat conduction coefficient. The heat flow out of the engine is . The power output of the engine is .
Side note: if one cycle of the engine takes time , and during this time, it is in contact with the hot side only for a time , then we can reduce to this case by replacing  with . Similar comments apply to the cold side.
By Carnot theorem, we have . This then gives us a problem of constraint optimization:<math display="block">\begin{cases}
\max_{T_H, T_L} \dot W \\
\frac{\dot W}{\dot Q_H} \leq 1 - \frac{T_L}{T_H}
\end{cases} at which point the engine is operating at efficiency .
In particular, if , then we have This is often the case with practical heat engines in power generation plants, where the work fluid can only spend a small amount of time with the hot bath (nuclear reactor core, coal furnance, etc), but a much larger amount of time with the cold bath (open atmosphere, a large body of water, etc).