Working Principles and Efficiency of Heat Engines

A heat engine is defined as any device that transforms heat into mechanical energy, also known as work.
The primary function of a heat engine is to convert thermal energy, derived from a heat source, into a useful mechanical output.

Heat engines operate by absorbing heat from a source at a relatively high temperature.
During the operation, the engine performs a specific amount of mechanical work.
Following the performance of work, the engine discards a portion of the heat at a low temperature.
The heat is rejected into a system known as a low-temperature reservoir or a sink.

Total energy entering the system from the high-temperature reservoir is denoted as $Q_1$ .
Energy rejected by the engine to the low-temperature reservoir or sink is denoted as $Q_2$ .
The mechanical work produced by the engine is the difference between the heat absorbed and the heat rejected.
The net work done, denoted as $W$ , is calculated using the following formula: $W = Q_1 - Q_2$

The heat engine treats any thermal energy that is not converted into useful work as waste.
This extra heat is rejected from the system through the exhaust mechanism, depositing the waste heat into the low-temperature reservoir (sink).

The thermal efficiency of a heat engine, represented by the Greek letter eta ( $\eta$ ), is defined as the ratio of the net work done ( $W$ ) in each cycle to the total heat absorbed ( $Q_1$ ) from the high-temperature reservoir during that same cycle.
The efficiency formula is expressed as: $\eta = \frac{W}{Q_1}$
Since $W = Q_1 - Q_2$ , the efficiency can also be calculated by substituting the work equation: $\eta = \frac{Q_1 - Q_2}{Q_1}$
This simplifies to: $\eta = 1 - \frac{Q_2}{Q_1}$