Heat Engines

What a Heat Engine Is

Heat engines are devices that transform heat partly into work or mechanical energy. They work by using cyclic processes and involve thermodynamic changes. An automobile engine is the familiar example.

A heat engine receives heat from a source called a reservoir and converts some of it into work. Not all the heat absorbed is converted into work; some is always lost as exhaust.

Elements of a Heat Engine

• Working substance: it is called the system. It can be an ideal gas for an ideal heat engine; for a practical engine it can be a mixture of fuel vapour and air in a petrol or diesel engine, or steam in a steam engine. It is the working substance that absorbs heat and does work.
• Hot and cold reservoirs: the hot reservoir, or source, is at a relatively high temperature $T_H$ and can supply a large amount of heat at that constant temperature. The cold reservoir, or sink, absorbs a large amount of heat from the working substance at the constant lower temperature $T_C$.
• Cylinder: the working substance is enclosed in a cylinder with a moving, frictionless, massless piston. The work done displaces the piston, and this displacement is transferred to the environment by an arrangement such as a crankshaft, which carries mechanical energy to the wheels of a vehicle.

The Three Basic Steps

• The working substance absorbs heat from a hot reservoir at higher temperature.
• Part of the heat absorbed by the working substance is converted into work.
• The remaining heat is transferred to a cold reservoir at lower temperature.

One single execution of these three steps is one operating cycle. Several such cycles are repeated when the engine runs; the number of repetitions per minute is the engine's RPM.

Two basic types, differing in how the working substance absorbs heat:

The Energy Flow Diagram

Figure 4.20 is the standard diagram. A circle in the middle represents the engine. A wide pipeline of width proportional to $Q_H$ enters the circle from the hot reservoir (a box at $T_H$) at the top. Two pipelines leave: one to the side, of width proportional to $W$, representing the heat converted into mechanical work; the other downwards, of width proportional to $|Q_C|$, into the cold reservoir (a box at $T_C$) at the bottom. For the most efficient engine the work pipeline is as wide as possible and the exhaust pipeline as narrow as possible.

$Q_H$ is the heat absorbed (positive) and $Q_C$ the heat rejected (negative) in one cycle. The net heat absorbed per cycle is

$$Q = Q_H + Q_C = |Q_H| - |Q_C|$$

Since the process is cyclic, $\Delta U = 0$, so by the first law the net work done in one operating cycle is

$$W = Q = Q_H + Q_C = |Q_H| - |Q_C|$$

Efficiency

Ideally we would want the engine to convert all of $Q_H$ into work. Practically that is impossible: there is always some heat lost, $Q_C \ne 0$. The thermal efficiency $\eta$ of a heat engine is defined as

$$\eta = \dfrac{W}{Q_H}$$

that is, the ratio of the work done by the working substance to the amount of heat absorbed by it — the ratio of output (work $W$) to input (heat supplied $Q_H$). In simple words, the efficiency of a heat engine is the fraction of the heat absorbed that is converted into work. Substituting $W = Q_H + Q_C$,

$$\eta = \dfrac{Q_H + Q_C}{Q_H} = 1 + \dfrac{Q_C}{Q_H} = 1 - \dfrac{|Q_C|}{Q_H}$$

Efficiency is a ratio of two energies, so it has no unit; but $W$, $Q_H$ and $Q_C$ must all be expressed in the same units. There is a fundamental limit on the efficiency of a heat engine, set by the second law of thermodynamics.

The Heat Engine Cycle on a p–V Diagram

The operation of a heat engine is a well-defined sequence of operations and is a cyclic thermodynamic process, so its $p$–$V$ diagram is a closed loop (Fig. 4.21). The loop for a typical heat engine is a rough rectangle traced clockwise through four points A, B, C, D.

• A to B: the gas absorbs heat at constant volume. No work is done by or on the gas. The pressure rises to the point B; the temperature and internal energy of the gas increase. This is the vertical left-hand side of the loop, drawn upwards.
• B to C: the gas expands, pushing the piston away; the volume increases and the pressure falls. The gas does work in this part of the cycle. This is the upper curve, drawn to the right.
• C to D: the excess heat, the heat not used to do work, is rejected. The gas cools and its internal energy decreases. This is again at constant volume, so the pressure falls to the point D. This is the vertical right-hand side, drawn downwards.
• D to A: the gas is compressed; its volume decreases and its pressure increases until the point A is reached. The cycle is complete and the system is ready for the next cycle. This is the lower curve, drawn to the left.

The area of the loop represents the work done during one complete cycle, since work is done only when the volume changes. Together with the temperatures obtained from the ideal gas law, the diagram also gives the changes in internal energy, and hence, through the first law, the heat added or rejected in each part. That is why a $p$–$V$ diagram is a visual tool for analysing the performance of any heat engine that uses a gas as its working substance.

Standard Results

$$Q = Q_H + Q_C = |Q_H| - |Q_C|, \qquad W = Q_H + Q_C$$

$$\eta = \dfrac{W}{Q_H} = 1 + \dfrac{Q_C}{Q_H} = 1 - \dfrac{|Q_C|}{Q_H}$$

• $\eta$ is dimensionless and always less than 1; multiply by 100 for a percentage.
• The area of the closed loop on the $p$–$V$ diagram is the work done per cycle. For $N$ cycles, total work $= N \times$ area.