Heat Engines and Their Efficiency

Heat into work, again and again

The chapter on the First Law showed that a gas can turn heat into work. Heat a gas under a piston and it expands and lifts the load. But one expansion is not an engine. The piston soon reaches the top of the cylinder and the work stops. A useful engine must do work over and over, for hours.

So the gas must be brought back to where it started, and then expand again. A series of processes that returns a system to its starting state is a cyclic process, or a cycle. On a p-V diagram a cycle is a closed loop. The gas that goes round the loop and does the work is the working substance.

Internal energy depends only on the state. After one full cycle the gas is back in its starting state, so its internal energy is unchanged: \(\Delta U = 0\). The first law, \(Q = \Delta U + W\), then gives \(Q_\text{net} = W_\text{net}\). Over a cycle, the net heat taken in equals the net work done. This is the whole idea of a heat engine.

Three parts of every heat engine

A heat engine takes heat \(Q_H\) from a hot body, the source, at temperature \(T_H\). It does work \(W\). It gives heat \(Q_C\) to a cold body, the sink, at temperature \(T_C\). Here \(Q_H\) and \(Q_C\) are both sizes, taken as positive numbers. Many books write \(Q_1\), \(Q_2\), \(T_1\) and \(T_2\) for the same four quantities. The workbook does so too.

The source and the sink are thermal reservoirs. A reservoir is so large that taking heat from it, or giving heat to it, does not change its temperature. The furnace of a power station is kept at a steady temperature by burning coal. The river or the air that carries heat away is so large that it hardly warms. So \(T_H\) and \(T_C\) stay fixed while the engine runs.

Over one cycle the working substance takes in \(Q_H\) and gives out \(Q_C\), so the net heat is \(Q_H - Q_C\). This equals the net work. So the heat engine obeys one energy balance, \(Q_H = W + Q_C\). The heat taken from the source is split into work and heat rejected. None of it is lost.

How good is an engine?

Heat costs money: coal, diesel, petrol or gas. Work is what we want. So the natural measure of an engine is the fraction of the heat taken in that comes out as work. This fraction is the thermal efficiency, written \(\eta\) (the Greek letter eta).

An engine takes \(1000\,\text{J}\) from the source and gives \(600\,\text{J}\) to the sink in each cycle. Its work is \(1000 - 600 = 400\,\text{J}\), and its efficiency is \(400/1000 = 0.40\), or \(40\,\%\). The second form of the formula shows the limit at once. The efficiency can be \(1\) only if \(Q_C = 0\), when nothing at all is given to the sink.

Try the sliders in the sim above. With the source at \(600\,\text{K}\) and the sink at \(300\,\text{K}\), even the best engine rejects half its heat. Raise \(T_H\) to \(900\,\text{K}\) and the work share grows to two thirds. Now raise \(T_C\) until it meets \(T_H\) at \(400\,\text{K}\). The work shrinks to nothing. The blue part never vanishes while the sink is warmer than absolute zero.

Real heat engines

A petrol engine takes a fresh charge of working substance in every cycle, a mixture of air and petrol vapour. The piston draws the mixture in and squeezes it. A spark burns it, and the hot gas pushes the piston out: this is the stroke that does the work. Then the piston pushes the used gas out through the exhaust. The hot exhaust carries away the rejected heat \(Q_C\), so the sink is the open air.

A coal power station keeps the same water in a closed loop. The furnace boils it to steam at a high pressure, and the steam spins a turbine. The spent steam must then be cooled back to water in a condenser before a pump returns it to the boiler. The condenser is the sink. Its heat goes into cooling water, and the cooling towers pass that heat to the air.

The best coal stations in India, such as Vindhyachal, turn about \(35\,\%\) of the heat of their coal into electricity. So a \(500\,\text{MW}\) unit must take in \(500/0.35 = 1429\,\text{MW}\) of heat. It rejects about \(929\,\text{MW}\). If all that heat went into evaporating water in the towers, about \(410\,\text{kg}\) of water would leave as vapour every second.

Now compare the turbine hall of the Bhakra dam on the Sutlej. Its turbines turn about \(90\,\%\) of the energy of the falling water into electricity. Is it a far better engine? No. It is not a heat engine at all. The water brings mechanical energy, which can in principle become work completely. Only friction and swirling water waste a little, as Fig 1.1 shows.

Now return to the cooling towers. The station takes in heat at the furnace and must give a large part of it to a cold sink. The towers are that sink at work. Lesson 2 shows that no design can remove them. A heat engine that gives no heat to a sink cannot exist.