Heat engines and heat pumps

The engine cycle

Every heat engine, petrol, diesel, steam turbine, is the same machine in outline. A working substance, usually a gas, is carried round a repeating cycle. Each cycle it absorbs heat Q_H from a hot source, converts part of that heat to work W, and must dump the remainder Q_C into a cold sink, the surroundings. Energy conservation fixes W = Q_H − Q_C, and the fraction of the input that became work is the efficiency:

Why must Q_C exist at all? Because, in the idealised closed cycle the theory works with, the working gas has to be returned to its starting state to go round again, and resetting it means compressing it, which demands rejecting heat somewhere colder. (A real open-flow engine expels its burnt charge and draws a fresh one, but the energy accounting comes out the same.) An engine that dumped nothing would need its exhaust as hot as its source, and no net work would be left. That is the second-law heart of the topic, and it caps the efficiency below one for any engine whatever. The cap has a clean form. The best possible cycle running between source temperature T_H and sink temperature T_C achieves

with both temperatures in kelvin. The formula is a ratio of absolute temperatures, so Celsius numbers wreck it; only the kelvin scale starts at the true zero the ratio needs. Real engines fall short of even this ceiling, through friction, turbulence and heat leaking where it should not.

Four strokes to one cycle

A piston engine runs that cycle in a cylinder, and it takes four strokes of the piston, two full turns of the crankshaft, to get round it once. The induction stroke fills the cylinder: the inlet valve opens and the piston travels down, drawing the charge in at roughly atmospheric pressure. The compression stroke shuts both valves and brings the piston back up, squeezing that charge into a small volume and heating it on the way. It happens quickly, so little heat escapes and the squeeze is close to adiabatic.

Then the power stroke, the only one that delivers work. The charge burns, the pressure jumps, and the hot gas drives the piston down, expanding and cooling while the crankshaft takes the work away. Last is the exhaust stroke: the exhaust valve opens, the pressure falls to atmospheric as the burnt gas escapes, and the rising piston pushes out what is left. The cylinder is back to its starting state, and one power stroke has arrived for every two revolutions.

Petrol and diesel engines differ in what gets compressed and in what lights it. A petrol engine draws in a mixture of air and fuel and compresses it by a factor of about ten, as far as it dare go before the mixture fires itself, and a spark plug ignites it at the top of the stroke. The burn is over before the piston has moved appreciably, so on the theoretical cycle the heat Q_H goes in at constant volume, drawn as a vertical climb in pressure at the smallest volume.

A diesel engine draws in air alone and compresses it far harder, by a factor of fifteen to twenty, which takes it to something like 900 K, far above the temperature at which the fuel ignites. Fuel is then sprayed in at the top of the stroke and burns as fast as it arrives, while the piston is already moving down, so on the theoretical cycle its heat goes in at constant pressure. No spark plug is needed, and the higher compression ratio is why a diesel's ceiling, and its measured efficiency, sit above a petrol engine's.

The two theoretical cycles agree on the rest. Expansion is adiabatic down to the largest volume; the exhaust valve opening is drawn as a fall in pressure at constant volume, and that fall is where Q_C leaves; and the induction and exhaust strokes run out and back along the bottom of the diagram at atmospheric pressure. Four strokes, two revolutions, one loop.

The indicator diagram

An indicator diagram is that loop measured rather than imagined, the pressure of the gas in one cylinder plotted against its volume while the engine actually runs. Reading it needs nothing new. Work is the area under a curve, rightward motion is expansion and does positive work, leftward motion is compression and costs it.

On the p-V diagram a full cycle is a closed loop, traversed clockwise for an engine. Expansion happens along the high-pressure top, compression along the low-pressure bottom, so the work out exceeds the work back in, and the difference, the net work per cycle, is the area enclosed by the loop.

The loop a textbook draws and the loop an indicator draws are not the same shape, and every difference between them is a loss. The theoretical diagram assumes valves that open instantly, a burn that finishes instantly at the top of the stroke, compression and expansion that are reversible and adiabatic, and gas that flows in and out against no resistance. In a real cylinder the valves take time, so the sharp corners round off; the burn takes time, so the peak pressure is lower and arrives after the piston has started down; and heat leaks into the cooled walls throughout.

The bottom of the real loop also splits in two. Drawing gas in through a valve needs the cylinder slightly below atmospheric pressure, and pushing it out needs it slightly above, so the induction and exhaust strokes trace a thin loop of their own, run the other way round, whose area is work spent on breathing rather than earned. Put those together and one sentence carries the comparison: the real diagram encloses less area than the theoretical one, so the work per cycle is smaller, typically by a fifth, before a single bearing has rubbed.

Auditing a real engine

The area of the measured loop feeds a chain of powers, and the audit runs from fuel to shaft. The input power is chemical energy arriving as fuel, the fuel's calorific value (joules per kilogram) times the rate it is burned. The indicated power is what the gas develops inside the cylinders, the loop area per cycle times the cycles per second, times the number of cylinders. The brake power is what actually reaches the output shaft, measured as P = Tω. What friction and moving parts eat between cylinder and shaft is the friction power:

Those powers are also what the efficiencies are made from, and there are three to keep apart. Thermal efficiency is the indicated power divided by the input power, and it asks how much of the fuel's energy the gas in the cylinders turned into work at all. That is the W/Q_H of the first section rewritten as powers, so it is the one the second law caps, and no amount of engineering will take it past (T_H − T_C)/T_H.

Mechanical efficiency is the brake power divided by the indicated power, and it asks something quite different: of the work the gas did on the pistons, how much survived the bearings, the valve gear and the oil pump to reach the shaft. No Carnot temperature ratio caps it, though energy conservation still holds it between 0 and 1, and what raises it is better lubrication rather than hotter gas.

Overall efficiency is the brake power divided by the input power, and it is the product of the other two, because the indicated power cancels between them. That product is worth writing out, since a disappointing overall figure has to be diagnosed before it can be cured: a thermal efficiency of 0.30 with a mechanical efficiency of 0.90 and a thermal 0.45 with a mechanical 0.60 both give 0.27, and they call for entirely different work.

Running it backwards

Drive the cycle anticlockwise, feeding work in, and heat is pumped the wrong way, from cold to hot. One machine, two names, chosen by which end you care about. A refrigerator is judged on the heat Q_C extracted from the cold space. A heat pump is judged on the heat Q_H = Q_C + W delivered to the hot space, a house in winter. Each is measured by a coefficient of performance, the useful heat divided by the work input:

Unlike efficiency, a COP can exceed one, and no law is broken. The work does not become most of the delivered heat; it drives the transfer of heat that was already there, outdoors. A domestic heat pump with a COP of 4 delivers four joules of heat per joule of electricity, against exactly one for the best possible electric fire, and both COPs climb as the temperature gap being pumped across narrows.

Selling the heat as well as the work

Look back at the audit and the biggest number in it is the one nobody wanted, the fifty-odd kilowatts leaving as hot exhaust and hot coolant. A power station tells the same story on a larger scale, rejecting perhaps sixty per cent of its fuel's energy into a river or a cooling tower. Combined heat and power is the decision to use that heat rather than dump it.

The change is one of sink temperature. Instead of rejecting Q_C to the coldest thing available, the plant rejects it into water at eighty or ninety degrees, which is then piped round the buildings nearby to heat them. The heat is doing a job now, so the share of the fuel's energy that ends up wanted climbs from something like forty per cent to something like eighty, and the plant is built in the town it heats rather than out by the river.

It is not free, and the second law says precisely what is lost. The ceiling on work is (T_H − T_C)/T_H, so lifting T_C from 300 K to 360 K lowers that ceiling, and a combined heat and power plant generates less electricity from the same fuel than one rejecting into a cold river. What that sacrifice gains is rejected heat leaving at a temperature high enough to be useful.

No law is bent by the eighty per cent, and it is worth being able to say why. The second law forbids turning all of Q_H into work; it says nothing against using Q_C as heat, which is what it already was. The figure is an energy utilisation rather than an efficiency, since it adds a quantity of work to a quantity of low-grade heat as though the two were interchangeable, and they are not: the work can drive a motor or a computer, while ninety-degree water can only ever warm something cooler than itself.