Physics · Engineering Physics · A2 Level

Heat Engines

Every petrol or diesel engine converts heat into motion by taking a trapped gas through a repeating cycle of expansion and compression. The p-V diagram is the single picture that captures everything about how much useful work that cycle actually produces.

10 min read

Reading a p-V diagram

A heat engine is any device that converts heat into useful mechanical work by repeatedly cycling a working fluid — almost always a gas — through a sequence of expansions and compressions. The standard way of visualising what's happening inside one is the p-V diagram: a graph of the gas's pressure $p$ plotted against its volume $V$ as the process unfolds. As the gas moves through a process — expanding, being compressed, being heated at constant volume, and so on — its state traces out a curve on this diagram, and every point on that curve represents one particular combination of pressure and volume the gas passes through.

The reason the p-V diagram is so useful is a simple geometric fact: the work done during a process is equal to the area beneath the curve on the p-V diagram, between the curve and the volume axis. This holds regardless of exactly how pressure varies with volume during the process.

As a sanity check, this rule reduces exactly to the already-familiar constant-pressure result. If pressure is held fixed at $p$ while volume changes by $\Delta V$, the "curve" is just a horizontal line, and the area beneath it is a simple rectangle of height $p$ and width $\Delta V$ — giving area $= p\Delta V$, exactly matching $W = p\Delta V$ from the previous lesson. The area rule is the general case; the constant-pressure formula is just its simplest special case.

Net work per cycle: the area of the loop

A real engine doesn't perform a single process and stop — it repeats a closed cycle over and over, returning the gas to its starting pressure and volume at the end of each cycle so the process can begin again. On a p-V diagram, a complete cycle traces out a closed loop. Because different parts of the loop involve expansion (the gas doing work on its surroundings) and compression (work being done on the gas), the net useful work done per cycle is the difference between these — and geometrically, this works out to be exactly the area enclosed by the loop.

Worked ExampleEstimating work done per cycle from a p-V loop

An idealised engine cycle traces out a loop on a p-V diagram that is approximately a parallelogram: volume varies between $2.0 \times 10^{-4}\ \text{m}^3$ and $5.0 \times 10^{-4}\ \text{m}^3$, while pressure varies between $1.0 \times 10^5\ \text{Pa}$ and $3.0 \times 10^5\ \text{Pa}$ over the cycle. Estimate the work done per cycle.

The enclosed area equals work done per cycle, so approximate the loop as a rectangle for an estimate: width $= \Delta V = (5.0-2.0)\times10^{-4} = 3.0\times10^{-4}\ \text{m}^3$, height $= \Delta p = (3.0-1.0)\times10^5 = 2.0\times10^5\ \text{Pa}$.

Real indicator diagrams are rarely simple rectangles, but treating the loop as a basic shape (rectangle, triangle, or a combination) is exactly how an "estimate the area" question is meant to be tackled.

$W_{\text{cycle}} \approx \Delta p \times \Delta V = (2.0\times10^5)\times(3.0\times10^{-4}) = 60\ \text{J}$.

Each cycle of this engine does approximately 60 J of net useful work.

Multiplying this by the number of cycles completed per second gives the power developed — exactly the quantity introduced below as indicated power.

Result

Real engine cycles and their indicator diagrams

The two most important practical engine cycles are the four-stroke petrol cycle and the diesel cycle, each with its own characteristic indicator diagram shape.

Four-stroke petrol engine. The cycle runs through four strokes of the piston: intake (the piston moves down, drawing in an air-fuel mixture at roughly constant low pressure), compression (the piston moves up, compressing the mixture, with pressure and temperature rising sharply), power (a spark ignites the compressed mixture, causing a near-instantaneous jump in pressure at roughly constant volume, followed by a rapid expansion that pushes the piston down and delivers the useful work of the cycle), and exhaust (the piston moves up again, expelling the spent gases at roughly constant low pressure). On an idealised indicator diagram, this appears as a loop with two near-vertical sides (the rapid pressure jump at ignition, and the pressure drop at the exhaust valve opening) connecting two curved sides (compression and power-stroke expansion).
Diesel engine. The diesel cycle compresses air alone (no fuel present) to a much higher pressure and temperature than a petrol engine — high enough that fuel injected at the top of the stroke ignites spontaneously from the heat of compression alone, without a spark. Combustion in a diesel engine happens more gradually, closer to constant pressure, rather than the near-instantaneous constant-volume jump of the petrol cycle, giving the diesel indicator diagram a visibly different top edge — a near-horizontal segment during combustion rather than a sharp vertical spike.

A pressure-volume indicator diagram showing a closed loop for a four-stroke petrol cycle. Starting at the bottom left, the loop traces a shallow intake curve, a steep compression curve rising to the top left, a near-vertical ignition jump followed by a power stroke curve bulging across the top of the loop, and finally a steep exhaust curve dropping back to the start. The interior of the loop is lightly hatched, representing the net work done per cycle. — Idealised indicator diagram for a four-stroke petrol cycle: the shaded area enclosed by the loop represents the net work done per cycle. A real, measured indicator diagram traces a similar but rounded-off loop of smaller enclosed area, owing to non-ideal gas behaviour, finite valve-timing, and other practical departures from this idealised shape.

Theoretical vs measured indicator diagrams

The idealised cycle diagrams drawn in textbooks — with sharp corners and perfectly vertical or horizontal sides — are never quite achieved in a real engine. A real, measured indicator diagram has rounded corners rather than sharp ones, and its sides are curved rather than perfectly straight, because of real gas behaviour, the finite time it takes for valves to open and close, and other practical effects. Crucially, the area enclosed by a real indicator diagram is always smaller than the idealised theoretical version, meaning a real engine always does less net work per cycle than the theoretical maximum the idealised cycle would suggest.

Power terminology: input power and indicated power

With the area-of-the-loop idea established, two power definitions used throughout the rest of this topic can now be introduced.

Input power is the rate at which chemical energy is supplied to the engine by its fuel:

$$\text{input power} = \text{calorific value} \times \text{fuel flow rate}$$

where the calorific value of a fuel is the energy released per unit mass (or volume) when it's burned, and the fuel flow rate is the mass (or volume) of fuel consumed per second. This is the total chemical energy supplied to the engine every second, before any of it has been converted to anything useful.

Indicated power is the power actually developed inside the cylinders by the expanding gas, calculated directly from the indicator diagram:

$$\text{indicated power} = (\text{area of } p\text{-}V \text{ loop}) \times (\text{cycles per second}) \times (\text{number of cylinders})$$

Since the area of the loop is the work done by the gas per cylinder per cycle, multiplying by the number of cycles completed each second converts this to a power (work done per second) for one cylinder, and multiplying again by the number of cylinders gives the total power developed across the whole engine. This is a genuinely measured quantity, obtained from real indicator diagrams recorded while the engine runs, not a theoretical prediction — and it's the natural bridge to the fuller efficiency picture covered in the next lesson.

Key takeaways

A heat engine converts heat into useful work by cycling a working gas through repeated expansions and compressions, visualised on a p-V diagram.
Work done during any process equals the area beneath its curve on a p-V diagram; for a constant-pressure process this reduces to the familiar $W = p\Delta V$.
Net work done per complete cycle equals the area enclosed by the closed loop traced on the p-V diagram.
The four-stroke petrol cycle (intake, compression, power, exhaust) and the diesel cycle each have characteristic, differently-shaped indicator diagrams, differing mainly in how combustion is represented.
Real, measured indicator diagrams have rounded corners and curved sides compared with the idealised theoretical cycle, and always enclose a smaller area — meaning less net work per cycle than the ideal case.
Input power = calorific value × fuel flow rate: the chemical energy supplied per second.
Indicated power = (area of p-V loop) × (cycles per second) × (number of cylinders): the power actually developed inside the cylinders, measured directly from the indicator diagram.

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