Fundamentals8 min read2 / 9

Work from pressure: the PV diagram

An engine is a machine for turning gas pressure into shaft work, and there is one picture that shows exactly how much it gets: the PV diagram, pressure plotted against cylinder volume. The area enclosed by the loop is the work per cycle — not a metaphor, an equality. This article derives that, then reads a real loop.

From pressure to work

Pressure is force per area, so the gas pushes on the piston crown with force $F = p\,A$ . Move the piston a small distance $dx$ and the gas does work $dW = F\,dx = p\,A\,dx$ . But $A\,dx$ is exactly the change in cylinder volume $dV$ , so:

dW = p\,dV

(1)

Integrate around one complete cycle and the piston returns to its start, so the net work is the closed loop integral:

W_{\text{net}} = \oint p\,dV = \text{area enclosed by the PV loop}

(2)

$p$: cylinder pressure [Pa]
$V$: cylinder volume (clearance + swept portion) [m³]
$W_{\text{net}}$: net work delivered to the piston per cycle [J]

The direction you travel the loop matters. Where the piston moves down under high pressure (the power stroke), $p\,dV$ is a big positive contribution. Where it moves back up at lower pressure (compression), the contribution is negative but smaller. Clockwise loops (in the usual orientation) produce net work; anticlockwise loops consume it.

Reading a real loop

Fig. 1. The pressure traces the demos actually animate, plotted as PV loops: Otto at r = 10.5 and Diesel at r = 18 (same bore and stroke). The Diesel reaches much higher compression pressure and holds a flat top through injection; the Otto spikes sharply at TDC.

Follow the Otto (blue) loop with the four strokes in mind: a flat run rightward near the bottom (intake, slightly below atmospheric), a rising sweep leftward (compression, steepening as the volume shrinks), a near-vertical spike at minimum volume (combustion at almost constant volume), a falling sweep rightward (expansion — the power stroke), and a flat run leftward just above atmospheric (exhaust).

Two features deserve names. The tall upper loop is the gross indicated work. The thin sliver at the bottom, traversed anticlockwise between the exhaust and intake lines, is the pumping loop — work the engine spends breathing. Throttle a petrol engine and the intake line drops well below atmospheric, fattening the pumping loop; this is the part-load throttling loss diesels famously avoid by never throttling their air.

The adiabatic legs

During compression and expansion both valves are shut and things happen far too fast for much heat to leave the gas, so the process is nearly adiabatic (no heat exchange) and, idealised as frictionless, follows:

p\,V^{\gamma} = \text{constant},\qquad \gamma = \frac{c_p}{c_v} \approx 1.4 \text{ for air}

(3)

That exponent is why the compression curve is steeper than the $\ 1/V$ of an isothermal squeeze: compressing the gas also heats it, and hot gas pushes back harder. The demos use exactly this relation for the valves-closed legs — with $\gamma = 1.4$ , as if the working gas were plain air throughout (the “air-standard” idealisation the next article builds on).