Going deeper9 min read4 / 9

The Diesel cycle

A diesel solves the knock problem by refusing to compress fuel at all: it squeezes pure air — as hard as it likes — and only then sprays fuel into gas already hot enough to ignite it. That one design choice changes the thermodynamic cycle, the PV loop’s shape, and the entire character of the engine.

Compression ignition

Squeeze air 18:1 and the isentropic relation $T_2 = T_1\,r^{\gamma-1}$ takes 300 K intake air to $300 \times 18^{0.4} \approx 950\,\text{K}$ — roughly 680 °C, comfortably past diesel fuel’s ~210 °C autoignition temperature. So there is no spark plug: the injector fires near TDC and each droplet ignites as it mixes. Because fuel arrives only at the moment of combustion, there is no end-gas waiting to knock, and the compression ratio can be whatever the block can withstand (14–24:1 in practice).

Combustion is also slower than a spark engine’s premixed flame — the fuel must find its air as it sprays. The classic idealisation: heat is added at constant pressure while the piston begins its descent, pressure held level by the expanding volume. Injection stops at the cutoff, and the rest of the stroke is ordinary isentropic expansion.

Fig. 1. The ideal Diesel cycle at r = 18. Compare the flat 2→3 top with Otto's constant-volume spike — this is the signature you can see in the demo's real PV traces too.

The efficiency, and the cutoff penalty

Define the cutoff ratio — how far the piston has travelled when injection stops:

r_c = \frac{V_3}{V_2}

(1)

Heat now enters at constant pressure ( $q_{in} = c_p\,\Delta T$ ) but still leaves at constant volume ( $q_{out} = c_v\,\Delta T$ ). Working through the same temperature bookkeeping as the Otto derivation (each state related to the last by its process law) gives:

\eta_{\text{Diesel}} = 1 - \frac{1}{r^{\gamma-1}}\;\underbrace{\left[\frac{r_c^{\gamma}-1}{\gamma\,(r_c-1)}\right]}_{>\,1 \text{ for } r_c>1}

(2)

$r$: compression ratio V₁/V₂ [–]
$r_c$: cutoff ratio V₃/V₂ (1 = no burn; ~2–3 at full load) [–]
$c_p,\;c_v$: specific heats at constant pressure / volume [J/(kg·K)]

This is exactly the Otto formula multiplied by a bracket that exceeds 1 whenever $r_c > 1$ . As $r_c \to 1$ (an infinitesimally short burn) the bracket → 1 and Diesel converges to Otto. So:

Try both levers below. Note how efficiency falls as you raise the cutoff ratio (more load, longer burn) — a genuinely diesel quirk; an Otto engine’s ideal efficiency does not care about load at all:

Compression ratio

r

18.0Cutoff ratio

r_c

2.00

\eta = 1 - \frac{1}{r^{\gamma-1}}\left[\frac{r_c^{\gamma}-1}{\gamma(r_c-1)}\right]

\eta = 63.2\,\%

The bracket is 1.171 — above 1 — so this Diesel gives up 5.4 points against an Otto cycle at the same $r$ — but no petrol engine can run $r = 18$ without knocking, which is why real diesels still win.

Why diesels feel the way they do

Torque-rich, rev-poor. Mixing-limited combustion simply runs out of time at high rpm, and the long stroke + heavy internals that survive 180+ bar peak pressures don’t like being hurried. Big pressure on every power stroke at low speed = the classic diesel shove.
Heavy and expensive. Roughly twice the peak cylinder pressure of a petrol engine must be contained, forever.
The clatter. Fuel injected during the ignition delay burns all at once when it finally lights — a small knock every cycle, loudest at cold idle. Modern pilot-injection strategies split the spray precisely to soften it.
Soot vs NOx. Droplet cores burn rich (soot) while the hot lean zones make NOx — and fixing one usually worsens the other. The particulate filters and urea systems on every modern diesel are the price of compression ignition’s efficiency.