Thermal Engineering
The Diesel Cycle
Compression ignition and constant-pressure heat addition — the most efficient reciprocating heat engine
The Diesel cycle is the ideal thermodynamic cycle of the compression-ignition (CI) engine: it compresses pure air adiabatically to a compression ratio of 14–25, adds heat at constant pressure as injected fuel autoignites near top dead center, expands the gas adiabatically, and rejects heat at constant volume. Its defining trait — the one that separates it from the Otto (spark-ignition) cycle — is that combustion occurs at roughly constant pressure rather than constant volume, and that ignition is caused by compression heat alone, with no spark plug. Because Diesels are not knock-limited they run far higher compression ratios than gasoline engines, giving brake thermal efficiencies of 40–46% in road vehicles and up to ~55% in large two-stroke marine engines. Rudolf Diesel patented the cycle in 1892 and demonstrated a working engine in 1897.
- IgnitionCompression — no spark plug
- Heat additionConstant pressure (2→3)
- Compression ratio r14–25
- Efficiency η1 − r1−γ·[(rcγ−1)/(γ(rc−1))]
- Brake efficiency40–46% road, ~55% marine
- Peak pressureUp to ~200 bar (common-rail)
Interactive visualization
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Watch the 60-second explainer
A condensed visual walkthrough — narrated, captioned, under a minute.
Why the Diesel cycle matters
The Diesel cycle is the reason a container ship can cross the Pacific on a single bunker fuelling and a long-haul truck can pull 40 tonnes for 1,000 km on one tank. It is the most thermally efficient single-cycle heat engine ever put into mass production: large low-speed marine two-strokes such as the Wärtsilä RT-flex96C and MAN B&W engines exceed 50% brake thermal efficiency, and automotive Diesels routinely reach 40–46%, against roughly 30–38% for a comparable gasoline (Otto) engine. That efficiency edge translates directly into lower fuel burn and lower CO₂ per unit of work.
- Heavy transport. Trucks, buses, locomotives, and ships — anywhere high torque at low RPM and long range dominate the design.
- Power generation. Standby and prime gensets, from 5 kW portable units to 20 MW stationary plants.
- Marine propulsion. Two-stroke crosshead Diesels drive nearly all large merchant shipping.
- Off-highway. Excavators, tractors, mining haul trucks, and generators where torque and durability rule.
- Efficiency benchmark. The cycle sets the practical ceiling that Atkinson-cycle hybrids and fuel cells are measured against.
How it works — the four processes
The ideal (air-standard) Diesel cycle assumes the working fluid is air behaving as an ideal gas, that compression and expansion are reversible and adiabatic (isentropic), and that combustion is replaced by external heat addition. On a pressure–volume (PV) diagram the cycle is bounded by two isentropes, one constant-pressure line, and one constant-volume line. Trace it in the order the piston moves:
- 1 → 2 — Isentropic compression. The piston moves from bottom dead center (BDC) to top dead center (TDC), compressing pure air by the compression ratio r = V₁/V₂. Temperature climbs from ~300 K to well over 850 K — above the fuel's autoignition point. No fuel is present yet, so there is nothing to knock.
- 2 → 3 — Constant-pressure heat addition. Fuel is injected into the hot air near TDC and autoignites after a short delay. As it burns, the piston has already begun to descend, so the expanding volume holds cylinder pressure roughly constant while heat qin = cp(T₃ − T₂) is released. The volume ratio across this step is the cutoff ratio rc = V₃/V₂.
- 3 → 4 — Isentropic expansion. The high-pressure gas pushes the piston to BDC, doing work on the crankshaft. This is the power stroke; the expansion ratio is V₄/V₃ = r/rc.
- 4 → 1 — Constant-volume heat rejection. The exhaust valve opens and hot gas is expelled at essentially fixed piston position, rejecting qout = cv(T₄ − T₁). In a real four-stroke engine this idealizes the exhaust and intake strokes.
On a temperature–entropy (TS) diagram the two isentropes appear as vertical lines, constant-pressure heat addition curves upward to the right, and constant-volume heat rejection returns down a slightly steeper curve. The enclosed area equals the net work per cycle in both the PV and TS planes.
The efficiency equation — every symbol defined
The ideal air-standard thermal efficiency of the Diesel cycle is:
η = 1 − r(1−γ) · [ (rcγ − 1) / ( γ (rc − 1) ) ]
| Symbol | Meaning | Typical value / units |
|---|---|---|
| η | Thermal efficiency (net work ÷ heat added) | dimensionless (0–1) |
| r | Compression ratio, V₁/V₂ | 14–25 |
| rc | Cutoff ratio, V₃/V₂ (volume ratio during combustion) | 1.5–3 (rises with load) |
| γ | Ratio of specific heats, cp/cv | ≈1.4 for air (cold-air standard) |
| cp, cv | Specific heats at constant pressure / volume | 1005, 718 J·kg⁻¹·K⁻¹ (air) |
| T₁…T₄ | State-point temperatures | K |
The whole bracketed term is the Diesel efficiency penalty factor. Because rc > 1, that bracket is always greater than 1, so at any given compression ratio a Diesel is less efficient than an Otto engine (whose efficiency is simply η = 1 − r1−γ). The apparent paradox resolves in the real world: gasoline knock caps the Otto compression ratio near 10–12, while the Diesel comfortably runs 16–22, and the r1−γ term rewards that higher compression far more than the bracket penalizes it. As rc → 1 (very brief injection, i.e. light load) the bracket → 1 and the Diesel efficiency approaches the Otto formula.
Otto vs Diesel — the constant-volume vs constant-pressure fork
| Attribute | Otto cycle (SI) | Diesel cycle (CI) |
|---|---|---|
| Ignition | Spark plug | Compression heat |
| Heat addition | Constant volume (2→3) | Constant pressure (2→3) |
| Charge compressed | Air + fuel premixed | Air only |
| Compression ratio | 8–12 (knock limited) | 14–25 |
| Power control | Throttle (air throttling) | Fuel quantity (unthrottled) |
| Ideal efficiency | 1 − r1−γ | 1 − r1−γ·[(rcγ−1)/(γ(rc−1))] |
| Brake efficiency (typical) | 30–38% | 40–46% (up to ~55% marine) |
| Peak pressure | ~40–60 bar | ~150–200 bar |
A useful mental model: the Otto engine burns fast and holds volume; the Diesel burns while expanding and holds pressure. The dual (Sabathé) cycle — a mix of constant-volume then constant-pressure heat addition — models real high-speed Diesels more accurately than the pure Diesel cycle, since fast common-rail injection produces a spike of near-constant-volume burn followed by a constant-pressure tail.
Worked example — efficiency of a truck Diesel
Take a heavy-duty Diesel with compression ratio r = 18 and cutoff ratio rc = 2, using the cold-air standard γ = 1.4.
- Compression factor: r1−γ = 18−0.4 = 0.3147
- Bracket: (rcγ − 1) / (γ(rc − 1)) = (21.4 − 1)/(1.4·(2 − 1)) = (2.639 − 1)/1.4 = 1.171
- Efficiency: η = 1 − 0.3147 × 1.171 = 1 − 0.3684 = 0.632, i.e. ~63%
For comparison, an Otto engine at the same r = 18 would give η = 1 − 0.3147 = 68.5% — the constant-pressure burn costs about 5 efficiency points here. But drop the Otto to a knock-realistic r = 10 and its efficiency falls to η = 1 − 10−0.4 = 60.2%, now below the Diesel. The ideal ~63% is not the number you measure on a dyno; friction, incomplete expansion, heat loss to the cylinder walls, finite combustion duration, and pumping bring real brake thermal efficiency down to about 42% for such an engine. The gap between ideal air-standard and brake efficiency is the engineer's whole design problem.
Common misconceptions and failure modes
- "Glow plugs ignite the fuel." They don't. Glow plugs only warm a cold combustion chamber so compression heat can reach autoignition on a cold start. Combustion is always compression-triggered.
- "Diesel is more efficient because of the constant-pressure burn." The opposite — constant-pressure heat addition costs efficiency versus constant-volume at equal r. Diesel wins on higher compression ratio and no throttling losses, not on the burn shape.
- "More cutoff (more fuel) is always better." Larger rc gives more work per cycle but lower efficiency, because late-burned fuel expands over a smaller ratio. This is why Diesel efficiency peaks at part load and why smoke appears at full fuelling (over-rich diffusion combustion).
- Diesel knock. A long ignition delay lets too much fuel accumulate before it lights, then it all burns at once — a steep dP/dt pressure rise heard as clatter. It is the mirror image of gasoline knock and is controlled by cetane number, pilot injection, and injection timing.
- The NOx–soot trade-off. Hot lean zones make NOx; fuel-rich diffusion-flame cores make soot. Retarding timing or cooling the charge (EGR) cuts NOx but raises soot and fuel consumption — the fundamental after-treatment driver behind DPFs and SCR (AdBlue/urea).
- Structural loading. Peak pressures of 150–200 bar demand stout bearings, crankshafts, and head bolts; a Diesel block is heavier than a gasoline block of equal displacement for exactly this reason.
Frequently asked questions
What is the Diesel cycle?
The Diesel cycle is the ideal air-standard cycle for a compression-ignition engine. It has four processes: adiabatic (isentropic) compression of pure air to a high compression ratio, constant-pressure heat addition as fuel injected near top dead center autoignites and burns while the piston starts moving down, adiabatic (isentropic) expansion of the combustion gases, and constant-volume heat rejection as the exhaust is expelled. Unlike the Otto cycle, there is no spark plug and no premixed charge — air alone is compressed until it is hot enough (about 700 to 900 kelvin) to ignite the fuel spray on contact.
How is the Diesel cycle different from the Otto cycle?
The key difference is how heat is added. In the Otto cycle heat is added at constant volume — a spark ignites a premixed air-fuel charge and it burns almost instantly at nearly fixed piston position. In the Diesel cycle heat is added at constant pressure — fuel is injected into hot compressed air and burns progressively as the piston descends, keeping cylinder pressure roughly flat. That difference forces a slightly lower efficiency for the same compression ratio, but the Diesel avoids knock, so it can run compression ratios of 14 to 25 versus 8 to 12 for gasoline, which more than compensates.
What is the efficiency of the Diesel cycle?
The ideal air-standard thermal efficiency is eta = 1 - r^(1-gamma) times [(rc^gamma - 1) / (gamma (rc - 1))], where r is the compression ratio, rc is the cutoff ratio (the volume ratio during constant-pressure combustion), and gamma is the specific-heat ratio, about 1.4 for air. For r = 18, rc = 2, gamma = 1.4 the ideal efficiency is roughly 63 percent. Real Diesel engines reach 40 to 46 percent brake thermal efficiency in cars and trucks and up to about 55 percent for large slow-speed two-stroke marine engines, the highest of any single-cycle heat engine in commercial use.
What is the cutoff ratio in the Diesel cycle?
The cutoff ratio rc is the ratio of the cylinder volume at the end of constant-pressure heat addition to the volume at the start of it (V3/V2). It measures how far into the power stroke fuel injection continues. A larger cutoff ratio means more fuel is burned and more work is produced per cycle, but efficiency falls because a larger fraction of the burn happens at lower expansion ratio. As rc approaches 1 (very short injection) the Diesel efficiency approaches the Otto efficiency for the same compression ratio. Cutoff ratio rises with engine load, which is why Diesel efficiency is highest at part load, not full load.
Why does a Diesel engine not need a spark plug?
Because the air is compressed so much that it becomes hot enough to ignite fuel on its own. Compressing air adiabatically from a compression ratio of 16 raises its temperature from about 300 kelvin to over 850 kelvin, well above diesel fuel's autoignition temperature of roughly 483 kelvin (210 celsius). When atomized fuel is sprayed into this hot air it ignites after a short ignition delay of about 0.5 to 2 milliseconds. Cold starts use glow plugs to preheat the chamber, but these are heaters, not igniters — combustion itself is triggered purely by compression heat.
Why are Diesel engines more efficient than gasoline engines?
Three reasons. First, the high compression ratio (14 to 25) gives a large expansion ratio, extracting more work from each unit of heat — efficiency rises with r^(1-gamma). Second, Diesels run unthrottled and control power by fuel quantity, so they avoid the pumping losses a gasoline throttle plate imposes at part load. Third, lean overall mixtures and a higher effective gamma reduce heat losses. Diesel fuel also carries about 12 percent more energy per litre than gasoline. Together these give Diesel vehicles roughly 25 to 35 percent better fuel economy on an energy basis than comparable gasoline engines.
What are the main failure modes and drawbacks of the Diesel cycle?
The dominant real-world limitations are emissions and mechanical stress. High-temperature lean combustion produces nitrogen oxides (NOx), and the diffusion flame produces particulate soot — the classic NOx-versus-soot trade-off that demands exhaust after-treatment (diesel particulate filters and selective catalytic reduction). High peak pressures, up to 200 bar in modern common-rail engines, require a heavy, robust structure. Long ignition delay can cause a sharp pressure rise (diesel knock or combustion roughness). Cold starts are harder because compression heat leaks away from a cold cylinder. Fuel injection must be precise to microsecond and microlitre tolerances.