The ideal Otto cycle, the pattern of a spark-ignition engine, has an efficiency of 1 − r^(1 − γ), where r is the compression ratio and γ the heat-capacity ratio of the charge. For γ = 1.35 it rises from 51.7% at a compression ratio of 8 to 58.1% at 12, in ever smaller steps, while the compression itself heats the charge to about 400 °C at 8 and 519 °C at 12 for γ = 1.4, and the practical ceiling is the knock that sets in when the unburned mixture ignites by itself.
Part of Internal Combustion
The idealised four-process cycle behind a spark-ignition engine, the arithmetic of compression ratio, efficiency and charge temperature, and how to measure the ratio of a real engine.
Four Ideal Processes Give One Formula
The ideal cycle has four processes. The charge is compressed without heat exchange (adiabatic), heat is added at constant volume when the spark ignites it, the hot gas expands without heat exchange and does the work, and heat is rejected at constant volume at the exhaust. Treating the charge as an ideal gas with a constant heat-capacity ratio γ, the efficiency depends only on the compression ratio r:
η = 1 − r^(1 − γ).
The corpus uses this formula in Engine Modifications, with γ = 1.35 as an assumption for a lean-ish mixture. Gas Laws in Practice uses γ = 1.4 for air, a recalled value. The formula is a recalled textbook result; the arithmetic below is mine and can be checked with a calculator.
What The Ratio Buys, And What It Heats
Compression ratio r is the cylinder volume with the piston at the bottom divided by the volume at the top: (swept volume + clearance volume) ÷ clearance volume. The adiabatic temperature rise is T2 = T1 × r^(γ − 1) and the pressure rises by r^γ, from the same relations the corpus uses for compressed air. Start at 293.15 K (20 °C) and 1 bar.
| r | Efficiency, γ = 1.35 | Efficiency, γ = 1.4 | End-of-compression temperature, γ = 1.4 | Pressure ratio, γ = 1.4 |
|---|---|---|---|---|
| 6 | 46.6% | 51.2% | 327 °C | 12.3 |
| 8 | 51.7% | 56.5% | 400 °C | 18.4 |
| 10 | 55.3% | 60.2% | 463 °C | 25.1 |
| 12 | 58.1% | 63.0% | 519 °C | 32.4 |
| 16 | 62.1% | 67.0% | 616 °C | 48.5 |
Check one row. For r = 8 and γ = 1.4: 8^0.4 = e^(0.4 × ln 8) = e^(0.4 × 2.0794) = e^0.8318 = 2.297. Temperature: 293.15 × 2.297 = 673.5 K = 400 °C. Pressure ratio: 8^1.4 = 18.4. Efficiency: 1 − 8^(−0.4) = 1 − 0.4353 = 0.565.
Two things stand out. The gain per unit of ratio shrinks. For γ = 1.35 the efficiency gains about 2.5 points per unit of ratio from 6 to 8, 1.8 from 8 to 10, 1.4 from 10 to 12 and 1.0 from 12 to 16. And the temperature at the end of compression climbs steadily. The carrier says compression alone heats the air in a diesel engine to 500+ °C, enough to ignite injected fuel. At r = 12 the ideal charge in a petrol engine is already at 519 °C, which is why a petrol mixture can begin to burn by itself before the spark. That is knock.
The temperature at which a mixture ignites on its own depends on the fuel, the pressure and the time; the table gives the heating, not a knock threshold. The knock limit is found by test, engine by engine. The carrier notes that ethanol has a higher octane rating than petrol (about 113 against 87 to 93), so an engine can run a higher compression ratio on it, which is the corpus reasoning behind the 12:1 figure in Engine Modifications.
Measuring The Compression Ratio Of A Real Engine
Step 1: Find the swept volume of one cylinder. π ÷ 4 × bore² × stroke, with the bore measured at several points and the stroke from the crank geometry.
Step 2: Measure the clearance volume. With the piston at top dead centre and the valves closed, fill the combustion space through the plug hole with a measured liquid from a burette or syringe. The liquid volume is the clearance volume, as the corpus does for the chamber in Engine Modifications.
Step 3: Compute r = (swept + clearance) ÷ clearance. The corpus case: a 500 cc cylinder with a chamber of 71.4 cc gives (500 + 71.4) ÷ 71.4 = 8.0.
Step 4: Compare with a compression gauge reading. A real engine reads lower than the ideal because of leakage past rings and valves, heat loss to the walls and the finite time of compression. A low reading against a healthy engine of the same type, or a large spread between cylinders, points to worn rings or leaking valves; no reference figure is given here, because the right reading belongs to the particular engine.
Worked Example: What A Little Metal Off The Head Does
Take the corpus engine, 500 cc per cylinder with a 71.4 cc chamber at r = 8, and remove metal until the chamber is 55.6 cc.
- New ratio: (500 + 55.6) ÷ 55.6 = 10.0.
- Ideal efficiency at γ = 1.35: from 51.7% to 55.3%, a gain of 3.6 points, which is 7% relative (55.3 ÷ 51.7 = 1.07).
- Ideal end-of-compression temperature at γ = 1.4: from 400 °C to 463 °C, which is 63 degrees hotter.
So the gain of 7% costs 63 degrees of extra charge heat and brings the engine closer to knock. That trade is the reason the corpus says to try it only when the head can be changed safely.
Common Mistakes
- Treating the ideal efficiency as what the engine will deliver. The carrier’s own figure for a real engine is about a quarter.
- Raising the ratio on a fuel that cannot take it. Low-octane fuel knocks and burns pistons.
- Measuring the chamber without closing the valves or sealing the rings. The liquid leaks away and the volume reads low.
- Forgetting that milling the head changes valve clearance and timing. See Valve Train.
Safety
Knock and over-compression destroy pistons and can throw the parts
Sustained knock overheats and cracks pistons and rings and can break a connecting rod. If the engine pings under load, back off the timing or the load at once. The fuel vapour used to fill and test a cylinder burns; measure the chamber with water or light oil, never with petrol, and keep ignition sources away.
What’s Next
- Learn next: Diesel Cycle — what changes when the mixture is lit by compression alone
- Combine with: Ignition Timing — where in the cycle the spark must fall