Otto and Diesel cycles explained with solved problems

Petrol and diesel engines both burn fuel inside a cylinder and push a piston, but they ignite the fuel differently, and that changes how efficient they can be. Engineers study them with two idealised cycles: the Otto cycle for spark-ignition (petrol) engines and the Diesel cycle for compression-ignition engines. The ideal cycles leave out friction and real combustion, but they explain why compression ratio matters so much.

The concept: four strokes

A four-stroke engine repeats the same sequence every two turns of the crankshaft:

1. Intake: the piston moves down and draws in air (and fuel, in a petrol engine).
2. Compression: the valves close and the piston squeezes the charge into a small volume.
3. Power: the charge burns, the pressure rises sharply and pushes the piston down.
4. Exhaust: the piston moves up and pushes the burnt gas out.

In a petrol engine a spark ignites a mixture of fuel and air near the top of the stroke and it burns almost instantly, so the heat is added at roughly constant volume. In a diesel engine, only air is compressed, so hot that fuel injected into it ignites by itself; the fuel burns as it is injected, while the piston starts to move down, so the heat is added at roughly constant pressure.

Assumptions (air-standard cycle)

• The working fluid is air, an ideal gas with constant specific heats (γ = 1.4, c_v = 0.718 kJ/kg·K).
• Combustion is replaced by heat added from outside; exhaust and intake are replaced by heat rejected at constant volume.
• Compression and expansion are reversible and adiabatic (isentropic).
• There is no friction and no heat loss to the cylinder walls.

The efficiency formulas

The compression ratio is $r = V_1 / V_2$, the cylinder volume at the bottom of the stroke divided by the volume at the top. For the Diesel cycle, the cut-off ratio is $\rho = V_3 / V_2$, how much the volume grows while fuel is burning.

$$\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}}$$ $$\eta_{Diesel} = 1 - \frac{1}{r^{\gamma - 1}} \left[ \frac{\rho^{\gamma} - 1}{\gamma\,(\rho - 1)} \right]$$

The bracket in the Diesel formula is always greater than 1, so at the same compression ratio the Otto cycle is more efficient. But petrol engines can’t use high compression ratios: above about 10–12 the mixture ignites by itself before the spark (knock). Diesel engines need high ratios, about 14–22, to ignite the fuel at all. So in practice diesels are more efficient.

Ideal versus real engines

Real engines reach roughly half to two-thirds of these ideal figures. Heat is lost to the cylinder walls and coolant, combustion takes time, valves open before the end of the stroke, pumping air in and out takes work, and friction in the bearings, rings and valve train takes more. Still, the ideal cycles predict the trends correctly: higher compression ratio, higher efficiency; and the earlier and faster the heat is added, the better.

Where this connects to design

The pressures in these cycles set the loads on the engine’s parts. In solved problem 1, the pressure after compression alone is 2.5 MPa; after combustion it is several times higher. Those gas forces act on the piston, push through the connecting rod and crankshaft (the slider-crank mechanism) and size the bearings, the bolts and the block. Our V-engine CAD assembly shows these parts together.