Otto vs Diesel Cycle: Air-Standard Efficiency with Worked Examples
Real engine cylinders contain a changing mixture of air, fuel and combustion products, with heat loss, friction and valve timing effects. To understand the basic trends, engineers replace this with an idealised air-standard cycle: air is the working fluid, combustion is replaced by heat addition, and all processes are reversible. The Otto cycle models the spark-ignition (petrol) engine and the Diesel cycle models the compression-ignition engine. Their efficiency formulas explain why compression ratio matters so much and why diesel engines use less fuel.
The Four Processes of Each Cycle
| Process | Otto cycle | Diesel cycle |
|---|---|---|
| 1 to 2 | Isentropic compression | Isentropic compression |
| 2 to 3 | Heat added at constant volume | Heat added at constant pressure |
| 3 to 4 | Isentropic expansion | Isentropic expansion |
| 4 to 1 | Heat rejected at constant volume | Heat rejected at constant volume |
The only difference is the heat addition process. In a petrol engine, the spark ignites a premixed charge that burns very quickly near TDC, which approximates constant volume. In a diesel engine, fuel is injected over a period of crank rotation while the piston starts moving down, so pressure stays roughly constant during burning.
Efficiency Formulas
Compression ratio r = V1 / V2 Cut-off ratio rho = V3 / V2 (Diesel only) gamma = cp/cv = 1.4 for air Otto: eta = 1 - 1 / r^(gamma - 1) Diesel: eta = 1 - [1 / r^(gamma - 1)] * [ (rho^gamma - 1) / (gamma * (rho - 1)) ] Isentropic relations: T2/T1 = r^(gamma-1), p2/p1 = r^gamma Mean effective pressure mep = W_net / (V1 - V2)
Otto efficiency depends only on compression ratio. Diesel efficiency has the same leading term multiplied by a bracket that is always greater than one when rho is greater than one. So at the same compression ratio, Otto is always more efficient than Diesel.
Worked Example 1: Comparing Efficiencies
Otto, r = 8: 8^0.4 = 2.297 eta = 1 - 1/2.297 = 56.5 %
Otto, r = 10: 10^0.4 = 2.512 eta = 1 - 1/2.512 = 60.2 %
Diesel bracket for rho = 2: (2^1.4 - 1) / (1.4 * 1) = (2.639 - 1)/1.4 = 1.171
Diesel, r = 8, rho = 2: eta = 1 - (1/2.297) * 1.171 = 49.0 %
Diesel, r = 18, rho = 2: 18^0.4 = 3.178
eta = 1 - (1/3.178) * 1.171 = 63.2 %At the same compression ratio of 8, the Otto cycle wins (56.5 versus 49.0 percent). But a petrol engine cannot raise its compression ratio much beyond about 10 to 12 without knocking, because the premixed charge self-ignites. A diesel engine compresses only air, so it can run at 16 to 22 and comfortably exceed the petrol engine's efficiency. This is the thermodynamic reason diesel trucks, tractors and generators use less fuel per kilowatt-hour.
| Cycle | r | rho | Air-standard efficiency |
|---|---|---|---|
| Otto | 8 | - | 56.5 % |
| Otto | 10 | - | 60.2 % |
| Diesel | 8 | 2 | 49.0 % |
| Diesel | 18 | 2 | 63.2 % |
Worked Example 2: Full Otto Cycle Analysis
An Otto cycle has compression ratio 8. At the start of compression air is at 1 bar and 300 K. Heat added is 1500 kJ/kg. Take cv = 0.718 kJ/kg K and R = 0.287 kJ/kg K. Find the temperatures, net work, efficiency and mean effective pressure.
State 2: T2 = 300 * 8^0.4 = 689 K
p2 = 1 * 8^1.4 = 18.4 bar
State 3: T3 = T2 + q_in/cv = 689 + 1500/0.718 = 2778 K
State 4: T4 = T3 / 8^0.4 = 1209 K
q_out = cv (T4 - T1) = 0.718 * (1209 - 300) = 653 kJ/kg
w_net = 1500 - 653 = 847 kJ/kg
eta = 847 / 1500 = 56.5 % (matches the formula)
v1 = R T1 / p1 = 0.287 * 300 / 100 = 0.861 m^3/kg
v2 = v1 / 8 = 0.1076 m^3/kg
mep = 847 / (0.861 - 0.1076) = 1124 kPa = 11.2 barMean effective pressure is the constant pressure that, acting over the whole stroke, would produce the same net work. It lets you compare engines of different sizes: a higher mep means more work from each litre of displacement.
Why Real Engines Are Less Efficient
- Specific heats increase with temperature, and combustion products have a lower gamma than air, which reduces efficiency.
- Heat is lost to the cylinder walls and coolant during combustion and expansion.
- Combustion takes time, so heat addition is neither perfectly constant-volume nor constant-pressure.
- Pumping work is needed to draw in charge and push out exhaust, especially at part throttle in petrol engines.
- Friction in piston rings, bearings and valve train reduces brake output.
The dual cycle (also called mixed or limited pressure cycle) adds heat partly at constant volume and partly at constant pressure. It represents modern high-speed diesel engines better than either ideal cycle alone.
Summary
Otto efficiency is 1 - 1/r^(gamma-1). Diesel efficiency is lower at the same compression ratio because of the cut-off ratio term, but diesel engines can use much higher compression ratios and so end up more efficient. Working through state temperatures and mean effective pressure gives a complete picture of the ideal cycle and a benchmark against which real engines can be judged.