Engineering
Otto & Diesel Cycle Thermal Efficiency formulas (Compression & Cutoff Ratio)
Calculate air-standard spark-ignition Otto efficiency 1 - 1 / r^(gamma - 1) and compression-ignition Diesel efficiency from compression ratio r, cutoff rc, and gamma.
Compare idealized constant-specific-heat cycles
These are cold-air-standard cycle models: ideal gas, constant specific-heat ratio gamma, reversible adiabatic compression/expansion, and ideal heat addition/rejection. Gamma 1.4 is an illustrative assumption, not a constant over all real engine temperatures. Results are not measured engine efficiency or fuel-economy forecasts.
The Otto expression uses constant-volume heat addition. The Diesel expression uses constant-pressure heat addition with cutoff ratio rc. Require compression r>1, gamma>1 and 1<rc<r for the nondegenerate Diesel cycle. At exactly rc=1, the formula below reports the mathematical limiting efficiency; heat input vanishes and this is not an operating engine.
The example uses different compression ratios for the two models. At the same compression ratio and gamma, the ideal Diesel efficiency for rc>1 is lower than the ideal Otto efficiency; this alone does not rank real engines.
| Engine Cycle Parameter | Cell | Gasoline Otto vs Diesel | Air-Standard Efficiency |
|---|---|---|---|
| Otto Compression Ratio (r_Otto) | B2 = 10 | gamma B5 = 1.4 -> 1 - 1/10^0.4 | 60.19% (0.6019) |
| Diesel Compression Ratio (r_Diesel) | B3 = 18 | Illustrative higher compression ratio | Base 1 - 1/18^0.4 = 68.53% |
| Diesel Cutoff Ratio (rc) | B4 = 2.0 | (2^1.4 - 1) / (1.4 * 1) = 1.1707 | Diesel Efficiency = 63.16% (0.6316) |
| Mathematical limit rc →1 | rc approaches 1 | Bracket approaches 1.0000 | Diesel converges to Otto at r = 18! |
Air-standard Otto Cycle thermal efficiency formula
Depends solely on compression ratio B2 and specific-heat ratio gamma B5.
=IF(OR(COUNT(B2)<>1,COUNT(B5)<>1,B2<=1,B5<=1),"Invalid inputs",ROUND(1 - 1 / (B2 ^ (B5 - 1)), 4))Air-standard Diesel Cycle thermal efficiency formula
Use the limit when rc is exactly 1. Very close to 1, use the first-order expansion 1+(gamma-1)(rc-1)/2 to avoid subtractive cancellation. The physical operating range still requires rc>1.
=IF(OR(COUNT(B3:B5)<>3,B3<=1,B4<1,B4>=B3,B5<=1),"Invalid inputs",LET(offset,B4-1,cutTerm,IF(ABS(offset)<0.000001,1+(B5-1)*offset/2,((B4^B5)-1)/(B5*offset)),ROUND(1-(1/(B3^(B5-1)))*cutTerm,4)))Try the example
Use the original values or change them to test your understanding.