Visualise the dual (Sabathé) cycle, the ideal model of a modern diesel engine. Adjust the compression ratio, pressure ratio, cutoff ratio and specific-heat ratio to see the thermal efficiency, state temperatures and pressures, and the constant-volume vs constant-pressure heat split update in real time — with an animated P-V diagram and an efficiency comparison against the Otto cycle.
Parameters
Compression ratio r
Ratio of maximum volume V₁ to minimum volume V₂ (V₁/V₂)
Pressure ratio (const. volume) α
Pressure ratio P₃/P₂ during constant-volume heating. 1 reduces to Diesel
Cutoff ratio (const. pressure) ρ
Volume ratio V₄/V₃ during constant-pressure heating. 1 reduces to Otto
Specific-heat ratio γ
c_p/c_v of the working air. About 1.40 for air at room temperature
Intake temperature T₁
K
Intake pressure P₁
kPa
About 100 kPa for natural aspiration; higher when turbocharged
Results
—
Thermal efficiency η (%)
—
Peak temperature T₄ (K)
—
Peak pressure P₃ (kPa)
—
Const.-volume heat (%)
—
Const.-pressure heat (%)
—
Cycle position
—
P-V diagram — cycle animation
1→2 isentropic compression, 2→3 constant-volume heat addition, 3→4 constant-pressure heat addition, 4→5 isentropic expansion, 5→1 constant-volume heat rejection. The enclosed area is the net work, and the marker traverses the five state points.
Thermal efficiency of the dual (Sabathé) cycle. r: compression ratio, α: pressure ratio (P₃/P₂ of constant-volume heating), ρ: cutoff ratio (V₄/V₃ of constant-pressure heating), γ: specific-heat ratio.
$$T_3=T_2\,\alpha,\qquad T_4=T_3\,\rho$$
Absolute temperatures at the state points. α is the constant-volume pressure ratio and ρ the constant-pressure cutoff ratio; ρ=1 gives the Otto cycle and α=1 the Diesel cycle.
Heat input q_in is the sum of the constant-volume term c_v(T₃−T₂) and the constant-pressure term c_p(T₄−T₃). c_v, c_p are the specific heats and R is the gas constant of air.
What is the Dual (Sabathé) Cycle?
🙋
I've learned the Otto and Diesel engine cycles. Is the "dual cycle" or "Sabathé cycle" yet another, separate one?
🎓
Less a separate one, more a "best of both". The Otto cycle idealises all heat addition as constant-volume, the Diesel cycle as constant-pressure. But if you look at the combustion in a real diesel engine closely, it is neither one nor the other. The dual cycle (Sabathé cycle) splits heat addition into two stages — part at constant volume, the rest at constant pressure. So it sits right between Otto and Diesel.
🙋
Two-stage heat addition…? What happens inside a real engine to make that occur?
🎓
In a diesel, fuel is injected just as the piston reaches top dead centre. The fuel does not burn instantly — there is a small "ignition delay". During that delay, fuel accumulates, and at the moment of ignition it all burns at once, in a flash. The piston is barely moving then, so volume is essentially constant — constant-volume heat addition. That is the pressure-ratio α part. The remaining fuel keeps burning slowly even after the piston starts to descend. That is roughly constant-pressure heat addition — the cutoff-ratio ρ part. Hence a dual cycle.
🙋
I see! Moving α and ρ on the left changes the shape of the P-V diagram. Raising α grows a vertical spike near the top.
🎓
Right — that vertical line is the constant-volume heat addition (2→3). Volume stays fixed while pressure jumps up, so on the P-V diagram it is a vertical straight line. The bigger α is, the longer that line and the higher the peak pressure P₃. The horizontal line that follows is the constant-pressure heat addition (3→4), where pressure stays fixed while volume grows; the bigger ρ, the longer it gets. The dual cycle's "vertical line plus horizontal line" P-V shape is exactly this two-stage heat addition.
🙋
So what happens if I set ρ to 1? If the cutoff ratio is 1, the constant-pressure horizontal line goes to zero?
🎓
Good observation. With ρ=1 the constant-pressure portion vanishes and all heat addition is at constant volume — that is the Otto cycle itself. Conversely, set α=1 and the constant-volume vertical line vanishes; all heat addition is at constant pressure, giving the Diesel cycle. The dual-cycle efficiency formula is built so that in these two limits it matches the Otto and Diesel formulas exactly. So the dual cycle is "a more general cycle with Otto and Diesel as its two endpoints".
🙋
Watching the "constant-volume heat" reading, efficiency rises as I increase α. What's the reasoning behind that?
🎓
For the same total heat input, adding it at constant volume is more efficient. Constant-volume heating pushes the peak pressure and peak temperature much higher, so the expansion stroke afterwards extracts more work. That is why a larger α and smaller ρ raise efficiency and move you toward Otto. The catch is that the peak pressure P₃ also shoots up, so in a real engine the strength of the bearings and cylinder head becomes the limit. Practical engine design picks α and ρ from a tug-of-war between efficiency and peak pressure.
Frequently Asked Questions
The dual-cycle efficiency is η = 1 − (1/r^(γ−1))·{(α·ρ^γ − 1)/((α−1) + γ·α·(ρ−1))}, where r is the compression ratio, α is the pressure ratio (P₃/P₂ during constant-volume heat addition), ρ is the cutoff ratio (V₄/V₃ during constant-pressure heat addition) and γ is the specific-heat ratio. When ρ = 1 the formula reduces to the Otto-cycle efficiency 1 − 1/r^(γ−1), and when α = 1 it reduces to the Diesel-cycle efficiency. The dual cycle sits between these two limits.
The pressure ratio α = P₃/P₂ is the pressure rise during constant-volume heat addition (the rapid burn just after top dead centre); it tells you what fraction of the heat is added at constant volume. The cutoff ratio ρ = V₄/V₃ is the volume ratio of the constant-pressure heat addition (the burn that continues while the piston descends); it corresponds to how long injection lasts. In a real diesel engine, part of the injected fuel burns almost at constant volume near top dead centre, and the rest keeps burning at roughly constant pressure as the piston starts to descend. α and ρ parameterise this two-stage combustion.
The Otto cycle idealises all heat addition as constant-volume, and the Diesel cycle idealises all of it as constant-pressure. The dual (Sabathé) cycle lies between them: part of the heat is added at constant volume and the rest at constant pressure. In a modern high-speed diesel engine, the fuel neither burns entirely at constant pressure nor entirely at constant volume — part burns rapidly (nearly at constant volume) near top dead centre, then the rest burns (nearly at constant pressure) as the piston descends. The dual cycle is the ideal model closest to this real combustion, reducing to the Otto cycle as ρ→1 and to the Diesel cycle as α→1.
For the same total heat input, a larger fraction of constant-volume heat addition (larger α, smaller ρ) gives higher thermal efficiency, while a larger fraction of constant-pressure heat addition (smaller α, larger ρ) lowers it. This is because constant-volume heating raises the peak pressure and peak temperature more, so the subsequent expansion extracts more work. However, adding more heat at constant volume sharply raises the peak pressure and runs into the engine's mechanical strength limits (bearings, cylinder head). In practice α and ρ are chosen to balance efficiency against peak pressure.
Real-World Applications
Modern high-speed diesel engines: The high-speed diesel engines in passenger cars, trucks and construction machinery are better described by the dual cycle than by the pure Diesel cycle (all heat added at constant pressure). At high speed there is less time for combustion, so fuel that accumulates during the ignition delay near top dead centre burns all at once, nearly at constant volume. Raise α in this tool and you can see this "premixed combustion" pressure spike as the vertical line on the P-V diagram.
Managing peak pressure in engine design: The dual-cycle parameter α directly governs the peak pressure P₃. Raising the intake pressure P₁ with a turbocharger and increasing α both raise efficiency, but P₃ then exceeds the engine's mechanical strength limits (crank bearings, connecting rods, cylinder-head gasket). Designers use a tool like this to vary α and P₁ and find the point that maximises efficiency within the allowable peak-pressure envelope.
Marine and stationary large diesels: Large low-speed two-stroke engines are also analysed with the dual cycle, because their combustion has both constant-volume and constant-pressure character. These are the internal-combustion engines that chase efficiency hardest, with some net thermal efficiencies above 50%. Set a high compression ratio r and a high α in this tool and you will see that efficiency can be optimised more flexibly than with Otto or Diesel alone.
Thermodynamics education and cycle comparison: The dual cycle is taught in university thermal engineering as the "most general internal-combustion cycle", with Otto and Diesel as special cases. Because you can verify numerically that it reduces to Otto as ρ→1 and to Diesel as α→1, it is an excellent teaching tool for understanding the three cycles in a unified way. Overlaying the efficiency curve against the Otto cycle in this tool makes a relationship that is hard to grasp from formulas alone intuitively clear.
Common Misconceptions and Pitfalls
The most common misconception is that "the dual cycle is just a simple average of the Otto and Diesel cycles". The dual cycle is not an average of the two; it is a more general cycle that contains both as special cases. Set ρ=1 and it matches the Otto efficiency formula exactly; set α=1 and it matches the Diesel efficiency formula exactly. The efficiency value varies continuously with the parameters α and ρ, and depending on how you allocate α and ρ it can lean toward Otto or toward Diesel. In this tool, bring ρ close to 1 or α close to 1 and confirm that the efficiency matches the corresponding limiting value.
Next, assuming that "the air-standard cycle efficiency is the real-engine efficiency". The η computed here is an ideal value that takes the working fluid to be an ideal gas (air), models combustion as external heating, and assumes constant specific heats. In a real engine, finite combustion time, heat loss to the walls, throttling losses on intake and exhaust (pumping loss) and friction loss all push the net thermal efficiency well below the ideal value. Use this tool's numbers as an "upper-bound guideline" and a "teaching aid for the effect of parameters", not as the actual fuel economy of a real engine.
Finally, the belief that "the larger α, the better". It is true that increasing the pressure ratio α raises the constant-volume heat fraction and lifts thermal efficiency. But at the same time the peak pressure P₃ rises sharply, and the mechanical load on the bearings, connecting rods and cylinder head exceeds its limits. Overly violent constant-volume combustion also increases combustion noise (diesel knock) and NOx emissions. Real engine design picks α and ρ by weighing peak pressure, noise and emissions, not just efficiency. While raising α in this tool, watch how far the peak pressure P₃ jumps as well.
How to Use
Set compression ratio (rCompNum): typical range 12–24 for diesel engines; higher ratios increase efficiency but reduce peak temperature margin.
Adjust cutoff ratio (rCutoffNum): defines the fraction of stroke where constant-pressure combustion occurs; 1.0 = Otto cycle, 2.0–3.0 = typical diesel dual-cycle operation.
Enter initial pressure (rPressNum) in kPa and gamma (gammaValNum, typically 1.4 for air); observe P₃, T₄, and thermal efficiency η updating on the P–V diagram and state-point table.
Worked Example
Modern heavy-duty diesel: compression ratio = 16, cutoff ratio = 2.2, initial pressure = 100 kPa, gamma = 1.4. Result: peak pressure P₃ ≈ 9500 kPa, peak temperature T₄ ≈ 1680 K, thermal efficiency η ≈ 54.8%. Constant-volume combustion contributes ~65% of total heat input; constant-pressure phase ~35%. Cycle footprint on P–V plot shows characteristic "dogleg" shape between Otto and Diesel limits.
Practical Notes
Increasing compression ratio above 18 in automotive diesels yields marginal efficiency gains (~0.5% per ratio point) but escalates injection pressure requirements and NOx emissions; balance with EGR strategy.
Cutoff ratio 1.8–2.4 optimizes modern common-rail injection: lower ratios approach Otto efficiency; higher ratios reduce peak pressure and improve load flexibility but lower peak temperature.
Verify gamma = 1.3–1.35 for high-temperature diesel exhaust; using 1.4 underestimates T₄ slightly, affecting thermal margin calculations for turbocharger inlet conditions.