Otto (gasoline) and Diesel Cycle Analyzer

Physics Thermodynamics • Heat Engines and Second Law

Written by STEM Calculators Team Published January 4, 2026 Updated February 24, 2026

3. Otto/Gasoline and Diesel Cycle Analyzer

Compute air-standard thermal efficiency for Otto (gasoline) and Diesel cycles using ideal-gas, reversible (isentropic) compression/expansion. Includes a PV loop plot with stages, pan/zoom, and an animated cycle tracer.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Cycle + Parameters

Cycle: \( \gamma \) (Cp/Cv): Compression ratio \( r = V_1/V_2 \): Cutoff ratio \( \alpha \) (Diesel/dual): Heat-add pressure ratio \( \beta \) (Otto/dual, for PV shape): Scale energies to units: \(n\) (mol): Baseline \(T_1\) (K): Energy unit: “Real engine” factor (optional):

Otto efficiency depends mainly on \(r\) and \(\gamma\). Diesel additionally depends on cutoff ratio \(\alpha\). \(\beta\) is used to shape the PV loop (and for dual-cycle efficiency).

Animation + plot controls

Progress \(\tau\in[0,1]\): Play speed: Shade PV loop area: Show stage labels: Mouse:

Wheel = zoom, drag = pan, double-click = reset.

Ready

Solution steps

Enter values and click Solve.

PV loop (air-standard, normalized)

PV diagram with stages + animated tracer

Hover: (V, P)=…

Simple piston schematic (animated)

Compression/expansion visualization (relative)

Phase: —

Note: “air-standard” efficiencies are upper bounds. Real engines suffer heat losses, finite-rate combustion, pumping losses, friction, etc. The optional real factor multiplies the air-standard \(\eta\) as a quick comparison.

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Theory — Otto (Gasoline) & Diesel Cycles

This tool uses the air-standard (ideal-gas) model with reversible isentropic compression/expansion to estimate an upper-bound thermal efficiency.

Key definitions

Compression ratio: \( r = \dfrac{V_1}{V_2} \) (with \(V_2\) the minimum volume).
Heat capacity ratio: \( \gamma = \dfrac{C_p}{C_v} \) (air ≈ 1.4 at room temperature).
Diesel cutoff ratio: \( \alpha = \dfrac{V_3}{V_2} \) during constant-pressure heating.
Otto pressure ratio (for PV shape): \( \beta = \dfrac{P_3}{P_2} \) during constant-volume heating.

In the PV plot, values are shown in normalized form \(V/V_1\) and \(P/P_1\) to emphasize the cycle shape.

Isentropic relations (ideal gas)

\[ P V^{\gamma} = \text{const}, \qquad T V^{\gamma-1} = \text{const}. \]

These apply to the compression and expansion strokes in the air-standard Otto/Diesel models.

Otto (Gasoline) cycle: efficiency

Otto cycle heat addition and rejection occur at constant volume. Using \( \eta = 1 - \dfrac{Q_{out}}{Q_{in}} \) and the isentropic temperature ratios yields:

\[ \eta_{Otto} = 1 - \frac{1}{r^{\gamma-1}}. \]

Important: \(\beta\) changes the PV “height” (combustion pressure rise) but the air-standard Otto efficiency depends only on \(r\) and \(\gamma\).

Diesel cycle: efficiency

Diesel cycle heat addition occurs at constant pressure, characterized by cutoff ratio \(\alpha\). The standard air-standard Diesel efficiency is:

\[ \eta_{Diesel} = 1 - \frac{1}{r^{\gamma-1}} \cdot \frac{\alpha^{\gamma}-1}{\gamma(\alpha-1)}. \]

As \(\alpha \to 1\) (no cutoff, heating becomes effectively constant-volume), the Diesel expression approaches the Otto form.

Dual cycle (university add-on)

Dual cycle combines constant-volume heating (\(\beta\)) followed by constant-pressure heating (\(\alpha\)). A common air-standard result is:

\[ \eta_{Dual} = 1 - \frac{\beta\alpha^{\gamma}-1}{r^{\gamma-1}\big((\beta-1)+\gamma\beta(\alpha-1)\big)}. \]

This is included mainly for university-level comparison and to see how adding a constant-pressure “tail” affects the cycle.

Work and heat per cycle

For a closed cycle:

\[ W_{net} = Q_{in} - Q_{out}, \qquad \eta = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}. \]

The calculator can show dimensionless \(q\) values, or scale them to Joules using \(nRT_1\) (where \(T_1\) is a baseline temperature).

Real-engine note

Real cycles are not perfectly reversible (entropy is produced).
Combustion is finite-rate; heat transfer occurs during compression/expansion.
Pumping losses, friction, and incomplete expansion reduce net work.

That’s why real gasoline engines may be ~25–35% efficient and diesels ~35–45% in typical operating conditions, even though air-standard efficiencies can be much higher.

Solution steps

PV loop (air-standard, normalized)

Simple piston schematic (animated)

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