Second Law Efficiency Calculator

Physics Thermodynamics • Advanced Applications and Extensions in Thermodynamics

Written by STEM Calculators Team Published January 8, 2026 Updated February 24, 2026

9. Second Law Efficiency Calculator

Compute second-law efficiency (exergy efficiency) $\eta_{II}=\dfrac{W_{\text{actual}}}{W_{\text{rev}}}$ for work-producing or work-consuming devices. Also supports the common heat-engine preview $\eta_{II}=\eta/\eta_{\text{Carnot}}$ . Includes a simple gauge visualization and a short “why <100%” explainer.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication. Enter efficiencies as % or fractions (e.g., 30% or 0.30).

Mode + Inputs

Mode: Device type: Actual work $W_{\text{actual}}$: Reversible work $W_{\text{rev}}$: Actual thermal efficiency $\eta$: Carnot efficiency $\eta_{\text{Carnot}}$: Explain why <100%: Efficiency gauge: Gauge fill $\tau\in[0,1]$ (animate): Play speed:

Second-law efficiency compares real performance to the reversible (maximum exergy) benchmark.

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Steps

Enter values and click Solve.

Second-law efficiency gauge

Gauge preview • $\eta_{II}$ (0–100%)

Gauge: $\eta_{II}$=…

The gauge fill uses $\tau$ as an animation parameter (0→target). Double-click the gauge to reset.

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Theory: Second Law Efficiency

Second-law efficiency (also called exergy efficiency) compares a real device to the best-possible reversible benchmark under the same inlet/outlet constraints. It measures how well a device converts available energy (exergy) into useful work, accounting for irreversibilities.

1) Core definition

A common work-based definition is: \[ \eta_{II}=\frac{W_{\text{actual}}}{W_{\text{rev}}}. \] Here, $W_{\text{rev}}$ is the reversible (maximum) work output for turbines/engines, or the reversible (minimum) work input for compressors/pumps, depending on your sign convention and definition.

This calculator uses the ratio exactly as entered. If you use the “work-consuming” device option, keep your signs consistent or use magnitudes intentionally.

2) Heat-engine shortcut

For a heat engine, a quick second-law perspective is: \[ \eta_{II}=\frac{\eta}{\eta_{\text{Carnot}}}. \] If the engine has $\eta=0.30$ and $\eta_{\text{Carnot}}=0.50$, then $\eta_{II}=0.60$.

3) Why is it usually < 100%?

Irreversibilities generate entropy: friction, turbulence, finite $\Delta T$ heat transfer, mixing, pressure drops.
Entropy generation destroys exergy, reducing achievable useful work (or increasing required work input).
Therefore $\eta_{II}$ typically lies between 0 and 1 for properly defined benchmarks.

4) Advanced note (university)

More general definitions use exergy rates (useful exergy output divided by exergy input) and can be visualized with a Grassmann diagram (exergy flow diagram) showing useful output, exergy destruction, and losses.

Website tip: The gauge is only a visualization of $\eta_{II}$; the real engineering work is selecting the correct reversible benchmark and consistent sign conventions.

Steps

Second-law efficiency gauge

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