Heat Engine or Refrigerator Cop Tool

Physics Thermodynamics • Heat Engines and Second Law

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

2. Heat Engine/Refrigerator COP Tool

Compute either heat-engine efficiency $\eta=\dfrac{W}{Q_h}$ or refrigerator/heat-pump performance $\mathrm{COP}_R=\dfrac{Q_c}{W}$ , $\mathrm{COP}_{HP}=\dfrac{Q_h}{W}$ . Includes the Carnot (maximum) benchmark from reservoir temperatures.

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

Mode + Inputs

Mode: Energy unit: $Q_h$ (to hot side / from hot side): $Q_c$ (to/from cold side): Work $W$: Auto-balance 1st law: Carnot benchmark (optional): $T_h$ (K): $T_c$ (K): Second-law ratio:

Mode guide: Engine uses $Q_h\rightarrow$ device, outputs $W$ and rejects $Q_c$. Refrigerator uses work input $W$ to move heat \$Q_c\$ from cold to hot (reject \$Q_h\$). Heat pump uses work input to deliver \$Q_h\$ to the hot side.

Animation + plot controls

Progress $\tau\in[0,1]$: Play speed: Arrow pulse: Shade feasible region: Plot variable: Temperature span (K):

Sankey diagram shows direction and relative magnitude of $Q_h$, $Q_c$, $W$. Carnot plot supports drag-to-pan, wheel-to-zoom, and double-click/reset.

Ready

Steps

Enter values and click Solve.

Energy-flow Sankey diagram (animated)

Heat/work flow: engine vs. refrigerator vs. heat pump

Phase: —

Carnot benchmark plot

Carnot maximum vs. temperature (mode-dependent)

Hover: (T, value)=…

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Heat Engine / Refrigerator COP — Theory

This tool compares three closely related devices that exchange heat with a hot reservoir (temperature $$T_h$$ ) and a cold reservoir (temperature $$T_c$$ ). Throughout, we use positive magnitudes for heat and work: $$Q_h>0$$ , $Q_c\ge 0$ , $$W>0$$ .

1) Energy balance (first law)

Heat engine (cycle): absorbs heat from hot side, outputs work, rejects to cold side

\[ Q_h = W + Q_c. \]

So $W = Q_h - Q_c$. For a physical engine, $Q_c \ge 0 \Rightarrow W \le Q_h$.

Refrigerator / heat pump: work input moves heat from cold to hot

\[ Q_h = Q_c + W. \]

So $W = Q_h - Q_c$. A larger temperature lift typically requires larger $W$.

2) Performance definitions

Heat engine efficiency

\[ \eta = \frac{W}{Q_h}. \]

Efficiency is a fraction (often displayed as a percent). Larger is better.

Refrigerator coefficient of performance

\[ \mathrm{COP}_R = \frac{Q_c}{W}. \]

COP can be greater than 1; it measures “cooling delivered per work input.”

Heat pump coefficient of performance

\[ \mathrm{COP}_{HP} = \frac{Q_h}{W}. \]

Measures “heat delivered to the hot space per work input.”

3) Carnot (maximum) benchmarks from the second law

A reversible (Carnot) device sets an upper bound for performance given reservoir temperatures. These are ideal limits; real devices satisfy $\eta \le \eta_{Carnot}$ and $\mathrm{COP} \le \mathrm{COP}_{Carnot}$.

Carnot heat engine

\[ \eta_{Carnot} = 1 - \frac{T_c}{T_h}. \]

Carnot refrigerator

\[ \mathrm{COP}_{R,\,Carnot} = \frac{T_c}{T_h - T_c}. \]

Carnot heat pump

\[ \mathrm{COP}_{HP,\,Carnot} = \frac{T_h}{T_h - T_c}. \]

4) Interpreting the Sankey diagram

Engine: arrow $Q_h$ points from hot reservoir to device, $W$ points out, $Q_c$ points to cold reservoir.
Refrigerator/heat pump: $W$ points into the device; heat is moved from cold to hot.
Arrow thickness scales with magnitude to help you “see” which term dominates.

5) Example (Refrigerator)

Given: $Q_c=1000\,\text{J}$, $W=200\,\text{J}$

\[ \mathrm{COP}_R = \frac{Q_c}{W} = \frac{1000}{200} = 5. \]

Carnot check: if $T_c=273\,\text{K}$, $T_h=303\,\text{K}$

\[ \mathrm{COP}_{R,\,Carnot} = \frac{273}{303-273} \approx 9.1. \]

Website tip: Use the mode selector to compare “pump vs. fridge” and relate these results to heat engines and the second law.