THM-02

Carnot thermal efficiency

η = 1 − Tc/Th, the second-law bound.

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CyclesCarnot 1824

Governing equation

\eta=1-\dfrac{T_c}{T_h}

$T_h \in \left[280, 2000\right]$

where

T_h: Hot temperature (K)
T_c: Cold temperature (K)
\eta: Efficiency (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\eta=1-\dfrac{T_c}{T_h}

$T_h \in \left[280, 2000\right]$

$T_h = 280$ $\eta = -0.071$

Along the curve

②

Domain of the variable — blue

T_h_{\min} = 280,\quad T_h_{\max} = 2000

$T_h \in \left[280, 2000\right]$

$T_h = 280$ $T_h = 280$

Along the curve

③

Response curve — green

\eta=1-\dfrac{T_c}{T_h}

$T_h \in \left[280, 2000\right]$

$T_h = 280$ $\eta = -0.071$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\eta}{\mathrm{d}T_h}

$T_h \in \left[280, 2000\right]$

$T_h = 290.8$ $\eta^{\prime} = 0.004$

Along the curve

Lecture brief

Historical brief

Carnot (1824), Clausius entropy, the first law and later van der Waals and Gibbs potentials turned heat into a state science. The sheets compute work, efficiency and vapour pressure. This sheet (THM-02 — Carnot thermal efficiency) is the form associated with Carnot 1824. Working symbols: $T_h$ , $T_c$ $\rightarrow$ $\eta$ . Any reversible engine between two reservoirs has the unique efficiency 1−Tc/Th; irreversible engines do worse.

Purpose

Purpose: compute

\eta

from

T_h

T_c

in Thermodynamics via

\eta=1-\dfrac{T_c}{T_h}

η = 1 − Tc/Th, the second-law bound. Use it when a real thermodynamics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

T_h = 600.000\,\mathrm{K}

T_c = 300.000\,\mathrm{K}

, the governing relation

\eta=1-\dfrac{T_c}{T_h}

yields

\eta = 0.5000\,\mathrm{—}

. Kelvin temperatures, two reservoirs only. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Hot temperature T_h600 KCold temperature T_c300 K

Outputs

Efficiency \eta0.5000 —

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Narration of this film

Kelvin temperatures, two reservoirs only.

Any reversible engine between two reservoirs has the unique efficiency 1−Tc/Th; irreversible engines do worse.

Reading speed

Watch on YouTube