MEC-04

Carnot efficiency

η = 1 − Tc/Th between two thermal reservoirs.

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Heat enginesCarnot 1824

Governing equation

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

$T_h \in \left[300, 2000\right]$

where

T_h: Hot reservoir (K)
T_c: Cold reservoir (K)
\eta: Carnot 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}

$T_h \in \left[300, 2000\right]$

$T_h = 300$ $\eta = 0$

Along the curve

②

Domain of the variable — blue

T_h_{\min} = 300,\quad T_h_{\max} = 2000

$T_h \in \left[300, 2000\right]$

$T_h = 300$ $T_h = 300$

Along the curve

③

Response curve — green

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

$T_h \in \left[300, 2000\right]$

$T_h = 300$ $\eta = 0$

Along the curve

④

Rate of change — amber

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

$T_h \in \left[300, 2000\right]$

$T_h = 310.7$ $\eta^{\prime} = 0.003$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-04 — Carnot efficiency) is the form associated with Carnot 1824. Working symbols: $T_h$ , $T_c$ $\rightarrow$ $\eta$ . Carnot proved that no engine between two temperatures can beat the reversible ratio 1−Tc/Th, the bound of the second law.

Purpose

Purpose: compute

\eta

from

T_h

T_c

in Mechanical via

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

η = 1 − Tc/Th between two thermal reservoirs. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

T_h = 800.000\,\mathrm{K}

T_c = 300.000\,\mathrm{K}

, the governing relation

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

yields

\eta = 0.6250\,\mathrm{—}

. Ideal reversible cycle, Kelvin temperatures. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Hot reservoir T_h800 KCold reservoir T_c300 K

Outputs

Carnot efficiency \eta0.6250 —

Reading speed

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Narration of this film

Ideal reversible cycle, Kelvin temperatures.

Carnot proved that no engine between two temperatures can beat the reversible ratio 1−Tc/Th, the bound of the second law.

Reading speed

Watch on YouTube