THM-08

Isentropic ideal-gas relation

T2/T1 = (P2/P1)^{(γ−1)/γ}.

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Ideal gasLaplace / Poisson

Governing equation

\dfrac{T_2}{T_1}=\left(\dfrac{P_2}{P_1}\right)^{(\gamma-1)/\gamma}

$P_2 \in \left[10, 8000\right]$

where

T_1: Inlet T (K)
P_1: Inlet P (kPa)
P_2: Exit P (kPa)
\gamma: Heat-capacity ratio (—)
T_2: Exit temperature (K)

The figure, part by part

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①

Governing law — red

\dfrac{T_2}{T_1}=\left(\dfrac{P_2}{P_1}\right)^{(\gamma-1)/\gamma}

$P_2 \in \left[10, 8000\right]$

$P_2 = 10$ $T_2 = 155.4$

Along the curve

②

Domain of the variable — blue

P_2_{\min} = 10,\quad P_2_{\max} = 8000

$P_2 \in \left[10, 8000\right]$

$P_2 = 10$ $P_2 = 10$

Along the curve

③

Response curve — green

\dfrac{T_2}{T_1}=\left(\dfrac{P_2}{P_1}\right)^{(\gamma-1)/\gamma}

$P_2 \in \left[10, 8000\right]$

$P_2 = 10$ $T_2 = 155.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T_2}{\mathrm{d}P_2}

$P_2 \in \left[10, 8000\right]$

$P_2 = 60.25$ $T_2^{\prime} = 1.525$

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-08 — Isentropic ideal-gas relation) is the form associated with Laplace / Poisson. Working symbols: $T_1$ , $P_1$ , $P_2$ , $\gamma$ $\rightarrow$ $T_2$ . For a reversible adiabatic of an ideal gas with constant γ, Poisson's relations link T, P and V.

Purpose

Purpose: compute

T_2

from

T_1

P_1

P_2

\gamma

in Thermodynamics via

\dfrac{T_2}{T_1}=\left(\dfrac{P_2}{P_1}\right)^{(\gamma-1)/\gamma}

T2/T1 = (P2/P1)^{(γ−1)/γ}. 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_1 = 300.000\,\mathrm{K}

P_1 = 100.000\,\mathrm{kPa}

P_2 = 400.000\,\mathrm{kPa}

\gamma = 1.400\,\mathrm{—}

, the governing relation

\dfrac{T_2}{T_1}=\left(\dfrac{P_2}{P_1}\right)^{(\gamma-1)/\gamma}

yields

T_2 = 445.798\,\mathrm{K}

. Calorically perfect gas, reversible adiabatic. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Inlet T T_1300 KInlet P P_1100 kPaExit P P_2400 kPaHeat-capacity ratio \gamma1.4 —

Outputs

Exit temperature T_2445.798 K

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Narration of this film

Calorically perfect gas, reversible adiabatic.

For a reversible adiabatic of an ideal gas with constant γ, Poisson's relations link T, P and V.

Reading speed

Watch on YouTube