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Engineering › Thermodynamics ›

Expansion and Compression of a Perfect Gas

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This section exmines the Comression and Expansion of a perfect Gas. The equations for AdiabatiC and Isothermal exoansions are derived

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Contents

Expansion And Compression Of Gases
Work Done During A Polytropic Compression.
The Heat Supplied During A Polytropic Expansion.
The Adiabatic Process.
An Isothermal Compression Or Expansion.
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Expansion And Compression Of Gases

Invariably Gases follow Polytropic expansions

$PV^n\;=\;Constant$

$\frac{PV}{T}\;=\;Constant$

Divide equation 1 by 2

$PV^n\times \frac{T}{PV}\;=\;Constant$

$\therefore\;\;\;\;\;TV^{n-1}\;=\;Constant$

From equation 2

$V\;=\;Constant\times \frac{T}{P}$

Substituting in equation 1

$\frac{PT^n}{P^n}\;=\;Constant$

$\therefore\;\;\;\;\;\frac{T}{P^\frac{n-1}{n}}\;=\;Constant$

NOTE n is called the index of Compression.

Work Done During A Polytropic Compression.

Work done = $\int_{V_1}^{V_2}P\.dV$

But $PV^n\;=\;Constant$

$\therefore\;\;\;\;\;P\;=\;\frac{Constant}{V^n}$

$\therefore\;\;\;\;\;Work\;Done\;=\;Constant\;\int_{V_1}^{V_2}V^{-n}\;dV$

$\therefore\;\;\;\;\;Work\;Done\;=\;Constant\;\left[\frac{V^{1-n}}{1-n} \right]_{V_1}^{V_2}$

$=\;\left(\frac{V_2^{1-n}\;-\;V_1^{1-n}}{1\;-\;n} \right)\times Constant$

$=\;\frac{P_2V_2^n\times V_2^{1-n}\;-\;P_1V_1^n\times V_1^{1-n}}{1-n}$

$\therefore\;\;\;\;\;Work\;Done\;for\;an\;expansion\;=\;\frac{P_2V_2\;-\;P_1V_1}{1-n}$

$\therefore\;\;\;\;\;Work\;Done\;for\;an\;compression\;=\;\frac{P_1V_1\;-\;P_2V_2}{1-n}$

$=\;\frac{wR(T_1\;-\;T_2)}{n\;-\;1}$

The Heat Supplied During A Polytropic Expansion.

$q\;=\;\delta\,U\;+\;w$

$=\;\frac{w\;(T_1\;-\;T_2)}{n-1}\;+\;w\;C_V(T_2\;-\;T_1)$

$=\;(T_1\;-\;T_2)\left(\frac{w\;R}{n\;-\;1}\;-\;w\;C_V \right)$

$But\;\;\;\;\;R\;=\;(\gamma \;-\;1)C_V$

$\therefore\;\;\;\;\;q\;=\;R\;w(T_1\;-\;T_2)\left(\frac{1}{n\;-\;1}\;-\;\frac{1}{\gamma \;-\;1} \right)$

$\therefore\;\;\;\;\;q\;=\;R\;w(T_1\;-\;T_2)\frac{(\gamma \;-\;n)}{(n\;-\;1)(\gamma \;-\;1)}$

$Thus\;\;\;\;\;q\;=\;\left(\frac{\gamma \;-\;n}{\gamma \;-\;1} \right)\times Work\;Done$

The Adiabatic Process.

In and Adiabatic Compression or expansion the heat supplied is zero.

$\left(\frac{\gamma \;-\;n}{n \;-\;1} \right)\;=\;0$

Also for a small change $PdV\;=\;-\;\delta\;U\;\;\;\;\;\;\;(The\; First\; Law)$

$\therefore\;\;\;\;\;P\,dV\;=\;-\;w\;C_V\.dt$

$But\;\;\;\;\;\;P\;=\;\frac{w\;R\;T}{V}$

$\frac{w\;R\;T}{V}dV\;=\;-\;wC_V\;dt$

$\therefore\;\;\;\;\;R\frac{dV}{V}\;=\;-\;C_V\;\frac{dt}{T}$

Integrating:-

$R\;Ln\;V\;+\;C_V\;Ln\;T\;=\;Constant$

$\therefore\;\;\;\;\;LnV^R\times T^{C_V}\;=\;Constant$

Taking Anti-Logs;-

$V^R\times T^{C_V}\;=\;Constant$

$T\times V^{\frac{R}{C_V}}\;=\;Constant$

$T\times V^{\gamma \;-\;1}\;=\;Constant$

Thus for an Adiabatic Expansion $\;n\;=\;\gamma$

$Work\;Done\;=\;\frac{wR(T_1\;-\;T_2)}{\gamma \;-\;1}\;=\;wC_V(T_1\;-\;T_2)\;=\;\delta \;U$

An Isothermal Compression Or Expansion.

$P\times V\;=\;Constant$

i.e. The Index of Compression "n" = 1

$Work\;Done\;=\;\int_{V_1}^{V_2}P\;dV\;=\;\int_{V_1}^{V_2}n\frac{RT}{V}\;dV$

Integrating:-

$Work\;Done\;=\;n\;R\;T\;Ln\frac{V_2}{V_1}$

$But\;\;\;\;\;q\;=\;\delta\;U\;+\;w$

$\delta\;U\;=\;0$

$\therefore\;\;\;\;\;Work \;Done\;=\;Heat\;Supplied$

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Last Modified: 2008-03-12 22:20:03 Page Rendered: 2010-09-25 04:56:00