Isentropic process

 ( Thermodynamic Processes ) 

Isentropic efficiency of turbines:

$\backslash eta\_\backslash text\{t\}\; =\; \backslash frac\{\backslash text\{actual\; turbine\; work\}\}\{\backslash text\{isentropic\; turbine\; work\}\}\; =\; \backslash frac\{W\_a\}\{W\_s\}\; \backslash cong\; \backslash frac\{h\_1\; -\; h\_\{2a\}\}\{h\_1\; -\; h\_\{2s\}\}.$

Isentropic efficiency of compressors:

$\backslash eta\_\backslash text\{c\}\; =\; \backslash frac\{\backslash text\{isentropic\; compressor\; work\}\}\{\backslash text\{actual\; compressor\; work\}\}\; =\; \backslash frac\{W\_s\}\{W\_a\}\; \backslash cong\; \backslash frac\{h\_\{2s\}\; -\; h\_1\}\{h\_\{2a\}\; -\; h\_1\}.$

Isentropic efficiency of nozzles:

$\backslash eta\_\backslash text\{n\}\; =\; \backslash frac\{\backslash text\{actual\; KE\; at\; nozzle\; exit\}\}\{\backslash text\{isentropic\; KE\; at\; nozzle\; exit\}\}\; =\; \backslash frac\{V\_\{2a\}^2\}\{V\_\{2s\}^2\}\; \backslash cong\; \backslash frac\{h\_1\; -\; h\_\{2a\}\}\{h\_1\; -\; h\_\{2s\}\}.$

For all the above equations:

$h\_1$ is the specific enthalpy at the entrance state,

$h\_\{2a\}$ is the specific enthalpy at the exit state for the actual process,

$h\_\{2s\}$ is the specific enthalpy at the exit state for the isentropic process.

Note: The isentropic assumptions are only applicable with ideal cycles. Real cycles have inherent losses due to compressor and turbine inefficiencies and the second law of thermodynamics. Real systems are not truly isentropic, but isentropic behavior is an adequate approximation for many calculation purposes.

Note that energy can be exchanged with the flow in an isentropic transformation, as long as it doesn't happen as heat exchange. An example of such an exchange would be an isentropic expansion or compression that entails work done on or by the flow.

For an isentropic flow, entropy density can vary between different streamlines. If the entropy density is the same everywhere, then the flow is said to be Homentropic flow.

$dU\; =\; \backslash delta\; W\; +\; \backslash delta\; Q.$

$\backslash delta\; W\; =\; -p\; \backslash ,dV,$

$dH\; =\; dU\; +\; p\; \backslash ,dV\; +\; V\; \backslash ,dp.$

Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), $\backslash delta\; Q\_\backslash text\{rev\}\; =\; 0$, and so $dS\; =\; \backslash delta\; Q\_\backslash text\{rev\}/T\; =\; 0$ All reversible adiabatic processes are isentropic. This leads to two important observations:

$dU\; =\; \backslash delta\; W\; +\; \backslash delta\; Q\; =\; -p\; \backslash ,dV\; +\; 0,$

$dH\; =\; \backslash delta\; W\; +\; \backslash delta\; Q\; +\; p\; \backslash ,dV\; +\; V\; \backslash ,dp\; =\; -p\; \backslash ,dV\; +\; 0\; +\; p\; \backslash ,dV\; +\; V\; \backslash ,dp\; =\; V\; \backslash ,dp.$

Next, a great deal can be computed for isentropic processes of an ideal gas. For any transformation of an ideal gas, it is always true that

$dU\; =\; n\; C\_v\; \backslash ,dT$, and $dH\; =\; n\; C\_p\; \backslash ,dT.$

Using the general results derived above for $dU$ and $dH$, then

$dU\; =\; n\; C\_v\; \backslash ,dT\; =\; -p\; \backslash ,dV,$

$dH\; =\; n\; C\_p\; \backslash ,dT\; =\; V\; \backslash ,dp.$

So for an ideal gas, the heat capacity ratio can be written as

$\backslash gamma\; =\; \backslash frac\{C\_p\}\{C\_V\}\; =\; -\backslash frac\{dp/p\}\{dV/V\}.$

For a calorically perfect gas $\backslash gamma$ is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get

$pV^\backslash gamma\; =\; \backslash text\{constant\},$

$\backslash frac\{p\_2\}\{p\_1\}\; =\; \backslash left(\backslash frac\{V\_1\}\{V\_2\}\backslash right)^\backslash gamma.$

Using the equation of state for an ideal gas, $p\; V\; =\; n\; R\; T$,

$TV^\{\backslash gamma-1\}\; =\; \backslash text\{constant\}.$

$\backslash frac\{p^\{\backslash gamma-1\}\}\{T^\backslash gamma\}\; =\; \backslash text\{constant\}$

also, for constant $C\_p\; =\; C\_v\; +\; R$ (per mole),

$\backslash frac\{V\}\{T\}\; =\; \backslash frac\{nR\}\{p\}$ and $p\; =\; \backslash frac\{nRT\}\{V\}$

$S\_2-S\_1\; =\; nC\_p\; \backslash ln\backslash left(\backslash frac\{T\_2\}\{T\_1\}\backslash right)\; -\; nR\backslash ln\backslash left(\backslash frac\{p\_2\}\{p\_1\}\backslash right)$

$\backslash frac\{S\_2-S\_1\}\{n\}\; =\; C\_p\; \backslash ln\backslash left(\backslash frac\{T\_2\}\{T\_1\}\backslash right)\; -\; R\backslash ln\backslash left(\backslash frac\{T\_2\; V\_1\}\{T\_1\; V\_2\}\backslash right\; )\; =\; C\_v\backslash ln\backslash left(\backslash frac\{T\_2\}\{T\_1\}\backslash right)+\; R\; \backslash ln\backslash left(\backslash frac\{V\_2\}\{V\_1\}\backslash right)$

Thus for isentropic processes with an ideal gas,

$T\_2\; =\; T\_1\backslash left(\backslash frac\{V\_1\}\{V\_2\}\backslash right)^\{(R/C\_v)\}$ or $V\_2\; =\; V\_1\backslash left(\backslash frac\{T\_1\}\{T\_2\}\backslash right)^\{(C\_v/R)\}$

>

$\backslash frac\{T\_2\}\{T\_1\}$	$=$	$\backslash left\; (\backslash frac\{P\_2\}\{P\_1\}\; \backslash right\; )^\backslash frac\; \{\backslash gamma-1\}\{\backslash gamma\}$	$=$	$\backslash left\; (\backslash frac\{V\_1\}\{V\_2\}\; \backslash right\; )^\{(\backslash gamma-1)\}$	$=$	$\backslash left\; (\backslash frac\{\backslash rho\_2\}\{\backslash rho\_1\}\; \backslash right\; )^\{(\backslash gamma\; -\; 1)\}$
$\backslash left\; (\backslash frac\{T\_2\}\{T\_1\}\; \backslash right\; )^\backslash frac\; \{\backslash gamma\}\{\backslash gamma-1\}$	$=$	$\backslash frac\; \{P\_2\}\; \{P\_1\}$	$=$	$\backslash left\; (\backslash frac\{V\_1\}\{V\_2\}\; \backslash right\; )^\{\backslash gamma\}$	$=$	$\backslash left\; (\backslash frac\{\backslash rho\_2\}\{\backslash rho\_1\}\; \backslash right\; )^\{\backslash gamma\}$
$\backslash left\; (\backslash frac\{T\_1\}\{T\_2\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma-1\}$	$=$	$\backslash left\; (\backslash frac\{P\_1\}\{P\_2\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma\}$	$=$	$\backslash frac\{V\_2\}\{V\_1\}$	$=$	$\backslash frac\{\backslash rho\_1\}\{\backslash rho\_2\}$
$\backslash left\; (\backslash frac\{T\_2\}\{T\_1\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma-1\}$	$=$	$\backslash left\; (\backslash frac\{P\_2\}\{P\_1\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma\}$	$=$	$\backslash frac\{V\_1\}\{V\_2\}$	$=$	$\backslash frac\{\backslash rho\_2\}\{\backslash rho\_1\}$

Derived from

$PV^\{\backslash gamma\}\; =\; \backslash text\{constant\},$

$PV\; =\; m\; R\_s\; T,$

$P\; =\; \backslash rho\; R\_s\; T,$

where:

$P$ = pressure,

$V$ = volume,

$\backslash gamma$ = ratio of specific heats = $C\_p/C\_v$,

$T$ = temperature,

$m$ = mass,

$R\_s$ = gas constant for the specific gas = $R/M$,

$R$ = universal gas constant,

$M$ = molecular weight of the specific gas,

$\backslash rho$ = density,

$C\_p$ = molar specific heat at constant pressure,

$C\_v$ = molar specific heat at constant volume.

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See also

• Gas laws
• Adiabatic process
• Isenthalpic process
• Isentropic analysis
• Polytropic process

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Notes

• Van Wylen, G. J. and Sonntag, R. E. (1965), Fundamentals of Classical Thermodynamics, John Wiley & Sons, Inc., New York. Library of Congress Catalog Card Number: 65-19470

Categories: Thermodynamic Processes, Thermodynamic Entropy