Isentropic process

In thermodynamics, an isentropic process or isoentropic process (ισον = "equal" (Greek); εντροπία entropy = "disorder"(Greek)) is one in which for purposes of engineering analysis and calculation, one may assume that the process takes place from initiation to completion without an increase or decrease in the entropy of the system, i.e., the entropy of the system remains constant. It can be proven that any reversible adiabatic process is an isentropic process.

Background

The Second law of thermodynamics states that, :$\backslash delta\; Q\; \backslash le\; TdS$ where $\backslash delta\; Q$ is the amount of energy the system gains by heating, $T$ is the temperature of the system, and $dS$ is the change in entropy. The equal sign will hold for a reversible process. For a reversible isentropic process, there is no transfer of heat energy and therefore the process is also adiabatic. For an irreversible process, the entropy will increase. Hence removal of heat from the system (cooling) is necessary to maintain a constant internal entropy for an irreversible process in order to make it isentropic. Thus an irreversible isentropic process is not adiabatic.

For reversible processes, an isentropic transformation is carried out by thermally "insulating" the system from its surroundings. Temperature is the thermodynamic conjugate variable to entropy, thus the conjugate process would be an isothermal process in which the system is thermally "connected" to a constant-temperature heat bath.

== Isentropic flow ==

An isentropic flow is a flow that is both adiabatic and reversible. That is, no heat is added to the flow, and no energy transformations occur due to friction or dissipative effects. For an isentropic flow of a perfect gas, several relations can be derived to define the pressure, density and temperature along a streamline.

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, with entail work done on or by the flow.

Derivation of the isentropic relations

For a closed system, the total change in energy of a system is the sum of the work done and the heat added, : $dU\; =\; dW\; +\; dQ\backslash ,\backslash !$ The reversible work done on a system by changing the volume is, :$dW\; =\; -pdV\backslash ,\backslash !$ where $p$ is the pressure and $V$ is the volume. The change in enthalpy ($H\; =\; U\; +\; pV\backslash ,\backslash !$) is given by, :$dH\; =\; dU\; +\; pdV\; +\; Vdp\backslash ,\backslash !$

Then for a process which is both reversible and adiabatic (i.e. no heat transfer occurs), $dQ\_\{rev\}\; =\; 0\backslash ,\backslash !$, and so $dS=dQ\_\{rev\}/T\; =\; 0\backslash ,\backslash !$. This leads to two important observations, : $dU\; =\; dW\; +\; dQ\; =\; -pdV\; +\; 0\backslash ,\backslash !$ , and

: $dH\; =\; dW\; +\; dQ\; +\; pdV\; +\; Vdp\; =\; -pdV\; +\; 0\; +\; pdV\; +\; Vdp\; =\; Vdp\backslash ,\backslash !$

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

: $dU\; =\; nC\_vdT\backslash ,\backslash !$, and $dH\; =\; nC\_pdT\backslash ,\backslash !$.

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

: $dU\; =\; nC\_vdT\; =\; -pdV\backslash ,\backslash !$, and : $dH\; =\; nC\_pdT\; =\; Vdp\backslash ,\backslash !$.

So for an ideal gas, the heat capacity ratio can be written as, :$\backslash gamma\; =\; \backslash frac\{C\_p\}\{C\_V\}\; =\; -\backslash frac\{dp/p\}\{dV/V\}\backslash ,\backslash !$

For an ideal gas $\backslash gamma\backslash ,\backslash !$ is constant. Hence on integrating the above equation, assuming a perfect gas, we get : $pV^\{\backslash gamma\}\; =\; \backslash mbox\{constant\}\; \backslash ,$ i.e.

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

Using the equation of state for an ideal gas, $p\; V\; =\; n\; R\; T\backslash ,\backslash !$, : $TV^\{\backslash gamma-1\}\; =\; \backslash mbox\{constant\}\; \backslash ,$

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

also, for constant $C\_p\; =\; C\_v\; +\; R$ (per mole), : $\backslash frac\{V\}\{T\}\; =\; \backslash frac\{nR\}\{p\}$ and $p\; =\; \backslash frac\{nRT\}\{V\}$

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

: $\backslash frac\{S\_2-S\_1\}\{n\}\; =\; C\_p\; \backslash ln\backslash left(\backslash frac\{T\_2\}\{T\_1\}\backslash right)\; -\; R\backslash ln\backslash left\; (\backslash frac\{T\_2V\_1\}\{T\_1V\_2\}\; \backslash right\; )\; =\; C\_v\backslash ln\backslash left(\backslash frac\{T\_2\}\{T\_1\}\backslash right)+R\; \backslash ln\backslash left(\backslash frac\{V\_2\}\{V\_1\}\backslash right)$

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

Table of isentropic relations for an ideal gas

:{| style="bgcolor:white" cellpadding=5 |- | align="center" | $\backslash frac\; \{p\_2\}\; \{p\_1\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{T\_2\}\{T\_1\}\; \backslash right\; )^\backslash frac\; \{\backslash gamma\}\{\backslash gamma-1\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{\backslash rho\_2\}\{\backslash rho\_1\}\; \backslash right\; )^\{\backslash gamma\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{V\_1\}\{V\_2\}\; \backslash right\; )^\{\backslash gamma\}$ |- | align="center" | $\backslash frac\; \{T\_2\}\; \{T\_1\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{p\_2\}\{p\_1\}\; \backslash right\; )^\backslash frac\; \{\backslash gamma-1\}\{\backslash gamma\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{\backslash rho\_2\}\{\backslash rho\_1\}\; \backslash right\; )^\{(\backslash gamma\; -\; 1)\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{V\_1\}\{V\_2\}\; \backslash right\; )^\{(\backslash gamma-1)\}$ |- | align="center" | $\backslash frac\; \{\backslash rho\_2\}\; \{\backslash rho\_1\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{T\_2\}\{T\_1\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma-1\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{p\_2\}\{p\_1\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash frac\{V\_1\}\{V\_2\}$ |- | align="center" | $\backslash frac\; \{V\_2\}\; \{V\_1\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{T\_1\}\{T\_2\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma-1\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash frac\{\backslash rho\_1\}\{\backslash rho\_2\}$ | align="center" | $=\backslash ,\backslash !$ | align="center" | $\backslash left\; (\backslash frac\{p\_1\}\{p\_2\}\; \backslash right\; )^\backslash frac\; \{1\}\{\backslash gamma\}$ |- |}

Derived from:

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

: $pV\; =\; m\; R\_s\; T\; \backslash ,\backslash !$

: $p\; =\; \backslash rho\; R\_s\; T\backslash ,\backslash !\; \backslash ,\backslash !$

::Where: :::$p\backslash ,\backslash !$ = Pressure :::$V\backslash ,\backslash !$ = Volume :::$\backslash gamma\backslash ,\backslash !$ = Ratio of specific heats = $C\_p/C\_v\backslash ,\backslash !$ :::$T\backslash ,\backslash !$ = Temperature :::$m\backslash ,\backslash !$ = Mass :::$R\_s\backslash ,\backslash !$ = Gas constant for the specific gas = $R/M\backslash ,\backslash !$ :::$R\backslash ,\backslash !$ = Universal gas constant :::$M\backslash ,\backslash !$ = Molecular weight of the specific gas :::$\backslash rho\backslash ,\backslash !$ = Density :::$C\_p\backslash ,\backslash !$ = Specific heat at constant pressure :::$C\_v\backslash ,\backslash !$ = Specific heat at constant volume

References

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

Notes

See also

Adiabatic process

Isenthalpic process

Polytropic process

Category:Thermodynamic processes Category:Thermodynamic entropy