Isentropic

Isentropic

In thermodynamics, an isentropic process (iso = "equal" (Greek); entropy = "disorder") is one during which the entropy of the system remains constant. ¹² It can be proved that any reversible adiabatic process is an isentropic process.

1 Background
2 Isentropic flow
- 2.1 Derivation of the isentropic relations
- 2.2 Table of isentropic relations for an ideal gas
3 References
- 3.1 Notes
4 See also

Background

Second law of thermodynamics states that,

$\delta Q \le TdS$

where $δ Q$ is the amount of energy the system gains by heating, $T$ is the temperature of the system, and $d S$ 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 so as 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, and so 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 energy is added to the flow, and no energy losses 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.

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,

$dU = dW + dQ\,\!$

The work done on a system by changing the volume is,

$dW = -pdV\,\!$

where $p$ is the pressure and $V$ the volume. The change in enthalpy ( $H = U + pV\,\!$ ) is given by,

$dH = dU + pdV + Vdp = n C_p dT\,\!$

Since a reversible process is adiabatic (i.e. no heat transfer occurs), so $dQ = 0, dS=0\,\!$ . This leads to two important observations,

$dU = -pdV\,\!$ , and

$dH = Vdp\,\!$ or $dQ = dH - Vdp = 0\,\!$

$dQ = TdS\,\!$ => $dS = (1/T) dH - (V/T) dp\,\!$

The heat capacity ratio can be written as,

$\gamma = \frac{C_p}{C_V} = -\frac{dp/p}{dV/V}\,\!$

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

$pV^{\gamma} = \mbox{constant} \,$ i.e.

$\frac{p_2}{p_1} = \left(\frac{V_1}{V_2}\right)^{\gamma}$

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

$TV^{\gamma-1} = \mbox{constant} \,$

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

also, for constant

C p = C v + R

(per mole),

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

$S_2-S_1 = nC_p ln\left(\frac{T_2}{T_1}\right) - nRln\left(\frac{p_2}{p_1}\right)$

$\frac{S_2-S_1}{n} = C_p ln\left(\frac{T_2}{T_1}\right) - Rln\left (\frac{T_2V_1}{T_1V_2} \right ) = C_vln\left(\frac{T_2}{T_1}\right)+R ln\left(\frac{V_2}{V_1}\right)$

Thus for insentropic processes with an ideal gas,

$T_2 = T_1\left(\frac{V_2}{V_1}\right)^{(R/C_v)}$ or $V_2 = V_1\left(\frac{T_2}{T_1}\right)^{(C_v/R)}$

Table of isentropic relations for an ideal gas

$\frac {P_2} {P_1} = \left (\frac{T_2}{T_1} \right )^\frac {k}{k-1} = \left (\frac{\rho_2}{\rho_1} \right )^{k} = \left (\frac{V_1}{V_2} \right )^{k}$

$\frac {T_2} {T_1} = \left (\frac{P_2}{P_1} \right )^\frac {k-1}{k} = \left (\frac{\rho_2}{\rho_1} \right )^{(k - 1)} = \left (\frac{V_1}{V_2} \right )^{(k-1)}$

$\frac {\rho_2} {\rho_1} = \left (\frac{P_2}{P_1} \right )^\frac {1}{k} = \left (\frac{T_2}{T_1} \right )^\frac {1}{k-1} = \frac{V_1}{V_2}$

$\frac {V_2} {V_1} = \left (\frac{P_1}{P_2} \right )^\frac {1}{k} = \left (\frac{T_1}{T_2} \right )^\frac {1}{k-1} = \frac{\rho_1}{\rho_2}$

Derived from:

$PV^{k} = constant \,\!$

$PV = m R_s T \,\!$

$P = \rho R_s T\,\! \,\!$

Where:

$P\,\!$ = Pressure

$V\,\!$ = Volume

${k}\,\!$ = 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

$\rho\,\!$ = Density

$C_p\,\!$ = Specific heat at constant pressure

$C_v\,\!$ = 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

^ Van Wylen, G.J. and Sonntag, R.E., Fundamentals of Classical Thermodynamics, Section 7.4
^ Massey, B.S. (1970), Mechanics of Fluids, Section 12.2 (2^nd edition) Van Nostrand Reinhold Company, London. Library of Congress Catalog Card Number: 67-25005

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