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Background

The second law of thermodynamics states⁸⁹ that

T surr d S ≥ δ Q , {\displaystyle T_{\text{surr}}dS\geq \delta Q,}

where δ Q {\displaystyle \delta Q} is the amount of energy the system gains by heating, T surr {\displaystyle T_{\text{surr}}} is the temperature of the surroundings, and d S {\displaystyle dS} is the change in entropy. The equal sign refers to a reversible process, which is an imagined idealized theoretical limit, never actually occurring in physical reality, with essentially equal temperatures of system and surroundings.¹⁰¹¹ For an isentropic process, if also reversible, there is no transfer of energy as heat because the process is adiabatic; δQ = 0. In contrast, if the process is irreversible, entropy is produced within the system; consequently, in order to maintain constant entropy within the system, energy must be simultaneously removed from the system as heat.

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 processes in thermodynamic systems

The entropy of a given mass does not change during a process that is internally reversible and adiabatic. A process during which the entropy remains constant is called an isentropic process, written Δ s = 0 {\displaystyle \Delta s=0} or s 1 = s 2 {\displaystyle s_{1}=s_{2}} .¹² Some examples of theoretically isentropic thermodynamic devices are pumps, gas compressors, turbines, nozzles, and diffusers.

Isentropic efficiencies of steady-flow devices in thermodynamic systems

Most steady-flow devices operate under adiabatic conditions, and the ideal process for these devices is the isentropic process. The parameter that describes how efficiently a device approximates a corresponding isentropic device is called isentropic or adiabatic efficiency.¹³

Isentropic efficiency of turbines:

η t = actual turbine work isentropic turbine work = W a W s ≅ h 1 − h 2 a h 1 − h 2 s . {\displaystyle \eta _{\text{t}}={\frac {\text{actual turbine work}}{\text{isentropic turbine work}}}={\frac {W_{a}}{W_{s}}}\cong {\frac {h_{1}-h_{2a}}{h_{1}-h_{2s}}}.}

Isentropic efficiency of compressors:

η c = isentropic compressor work actual compressor work = W s W a ≅ h 2 s − h 1 h 2 a − h 1 . {\displaystyle \eta _{\text{c}}={\frac {\text{isentropic compressor work}}{\text{actual compressor work}}}={\frac {W_{s}}{W_{a}}}\cong {\frac {h_{2s}-h_{1}}{h_{2a}-h_{1}}}.}

Isentropic efficiency of nozzles:

η n = actual KE at nozzle exit isentropic KE at nozzle exit = V 2 a 2 V 2 s 2 ≅ h 1 − h 2 a h 1 − h 2 s . {\displaystyle \eta _{\text{n}}={\frac {\text{actual KE at nozzle exit}}{\text{isentropic KE at nozzle exit}}}={\frac {V_{2a}^{2}}{V_{2s}^{2}}}\cong {\frac {h_{1}-h_{2a}}{h_{1}-h_{2s}}}.}

For all the above equations:

h 1 {\displaystyle h_{1}} is the specific enthalpy at the entrance state, h 2 a {\displaystyle h_{2a}} is the specific enthalpy at the exit state for the actual process, h 2 s {\displaystyle h_{2s}} is the specific enthalpy at the exit state for the isentropic process.

Isentropic devices in thermodynamic cycles

Cycle	Isentropic step	Description
Ideal Rankine cycle	1→2	Isentropic compression in a pump
Ideal Rankine cycle	3→4	Isentropic expansion in a turbine
Ideal Carnot cycle	2→3	Isentropic expansion
Ideal Carnot cycle	4→1	Isentropic compression
Ideal Otto cycle	1→2	Isentropic compression
Ideal Otto cycle	3→4	Isentropic expansion
Ideal Diesel cycle	1→2	Isentropic compression
Ideal Diesel cycle	3→4	Isentropic expansion
Ideal Brayton cycle	1→2	Isentropic compression in a compressor
Ideal Brayton cycle	3→4	Isentropic expansion in a turbine
Ideal vapor-compression refrigeration cycle	1→2	Isentropic compression in a compressor
Ideal Lenoir cycle	2→3	Isentropic expansion
Ideal Seiliger cycle	1→2	Isentropic compression
Ideal Seiliger cycle	4→5	Isentropic compression

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.

Isentropic flow

In fluid dynamics, an isentropic flow is a fluid 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 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.

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:

d U = δ W + δ Q . {\displaystyle dU=\delta W+\delta Q.}

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

δ W = − p d V , {\displaystyle \delta W=-p\,dV,}

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

d H = d U + p d V + V d p . {\displaystyle dH=dU+p\,dV+V\,dp.}

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

d U = δ W + δ Q = − p d V + 0 , {\displaystyle dU=\delta W+\delta Q=-p\,dV+0,} d H = δ W + δ Q + p d V + V d p = − p d V + 0 + p d V + V d p = V d p . {\displaystyle dH=\delta W+\delta Q+p\,dV+V\,dp=-p\,dV+0+p\,dV+V\,dp=V\,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

d U = n C v d T {\displaystyle dU=nC_{v}\,dT} , and d H = n C p d T . {\displaystyle dH=nC_{p}\,dT.}

Using the general results derived above for d U {\displaystyle dU} and d H {\displaystyle dH} , then

d U = n C v d T = − p d V , {\displaystyle dU=nC_{v}\,dT=-p\,dV,} d H = n C p d T = V d p . {\displaystyle dH=nC_{p}\,dT=V\,dp.}

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

γ = C p C V = − d p / p d V / V . {\displaystyle \gamma ={\frac {C_{p}}{C_{V}}}=-{\frac {dp/p}{dV/V}}.}

For a calorically perfect gas γ {\displaystyle \gamma } is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get

p V γ = constant , {\displaystyle pV^{\gamma }={\text{constant}},}

that is,

p 2 p 1 = ( V 1 V 2 ) γ . {\displaystyle {\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 {\displaystyle pV=nRT} ,

T V γ − 1 = constant . {\displaystyle TV^{\gamma -1}={\text{constant}}.}

(Proof: P V γ = constant ⇒ P V V γ − 1 = constant ⇒ n R T V γ − 1 = constant . {\displaystyle PV^{\gamma }={\text{constant}}\Rightarrow PV\,V^{\gamma -1}={\text{constant}}\Rightarrow nRT\,V^{\gamma -1}={\text{constant}}.} But nR = constant itself, so T V γ − 1 = constant {\displaystyle TV^{\gamma -1}={\text{constant}}} .)

p γ − 1 T γ = constant {\displaystyle {\frac {p^{\gamma -1}}{T^{\gamma }}}={\text{constant}}}

also, for constant C p = C v + R {\displaystyle C_{p}=C_{v}+R} (per mole),

V T = n R p {\displaystyle {\frac {V}{T}}={\frac {nR}{p}}} and p = n R T V {\displaystyle p={\frac {nRT}{V}}} S 2 − S 1 = n C p ln ⁡ ( T 2 T 1 ) − n R ln ⁡ ( p 2 p 1 ) {\displaystyle S_{2}-S_{1}=nC_{p}\ln \left({\frac {T_{2}}{T_{1}}}\right)-nR\ln \left({\frac {p_{2}}{p_{1}}}\right)} S 2 − S 1 n = C p ln ⁡ ( T 2 T 1 ) − R ln ⁡ ( T 2 V 1 T 1 V 2 ) = C v ln ⁡ ( T 2 T 1 ) + R ln ⁡ ( V 2 V 1 ) {\displaystyle {\frac {S_{2}-S_{1}}{n}}=C_{p}\ln \left({\frac {T_{2}}{T_{1}}}\right)-R\ln \left({\frac {T_{2}V_{1}}{T_{1}V_{2}}}\right)=C_{v}\ln \left({\frac {T_{2}}{T_{1}}}\right)+R\ln \left({\frac {V_{2}}{V_{1}}}\right)}

Thus for isentropic processes with an ideal gas,

T 2 = T 1 ( V 1 V 2 ) ( R / C v ) {\displaystyle T_{2}=T_{1}\left({\frac {V_{1}}{V_{2}}}\right)^{(R/C_{v})}} or V 2 = V 1 ( T 1 T 2 ) ( C v / R ) {\displaystyle V_{2}=V_{1}\left({\frac {T_{1}}{T_{2}}}\right)^{(C_{v}/R)}}

Table of isentropic relations for an ideal gas

T 2 T 1 {\displaystyle {\frac {T_{2}}{T_{1}}}}	= {\displaystyle =}	( P 2 P 1 ) γ − 1 γ {\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}}	= {\displaystyle =}	( V 1 V 2 ) ( γ − 1 ) {\displaystyle \left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}}	= {\displaystyle =}	( ρ 2 ρ 1 ) ( γ − 1 ) {\displaystyle \left({\frac {\rho _{2}}{\rho _{1}}}\right)^{(\gamma -1)}}
( T 2 T 1 ) γ γ − 1 {\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)^{\frac {\gamma }{\gamma -1}}}	= {\displaystyle =}	P 2 P 1 {\displaystyle {\frac {P_{2}}{P_{1}}}}	= {\displaystyle =}	( V 1 V 2 ) γ {\displaystyle \left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }}	= {\displaystyle =}	( ρ 2 ρ 1 ) γ {\displaystyle \left({\frac {\rho _{2}}{\rho _{1}}}\right)^{\gamma }}
( T 1 T 2 ) 1 γ − 1 {\displaystyle \left({\frac {T_{1}}{T_{2}}}\right)^{\frac {1}{\gamma -1}}}	= {\displaystyle =}	( P 1 P 2 ) 1 γ {\displaystyle \left({\frac {P_{1}}{P_{2}}}\right)^{\frac {1}{\gamma }}}	= {\displaystyle =}	V 2 V 1 {\displaystyle {\frac {V_{2}}{V_{1}}}}	= {\displaystyle =}	ρ 1 ρ 2 {\displaystyle {\frac {\rho _{1}}{\rho _{2}}}}
( T 2 T 1 ) 1 γ − 1 {\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)^{\frac {1}{\gamma -1}}}	= {\displaystyle =}	( P 2 P 1 ) 1 γ {\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{\frac {1}{\gamma }}}	= {\displaystyle =}	V 1 V 2 {\displaystyle {\frac {V_{1}}{V_{2}}}}	= {\displaystyle =}	ρ 2 ρ 1 {\displaystyle {\frac {\rho _{2}}{\rho _{1}}}}

Derived from

P V γ = constant , {\displaystyle PV^{\gamma }={\text{constant}},} P V = m R s T , {\displaystyle PV=mR_{s}T,} P = ρ R s T , {\displaystyle P=\rho R_{s}T,}

where:

P {\displaystyle P} = pressure, V {\displaystyle V} = volume, γ {\displaystyle \gamma } = ratio of specific heats = C p / C v {\displaystyle C_{p}/C_{v}} , T {\displaystyle T} = temperature, m {\displaystyle m} = mass, R s {\displaystyle R_{s}} = gas constant for the specific gas = R / M {\displaystyle R/M} , R {\displaystyle R} = universal gas constant, M {\displaystyle M} = molecular weight of the specific gas, ρ {\displaystyle \rho } = density, C p {\displaystyle C_{p}} = molar specific heat at constant pressure, C v {\displaystyle C_{v}} = molar specific heat at constant volume.

See also

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

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

References

1. Partington, J. R. (1949), An Advanced Treatise on Physical Chemistry., vol. 1, Fundamental Principles. The Properties of Gases, London: Longmans, Green and Co., p. 122. /wiki/J._R._Partington ↩

2. Kestin, J. (1966). A Course in Thermodynamics, Blaisdell Publishing Company, Waltham MA, p. 196. ↩

3. Münster, A. (1970). Classical Thermodynamics, translated by E. S. Halberstadt, Wiley–Interscience, London, ISBN 0-471-62430-6, p. 13. /wiki/ISBN_(identifier) ↩

4. Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081, p. 71. ↩

5. Borgnakke, C., Sonntag., R.E. (2009). Fundamentals of Thermodynamics, seventh edition, Wiley, ISBN 978-0-470-04192-5, p. 310. /wiki/ISBN_(identifier) ↩

6. Massey, B. S. (1970), Mechanics of Fluids, Section 12.2 (2nd edition) Van Nostrand Reinhold Company, London. Library of Congress Catalog Card Number: 67-25005, p. 19. ↩

7. Çengel, Y. A., Boles, M. A. (2015). Thermodynamics: An Engineering Approach, 8th edition, McGraw-Hill, New York, ISBN 978-0-07-339817-4, p. 340. /wiki/ISBN_(identifier) ↩

8. Mortimer, R. G. Physical Chemistry, 3rd ed., p. 120, Academic Press, 2008. ↩

9. Fermi, E. Thermodynamics, footnote on p. 48, Dover Publications,1956 (still in print). ↩

10. Guggenheim, E. A. (1985). Thermodynamics. An Advanced Treatment for Chemists and Physicists, seventh edition, North Holland, Amsterdam, ISBN 0444869514, p. 12: "As a limiting case between natural and unnatural processes[,] we have reversible processes, which consist of the passage in either direction through a continuous series of equilibrium states. Reversible processes do not actually occur..." /wiki/Edward_A._Guggenheim ↩

11. Kestin, J. (1966). A Course in Thermodynamics, Blaisdell Publishing Company, Waltham MA, p. 127: "However, by a stretch of imagination, it was accepted that a process, compression or expansion, as desired, could be performed 'infinitely slowly'[,] or as is sometimes said, quasistatically." P. 130: "It is clear that all natural processes are irreversible and that reversible processes constitute convenient idealizations only." ↩

12. Cengel, Yunus A., and Michaeul A. Boles. Thermodynamics: An Engineering Approach. 7th Edition ed. New York: Mcgraw-Hill, 2012. Print. ↩

13. Cengel, Yunus A., and Michaeul A. Boles. Thermodynamics: An Engineering Approach. 7th Edition ed. New York: Mcgraw-Hill, 2012. Print. ↩