Contents

Isentropic process

Thermodynamic process that is reversible and adiabatic

Thermodynamics
The classical Carnot heat engine
Branches Classical Statistical Chemical Quantum thermodynamics Equilibrium / Non-equilibrium
Laws Zeroth First Second Third
Systems Closed system Open system Isolated system State Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments Processes Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility Cycles Heat engines Heat pumps Thermal efficiency
System properties Note: Conjugate variablesinitalics Property diagrams Intensive and extensive properties Process functions Work Heat Functions of state Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties
Material properties Property databases Specific heat capacity  ${\displaystyle c=}$ ${\displaystyle T}$ ${\displaystyle \partial S}$ ${\displaystyle N}$ ${\displaystyle \partial T}$ Compressibility  ${\displaystyle \beta =-}$ ${\displaystyle 1}$ ${\displaystyle \partial V}$ ${\displaystyle V}$ ${\displaystyle \partial p}$ Thermal expansion  ${\displaystyle \alpha =}$ ${\displaystyle 1}$ ${\displaystyle \partial V}$ ${\displaystyle V}$ ${\displaystyle \partial T}$
Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Onsager reciprocal relations Bridgman's equations Table of thermodynamic equations
Potentials Free energy Free entropy Internal energy ${\displaystyle U(S,V)}$ Enthalpy ${\displaystyle H(S,p)=U+pV}$ Helmholtz free energy ${\displaystyle A(T,V)=U-TS}$ Gibbs free energy ${\displaystyle G(T,p)=H-TS}$
History Culture History General Entropy Gas laws "Perpetual motion" machines Philosophy Entropy and time Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications An Experimental Enquiry Concerning ... Heat On the Equilibrium of Heterogeneous Substances Reflections on the Motive Power of Fire Timelines Thermodynamics Heat engines Art Education Maxwell's thermodynamic surface Entropy as energy dispersal
Scientists Bernoulli Boltzmann Bridgman Carathéodory Carnot Clapeyron Clausius de Donder Duhem Gibbs von Helmholtz Joule Kelvin Lewis Massieu Maxwell von Mayer Nernst Onsager Planck Rankine Smeaton Stahl Tait Thompson van der Waals Waterston
Other Nucleation Self-assembly Self-organization Order and disorder
Category
v t e

Anisentropic process is an idealized thermodynamic process that is both adiabatic and reversible.^{[1][2][3][4][5][6][excessive citations]} The work transfers of the system are frictionless, and there is no net transfer of heatormatter. Such an idealized process is useful in engineering as a model of and basis of comparison for real processes.^[7] This process is idealized because reversible processes do not occur in reality; thinking of a process as both adiabatic and reversible would show that the initial and final entropies are the same, thus, the reason it is called isentropic (entropy does not change). Thermodynamic processes are named based on the effect they would have on the system (ex. isovolumetric: constant volume, isenthalpic: constant enthalpy). Even though in reality it is not necessarily possible to carry out an isentropic process, some may be approximated as such.

The word "isentropic" derives from the process being one in which the entropy of the system remains unchanged. In addition to a process which is both adiabatic and reversible, this can also occur in a system where the work done on the system includes friction internal to the system, and heat is withdrawn from the system sufficient to compensate for it so as to leave the entropy unchanged.^[8] However, in relation to the Universe, its entropy would increase as a result, in agreement with the Second Law of Thermodynamics.^{[citation needed]}

Background[edit]

The second law of thermodynamics states^[9][10] that

${\displaystyle T_{\text{surr}}dS\geq \delta Q,}$

where ${\displaystyle \delta Q}$ is the amount of energy the system gains by heating, ${\displaystyle T_{\text{surr}}}$is the temperature of the surroundings, and ${\displaystyle dS}$ 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.^[11][12] 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[edit]

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 ${\displaystyle \Delta s=0}$or${\displaystyle s_{1}=s_{2}}$.^[13] Some examples of theoretically isentropic thermodynamic devices are pumps, gas compressors, turbines, nozzles, and diffusers.

Isentropic efficiencies of steady-flow devices in thermodynamic systems[edit]

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:

${\displaystyle \eta _{\text{t}}={\frac {\text{actual turbine work}}{\text{isentropic turbine work}}}={\frac {W_{a}}{W_{s}}}\cong {\frac {h_{1}-h_{2a}}{h_{1}-h_{2s}}}.}$

Isentropic efficiency of compressors:

${\displaystyle \eta _{\text{c}}={\frac {\text{isentropic compressor work}}{\text{actual compressor work}}}={\frac {W_{s}}{W_{a}}}\cong {\frac {h_{2s}-h_{1}}{h_{2a}-h_{1}}}.}$

Isentropic efficiency of nozzles:

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

For all the above equations:

${\displaystyle h_{1}}$ is the specific enthalpy at the entrance state,

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

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

Isentropic devices in thermodynamic cycles[edit]

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[edit]

Influid 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 frictionordissipative 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[edit]

For a closed system, the total change in energy of a system is the sum of the work done and the heat added:

${\displaystyle dU=\delta W+\delta Q.}$

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

${\displaystyle \delta W=-p\,dV,}$

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

${\displaystyle dH=dU+p\,dV+V\,dp.}$

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

${\displaystyle dU=\delta W+\delta Q=-p\,dV+0,}$

${\displaystyle dH=\delta W+\delta Q+p\,dV+V\,dp=-p\,dV+0+p\,dV+V\,dp=V\,dp.}$

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

${\displaystyle dU=nC_{v}\,dT}$, and ${\displaystyle dH=nC_{p}\,dT.}$

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

${\displaystyle dU=nC_{v}\,dT=-p\,dV,}$

${\displaystyle dH=nC_{p}\,dT=V\,dp.}$

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

${\displaystyle \gamma ={\frac {C_{p}}{C_{V}}}=-{\frac {dp/p}{dV/V}}.}$

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

${\displaystyle pV^{\gamma }={\text{constant}},}$

that is,

${\displaystyle {\frac {p_{2}}{p_{1}}}=\left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }.}$

Using the equation of state for an ideal gas, ${\displaystyle pV=nRT}$,

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

(Proof: ${\displaystyle PV^{\gamma }={\text{constant}}\Rightarrow PV\,V^{\gamma -1}={\text{constant}}\Rightarrow nRT\,V^{\gamma -1}={\text{constant}}.}$ But nR = constant itself, so ${\displaystyle TV^{\gamma -1}={\text{constant}}}$.)

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

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

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

${\displaystyle S_{2}-S_{1}=nC_{p}\ln \left({\frac {T_{2}}{T_{1}}}\right)-nR\ln \left({\frac {p_{2}}{p_{1}}}\right)}$

${\displaystyle {\frac {S_{2}-S_{1}}{n}}=C_{p}\ln \left({\frac {T_{2}}{T_{1}}}\right)-R\ln \left({\frac {T_{2}V_{1}}{T_{1}V_{2}}}\right)=C_{v}\ln \left({\frac {T_{2}}{T_{1}}}\right)+R\ln \left({\frac {V_{2}}{V_{1}}}\right)}$

Thus for isentropic processes with an ideal gas,

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

Table of isentropic relations for an ideal gas[edit]

${\displaystyle {\frac {T_{2}}{T_{1}}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{\frac {\gamma -1}{\gamma }}}$	${\displaystyle =}$	${\displaystyle \left({\frac {V_{1}}{V_{2}}}\right)^{(\gamma -1)}}$	${\displaystyle =}$	${\displaystyle \left({\frac {\rho _{2}}{\rho _{1}}}\right)^{(\gamma -1)}}$
${\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)^{\frac {\gamma }{\gamma -1}}}$	${\displaystyle =}$	${\displaystyle {\frac {P_{2}}{P_{1}}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {V_{1}}{V_{2}}}\right)^{\gamma }}$	${\displaystyle =}$	${\displaystyle \left({\frac {\rho _{2}}{\rho _{1}}}\right)^{\gamma }}$
${\displaystyle \left({\frac {T_{1}}{T_{2}}}\right)^{\frac {1}{\gamma -1}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {P_{1}}{P_{2}}}\right)^{\frac {1}{\gamma }}}$	${\displaystyle =}$	${\displaystyle {\frac {V_{2}}{V_{1}}}}$	${\displaystyle =}$	${\displaystyle {\frac {\rho _{1}}{\rho _{2}}}}$
${\displaystyle \left({\frac {T_{2}}{T_{1}}}\right)^{\frac {1}{\gamma -1}}}$	${\displaystyle =}$	${\displaystyle \left({\frac {P_{2}}{P_{1}}}\right)^{\frac {1}{\gamma }}}$	${\displaystyle =}$	${\displaystyle {\frac {V_{1}}{V_{2}}}}$	${\displaystyle =}$	${\displaystyle {\frac {\rho _{2}}{\rho _{1}}}}$

Derived from

${\displaystyle PV^{\gamma }={\text{constant}},}$

${\displaystyle PV=mR_{s}T,}$

${\displaystyle P=\rho R_{s}T,}$

where:

${\displaystyle P}$ = pressure,

${\displaystyle V}$ = volume,

${\displaystyle \gamma }$ = ratio of specific heats = ${\displaystyle C_{p}/C_{v}}$,

${\displaystyle T}$ = temperature,

${\displaystyle m}$ = mass,

${\displaystyle R_{s}}$ = gas constant for the specific gas = ${\displaystyle R/M}$,

${\displaystyle R}$ = universal gas constant,

${\displaystyle M}$ = molecular weight of the specific gas,

${\displaystyle \rho }$ = density,

${\displaystyle C_{p}}$ = specific heat at constant pressure,

${\displaystyle C_{v}}$ = specific heat at constant volume.

See also[edit]

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

Notes[edit]

References[edit]

• 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