Editing Isentropic process

{{Short description|Thermodynamic process that is reversible and adiabatic}}
{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}

An '''isentropic process''' is an idealized  [[thermodynamic process]] that is both [[Adiabatic process|adiabatic]] and [[Reversible process (thermodynamics)|reversible]].<ref>{{Citation
 | last = Partington
 | first = J. R.
 | author-link = J. R. Partington
 | title = An Advanced Treatise on Physical Chemistry.
 | place = Fundamental Principles. The Properties of Gases, London
 | publisher = [[Longman|Longmans, Green and Co.]]
 | volume = 1
 | year = 1949
 | page = 122}}.
</ref><ref>Kestin, J. (1966). ''A Course in Thermodynamics'', Blaisdell Publishing Company, Waltham MA, p. 196.</ref><ref>Münster, A. (1970). ''Classical Thermodynamics'', translated by E. S. Halberstadt, Wiley–Interscience, London, {{ISBN|0-471-62430-6}}, p. 13.</ref><ref>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.</ref><ref>Borgnakke, C., Sonntag., R.E. (2009). ''Fundamentals of Thermodynamics'', seventh edition, Wiley, {{ISBN|978-0-470-04192-5}}, p. 310.</ref><ref>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.</ref>{{Excessive citations inline|date=February 2024}} The [[work (physics)|work]] transfers of the system are [[friction|frictionless]], and there is no net transfer of [[heat]] or [[matter]].   Such an idealized process is useful in engineering as a model of and basis of comparison for real processes.<ref>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.</ref> 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). [[Thermodynamics|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.

== Background ==

The [[second law of thermodynamics]] states<ref name="MortimerBook">Mortimer, R. G. ''Physical Chemistry'', 3rd ed., p. 120, Academic Press, 2008.</ref><ref name="FermiBook">Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print).</ref> that
:<math>T_\text{surr}dS \ge \delta Q,</math>

where <math>\delta Q</math> is the amount of energy the system gains by heating, <math>T_\text{surr}</math> is the [[temperature]] of the surroundings, and <math>dS</math> is the change in entropy. The equal sign refers to a [[Reversible process (thermodynamics)|reversible process]], which is an imagined idealized theoretical limit, never actually occurring in physical reality, with essentially equal temperatures of system and surroundings.<ref>[[Edward A. Guggenheim|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..."</ref><ref>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."</ref> For an isentropic process, if also reversible, there is no transfer of energy as heat because the process is [[adiabatic process|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 variables (thermodynamics)|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 ==
[[File:Isentropic.jpg|thumb|T–s (entropy vs. temperature) diagram of an isentropic process, which is a vertical line segment]]
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 <math> \Delta s=0 </math> or <math> s_1 = s_2 </math>.<ref name="Cengel Boles 2012">Cengel, Yunus A., and Michaeul A. Boles. Thermodynamics: An Engineering Approach. 7th Edition ed. New York: Mcgraw-Hill, 2012. Print.</ref> Some examples of theoretically isentropic thermodynamic devices are [[pump]]s, [[gas compressor]]s, [[turbine]]s, [[nozzle]]s, and [[Diffuser (thermodynamics)|diffuser]]s.

===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.<ref name="Cengel Boles 2012"/>

Isentropic efficiency of turbines:
: <math> \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}}. </math>

Isentropic efficiency of compressors:
: <math> \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}. </math>

Isentropic efficiency of nozzles:
: <math> \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}}. </math>

For all the above equations:
: <math> h_1 </math> is the specific [[enthalpy]] at the entrance state,
: <math> h_{2a}</math> is the specific enthalpy at the exit state for the actual process,
: <math> h_{2s}</math> is the specific enthalpy at the exit state for the isentropic process.

=== Isentropic devices in thermodynamic cycles===

{| class="wikitable"
|-
! 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 [[gas compressor|compressor]]
|-
| Ideal [[Brayton cycle]] || 3→4 || Isentropic expansion in a [[turbine]]
|-
| Ideal [[vapor-compression refrigeration]] cycle || 1→2 || Isentropic compression in a [[gas compressor|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 ==<!-- This section is linked from [[Shock wave]] -->

In [[fluid dynamics]], an '''isentropic flow''' is a [[Fluid dynamics|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 [[dissipation|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 flow|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:
: <math>dU = \delta W + \delta Q.</math>
The reversible work done on a system by changing the volume is
:<math>\delta W = -p \,dV,</math>
where <math>p</math> is the [[pressure]], and <math>V</math> is the [[Volume (thermodynamics)|volume]]. The change in [[enthalpy]] (<math>H = U + pV</math>) is given by
:<math>dH = dU + p \,dV + V \,dp.</math>

Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), <math> \delta Q_\text{rev} = 0</math>, and so <math>dS = \delta Q_\text{rev}/T = 0 </math> All reversible adiabatic processes are isentropic.  This leads to two important observations:
: <math>dU = \delta W + \delta Q = -p \,dV + 0,</math>
:<math>dH = \delta W + \delta Q + p \,dV + V \,dp = -p \,dV + 0 + p \,dV + V \,dp = V \,dp.</math>

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
:<math>dU = n C_v \,dT</math>, and <math>dH = n C_p \,dT.</math>

Using the general results derived above for <math>dU</math> and <math>dH</math>, then
: <math>dU = n C_v \,dT = -p \,dV,</math>
: <math>dH = n C_p \,dT = V \,dp.</math>

So for an ideal gas, the [[heat capacity ratio]] can be written as
:<math>\gamma = \frac{C_p}{C_V} = -\frac{dp/p}{dV/V}.</math>

For a calorically perfect gas <math>\gamma</math> is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get
: <math> pV^\gamma = \text{constant},</math>
that is,
: <math>\frac{p_2}{p_1} = \left(\frac{V_1}{V_2}\right)^\gamma.</math>

Using the [[Equation of state#Classical ideal gas law|equation of state]] for an ideal gas, <math>p V = n R T</math>,
: <math> TV^{\gamma-1} = \text{constant}.</math>     
(Proof: <math>PV^\gamma = \text{constant} \Rightarrow PV\,V^{\gamma-1} = \text{constant} \Rightarrow nRT\,V^{\gamma-1} = \text{constant}.</math> But ''nR'' = constant itself, so <math>TV^{\gamma-1} = \text{constant}</math>.)

: <math> \frac{p^{\gamma-1}}{T^\gamma} = \text{constant} </math>

also, for constant <math>C_p = C_v + R</math> (per mole),
: <math> \frac{V}{T} = \frac{nR}{p}</math> and <math>p = \frac{nRT}{V}</math>

: <math> S_2-S_1 = nC_p \ln\left(\frac{T_2}{T_1}\right) - nR\ln\left(\frac{p_2}{p_1}\right)</math>

: <math> \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)</math>

Thus for isentropic processes with an ideal gas,
: <math> T_2 = T_1\left(\frac{V_1}{V_2}\right)^{(R/C_v)}</math> or <math> V_2 = V_1\left(\frac{T_1}{T_2}\right)^{(C_v/R)}</math>

=== Table of isentropic relations for an ideal gas ===

:{| style="bgcolor:white" cellpadding=5
|-
| align="center" | <math> \frac{T_2}{T_1} </math>
| align="center" | <math>=</math>
| align="center" | <math> \left (\frac{P_2}{P_1} \right )^\frac {\gamma-1}{\gamma}</math>
| align="center" | <math>=</math>
| align="center" | <math>\left (\frac{V_1}{V_2} \right )^{(\gamma-1)}</math>
| align="center" | <math>=</math>
| align="center" | <math> \left (\frac{\rho_2}{\rho_1} \right )^{(\gamma - 1)}</math>
|-
| align="center" | <math> \left (\frac{T_2}{T_1} \right )^\frac {\gamma}{\gamma-1}</math>
| align="center" | <math>=</math>
| align="center" | <math> \frac {P_2} {P_1} </math>
| align="center" | <math>=</math>
| align="center" | <math>\left (\frac{V_1}{V_2} \right )^{\gamma}</math>
| align="center" | <math>=</math>
| align="center" | <math> \left (\frac{\rho_2}{\rho_1} \right )^{\gamma}</math>
|-
| align="center" | <math> \left (\frac{T_1}{T_2} \right )^\frac {1}{\gamma-1}</math>
| align="center" | <math>=</math>
| align="center" | <math> \left (\frac{P_1}{P_2} \right )^\frac {1}{\gamma}</math>
| align="center" | <math>=</math>
| align="center" | <math> \frac{V_2}{V_1} </math>
| align="center" | <math>=</math>
| align="center" | <math>\frac{\rho_1}{\rho_2}</math>
|-
| align="center" | <math> \left (\frac{T_2}{T_1} \right )^\frac {1}{\gamma-1}</math>
| align="center" | <math>=</math>
| align="center" | <math> \left (\frac{P_2}{P_1} \right )^\frac {1}{\gamma}</math>
| align="center" | <math>=</math>
| align="center" | <math>\frac{V_1}{V_2}</math>
| align="center" | <math>=</math>
| align="center" | <math> \frac{\rho_2}{\rho_1} </math>
|-
|}

Derived from
: <math>PV^{\gamma} = \text{constant},</math>
: <math>PV = m R_s T,</math>
: <math>P = \rho R_s T,</math>
where:
: <math>P</math> = pressure,
: <math>V</math> = volume,
: <math>\gamma</math> = ratio of specific heats = <math>C_p/C_v</math>,
: <math>T</math> = temperature,
: <math>m</math> = mass,
: <math>R_s</math> = gas constant for the specific gas = <math>R/M</math>,
: <math>R</math> = universal gas constant,
: <math>M</math> = molecular weight of the specific gas,
: <math>\rho</math> = density,
: <math>C_p</math> = molar specific heat at constant pressure,
: <math>C_v</math> = molar specific heat at constant volume.

== See also ==
* [[Gas laws]]
* [[Adiabatic process]]
* [[Isenthalpic process]]
* [[Isentropic analysis]]
* [[Polytropic process]]

==Notes==
{{reflist}}

==References==
* 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

[[Category:Thermodynamic processes]]
[[Category:Thermodynamic entropy]]