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thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws o ...
, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
or matter. Such an idealized process is useful in engineering as a model of and basis of comparison for real processes. 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" can be interpreted in another way, since its meaning is deducible from its
etymology Etymology () The New Oxford Dictionary of English (1998) – p. 633 "Etymology /ˌɛtɪˈmɒlədʒi/ the study of the class in words and the way their meanings have changed throughout time". is the study of the history of the form of words ...
. It means a process in which the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
of the system remains unchanged; as mentioned, this could occur if the process is both adiabatic and reversible. However, this could also occur in a system where the work done on the system includes friction internal to the system, and heat is withdrawn from the system in just the right amount to compensate for the internal friction, so as to leave the entropy unchanged. However, in relation to the universe, the entropy of the universe would increase as a result, in agreement with the Second Law of Thermodynamics.


Background

The
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
statesMortimer, R. G. ''Physical Chemistry'', 3rd ed., p. 120, Academic Press, 2008.Fermi, E. ''Thermodynamics'', footnote on p. 48, Dover Publications,1956 (still in print). that :T_\textdS \ge \delta Q, where \delta Q is the amount of energy the system gains by heating, T_\textis the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on ...
of the surroundings, and 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.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." 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 thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...
, 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 \Delta s=0 or s_1 = s_2 .Cengel, Yunus A., and Michaeul A. Boles. Thermodynamics: An Engineering Approach. 7th Edition ed. New York: Mcgraw-Hill, 2012. Print. Some examples of theoretically isentropic thermodynamic devices are
pump A pump is a device that moves fluids ( liquids or gases), or sometimes slurries, by mechanical action, typically converted from electrical energy into hydraulic energy. Pumps can be classified into three major groups according to the method the ...
s,
gas compressor A compressor is a mechanical device that increases the pressure of a gas by reducing its volume. An air compressor is a specific type of gas compressor. Compressors are similar to pumps: both increase the pressure on a fluid and both can tra ...
s, turbines,
nozzle A nozzle is a device designed to control the direction or characteristics of a fluid flow (specially to increase velocity) as it exits (or enters) an enclosed chamber or pipe. A nozzle is often a pipe or tube of varying cross sectional area, ...
s, 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. Isentropic efficiency of turbines: : \eta_\text = \frac = \frac \cong \frac. Isentropic efficiency of compressors: : \eta_\text = \frac = \frac \cong \frac. Isentropic efficiency of nozzles: : \eta_\text = \frac = \frac \cong \frac. For all the above equations: : h_1 is the specific enthalpy at the entrance state, : h_ is the specific enthalpy at the exit state for the actual process, : h_ is the specific enthalpy at the exit state for the isentropic process.


Isentropic devices in thermodynamic cycles

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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
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: : dU = \delta W + \delta Q. The reversible work done on a system by changing the volume is :\delta W = -p \,dV, where p is the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
, and V is the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
. The change in enthalpy (H = U + pV) is given by :dH = dU + p \,dV + V \,dp. Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), \delta Q_\text = 0, and so dS = \delta Q_\text/T = 0 All reversible adiabatic processes are isentropic. This leads to two important observations: : dU = \delta W + \delta Q = -p \,dV + 0, :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 :dU = n C_v \,dT, and dH = n C_p \,dT. Using the general results derived above for dU and dH, then : dU = n C_v \,dT = -p \,dV, : dH = n C_p \,dT = V \,dp. So for an ideal gas, the heat capacity ratio can be written as :\gamma = \frac = -\frac. For a calorically perfect gas \gamma is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get : pV^\gamma = \text, that is, : \frac = \left(\frac\right)^\gamma. Using the equation of state for an ideal gas, p V = n R T, : TV^ = \text. (Proof: PV^\gamma = \text \Rightarrow PV\,V^ = \text \Rightarrow nRT\,V^ = \text. But ''nR'' = constant itself, so TV^ = \text.) : \frac = \text also, for constant C_p = C_v + R (per mole), : \frac = \frac and p = \frac : S_2-S_1 = nC_p \ln\left(\frac\right) - nR\ln\left(\frac\right) : \frac = C_p \ln\left(\frac\right) - R\ln\left(\frac\right ) = C_v\ln\left(\frac\right)+ R \ln\left(\frac\right) Thus for isentropic processes with an ideal gas, : T_2 = T_1\left(\frac\right)^ or V_2 = V_1\left(\frac\right)^


Table of isentropic relations for an ideal gas

: Derived from : PV^ = \text, : PV = m R_s T, : P = \rho R_s T, where: : P = pressure, : V = volume, : \gamma = 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.


See also

*
Gas laws The gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases. Boyl ...
* Adiabatic process * Isenthalpic process * Isentropic analysis * Polytropic process


Notes

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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 Thermodynamic processes Thermodynamic entropy