On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites

This preprint concerns polytropic processes, a fundamental process type in engineering thermodynamics. An etymology is presented for the term, and the ties to its usefulness are identiﬁed. The seemingly new support concept of ‘logical’ thermodynamic process, as well as the seemingly new working concept of ‘exact’ polytropic process, and a statement for ‘local’ polytropic process, are herein provided. The proposition of employing local polytropic processes as computational discrete elements for generic engineering thermodynamics process modeling is made. Finally, theoretical requisites for a process to be an exact polytropic one, including the deduction of the most general equation of state of the underlying substance, subject to u : u ( T ) , are discussed beyond a reference.


. Introduction
Many equilibrium engineering thermodynamics processes are taken to follow a polytropic relationship, in which P is the system pressure, usually in kPa, v is the system speci c volume, usually in m 3 /kg, and n is the dimensionless polytropic exponent [ ]-which, for a ' -' process, with initial and nal end states labeled as ' ' and ' ', can also be written in terms of end state properties, as P 1 v n 1 = P 2 v n 2 , which also sets the particular value of the c constant.
Some mainstream thermodynamics textbooks introduce polytropic processes in the context of closed system boundary work, as a P : P(v) relationship to plug into the boundary work integral, which contains a P dv integrand [ , , ].In such texts, the polytropic relationship is frequently said to nd support in measurements, while no speci c theoretical derivation is presented at the point of introduction.
On the other hand, other texts [ , , ] include derivations that lead to a polytropic process, or at least to an isentropic version of it, in which the exponent n has a determined value.
Moreover, Bejan [ , p. ] indicates that a constant Pv n relation only holds locally if the process is such that n is a function of either P, v, or both.
A paper due to Christians [ ] discusses the topic from a perspective of teaching polytropic processes themselves, placing emphasis on the heat-to-work transfer ratio-named by that author as "energy transfer ratio"-and how its constancy not only yields, but constitutes a prerequisite for a process to be polytropic, besides, naturally, the constancy of the caloric properties of the working pure substance.
Starting with an etymological presentation of polytropic processes, this work proposes and develops the concepts of exact polytropic and local polytropic processes, as well as the supporting concept of logical thermodynamic processes, and presents theory-derived requisites for a process to be exactly polytropic beyond reference [ ].Moreover, local polytropic processes are proposed as discrete building blocks for general engineering thermodynamics process modeling.

. Etymology of the 'Polytropic' Term
e 'polytropic' term has its etymology (origin) in the Greek language.is author's sources on Greek are mostly based on modern romance languages, such as French [ , ] and his native Portuguese [ ]; therefore, the etymology herein brought forth will include intermediate French terms.
In fact, from a mathematical standpoint, Equation ( ), there are really in nitely many allowable (possible) values e lexical form of Greek adjectives is the nominative, singular, masculine.e nominative, singular, neuter of 'πολύς' is 'πολύ'.
for the real polytropic exponent n and for the c constant; thence, uncountably many processes departing from uncountably many initial states.It is this kind of exibility that is encoded in the etymology of the process name.

. Exact and Local Polytropic Processes
. Logical Processes In equilibrium engineering thermodynamics, a processmore properly a quasi-static or quasi-equilibrium one-is de ned in terms of changes from a certain equilibrium state of a system to another [ ], with process path being the (in nite) sequence of (quasi-)equilibrium states visited by the system during the process.A process can be referred to by its path, with implicit or explicit end states.
It is worth noting that no constraints are stated for the end states of a process in its de nition.is allows for the needed exibility in describing the variety of transformations systems and control volumes can undergo in engineering thermodynamics.
is lack of end state constraints in the de nition of a process allows them to be splitted into multiple, successive 'sub-processes' that still t the de nition of a process, as well as to merge multiple successive ones together into 'super-processes' that also t the de nition of a process.is ability is extremely useful in grouping and splitting systems and control volumes along with their underlying processes-a common practice in engineering thermodynamics.
However, in order to make the intended distinction between proposed 'exact' and 'local' polytropic processes, additional constraints need to be made to process end states.us, the following de nes a logical thermodynamic process, which is a thermodynamic process with constrained end states.In context, i.e., in thermodynamics texts, what's de ned next can be simply called a 'logical process':

De nition (logical process).
A logical process is one in which its stated defining conditions, which determines all of the allowed interactions or property relations for the underlying system or control volume, uniformly and continually apply to the entirety of its path from-but not earlier than-its initial state until-but not later than-its end state.erefore, for a simple compressible system-one admitting only work and heat interactions-either stated heat and work interactions, or system property speci cations, or combinations of the two, de ne possible logical processes.
Moreover, the stated de ning conditions of a logical process can also carry a 'logical' quali er, as to make it explicit they're being used in the de nition of a logical process, as in the 'logical conditions,' or 'de ning logical conditions' expressions.Furthermore, the conditions themselves can carry the 'logical' quali er, for the same purpose.

. Exact Polytropic Processes
One is now in a position to de ne exact polytropic processes: De nition (exact polytropic process).An exact polytropic process is a logical process in which either (i) a polytropic relation Pv n = const., with a unique polytropic exponent n, or (ii) an isochoric logical condition, can be its sole logical defining condition, provided that no state in its path is visited more than once or serves simultaneously as initial and final end states in a given execution.
It is worth noting that isochoric processes are equivalent to polytropic processes with n → ±∞ between stated end states.De nition accounts for the isochoric process non-unique polytropic exponent by explicitly including it as a valid exact polytropic process.

Lemma . Any logical process defined by a single polytropic relation with a unique polytropic exponent, with non-identical end states can only be an exact polytropic process in the absence of reversals in its path.
Proof.Let a logical process P 1,2 be de ned by a single polytropic relation with a unique polytropic exponent n, with non-identical end states '1' and '2' that contains a set of reversal states 'r i ', with {i ∈ Z|i 1} in its path.
Without loss of generality, let further P r 1 < P 1 , if n = 0, i.e., an initially pressure-decreasing process-since the following argument can be otherwise ipped, or laid out in either direction in v, for non-zero n: Let state 'ε', de ned as (P ε , v ε ), belong to the path of a sub-process P 1,r 1 of the original logical process P 1,2 , so that , it follows from the de nition of P 1,2 : Since at 'r 1 ', the process experiments a reversal, thus proceeding with increasing pressures.Let state 'ζ ', de ned as (P ζ , v ζ ), belong to the path of a sub-process P r 1 ,2 of the original logical process P 1,2 , so that so that process P r 1 ,2 is guaranteed to contain state 'ζ ', since state 'r 1 ' is a reversal one, rather than a stopping one, therefore: 'ζ ' ∈ P r 1 ,2 ∈ P 1,2 , implying: From Eqs. ( ) and ( ), one has states 'ε' and 'ζ ' being identical: 'ε' ≡ 'ζ ', with 'ε' ∈ P 1,2 and 'ζ ' ∈ P 1,2 .erefore, by the exact polytropic process de nition, De nition , logical process P 1,2 cannot be an exact polytropic process, since there is at least one state-state 'ε'that is visited more than once in a given execution.
Remarks.Given that the polytropic exponent is unique, any reversal would cause the underlying system or control volume to re-visit states already covered, thus violating the exact polytropic process de nition.

eorem (Cycle). No cycle is an exact polytropic process.
Proof.
e de ning feature of a cycle of same end states directly violates the restriction of no same initial and end states for an exact polytropic process.
Remarks.Even if a cycle can be collapsed down as to be described in terms of a single polytropic process, such as an ideal Diesel cycle with fuel cut ratio r c ≡ v 3 /v 2 = 1, the fact that such a cycle must invariably make at least one reversal, as to cycle back to previous states, Lemma would constitute a second reason to disqualify the cycle as a (single) exact polytropic process.
For an exact polytropic process, one has, between its end states extrema: ) is an a ne relationship between log P and log v-a line segment that links the process end states extrema-in which the polytropic exponent n gures as the negative of the line segment slope.Figure is drawn in log P × log v coordinates, and depicts one line segment between each adjacent labeled state pairs ' -', ' -', ' -', and ' -', in which all labeled states are extrema of each line segment, and there are no reversals.
erefore, all such processes, enumerated on Example with roman numerals (i)-(iv), are exact polytropic ones.

. Local Polytropic Processes
Figure illustrates a process that plots as a curved segment in log P × log v coordinates.Basic derivative knowledge allows us to think of such process as one with a continuously variable slope in double logarithmic P × v coordinates, which, allied to the conclusions drawn from Equation ( ), as one with a polytropic-like relationship with continuously varying exponent n.
Attempting to exactly represent the process as a set of straight line segments would result in an in nite set of segments, with each one being of vanishing length-hence, having same end states.e de nition of exact polytropic processes excludes all processes in such a set from being exact polytropic ones; thus, the following Lemma: Lemma (continuously curved process).A process whose path is continuously curved in log P × log v coordinates has no exact polytropic sub-process segments.If, however, one allows for approximations, as is common practice in engineering, the curved process can be represented by an increasing but nite amount of line segments with decreasing amounts of deviation (errors).e beginning of such a process, in which the amount of line segments is doubled at each step, is depicted, with a shi in v for the sake of improved visualization, on Figure .e rationale behind such approximation process is in support of the following de nition:

De nition (local polytropic process). A local polytropic process is a polytropic process that approximates or models a subset of or another process to within suitable error intervals, while sharing common end states with it.
Regarding envisioned capabilities of local polytropic processes in the context of equilibrium thermodynamics, one herein states:

Conjecture (general approximability). Any continuous, quasi-equilibrium process set can be approximated or modeled, to within finite error intervals, by a finite set of local polytropic processes.
Remarks.It is worth noting that all equilibrium thermodynamic cycles can be trivially subdivided into the process set stated on the general approximability conjecture, since (sub-)process end states can be arbitrarily placed within the larger cycle path.e rationale behind the general approximability conjecture is the very de nition of local polytropic process, stated on De nition .
Numerical methods in engineering typically employ a form of discretization of the underlying quantities.Numerical schemes geared towards solving equilibrium engineering thermodynamic cycles and processes may also apply the concept to processes.Proposition (process modeling).It is proposed that local polytropic processes, as herein defined, to be employed as discrete building blocks for general equilibrium engineering thermodynamics process modeling.
A validated model description published by the author before the present formalization, has successfully employed the proposed strategy in solving Finite-Time Heat-Addition (FTHA) air-standard Otto cycles, using local polytropic processes as computational elements [ ].
Since unsteady, non-instantaneous addition of heat in an Otto cycle invariably leads to simultaneous heat and work interactions and result in non-trivial thermodynamic processes, even under quasi-equilibrium hypotheses, study [ ] concluded that a set of discrete polytropic subprocesses was the theoretical tool to provide the needed generality in the model formulation, and thus, inspired the current formulations.

. Requisites for Exact Polytropic Processes
One can show that the polytropic relation, Equation ( ), is the solution of with n not a function of either P or v: thus recovering Equation (). e work of Christians [ ] shows that internally reversible processes in closed systems with a calorically perfect gas with a Pv = ZRT equation of state with constant compressibility factor Z assuming negligible changes in system kinetic and potential energies, absence of chemical reactions, and constant energy transfer ratio, yield a polytropic relation between P and v with a constant exponent n = (1 − γ)K + γ.In this work, K is also referred to as heat-to-work transfer ratio.e employed Pv = ZRT equation of state is indicative of real gases; however, the constant Z assumption narrows down the scope of the result.is prompts the question of whether this nding is actually only applicable to ideal gases, for which Z = 1, or whether other constants work.
e energy balance equation for a closed unreactive system reads, in the intensive di erential form, as where δ q and δ w are the di erential heat and work interactions, following the historical thermodynamic sign convention of positive heat interactions being into the system and positive work interactions being out of the system; u is the system speci c internal energy, with e k and e p being the system speci c kinetic and potential (macroscopic) energies, respectively.Neglecting the variation of the system macroscopic energy forms, simpli es Equation ( ), giving: in which Equation ( ) is used.At this point, reference [ ] replaces du by c v dTwhere c v and c p are the constant-volume and constantpressure speci c heats, respectively-in order to be able to arrive at a polytropic relation between P and v, and continues the analysis with a Pv = ZRT equation of state with the assumption of constant Z.
Shortly a er, the ZR term is replaced by (c p − c v ), which can recover an ideal gas result if Z is further restricted to 1. His analysis seems to request the adoption of an ideal gas model, rather than a real gas one, as the ZR = (c p − c v ) relation cannot hold in general for a real gas.Take, for instance, the close vicinity of the critical state, predictable by a real gas model, in which c p → +∞, even if approached from the monophase region, while all other quantities remain nite, indicating the ZR = (c p − c v ) relation is subject to arbitrarily large errors in the real gas model domain.
Moreover, from the perspective of real gas model, constant Z in general must be based either on (i) negligible reduced pressure, temperature, and speci c volume variations in the process-recall the generalized compressibility chart [ ]-or on (ii) the substance being in the ideal gas limit during the entirety of the process.
Following reference [ ], du is replaced by c v dT , yielding a u : u(T )-only substance, whose property relations are investigated.In the following, it is theoretically demonstrated that Z : Z(v) for such a substance-a useful intermediate result-among other outcomes.
. Substance Equation of State Yielding u : u(T ) For u : u(T ) to hold, one must have: for any choice of property i other than T .
Perhaps the easiest way of deriving the outcomes of Eq. ( ) is by rewriting it in terms of Bridgman's relations [ ], which are expressed in terms of a peculiar notation.erefore, rewriting Eq. ( ) in Bridgman's notation yields It is worth noting than the ≡ sign on Eq. ( ) indicates the de nition of the ratio between Bridgman's primitives (∂ u) T and (∂ i) T in terms of (∂ u/∂ i) T , rather than the other way around.
Bridgman's relations are tabulated expressions for its individual primitives in terms of thermodynamic properties that are easily obtainable from physical measurements [ ], which makes them ingeniously useful.
Bridgman's peculiar notation allowed Eq. ( ) to be expressed in terms of Bridgman's (∂ u) T primitive only, thus eliminating the role of Bridgman's (∂ i) T primitive, irrespective of the choice of property i. is greatly simpli es the analysis of Eq. ( )'s outcomes.
From reference [ ], one has

Figure .
Figure .Air-standard ideal Diesel cycle in log P × log v coordinates, in support for the Example .

Figure .
Figure .A process displaying a curve in logarithmic P × v coordinates.

Figure .
Figure .Successive approximations to the curved process, in dark blue solid line, by means of , , and straight line segment sub-processes, in dark yellow solid lines, in log P × log v coordinates (shi ed in v for easier visualization) β is the volumetric coe cient of thermal expansion, or the volume expansivity [ ], or the coe cient of volume expansion [ ], and κ is the isothermal compressibility [ ], also denoted by some authors as κ T , as to distinguish it from the isentropic compressibility, κ s [ ].Put di erently, Eqs. ( )-( ) indicate that any substance for which β T = κP will have u : u(T ) only.Eq. ( ) brings forth de nitions of β and κ: v) arising from the partial integrations.erefore, this result expresses a substance in whichT ∝ P at constant volume.Letting f(v) ≡ f (v)/R, yields P f (v) = RT, ( ) with f : f (v)being an arbitrary function of v only.If this equation of state is represented as Pv = ZRT , then Z = v/ f (v), as previously announced.Equation ( ) is the most general equation of state for a substance that has u : u(T ), and consequently du = c v dT . .Speci c Heats of a u : u(T ) Substance Writing s : s(T, v) and di erentiating, with du = T ds − P dv and the Maxwell relation based on the Helmholtz energy, (∂ s/∂ v) T = (∂ P/∂ T ) v , one arrives at s(T, P) and di erentiating, with dh = T ds+ v dP and the Maxwell relation based on the Gibbs energy, (∂ s/∂ P) T = −(∂ v/∂ T ) P , one arrives at derivatives from the dT and dv (or dP) coe cients on Eqs. ( ) and ( ), yields [ ]: