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

engrXiv 2020-05-04 16:18:54 UTC revision

On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites C. Naaktgeboren1?

Abstract is 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 identied. e 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. e 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. Keywords ermodynamics — Polytropic Processes — Logical Processes — Etymology Remarks Provides etymolo for the ‘polytropic’ term — Denes logical thermodynamic process — Denes exact polytropic process — Presents a statement for local polytropic process — Denes theoretical requisites for exact polytropic process

1Universidade Tecnologica´ Federal do Parana´ – UTFPR, Campusˆ Guarapuava. Grupo de Pesquisa em Cienciasˆ Termicas.´ ?Corresponding author: NaaktgeborenC·PhD@gmail·com

License 1. Introduction Many equilibrium engineering thermodynamics processes are taken to follow a polytropic relationship, https://creativecommons.org/licenses/by/4.0/

Pvn = c = const., (1) Contents in which P is the system pressure, usually in kPa, v is the 1 Introduction1 system specic volume, usually in m3/kg, and n is the 2 Etymology of the ‘Polytropic’ Term2 dimensionless polytropic exponent [13]—which, for a ‘1–2’ 3 Exact and Local Polytropic Processes2 process, with initial and nal end states labeled as ‘1’ and 3.1 Logical Processes...... 2 ‘2’, can also be written in terms of end state properties, as n n P1v = P2v , which also sets the particular value of the c 3.2 Exact Polytropic Processes...... 3 1 2 constant. 3.3 Local Polytropic Processes...... 4 Some mainstream thermodynamics textbooks intro- 4 Requisites for Exact Polytropic Processes5 duce polytropic processes in the context of closed system P : P(v) 4.1 Substance Equation of State Yielding u:u(T) ...6 boundary work, as a relationship to plug into the boundary work integral, which contains a Pdv inte- u:u(T) 4.2 Specic Heats of a Substance...... 6 grand [13, 10, 12]. In such texts, the polytropic relationship 4.3 Energy Balance with a u:u(T) Model...... 7 is frequently said to nd support in measurements, while 5 Summary and Conclusions8 no specic theoretical derivation is presented at the point of introduction. References9 On the other hand, other texts [7, 4, 8] include deriva- tions that lead to a polytropic process, or at least to an

1 On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 2/9 isentropic version of it, in which the exponent n has a for the real polytropic exponent n and for the c constant; determined value. thence, uncountably many processes departing from un- Moreover, Bejan [4, p. 175] indicates that a constant countably many initial states. It is this kind of exibility Pvn relation only holds locally if the process is such that n that is encoded in the etymology of the process name. is a function of either P, v, or both. A paper due to Christians [6] discusses the topic from 3. Exact and Local Polytropic Processes a perspective of teaching polytropic processes themselves, 3.1 Logical Processes placing emphasis on the heat-to-work transfer ratio—named In equilibrium engineering thermodynamics, a process— by that author as “energy transfer ratio”—and how its con- more properly a quasi-static or quasi-equilibrium one—is stancy not only yields, but constitutes a prerequisite for a dened in terms of changes from a certain equilibrium process to be polytropic, besides, naturally, the constancy state of a system to another [13], with process path being of the caloric properties of the working pure substance. the (innite) sequence of (quasi-)equilibrium states visited Starting with an etymological presentation of polytropic by the system during the process. A process can be referred processes, this work proposes and develops the concepts to by its path, with implicit or explicit end states. of exact polytropic and local polytropic processes, as well as It is worth noting that no constraints are stated for the supporting concept of logical thermodynamic processes, the end states of a process in its denition. is allows and presents theory-derived requisites for a process to be ex- for the needed exibility in describing the variety of trans- actly polytropic beyond reference [6]. Moreover, local poly- formations systems and control volumes can undergo in tropic processes are proposed as discrete building blocks engineering thermodynamics. for general engineering thermodynamics process modeling. is lack of end state constraints in the denition of a process allows them to be splitted into multiple, successive 2. Etymology of the ‘Polytropic’ Term ‘sub-processes’ that still t the denition of a process, as e ‘polytropic’ term has its etymology (origin) in the well as to merge multiple successive ones together into Greek language. is author’s sources on Greek are mostly ‘super-processes’ that also t the denition of a process. based on modern romance languages, such as French [5, 2] is ability is extremely useful in grouping and splitting and his native Portuguese [9]; therefore, the etymology systems and control volumes along with their underlying herein brought forth will include intermediate French processes—a common practice in engineering thermody- terms. namics. e word ‘polytropic’ stems from the Greek word ‘πολύ- However, in order to make the intended distinction τροπος’ that itself is composed of two Greek words: (i) ‘πο- between proposed ‘exact’ and ‘local’ polytropic processes, λύς’1, and (ii) ‘τρόπος’. additional constraints need to be made to process end e Greek adjective ‘πολύς’ includes meanings such as states. us, the following denes a logical thermodynamic abondant, nombreux, vaste [5], i.e., abundant, numer- process, which is a thermodynamic process with constrained ous, vast [3, 1]. e Greek noun ‘τρόπος’ includes meanings end states. In context, i.e., in thermodynamics texts, what’s such as maniere,` façon, mode [2], i.e., manner, way, fash- dened next can be simply called a ‘logical process’: ion, and mode [3, 1]—hence, meaning: numerous forms, or Denition 1 (logical process). A logical process is one in many ways. which its stated defining conditions, which determines all of the e composed Greek noun ‘πολύτροπος’ is also lexical, allowed interactions or property relations for the underlying sys- and includes gurative meanings such as souple, habile, tem or control volume, uniformly and continually apply to the industrieux [2], i.e., exible, adaptive, able, laborious [1]; entirety of its path from—but not earlier than—its initial state as well as meanings such as versatilite,´ tres` divers, tres` until—but not later than—its end state. varie´ [2], i.e., versatility, very diverse, a great variety— erefore, for a simple compressible system—one ad- thus indicating the wide-range of processes that it is capable mitting only work and heat interactions—either stated of representing. heat and work interactions, or system property specica- In fact, from a mathematical standpoint, Equation (1), tions, or combinations of the two, dene possible logical there are really innitely many allowable (possible) values processes. 1e lexical form of Greek adjectives is the nominative, singular, Moreover, the stated dening conditions of a logical masculine. e nominative, singular, neuter of ‘πολύς’ is ‘πολύ’. process can also carry a ‘logical’ qualier, as to make it On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 3/9 explicit they’re being used in the denition of a logical Lemma 1. Any logical process defined by a single polytropic process, as in the ‘logical conditions,’ or ‘dening logical relation with a unique polytropic exponent, with non-identical conditions’ expressions. Furthermore, the conditions them- end states can only be an exact polytropic process in the absence selves can carry the ‘logical’ qualier, for the same purpose. of reversals in its path.

Example 1. e well-known air-standard ideal Diesel power Proof. Let a logical process P1,2 be dened by a single cycle, illustrated on Figure1, with ‘intake’ state (of lowest tem- polytropic relation with a unique polytropic exponent n, perature and pressure and maximum speciﬁc volume) labeled with non-identical end states ‘1’ and ‘2’ that contains a set as ‘1’, can be divided in dierent ways using only logical proof reversal states ‘ri’, with {i ∈ Z|i > 1} in its path. cesses. One such division is: (i) ‘logical isentropic compression’, Without loss of generality, let further Pr1 < P1, if n 6= (∆s=0 ∴ Q=0, W<0), (ii) ‘logical isobaric heating’, (Q>0, 0, i.e., an initially pressure-decreasing process—since the ∆P = 0 ∴ W > 0), (iii) ‘logical isentropic expansion’, (∆s = following argument can be otherwise ipped, or laid out 0 ∴ Q=0, W>0), and (iv) ‘logical isochoric cooling’, (Q<0, in either direction in v, for non-zero n: ∆v = 0 ∴ W = 0), which correspond, respectively, to the ‘1–2’, Let state ‘ε’, dened as (Pε ,vε ), belong to the path of

‘2–3’, ‘3–4’, and ‘4–1’ processes, i.e., to the canonical processes a sub-process P1,r1 of the original logical process P1,2, for this cycle, excluding sub-processes thereof. Another possible so that division is: (a) ‘logical isentropic compression’, (∆s=0 ∴ Q=0, P − P = ∆P → 0. W<0), (b) ‘logical no-heat-removal expansion’, (Q>0, W>0), ε r1 (2) and (c) ‘logical isochoric cooling’, (Q < 0, ∆v = 0 ∴ W = 0), Since ‘ε’ ∈ P ,r ∈ P , , it follows from the denition of which correspond, respectively, to the ‘1–2’, ‘2–4’ (through ‘3’), 1 1 1 2 P , : and ‘4–1’ processes, excluding sub-processes thereof, since these 1 2 encompass the farthermost end states that uniformly and contin- P vn = P vn = P vn . ε ε 1 1 r1 r1 (3) ually match the stated deﬁning logical conditions (a), (b), and (c). Since at ‘r1’, the process experiments a reversal, thus pro- ceeding with increasing pressures. Let state ‘ζ’, dened as

3 (Pζ ,vζ ), belong to the path of a sub-process Pr1,2 of the log P 2 original logical process P1,2, so that

4 Pζ ≡ Pε = lim Pr1 + ∆P, (4) ∆P→0 1 so that process Pr ,2 is guaranteed to contain state ‘ζ’, log v 1 since state ‘r1’ is a reversal one, rather than a stopping one, Figure 1. Air-standard ideal Diesel cycle in logP × logv therefore: ‘ζ’ ∈ Pr ,2 ∈ P1,2, implying: coordinates, in support for the Example1. 1 P vn = P vn = P vn + ζ ζ r1 r1 ε ε (by Eq. (3)) (5) n n 3.2 Exact Polytropic Processes Pε vζ = Pε vε + (6) One is now in a position to dene exact polytropic processes: n n vζ = vε + (7)

Denition 2 (exact polytropic process). An exact polytropic vζ = vε . (8) process is a logical process in which either (i) a polytropic relation Pvn = const., with a unique polytropic exponent n, or From Eqs. (4) and (8), one has states ‘ε’ and ‘ζ’ being iden- (ii) an isochoric logical condition, can be its sole logical deﬁning tical: ‘ε’ ≡ ‘ζ’, with ‘ε’ ∈ P1,2 and ‘ζ’ ∈ P1,2. condition, provided that no state in its path is visited more than erefore, by the exact polytropic process denition, once or serves simultaneously as initial and ﬁnal end states in Denition2, logical process P1,2 cannot be an exact poly- a given execution. tropic process, since there is at least one state—state ‘ε’— that is visited more than once in a given execution. It is worth noting that isochoric processes are equiva- lent to polytropic processes with n → ±∞ between stated Remarks. Given that the polytropic exponent is unique, end states. Denition2 accounts for the isochoric process any reversal would cause the underlying system or control non-unique polytropic exponent by explicitly including it volume to re-visit states already covered, thus violating the as a valid exact polytropic process. exact polytropic process denition. On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 4/9

eorem 1 (Cycle). No cycle is an exact polytropic process. log P 2 Proof. e dening feature of a cycle of same end states directly violates the restriction of no same initial and end states for an exact polytropic process. 1 Remarks. Even if a cycle can be collapsed down as to be log v described in terms of a single polytropic process, such as Figure 2. A process displaying a curve in logarithmic P × v an ideal Diesel cycle with fuel cut ratio rc ≡ v3/v2 = 1, coordinates. the fact that such a cycle must invariably make at least one reversal, as to cycle back to previous states, Lemma1 If, however, one allows for approximations, as is com- would constitute a second reason to disqualify the cycle as mon practice in engineering, the curved process can be a (single) exact polytropic process. represented by an increasing but nite amount of line seg- For an exact polytropic process, one has, between its ments with decreasing amounts of deviation (errors). e end states extrema: beginning of such a process, in which the amount of line segments is doubled at each step, is depicted, with a shi n Pv = c1 = const. + (9) in v for the sake of improved visualization, on Figure3. n log(Pv ) = log(c1) ≡ c2 + (10) log P 2 logP + nlogv = c2 + (11)

logP = −nlogv + c2. (12)

Equation (12) is an ane relationship between logP and 1 logv—a line segment that links the process end states log v n extrema—in which the polytropic exponent gures as Figure 3. Successive approximations to the curved process, the negative of the line segment slope. in dark blue solid line, by means of 2, 4, and 8 straight logP × logv Figure1 is drawn in coordinates, and de- line segment sub-processes, in dark yellow solid lines, in picts one line segment between each adjacent labeled state logP × logv coordinates (shied in v for easier pairs ‘1–2’, ‘2–3’, ‘3–4’, and ‘4–1’, in which all labeled states visualization) are extrema of each line segment, and there are no reversals. erefore, all such processes, enumerated on Example1 e rationale behind such approximation process is in with roman numerals (i)–(iv), are exact polytropic ones. support of the following denition: 3.3 Local Polytropic Processes Denition 3 (local polytropic process). A local polytropic Figure2 illustrates a process that plots as a curved segment process is a polytropic process that approximates or models a in logP×logv coordinates. Basic derivative knowledge al- subset of or another process to within suitable error intervals, lows us to think of such process as one with a continuously while sharing common end states with it. variable slope in double logarithmic P × v coordinates, Regarding envisioned capabilities of local polytropic which, allied to the conclusions drawn from Equation (12), processes in the context of equilibrium thermodynamics, as one with a polytropic-like relationship with continu- one herein states: ously varying exponent n. Attempting to exactly represent the process as a set Conjecture 1 (general approximability). Any continuous, of straight line segments would result in an innite set of quasi-equilibrium process set can be approximated or modeled, segments, with each one being of vanishing length—hence, to within ﬁnite error intervals, by a ﬁnite set of local polytropic having same end states. e denition of exact polytropic processes. processes excludes all processes in such a set from being Remarks. It is worth noting that all equilibrium thermo- exact polytropic ones; thus, the following Lemma: dynamic cycles can be trivially subdivided into the process Lemma 2 (continuously curved process). A process whose set stated on the general approximability conjecture, since path is continuously curved in logP×logv coordinates has no (sub-)process end states can be arbitrarily placed within exact polytropic sub-process segments. the larger cycle path. On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 5/9

e rationale behind the general approximability con- system kinetic and potential energies, absence of chemical jecture is the very denition of local polytropic process, reactions, and constant energy transfer ratio, stated on Denition3. δq Numerical methods in engineering typically employ K ≡ , (19) a form of discretization of the underlying quantities. Nu- δw merical schemes geared towards solving equilibrium en- yield a polytropic relation between P and v with a constant gineering thermodynamic cycles and processes may also exponent n = (1−γ)K +γ. In this work, K is also referred apply the concept to processes. to as heat-to-work transfer ratio. e employed Pv = ZRT equation of state is indicative Proposition 1 (process modeling). It is proposed that local of real gases; however, the constant Z assumption narrows polytropic processes, as herein deﬁned, to be employed as discrete down the scope of the result. is prompts the question building blocks for general equilibrium engineering thermody- of whether this nding is actually only applicable to ideal namics process modeling. gases, for which Z = 1, or whether other constants work. A validated model description published by the au- e energy balance equation for a closed unreactive thor before the present formalization, has successfully em- system reads, in the intensive dierential form, as ployed the proposed strategy in solving Finite-Time Heat- δq − δw = du + de + de , (20) Addition (FTHA) air-standard Otto cycles, using local k p polytropic processes as computational elements [11]. where δq and δw are the dierential heat and work in- Since unsteady, non-instantaneous addition of heat teractions, following the historical thermodynamic sign in an Otto cycle invariably leads to simultaneous heat convention of positive heat interactions being into the sys- and work interactions and result in non-trivial thermody- tem and positive work interactions being out of the system; namic processes, even under quasi-equilibrium hypotheses, u is the system specic internal energy, with ek and ep be- study [11] concluded that a set of discrete polytropic sub- ing the system specic kinetic and potential (macroscopic) processes was the theoretical tool to provide the needed energies, respectively. Neglecting the variation of the sys- generality in the model formulation, and thus, inspired tem macroscopic energy forms, simplies Equation (20), the current formulations. giving:

4. Requisites for Exact Polytropic Processes δq − δw = du, + (21) One can show that the polytropic relation, Equation (1), is (K − 1)δw = du, (22) the solution of in which Equation (19) is used. dP dv At this point, reference [6] replaces du by c dT— = −n , (13) v P v where cv and cp are the constant-volume and constant- pressure specic heats, respectively—in order to be able n P v with not a function of either or : to arrive at a polytropic relation between P and v, and Z dP Z dv continues the analysis with a Pv = ZRT equation of state = −n + (14) P v with the assumption of constant Z. Shortly aer, the ZR term is replaced by (cp − cv), logP + c1 = −nlogv + c2 + (15) which can recover an ideal gas result if Z is further re- logP = −nlogv + c + (16) 3 stricted to 1. His analysis seems to request the adoption logP + nlogv = c3 + (17) of an ideal gas model, rather than a real gas one, as the n c3 Pv = e ≡ c, (18) ZR = (cp − cv) relation cannot hold in general for a real gas. Take, for instance, the close vicinity of the critical state, thus recovering Equation (1). predictable by a real gas model, in which cp → +∞, even e work of Christians [6] shows that internally re- if approached from the monophase region, while all other versible processes in closed systems with a calorically per- quantities remain nite, indicating the ZR = (cp − cv) fect gas with a Pv = ZRT equation of state with constant relation is subject to arbitrarily large errors in the real gas compressibility factor Z assuming negligible changes in model domain. On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 6/9

Moreover, from the perspective of real gas model, con- Put dierently, Eqs. (23)–(26) indicate that any sub- stant Z in general must be based either on (i) negligible stance for which βT = κP will have u:u(T) only. reduced pressure, temperature, and specic volume varia- Eq. (27) brings forth denitions of β and κ: tions in the process—recall the generalized compressibility 1 ∂v −1 ∂v chart [13]—or on (ii) the substance being in the ideal gas β ≡ , and κ ≡ . (27) v ∂T v ∂P limit during the entirety of the process. P T Following reference [6], du is replaced by cv dT, yield- Pluing in Eq. (27) on Eq. (26) and using the cyclic ing a u:u(T)-only substance, whose property relations are relationship, investigated. In the following, it is theoretically demon- ∂ j ∂` ∂i strated that Z :Z(v) for such a substance—a useful inter- −1 = , (28) ∂i ∂ j ∂` mediate result—among other outcomes. ` i j gives, aer some manipulation, 4.1 Substance Equation of State Yielding u:u(T) ∂T T For u:u(T) to hold, one must have: = + (29) ∂P v P ∂u = 0. (23) ∂T ∂P ∂i = + (30) T T v P v for any choice of property i other than T. T = f(v)P, (31) Perhaps the easiest way of deriving the outcomes of with f(v) arising from the partial integrations. erefore, Eq. (23) is by rewriting it in terms of Bridgman’s rela- this result expresses a substance in which T ∝ P at constant tions [4], which are expressed in terms of a peculiar nota- volume. Letting f(v) ≡ f (v)/R, yields tion. erefore, rewriting Eq. (23) in Bridgman’s notation yields P f (v) = RT, (32) f : f (v) v (∂u)T ∂u with being an arbitrary function of only. If this ≡ = 0 + (∂u)T = 0. (24) equation of state is represented as Pv = ZRT, then Z = (∂i)T ∂i T v/ f (v), as previously announced. Equation (32) is the most It is worth noting than the ≡ sign on Eq. (24) indicates general equation of state for a substance that has u:u(T), the denition of the ratio between Bridgman’s primitives and consequently du = cv dT. (∂u)T and (∂i)T in terms of (∂u/∂i)T , rather than the other way around. 4.2 Specic Heats of a u:u(T) Substance Bridgman’s relations are tabulated expressions for its Writing s:s(T,v) and dierentiating, with du = T ds − individual primitives in terms of thermodynamic properties Pdv and the Maxwell relation based on the Helmholtz that are easily obtainable from physical measurements [4], energy, (∂s/∂v)T = (∂P/∂T)v, one arrives at which makes them ingeniously useful. c ∂P ds = v dT + dv. Bridgman’s peculiar notation allowed Eq. (23) to be (33) T ∂T v expressed in terms of Bridgman’s (∂u)T primitive only, Writing s:s(T,P) and dierentiating, with dh = T ds+ thus eliminating the role of Bridgman’s (∂i)T primitive, vdP irrespective of the choice of property i. is greatly simplies and the Maxwell relation based on the Gibbs energy, ( s/ P) = −( v/ T) the analysis of Eq. (23)’s outcomes. ∂ ∂ T ∂ ∂ P, one arrives at From reference [4], one has cp ∂v ds = dT − dP. (34) T ∂T P (∂u)T = v(βT − κP) = 0 + (25) Equating the cross derivatives from the dT and dv (or βT = κP, (26) dP) coecients on Eqs. (33) and (34), yields [13]: where β is the volumetric coecient of thermal expansion, or 2 ∂cv ∂ P = +T , and (35) the volume expansivity [4], or the coecient of volume expan- ∂v ∂T 2 sion [7], and κ is the isothermal compressibility [4], also de- T v 2 ∂cP ∂ v noted by some authors as κT , as to distinguish it from the = −T . (36) P T 2 isentropic compressibility, κs [7]. ∂ T ∂ P On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 7/9

Eqs. (35) and (36) are to be used in determining whether thus making h : h(T) for an unreactive substance with the specic heats of a substance with equation of state P f (v) = RT equation of state. given by Eq. (32) are functions of other thermodynamic e ndings of this subsection are thus summarized properties. in an already proven theorem: erefore, from Eq. (32), one has: eorem 2 (u:u(T) unreactive substance). An unreactive RT substance for which u : u(T), necessarily has cv : cv(T), cP : P = + (37) f (v) cP(T), and γ :γ(T) at best, and also h:h(T). Moreover, it has ∂P R a P f (v) = RT equation of state, with f (v) being an arbitrary = + (38) function of v. ∂T v f (v) 2 ∂ P 4.3 Energy Balance with a u:u(T) Model 2 = 0. (39) ∂T v Pluing in du = cv dT on Eq. (22) and dierentiating Eq. (32) will make a dT appear on both equations. Equating erefore, (∂cv/∂v)T = 0 making cv : cv(T) at best the common dT leads to for a Eq. (32) substance. K − 1 P f 0(v)dv + f (v)dP Let dT = Pdv = , (50) cv R g(v) ≡ f (v)/v. (40) where f 0(v) ≡ d f /dv. then, from Eq. (32), one has: e goal now is to write Eq. (50) in the form of Eq. (13), with constant n, which yields RT v = + (41) dP dv Pg(v) = −[ f 0(v) + (1 − K)(γ − 1)] + (51) P f (v) ∂v R = + (42) dP f 0(v) + (1 − K)(γ − 1) dv ∂T Pg(v) = − , (52) P P g(v) v ∂ 2v = 0. 2 (43) in which g(v) is dened by Eq. (40). By inspection, the ∂T P polytropic exponent is identied as: erefore, (∂cP/∂P)T = 0 making cP :cP(T) at best f 0(v) + (1 − K)(γ − 1) for a Eq. (32) substance. Moreover, since n ≡ + (53) g(v) cp 0 γ ≡ , v f (v) v (44) n = + (1 − K)[γ(T) − 1] . (54) cv f (v) f (v) one also has γ :γ(T) at best for this substance. e constancy of n depends on the constancy of all e textbook expression for dh is [13]: of Eq. (54) terms. Enforcing this condition upon the rst ∂h ∂h term leads to the following ODE: dh = dT + dP (45) ∂P T ∂T P v d f = c1 + (55) ∂v f dv dh = c dT + v − T dP. (46) p ∂T d f dv P = c + (56) f 1 v From Eq. (42), the (∂h/∂T)P term can be written as c1 f (v) = v + c2. (57) ∂h ∂v = v − T + (47) e second term of Eq. (54) contains v and T functions. ∂T P ∂T P Enforcing constancy separately in v terms yields: ∂h RT = v − + (48) v T Pg(v) = c3 + (58) ∂ P f (v) ∂h v = v − v = 0, (49) c = c3, (59) ∂T P v 1 + c2 On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 8/9

which yields c1 = 1, c2 = 0, with c3 = 1 being the only e concept of a logical thermodynamic process, idealized possibility for constancy in v terms. is result nally xes by the author as a process with restricted end states, has the substance equation of state by xing f (v) as been laid out by Denition1. With logical process, or logical conditions, processes and associated end states can f (v) = v. (60) be uniquely referred to. Example1 works out the concept Consequently, the P f (v) = RT equation of state, with on an ideal, air-standard Diesel cycle. f (v) = v, exactly recovers the ideal gas equation of state, which e concept of exact polytropic process, idealized by the reads author as being both logical, polytropic, and non-reversing, see Lemma1, has been laid out in Section 3.2. A ‘Cycle’ Pv = RT. (61) theorem relating to exact polytropic processes, eorem1, Moreover, the constancy of the (1−K)[γ(T)−1] term is provided. e way towards local polytropic process is must account for the fact that K is dened only in terms paved with the discussion of processes with continuously of process ener interactions, Eq. (19), while γ(T) is dened curved paths in double logarithmic P × v coordinates, see only in terms of system substance properties, Eq. (44). ere- Lemma2. fore, the analysis is two-fold: A denition of local polytropic process, formulated by (i) For interactions that yield a constant value of K, as the author—although the concept of process relations ap- assumed on reference [6], one must have constant γ(T) = plying locally is found in previous literature, as, for in- 1 + R/cv(T), for an ideal gas, which requires constancy stance, in [4, p. 175]—that explicitly mentions the purpose in cv, which, in turn, also means constancy in cp, since of approximation of another process, is provided by De- cp = cv + R for ideal gases. nition3 in Section 3.3. (ii) For Eq. (54)’s second term constancy, with v/ f (v) = By means of the Conjecture1, it is claimed that nitely 1, interactions must be made system temperature depen- many local polytropic process can be used to model any dent, i.e., K:K(T), resulting in [1−K(T)] ∝ cv(T), which given equilibrium thermodynamics process or cycle to is hardly practical and is le as a note. within nite errors, which leads to Proposition1, of equi- Pluing the results back in Eq. (54), yields the value librium thermodynamics numerical schemes to use them of the resulting polytropic exponent: as discrete building blocks for general process modeling. n = 1 + (1 − K)(γ − 1) = γ + K(1 − γ), (62) A validated model description that employs discrete local polytropic processes as building blocks in general which is in agreement with Christian’s results [6]. process modeling [11], which has been published by the is section’s outcomes are the base for the following, author before the present formalization of concepts, is ref- ‘K-polytropic’, theorem: erenced in Section 3.3 as a successful case in implementing Proposition1. eorem 3 (K-polytropic). Internally reversible processes in constant-speciﬁc-heat unreactive closed systems with negligible In Section4, a statement that under certain conditions kinetic and potential ener changes and constant heat-to-work a gas with u:u(T) and a Pv = ZRT equation of state with ratio interactions, K, are exact polytropic processes only if the constant Z displays a constant exponent polytropic process, substance is an ideal gas. found in [6], is investigated using thermodynamic property relations, seeking theoretical requisites for the occurrence erefore, the K-polytropic theorem answers the ques- of exact polytropic processes. tion herein raised in light of Christian’s work [6], that the Section 4.1 shows that a u:u(T) substance must have constant-Z, u:u(T) real gas working substance yielding a Z :Z(v). Moreover, Section 4.2 determines whether cv, an exact polytropic process under a reversible, constant- cp, γ, and h of the u:u(T) substance depend on properties K process in a closed system, needs to be further narrow other than T. e ndings are summarized in eorem2. down to an ideal gas, i.e., one for which Z = 1. It is nally concluded, in Section 4.3, that the most u : u(T) 5. Summary and Conclusions general substance to yield an exact polytropic process under constant heat-to-work transfer ratio is, in is work has concerned itself with polytropic processes. fact, an ideal gas—see eorem3. An etymolo of the ‘polytropic’ term that relates to its mathematical possibilities has been given on Section2. On Exact and Local Polytropic Processes: Etymology, Modeling, and Requisites — 9/9

Acknowledgments [12] Gordon Wylen. Fundamentals of classical thermodynamics. Wiley, New York, fourth edition, 1985. is research received no specic grant from any funding [13] agency in the public, private, or not-for-prot sectors. Y. A. C¸ engel and M. A. Boles. Termodinamicaˆ . AMGH, To YHWH God be the glory! Porto Alegre, 7th edition, 2013.

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