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 ‘poλύ- However, in order to make the intended distinction tropoc’ that itself is composed of two Greek words: (i) ‘po- between proposed ‘exact’ and ‘local’ polytropic processes, λύς’1, and (ii) ‘τρόπος’. additional constraints need to be made to process end e Greek adjective ‘poλύς’ 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,` fac¸on, 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 ‘poλύτροpoc’ 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 ‘poλύc’ is ‘poλύ’. 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 pro- of reversal states ‘ri’, with fi 2 Zji > 1g in its path. cesses. One such division is: (i) ‘logical isentropic compression’, Without loss of generality, let further Pr1 < P1, if n 6= (Ds=0 ) Q=0, W<0), (ii) ‘logical isobaric heating’, (Q>0, 0, i.e., an initially pressure-decreasing process—since the DP = 0 ) W > 0), (iii) ‘logical isentropic expansion’, (Ds = 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: Dv = 0 ) W = 0), which correspond, respectively, to the ‘1–2’, Let state ‘e’, dened as (Pe ;ve ),