Association of Finite-Time Thermodynamics and a Bond-Graph Approach for Modeling an Endoreversible Heat Engine

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1 Entropy, 4, ; doi:.339/e4464 Artile OPEN ACCE entropy IN Assoiation of Finite-ime ermodynamis and a Bond-Grap Approa for Modeling an Endoreversible Heat Engine Yuxiang Dong,, *, Amin El-Bakkali, Georges Desombes, Miel Feidt 3 and Cristelle Périlon 3 RENAUL, Advaned Eletronis and enologies Division, enoentre Renault, avenue de Golf 7888 Guyanourt, Frane; amin.el-bakkali@renault.om Laboratoire du génie des proédés pour l environnement, l énergie et la santé (LGPE-EA), Cnam-Cemagref, ase D3R, 9 rue saint Martin 753 Paris, Frane; s: georges.desombes@nam.fr (G.D.); ristelle.perilon@nam.fr (C.P.) Laboratoire d Energétique et de Méanique éorique et Appliquée, ENEM,, avenue de la Forêt de Haye 5456 Vandoeuvre, Frane; miel.feidt@ensem.inpl-nany.fr * Autor to wom orrespondene sould be addressed; yuxiang.dong@renault.om. Reeived: 6 January ; in revised form: 3 Mar / Aepted: 3 Mar / Publised: 8 Mar Abstrat: In reent deades, te approa known as Finite-ime ermodynamis as provided a fruitful teoretial framework for te optimization of eat engines operating between a eat soure (at temperature s ) and a eat sink (at temperature s ). e aim of tis paper is to propose a more omplete approa based on te assoiation of Finite-ime ermodynamis and te Bond-Grap approa for modeling endoreversible eat engines. is approa makes it possible for example to find in a simple way te arateristis of te optimal operating point at wi te maximum meanial power of te endoreversible eat engine is obtained wit entropy flow rate as ontrol variable. Furtermore it provides te analytial expressions of te optimal operating point of an irreversible eat engine were te energy onversion is aompanied by irreversibilities related to internal eat transfer and eat dissipation penomena. is original approa, applied to an analysis of te performane of a termoeletri generator, will be te objet of a future publiation. eywords: finite-time termodynamis; bond grap approa; entropy generation; endoreversible eat engine; Cambadal-Novikov-Curzon-Alborn effiieny

2 Entropy, Nomenlature Variable Unit Desription emperature of te eat soure s emperature of te eat sink s Hot side temperature of te endoreversible onverter Cold side temperature of te endoreversible onverter s W/ Entropy flow rate transferred from eat soure to endoreversible onverter s W/ Entropy flow rate transferred from endoreversible onverter to eat sink W/ Entropy flow rate involved in energy onverter W/ Cutoff entropy flow rate s W/ Rate of total entropy generation witin te endoreversible eat engine W/ Global termal ondutane of te eat exanger at ot side W/ Global termal ondutane of te eat exanger at old side Q W ermal power exanged between te eat soure and te endoreversible onverter Q W ermal power exanged between te eat sink and te endoreversible onverter W W Meanial power W W Maximum meanial power max -- Energy onversion effiieny -- Carnot effiieny C -- Cambadal-Novikov-Curzon-Alborn effiieny CNCA. Introdution e energy onversion effiieny of a two-reservoir eat engine is generally ompared wit te teoretial effiieny of te Carnot engine. e Carnot engine assumes tat te eat transfers at te eat soure and at te eat sink our witout entropy prodution wi exludes any termal gradient (f. Figure ). Figure. Carnot yle in - diagram. s Q CARNO CYCLE W Q Q s Q

3 Entropy, Witout termal gradients between te working fluid and te termostats, te eat flow rate involved are zero wi aording to te first law of termodynamis leads to te paradoxial fat tat s te Carnot eat engine produes zero meanial power but wit a maximum effiieny C! s e question is weter tere exists a more realisti limit of te energy onversion effiieny of bi-termal eat engines besides te teoretial one of te Carnot engine. For tis purpose, we ave to admit tat te eat exange area and tus te global termal ondutanes between te working fluid and te termostats are finite. We ten enter into te framework of a new approa of termodynamis known as Finite-ime ermodynamis, initiated independently by Cambadal [] and Novikov [] in 957 and ten larified by Feidt [3] and oters [4 8]. e main idea of tis approa is te oupling of a reversible onverter wit two eat exangers wi onnet te reversible onverter to termostats (f. Figure ). e eat exangers are modeled by termal ondutanes wit finite values. e system ten forms wat we all an Endoreversible Heat Engine were only te irreversibilities related to external eat transfer between termostats and te onverter are taken into aount. But in reality, for a real eat engine, in addition to irreversibilities seen previously, tere are also internal irreversibilities witin te onverter (internal eat transfer, meanial dissipation, et.) [9]. Obtaining an analytial solution of su maines is very important for engineers seeking to optimize te design and te ontrol of tem. is will onstitute te subjet of our future paper. Figure. Diagram of endoreversible eat engine. eat soure ondutane Endoreversible Heat engine ondutane eat sink everal studies ave been onduted on tis basis to optimize eat engines by taking finite onstraints into aount [ 5]. Besides te riterion of maximum onversion effiieny [6], oter optimization riteria ave been proposed: maximum power [7,8], maximum of power density [9], termo-eonomi optimization [,] or eologial optimization [,3]. Our approa detailed below is based on te assoiation of te Finite-ime ermodynamis approa and te Bond Grap Modeling approa [4]. e Bond-Grap Modeling approa

4 Entropy, (f. Figure 3) onsists in expressing all te powers exanged between two systems as te produt of an effort variable e by a flow variable f (f. able ). From a fundamental point of view, it is mainly based on te Onsager work [5,6]. In partiular, a eat flow rate is te produt of te temperature (effort variable) by te entropy flow rate (flow variable). e bond between two elements exanging te power is ompleted by a alf arrow indiating te positive diretion of te power transfer (e.g., A to B, f. Figure 3) and a so-alled ausal line indiating te element wi imposes te effort on te oter and reeives te flow reation. Figure 3. Bond-Grap representation of power exange between two pysial systems. bond ausal stroke A e B f sign onvention : P = e.f > able. Effort variables and flow variables in te multi-pysial bond grap. Power Effort Flow Eletrial Voltage [V] Current [A] Meanial, translation Fore [N] Linear veloity [m/s] Meanial, rotation orque [N.m] Angular veloity [rad/s] Fluid Pressure [N/m²] Volumetri flow rate [m 3 /s] ermal emperature [] Entropy flow rate [W/] Cemial Cemial potential [J/mol] molar flux [mol/s] We will sow in tis paper tat te ombination of tese two approaes allows one to find te natural ontrol variable of te endoreversible eat engine and simplify te modeling of te system. In a subsequent paper we will give te analytial expressions of te arateristis of te optimal operating point of an irreversible eat engine, in wi te energy onversion is aompanied by irreversibilities related to internal eat transfer and eat dissipation penomena. By applying tis new approa to a termoeletri generator [7], te energy reovery potential an be expressed aording to te pysial parameters of te system. e optimization riterion used ere is based on te maximum meanial power as it is a relevant riterion for eat reovery systems in wi te eat soure is onsidered free su as te exaust gases of a motor veile. ese eat reovery systems (ORC system, termoeletri generator) are potentially interesting in view of te tenial solutions designed to redue te (otal Cost of Ownersip) CO of veiles and greenouse gas emissions.

5 Entropy, Modeling of Endoreversible Heat Engine at teady tate Figure 4 sows te Bond-Grap diagram of endoreversible eat engine. We assume tat te maine works at steady state ondition and tus, aording to te seond law of termodynamis, we obtain te onservation of entropy flow rate troug te reversible part of eat engine. By onvention, te arrows next to te flow variables indiate te positive diretion of power transfer. Figure 4. Bond-Grap diagram of an endoreversible eat engine at steady state. Heat soure s s Condutane Heat apaity Energy Conversion Heat apaity f e Endoreversible Heat Engine e.f Q Condutane s Heat sink s Endoreversible Heat Engine Q W.. Energy Balane wit Finite Heat ransfer Constraint We make te following assumptions: e temperatures of eat soure and eat sink are onstant ( s and s). Heat exangers ave onstant global termal ondutanes ( and ). e maine operates at steady state. In tis ase, from te energy balanes, we obtain: Q Q s s s s s W e. f Q Q It an be seen tat te meanial power W s (3) () () is expressed as te produt of a urrent e (entropy flow rate) by a differene of potential f (temperatures). is is te advantage of te Bond-Grap representation wi establises analogies between different forms of energy.

6 Entropy, Coie of te Control Variable of te Endoreversible Heat Engine As sown in Figure 4, te endoreversible eat engine reeives a flow f from te external environment and reats wit an effort e wi depends on tis flow, te ondutanes, and te temperatures s and s sale. us, te flow f represents te natural variable to ontrol te endoreversible eat engine. In te ase of termoeletri onversion, te flow f represents te urrent imposed to te maine wi reats by te voltage (f. Figure 5). Figure 5. Diagram of termoeletri onversion. I U I U i U U o U i U U o I I ELECRICAL HERMAL In tis ase, we ave te relations. f and e were is te eebek oeffiient [7]. As a result, te entropy flow rate an at as ontrol variable sine it is te image of te flow variable f. is oie as te advantage of desribing te operating point of te maine only by means of internal variables, and. We an now, tanks to te relations () to (3), express te temperatures and ten te termal flux and te meanial power as a funtion of te entropy flow rate inside te onverter: Q s, s, Q s (4) s (5) W s s (6) e energy onversion effiieny an also be expressed in terms of entropy flow rate : W Q s Q Q (7) Wen we vary te entropy flow rate, te operating point of te maine moves. e relations (6) and (7) ten form te parametri equations of te operating urve of te endoreversible eat engine tat we are going to study in te [W, ] diagram. s

7 Entropy, Operating Range of te Endoreversible Heat Engine e range of variation of te entropy flow rate for an endoreversible eat engine is impliitly defined by te inequality W. Note tat te meanial power W is zero at two remarkable points: Point (A): zero entropy flow rate: Point (B): utoff entropy flow rate: s, s Q Q W, Q s Q s Q s W, s s s s s s s s s s At point (A), te entropy flow rate is zero, te maine produes zero meanial power but gives te maximum energy onversion effiieny, i.e., Carnot effiieny. At point (B), wit te utoff entropy flow rate s, te maine beaves as two termal resistanes in series ( / and / ) wit te only effet tat te termal power Q is transferred from te eat soure to te eat sink witout any s prodution of meanial power. Finally, te parametri representation of te operating urve of te endoreversible eat engine (f. Figure 6) in te [W, ] diagram is given by te following relations: s s s (8) (9) W s s s s () By deriving te meanial power given by Equation () wit regard to te entropy flow rate aross te onverter, we obtain: dw d s s, d W d s s 3 3 () Between points (A) and (B), tere exists an optimal point (O) orresponding to te maximum meanial power W. max

8 Entropy, Figure 6. Operating urve of te endoreversible eat engine. Energy onversion effiieny C A O Entropy flow rate (a) B meanial power Energy onversion effiieny A B Meanial power W (b) O Optimal operating zone.4. Determination of te Maximum Meanial Power and te Assoiated Effiieny e value of te entropy flow rate o at te optimal point (O) an be obtained by solving te dw equation [f. relation ()]. Given te expressions (4) to (7), we obtain all te arateristis d of te point (O): o s s s s Point (O): maximum meanial power:,o Q W s,o s max Q,o s,o s s s s s s s () W max s s Finally, we obtain in a new way all te lassial results and in partiular te assoiated energy s onversion effiieny at te optimal point CNCA wi depends only on te temperatures of s eat soure and eat sink [8]. e arateristis of tis optimal point (O) an be determined troug oter ways. For example, Cambadal and Novikov ose te temperature as ontrol variable and troug te energy and entropy balanes, tey obtained te meanial power as funtion of te temperature and ten te optimal point is determined by a simple derivation [,]. e drawbak of tis metod is tat one an t diretly ontrol te temperature. Oter autors [8] used a more general metod of Lagrange

9 Entropy, 4 65 multipliers to determine te optimal point but te inonvenient of tis approa is tat it doesn t give diretly te arateristis of te operating urve of te maine. e main advantage of our approa wi onsists of expressing all te variables of te eat engine as funtion of te entropy flow rate is tat te operating range of te maine is expliitly defined by te utoff entropy flow rate and tis oie of ontrol variable allows in a simple way to draw te operating urve of te maine and to determine te arateristis of te optimal point..5. Analysis of te Rate of Entropy Generation e expressions of te rate of entropy generation attaed to eat transfers between te eat soure/ endoreversible onverter and te endoreversible onverter/eat sink an be obtained from simple entropy balane applied to te two ondutanes [f. relation (5)]: s Q (3) s Q (4) e rate of entropy generation an be written as te produt of an extensive variable ( Q ) wit te gradient of an intensive variable assoiated ( ). Ea transformation is aompanied by ertain entropy generation. In te absene of gradient of intensive variable (temperature, pressure, onentration, et.), no transformation is possible (Onsager teory). e total rate of entropy generation in te ase of an endoreversible eat engine beomes: (5) By deriving twie te rate of total entropy generation wit regard to te entropy flow rate, we easily see tat it inreases faster and faster wit te entropy flow rate (f. Figure 7): s s ds d, 3 3 d d (6) In partiular, at point (A) orresponding to zero entropy flow rate, te rate of entropy generation as well as its derivative are zero. By applying a seond order aylor expansion of te meanial power and te rate of entropy generation around, we obtain from Equations () and (6): W s s, (7) By eliminating te entropy flow rate between te two expressions (7), we obtain te relation: s s W W (8) wi sows tat in te neigborood of te origin, te rate of entropy generation is of te seond order wit regard to te meanial power produed. is result provides an explanation for te

10 Entropy, 4 65 paradoxial arater of te Carnot engine wi gives te best energy onversion effiieny wit zero prodution of meanial power! Figure 7. Evolution of te meanial power and te rate of entropy generation as a funtion of entropy flow rate. meanial power rate of entropy generation O A Entropy flux d entropie flow rate B 3. Conlusions and Outlook By assoiating Finite-ime ermodynamis and te Bond-Grap approa, we ave sown in tis paper te interest of seleting te entropy flow rate involved in reversible energy onversion as te ontrol variable of endoreversible eat engines. Indeed, tis oie an be onsidered as a natural and onvenient one to draw te operating urve of te maine. Furtermore, tis oie allows one to obtain in a simple way: e expressions of all te variables as a funtion of te ontrol variable. e range of variation of te ontrol variable. e parametri equations of te operating urve of te maine. e arateristis of te optimal point aording to te maximum meanial power (CNCA effiieny, ). is approa is proposed in opposition to te great majority of existing results, using temperature as ontrol variable (tat is to say intensive variable and not extensive one). In a seond paper we will give te analytial expressions of te arateristis of te optimal operating point of an irreversible eat engine in wi te energy onversion is aompanied by irreversibilities related to internal eat transfer and eat dissipation penomena. By applying tis proposed approa to a termoeletri generator, te eat reovery potential an be estimated wit regard to te pysial parameters of termoeletri ells and eat exangers. Referenes. Cambadal, P. Les Centrales Nuléaires; Armand Colin: Paris, Frane, 957; pp Novikov, I.I. e effiieny of atomi power station. J. Nul. Energy 958, 7, Feidt, M. ermodynamique et Optimisation Energétique des ystèmes et Proédés; Lavoisier: Paris, Frane, 987.

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12 Entropy, Vijay, P.; amantaray, A..; Mukerjee, A. A bond grap model-based evaluation of a ontrol seme to improve te dynami performane of a solid oxide fuel ell. Meatronis 9, 9, Onsager, L. Reiproal relations in irreversible proesses. I. Pys. Rev. 93, 37, Onsager, L. Reiproal relations in irreversible proesses. II. Pys. Rev. 93, 38, Lenoir, B.; Mienaud, J.; Dauser, A. ermoéletriité: Des Prinipes aux Appliations. enique de l ingénieur, Référene 73. by te autors; liensee MDPI, Basel, witzerland. is artile is an open aess artile distributed under te terms and onditions of te Creative Commons Attribution liense (ttp://reativeommons.org/lienses/by/3./).