MODEL OF A CONTINUOUS STIRRED TANK REACTOR USING BOND GRAPH FORMALISM

Transcription

1 4th MAHMOD Venna- Fourth Internatonal Symposum on Mathematcal Modellng February 5-7, 00 Venna Unversty of echnology MODEL OF A CONINUOUS SIRRED ANK REACOR USING BOND GRAPH FORMALISM P. Breedveld, F. Couenne,, C. Jallut, B. Maschke and M. ayakout Correspondng Author : F. Couenne Unversty of wente, Cornels J. Drebbel Insttute for Mechatroncs, Control Laboratory, EL/N 850, P.O Box 7, 7500 AE Enschede, Netherlands Phone.: , fax: , emal: p.c.breedveld@el.utwente.nl Unversté Claude Bernard Lyon-, Laboratore d'automatque et de Géne des Procédés, UCB Lyon - CNRS UMR 5007, CPE Lyon - Bâtment 08 G, 4, bd du Novembre 98, F-696 Vlleurbanne cedex, France Phone.: + ( , fax : + ( , emal: couenne@lagep.unv-lyon.fr Abstract. hs paper gves the generalzed bond graph model of a chemcal reactor whch demonstrates the way to derve models n chemcal engneerng, has then to be adapted. he dscusson s based on a classcal example: the model of a non sothermal gas phase Contnuous Strred ank Reactor (CSR, specfcally the balanced exothermc reacton of Hydrogen and Iodne n Hydrogen Iodde.. Introducton he am of ths paper s to present an alternatve approach to modellng n chemcal engneerng usng port based modellng approach. One of the expected advantages of such an approach s the development of reusable models. hs concern s shared wth E. D. Glles [5], [9] but the approach developed hereafter, s dfferent snce uses port-based modellng and the generalzed bond-graph formalsm as proposed by P. Breedveld []. It combnes the axomatc relaton of the thermodynamcs wth pars of conjugate varables (ntensve and extensve whose the product gves a power term. But secondly t s a graphcal network language whose the model s composed of multport elements nterconnected by a power contnuous network. hs paper s ntroductory and ams to provde some tentatve answers to ths problem by analysng n what extent, n our partcular example, t s true. he port based modellng approach mples the use of the basc thermodynamc axoms. So the ntensve or extensve nature of thermodynamcs varables s thoroughly used. It leads to wrte the entropy balance, to formulate the rreversble entropy producton term and to rewrte the consttutve relatons n such a way the ntensve varables are expressed as functon of the extensve one. he ntensve varables encountered n thermodynamcs are the temperature, the pressure and the chemcal potentals of components of the systems. he extensve ones are the entropy, the volume and the mole number of each spece. he extensve varables are not classcally used by chemcal engneers to descrbe the process. Instead the measurable varables such as the temperature, the pressure and the mole numbers and generally the energy balance are used. As mentoned n [9], these models are n general not reusable and a slght modfcaton of hypotheses of functonng leads to a complete rewrtng of the model. A thrd consequence of the port based modellng s the rewrtng of the knetcs rates n functon of ts conjugate varables, the forward and reverse affntes, nstead of concentratons of reactants. If bond-graph models are extensvely used n mechancal or electrcal areas, t s no more true n chemcal processes and more precsely for chemcal reactors. he bblography s qute reduced n ths doman. In [0], Oster et al. propose the descrpton of the general balanced chemcal reacton n a closed wellstrred reactor n usng the bond-graph formalsm. hey assume moreover that constant nternal temperature and pressure and consequently are nterested n the representaton of the Gbbs free energy G. From the Gbbs relaton and the entropy balance, the authors propose a bond-graph model of the reacton. As a consequence, the rreversble entropy producton n ths bond-graph s not represented. In the case far from equlbrum, t appears that the affnty has to be splt n two terms, f r that s to say a forward affnty A (correspondng to the drect reacton and a reverse affnty A (correspondng to the reverse reacton and a two port element s necessary to descrbe the dsspaton f r due to the reacton. he reacton rate r s then expressed wth respect to A and A. hs V

2 representaton does not take nto account the thermal doman but only the materal one. It s the reason why a resstve element s suffcent. Nevertheless the work of Oster et al. paves the way of the present paper. In [], P. C. Breedveld completes the prevous model of the reacton by couplng the materal doman and the thermal one va a RS element. Hence the model consders the nternal energy and allows to represent the frst prncple (the conservaton of energy.hs wll be done n our example. Let us remark that D. Karnopp extends these models to electrochemcal systems n [7]. On the other hand, there exsts a lterature concernng pseudo-bond graph models of contnuous strred reactor : see for nstance [], [6]. In general, the authors do not wrte down the entropy balance but the enthalpy balance (they take conjugate par as enthalpy flow and temperature and for the materal doman, the conjugate par s gven by the molar flow of consttuents and the concentratons. he reacton knetcs s then wrtten n term of concentratons. he authors keep the classcal approach of chemcal engneerng and are concerned about smulaton and not structured port-based modellng. In ths paper, we provde a bond graph model of a complete chemcal reactors usng systematcally the power conjugate varables. Furthermore, the prevously true bond graph models of chemcal reactons proposed for nstance by [0] have been augmented n order to model a complete open contnuous strred chemcal reactor. For chemcal engneerng ths approach s novel n the sense that the elementary phenomena are represented as basc mult-port elements whch can be locally connected. Furthermore the local use of port-based concepts that satsfy the regular constrants guarantee that the global model consstng of connected basc models also satsfes the basc balance equatons (energy, due to the bond graph grammar. he reacton under consderaton s the gas phase balanced hydrogen odne reacton H +I HI. For smplcty, we dentfy the consttuent H to P, I to P and HI to P. he chemcal reacton and the jacketed reactor n whch the reacton takes place are modelled wth the followng assumptons: - the consttuent are perfect gas. - the knetcs of the forward and reverse reacton satsfy the hypothess of mass acton consttutve relaton.he global reacton rate (n mol.m - s - s then gven by r = k (C C k ( C wth = = and = and C, C, C represent the respectve molar concentraton of the consttuents P, P, P of the reacton. Moreover they verfy the Arrhenus law. - the molar heat capacty cp for any spece P s supposed to be constant. - the reactor s contnuous and perfectly strred. - he reacton volume V s constant. he outlet flow rate s set up such that pressure P s constant and equal to atmospherc pressure. - hermal exchanges Q & between the reactor and the neghbourhood represent the effect of a jacket. - At the nlets, the three pure components are njected separately. he organzaton of the paper s the followng : n secton we present the C element. We formulate the consttutve relatons of the energy storage element n terms of chemcal potentals for the gas mxture that s nvolved n the reacton n such a way that the correspondng energy-storng element may admt ntegral causalty assgnment. he most dffcult task wll be the wrtng of the thermodynamcal propertes wth respect to extensve varables as already emphaszed n [], [4]. In secton, the balance equatons correspondng to 0 juncton are detaled. he chemcal reacton s descrbed n secton 4; t corresponds to the RS element and the FSto transformer. Secton 5 s devoted to the transport phenomena; they correspond to the mxng element and the MF elements of the model. Fnally the boundary conons are analysed n secton 6. And we conclude n secton 7. Fgure below gve the complete bond-graph model of the reactor.. he C element : the thermodynamc propertes of the mxture As defned n classcal thermodynamcs, the states of the reactor are descrbed usng to the extensve varables whch are the volume V, the entropy S and the mole number of each chemcal component, the n s. he concept of local thermodynamc equlbrum mples that whle the system as a whole s not n equlbrum, a local equlbrum exsts n each small element of the medum. hs mples that at least locally the Gbbs law, gven below, s vald : du = ds PdV + µ dn ( = V

3 where U s the nternal energy, P the pressure, the temperature, and µ the chemcal potental of the consttuent P. out F out h µ µ 0 ex n h n F Sf MF Pressure_constrant conv_entr_outle t P dv dn C µ ds MIXING σ mx conv_entropy_nle t MF 0 Q & Sf σ reac C FSto Fgure he bond graph of the reactor In the sequel, we shall derve the consttutve relatons of the mxture by expressng the ntensve varables, P and µ as functons of the extensve varables S, V and n. herefore we shall frst express the dfferent molar propertes. In the sequel we note the total number of moles by N = n = wth n the number of moles of consttuent P. Moreover we shall use the superscrpt n order to denote propertes for the pure speces... Enthalpy calculaton RS Assumng that each reactant s a perfect gas, leads to the followng dfferental for the molar enthalpy for pure speces dh = cp d. he specfc enthalpy of pure consttuent P at temperature s gven by h ( = c p ( ref + h ref. ref denotes the reference temperature and h ref denotes the molar reference enthalpy of consttuent P. Let us remark that n the case of a reacton, the h ref s have to be chosen wth regard of the reacton under consderaton. Hence the enthalpy H of the mxture of perfect gas n the reactor s gven by ( c ( +, n, n = n h ( = n h ( = n p ref h ref = = = H (, n (.. Entropy calculaton From the perfect gas hypothess, the molar entropy of pure component P, at and P, s gven by :

4 P s = + where R represent the perfect gas constant and Pref the reference cp ln( R ln( sref ref Pref pressure. For the mxture and a molar fracton = y = n, the molar entropy s then gven by the vector: N s (, P, y = s (, P R ln(y,, ( and the entropy of the mxture S by :, n, n = n s (, P, y = S (, P, n ( 4.. Chemcal potental calculaton For each consttuent P, the expresson of the molar chemcal potental s In the mxture, we have ( + = h s µ = h s µ, P, y = µ (, P R ln(y wth µ ( 5.4. he expresson of the consttutve relatons By expressng from relaton ( 4 and from the perfect gas equaton PV = NR ( 6 we obtan : = + ( + NRref ref exp S R n ln y RN ln n sref n cp RN = VP ( 7 ref = = hen the expresson of P wth respect to V, S and n s may be deduced from equatons (6 and (7 he expresson of the chemcal potentals µ s follows by ( 5.. he balance equatons (the 0 junctons Let us recall the expresson of the materal balance for the spece P : dn = Fe Fs + rvv ( 8 where F e, Fs respectvely represent the nlet and outlet molar flows and s the sgned stochometrc coeffcent (- f t appears on the left sde of the reacton scheme, + n the other case. he ntensve common varable assocated to ths balance s the chemcal potental vector. he energy balance s gven by : du Feh e + Q = & F h or s s Fe h e + Q& = Fsh s + where h e, h s respectvely represent the nlet and outlet specfc molar enthalpy. Substtutng F s n the thermal balance by ts value n ( 8, we obtan the classcal thermal balance : d c p Fe (e + Q& = rvv( h + ( n cp ( 9 wth e the nlet flow temperature and Q & the heat comng from the jacket. he entropy balance s gven by ds convectve terms = Fes e Fss s Q& + ext + σ where s s the rreversble entropy producton and s e, ss respectvely represent the nlet and outlet molar entropy. Note that the entropy balance s wrtten at the temperature of the mxture (the common effort varable of the 0-juncton. Wth the local equlbrum hypothess, we obtan by substtutng ( 8 and ( 9 n relaton ( 0: ( 0 dh σ = Q& Q Fe & ( ( h + µ e s e Fs h s s s ext dn (

5 and snce the mxture s supposed to be homogeneous, σ = mxng jacket reacton & & Q Q Fe ( ex e rvv µ + + ext µ = = Wth the exergy ex e = h e (e s e (e, Pe. Each term of ths expresson wll be detaled n the next two sectons. 4. he reacton RS element and F due to stochometry hs RS element gves the expressons of the rreversble entropy producton and of the reacton rates. he F element s qute classcal snce t expresses the stochometrc relaton : µ rv 0 A f 0 rvf = µ and A r 0 0 rv = 0 ( µ rvr rv 0 wth r = k (C C, r = k ( C. From the expresson ( 5 of the chemcal potentals, the Vf Vr expresson of the perfect gas and of the reacton rates wth respect to concentratons, we obtan : ( + V A f ( µ (, P + µ (, P r = k ( exp( exp( vf N R R ( 4 V A ( µ (, P r = k ( exp( r exp( vr N R R And the rreversble entropy term due to reacton can also be expressed by: σ = V rvf reac [ A f A r ] ( 5 rvr Equaton ( 5 s the reacton term n (. ( 5. he transport phenomena 5.. he convectve terms Frst let us analyse the convectve terms (n expresson ( 0. hey are represented by two MF elements whch respectvely express: G = s s s formed by specfc entropy at the nlet, the convected - At the nlet, wth the gan e ( e e e entropy term F e s e s equal to G ( e Fe Fe Fe. he conjugate varable s the temperature of the mxture. By dualty, the expresson of the ntensve varable at the materal mult- port of the MF element s nlet. e G. Note that the conjugate varable s the vector of molar flows ( F F F - At the outlet, he same reasonng s appled. he convected entropy term F s s G ( F F F wth the gan ( s s e e e at the s equal to s s s G = s formed by specfc entropy (functon of the state varables n the reactor. he conjugate varable s the vector of molar flows ( F s Fs Fs at the outlet. 5.. he mxng element Recall that at the nlet, contrary to the outlet, the speces are not n thermodynamcal equlbrum wth the speces n the mxture. hs s expressed by the fact that the term ( ex e µ n equaton ( s not zero. hs term s the drvng force for the rreversble entropy producton due to the mxng (the mxng term n (.

6 he consttutve relaton conssts frst n the expresson of the rreversble entropy creaton σ mxng = = F e ( ex µ e. On the materal part, the effort varable s the term ( ex µ e whch s computed, not usng the nlet molar flow F e but the specfc enthalpes and entropes of the speces n the reactor and at the nlet. hese varables are computed from the states of the speces. 6. he Boundary conons 6.. he pressure constrant he bond graph s termnated on the spatal doman by makng the followng assumptons. Frstly the volume of the gas s the volume of the reactor whch s supposed constant. dv = 0 ( 6 Secondly, the pressure remans constant n the reactor. hs s expressed as a constrant P = Pa. hs s realzed by a port element couplng the spatal port of the C element wth the materal bond at the outlet. he mult port s on one sde a zero source flow for the volume rate and on the other sde, the outlet molar flow s computed such that the Lagrange multpler satsfes the pressure constrant. Indeed, t can be shown that the pressure constrant s of ndex and actually usng ( 6, one can compute the flow: ( 7 ( N F = F r V + + F c ( Q& s e v r V c ( e p e v p ref = = = n cp 4 4 = H 6.. he heat supply he jacket s represented as a heat supply by couplng the thermal port of the C element to a entropy flow source. 6.. he nlet he nlet s represented by a source of molar flow and a degenerate C element whch provdes the state functons of the reactants at the nlet :, h, s. e e e 7. Concluson We have proposed a generalzed Bond graph model of the contnuous strred tank reactor for the balanced hydrogen odne reacton. he thermodynamcal propertes of the reactants have been wrtten as qualtatve relatons gven ntensve varables as functons of the extensve ones. he reacton has been represented by an RS element wth port varables beng the global reacton rates and forward and reverse affnty. Its consttutve relatons have been wrtten n term of these port varables. hs reactor s an open system and a specal attenton was devoted to the mass and heat transport phenomena. In partcular two-port element represents the phenomena arsng from the non-equlbrum between the reactants n the reactor and at the nlet. In ths paper, a smple case has been treated for whch all the computatons can be done analytcally. In future work we shall nvestgate more complex stuatons and how the numercal models of the thermodynamc propertes and the reacton knetcs may be ncluded n such port based models. It remans also to demonstrate how such models may be adapted to changes n the modellng assumptons : non constant heat capacty, non perfect gas, cascade of reactors, varyng pressure nsde the reactor, Acknowledgements: hs work has been done n the context of the European sponsored project GeoPlex wth reference code IS Further nformaton s avalable at 8. References. Auslander D.M., Oster G.F., Perelson A. and Clfford G., On systems wth coupled chemcal reacton and dffuson, rans. ASME, J. Dyn. Syst. Meas. Control. ( Breedveld P. C., Physcal systems theory n terms of bond graphs, PHD hess, 984.

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