1 Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is n open ccess repository tht collects the work o Toulouse reserchers nd mkes it reely vilble over the web where possible. This is n uthor-deposited version published in : Eprints ID : To link to this rticle : DOI : /cjce URL : To cite this version : Mrin Gllego, Jun Crlos nd Olivier-Mget, Nelly nd Hétreux, Gilles nd Gbs, Ndine nd Cbssud, Michel Towrds the modelling o het-exchnger rector by dynmic pproch. (2015) The Cndin Journl o Chemicl Engineering, vol. 93 (n 2). pp ISSN Any correspondence concerning this service should be sent to the repository dministrtor:
2 Towrds the Modelling o Het Exchnger Rector by Dynmic Approch Mrin Gllego Jun Crlos,* Olivier Mget Nelly, Hetreux Gilles, Gbs Ndine nd Cbssud Michel Lbortoire de Génie Chimique. UMR 5503 (INPT/CNRS/UPS). 4, Allée Emile Monso, BP F 31432, Toulouse, Frnce The im o this pper is to present the development o simultion tool in order to ssess the inherently se chrcteristics o het exchnger rector (HEX) operting rection systems. The modelling o stedy nd trnsient sttes o HEX rector is perormed ollowing hybrid dynmic pproch. The globl dynmic behviour o this rector cn be represented by severl continuous models, which re bounded by stte or time events. Ech continuous model is deined s system o prtil dierentil lgebric equtions. The numericl scheme is bsed on the method o lines. Specil ttention is pid to the model initiliztion nd simultion strtegy o the strt up phse is presented. The vlidtion o the model is mde by numerous exmples, such s the simultion o n exothermic rection. Keywords: Dynmic Hybrid Simultion, xil dispersion model, method o lines, het exchnger rector INTRODUCTION During the lst decdes, dvnces in rector design hve mde possible to trnspose trditionl btch chemicl processes to continuous intensiied systems. As mtter o ct, discontinuous rectors present technologicl limittions tht my result in sety nd productivity constrints. These drwbcks re minly due to their poor het exchnging perormnces. In process intensiiction, het exchnger rectors (HEX rectors) re well known or their therml nd hydrodynmic perormnces  nd re well suited or highly exothermic rections.  The ppliction o HEX rectors or multiphse rections is subject o interest in process sety. Even though the use o this type o processes is n interesting lterntive to btch systems, the number o prmeters to tke into ccount or their design (system kinetics, hydrodynmics, het nd mss trnser dt nd multiple chnnel conigurtions) mkes diicult their ppliction to multiphse systems. In this context, dynmic simultion is useul tool to study the system rom process sety point o view nd to nlyze the inluence o the dierent operting prmeters. In this pper, we ocus on the irst steps o the modelling o intensiied HEX rectors or multiphse systems. Chemicl rections in multiphse systems led to highly non liner problems. Some dynmic models or homogeneous systems re studied in literture. [2,3] Recent studies on stedy stte modelling or multiphse pplictions hve been proven useul or the design o HEX rectors.  However, studying the system dynmics is n essentil step towrds complete understnding o ny new equipment in terms o process sety. [5,6] The min objective o this study is to vlidte the hydrodynmic nd therml model o the HEX rector. This work is then orgnized s ollows. In the section Rector Model, brie description o the rector is mde nd the model equtions re written. In the section Resolution Methods, the model structure is presented nd the pplied numericl resolution strtegy is discussed. The hybrid dynmic pproch is then explined vi the simultion o the strtup o the rector in the section Dynmic Hybrid Simultion. In the section Simultion Results, some preliminry results or the vlidtion o hydrodynmics equtions re presented nd discussed. Firstly, the simultion o the strt up nd the illing o the rector illustrtes the hybrid pproch. Next, the hydrodynmic model is vlidted thnks to the simultion o residence time distribution experiment nd the comprison o the results with experimentl dt; the simultion o chnge o the properties o the inlet luid. Then, the therml prt o the model is studied through the simultion o three exmples: heting by the wll t constnt temperture, n exothermic system with n dibtic behviour nd n exothermic rection with het exchnge with wll t constnt temperture. Finlly, Conclusion nd Perspectives presents some conclusions nd perspectives. REACTOR MODEL Generl Description o the Rector Designed with plte het exchnger modulr structure, HEX rectors re vilble ollowing wide vriety o conigurtions.  Rection nd cooling pltes re seprted by therml conducting plte. The number o pltes nd the geometric conigurtion re chosen ccording to mixing nd therml requirements, luid properties, rection prmeters nd sety considertions. Process nd utility lows re mostly circulting perpendiculrly. For simultion purpose, it is only possible to deine co current or counter current low between two contiguous pltes by considering the min low direction o ech chnnel. To completely deine the low in recting chnnel, three sptil coordintes would be needed. *Author to whom correspondence my be ddressed. E mil ddress: DOI /cjce.22119
3 source term or chemicl rections within the phse, clculted s H M k ¼ XN r r¼1 y k;r R r source term representing mss trnser between phses, given by QM k ¼ XN Q ðgþ k g¼16¼ The velocity is clculted thnks to this eqution: Figure 1. Chnnel nd low conigurtion.  Figure 1 shows one possible low conigurtion or the recting pltes. The utility low presents Z type rrngement nd the process low circultes in single chnnel in order to oer the highest possible residence time or rectnts. N p is the totl number o pltes nd P 1, P 2, P Np re reltive to the Np pltes. F eed nd F out represent the inlet nd outlet low rte respectively. Moreover, the utility low rte is widely superior to the process low rte. This ct implies tht the dierent geometric conigurtions o the utility low cn be neglected or the evlution o the therml trnser. Model Equtions Even i the low structure within the chnnel hs threedimensionl nture, clssicl models in chemicl rection engineering dmit geometricl simpliictions. Mss nd het blnces within the rector re written s system o Prtil Dierentil nd Algebric Equtions (PDAE) in one dimension. HEX rector hydrodynmics hs been chrcterized during the lst decde nd previous studies show tht the single phse low is well represented by the xilly dispersed plug low rector model. [1,8,7] Hydrodynmic Model For multiphse low, simpliied one dimensionl low model hs been developed. As the homogeneous xilly dispersed plug low model, it tkes into ccount the xil dispersion eects in continuous phses vi dispersion term. Consider one luid phse, lowing throughout the chnnel length. Let z be the sptil coordinte, ollowed by the men low. The prtil molr blnce o component k within this phse presenting xil dispersion is written ðe x k C Þ D x k C ðe x k C u Þ þ HM k þ QM k : At ny point within the rector, the time rte chnge o the molr concentrtion depends on our terms given by the right hnd side o Eqution (1): molr diusion lux, convective lux, ð1þ W u ¼ Ve C M : The phse rction nd the phse velocity re two vribles intrinsiclly relted. The complexity o the multiphse interctions mkes mndtory the cquisition o some experimentl observtion nd dt to eed mcroscopic one dimensionl model. The phse velocity nd the phse volume rction re implicitly computed with the phse mteril blnce, nd, in order to complete the PDAE system, speciic constrints on phse volume rctions re needed. These constrints depend on the ctul multiphse low regime. We cn ssume s irst pproximtion tht ll phses low t the sme velocity, s in slug low regime. The phse mteril blnce is given by the ðe C Þ D C ðe C u Þ þ X k H M k þ X k A constrint on the phse volume rction is given by 1 ¼ XNw ¼1 e : Therml Model ð2þ Q M k : Temperture grdients between phses cn be neglected. The inluence o the riction loss on the energy blnce is neglected. As usully done in chemicl rector modelling, we cn ssume n isobric system. The energy blnce is X e C h H T ð3þ ð4þ! X e C h u þ H T þq T : is source term representing het production due to chemicl rections: X H T R r DH r: r Q T represents the het trnser between the luid nd the rector wll: Q T ¼ UðT T w Þ: ð5þ
4 As or most pplictions, enthlpy dependences on the pressure nd chemicl potentils re negligible. For n isobric system, enthlpy or ech phse is then clculted s ollows: Z u h ðtþ ¼ h re þ Cp du: T re Boundry Conditions Boundry conditions pplied to the irst nd lst cells o recting plte o length L re deined in Tble 1. Ech boundry cn either be opened or closed to dispersion. In our cse, n opened boundry is deined to link rection pltes. Closed boundries re pplied to the inlet nd outlet o the rector. Some constitutive equtions or the clcultion o physicl properties such speciic het, therml conductivity, nd other model prmeters such s the xil dispersion coeicient, nd the het trnser coeicient re lso included in the PDAE system. RESOLUTION METHODS The hybrid dynmic model hs been developed in MATLAB. For the prtil dierentil Equtions (1) nd (5), the chosen resolution scheme is derived by pplying the method o lines.  This method proceeds in two min steps. Sptil derivtives re irst pproximted using discretiztion method (inite dierences, inite volumes, or inite elements). The resulting system o semi discrete (discrete in spce nd continuous in time) equtions cn be integrted in time using one o the Ordinry Dierentil Equtions (ODE) solvers rom MATLAB. A suitble solver or sti odes, ode15s, hs been used in this work. Ode15s is vrible order solver bsed on the numericl dierentition ormule (NDF), which re vrint o bckwrd dierentition ormule (BDFs or Ger s method). [10,11] The inite volume method hs been used s it is conservtive rom construction or the modelling o the hybrid system. The choice o conservtive scheme is n essentil step towrds the chievement o generic cell model. In order to illustrte these spects, the strt up nd the illing o the rector re simulted. Then, the inite dierence method is used or the simultion o the hydrodynmic, therml nd rective behviour o the rector. Finite Dierence Approximtion The semi discrete equtions re obtined by replcing derivtive terms in Equtions (1) nd (3) by convenient choice o inite ð6þ Figure 2. Finite dierence grid. dierence pproximtion. Figure 2 represents the regulr grid o the inite dierence used in this work. As presented by Vnde Wouwer et l.,  dierentition mtrixes cn be used or computing derivtive pproximtions. This opertion is done strightorwrd by multiplying vector vlues by the mtrix corresponding to the chosen pproximtion. Specil ttention should be given to the numericl stbility o the resulting scheme. Detiled inormtion on its implementtion nd stbility is vilble in literture. [13,14] Consider the homogeneous version o Eqution (1) with constnt D x. When using n upwind irst order pproximtion or the irst order derivtive nd second order centred pproximtion or the second order derivtive, the resulting discretized scheme or Eqution (1) is given s i C C iþ1 2C i þ C i 1 ¼ D x x i þ 2 C i C i 1 Dz 2 Dz x k;i x k;i 1 x iþ1 2x i þ x i 1 þ C i Dz Dz 2 x k;i x k;i 1 C i C i 1 u i u i 1 u i C i þ u i x k;i þ C i x k;i þ H M : Dz Dz Dz Finite Volume Semi Discretiztion Figure 3 illustrtes the structure o the model. In this cse the physicl system corresponds to recting plte with mendering squred chnnels, s studied by Anxionnz et l.  The chnnel is etched inside the conducting mteril. By ollowing inite volume pproch, the recting chnnel cn be discretized into N unitry cells, which re plced in such wy tht they represent the conigurtion o the ctul system. The inlet low rte is F eed nd the outlet low rte corresponding to the low leving the lst discretiztion cell is F out. Ech cell is ble to trnser mss nd het with other neighboring cells. Interctions o the i th cell re considered only with the cells tht shre boundry surce. Mss luxes re exchnged by two sides, while therml luxes cn be exchnged by the 6 sides o the cell. ð7þ Tble 1. Boundry conditions or the multiphse dispersion model Boundry conditions t z ¼ 0 Boundry condition t z ¼ L Closed Closed ðu e x k C Þ in ¼ u e x k C x k C Þ x ðu e x k C Þ out ¼ u e x k C x k C Þ x z¼0 þ ðu e h C Þ in ¼ u e h C x ðu e h C Þ out ¼ u e h C x z¼0 þ z¼l z¼l Opened u e x k C x k C Þ x ¼ u e x k C x k C Þ x u e h C h C Þ x ¼ u e h C x z¼0 þ z¼0 þ Opened u e x k C x k C Þ x ¼ u e x k C D x out u e x k C h C Þ x ¼ out u e h C x k C Þ z¼l z¼l
5 Figure 3. Geometric structure nd discretiztion o single recting plte. The integrl orms o Equtions (1) nd (5) result in the ollowing @t e i C i x k;i ¼ V M k;iþ 1 M k;i 1 2 2! X e C h ¼ V X T iþ 1 2 X þ V HM k;i þq M k;i ð8þ! T þ VðH i 1 Ti þq Ti Þ 2 terms re deined s numericl luxes crossing the control volume boundries. Notice tht the irst term o right hnd sides o Equtions (8) nd (9) cn be regrded s inite dierence pproximtions o diusive nd dispersive terms in Equtions (1) nd (3). The intercell luxes cn tke two orms. The purely convective luxes represent the idel plug low eture o the rector nd re written s M k;i ¼ u i e i C i x k;i ð10þ T i ¼ u i e i C i h i ð11þ wheres convective dispersive luxes tkes into ccount the xil dispersion o the low. Fluxes or mteril nd het blnces re respectively given by M k;i ¼ u i e i C i x D i C i x k;i Þ x;i dz T i ð9þ ð12þ ¼ u i e i C i h i i x;i : ð13þ For the ollowing sections, the chosen representtion o these luxes is shown in Figure 4. As previously mentioned, the rector presents modulr structure. It is composed o pltes, nd pltes re composed o elementry cells (Figure 3). The properties o ech cell re determined by het nd mss conservtion; however, the cell model structure my not be the sme or inner nd boundry cells ccording to the chosen numericl scheme. In ddition, the pproximtion method or i þ 1/2 my result in dierent lux deinition or the two consecutive cells (i nd i þ 1). Consider tht the luid is lowing rom let to right. The irst order upwind estimtion or the intercell luxes hs been chosen despite its numericl dissiption s it hs the dvntge o being unconditionlly stble in the presence o steep ronts.  A second order centred inite dierence pproximtion is chosen to clculte the derivtive term or dispersion. M k;i 1 2 ¼ u i 1 e i 1 C i 1 x i 1 D x;i 1 e i C i x k;i e i 1 C i 1 x k;i 1 Dz ð14þ As or the irst nd lst luxes, the pproximtion is stggered in order to keep the sme order o ccurcy. Thnks to the conservtive inite volume semi discretiztion, it is possible to dopt one single eqution or dierent cell models. A generic multiphse model is then given ðb ie i C i x k;i Þ ¼ V M;in X b i e C h ¼ V M;out k;i X þ Vb i ðh M k;i þq M k;i Þ T;in X þ Vb i ðh Ti þq Ti Þ T;out! ð15þ ð16þ Figure 4. Deinition o intercell luxes. The vribles M;in k;i nd M;out k;i re molr luxes o component k tht goes rom nd to the i th cell, respectively. T;in nd t;out re the therml luxes entering nd leving the i th cell. Here, b i is the rction o the totl volume occupied by the luid phses in the ith control volume. b i is the sme or ll the phses nd is equl to one once the cell is ull. This stte vrible hs been introduced to tke
6 Figure 5. Hybrid dynmic system. into ccount the representtion o empty or prtilly illed cells with the sme model. The globl model cn be modiied by exclusively chnging the lux deinition t ech stte. DYNAMIC HYBRID SIMULATION The continuous equtions described previously correspond to the modelling o the mx stte o cell (i.e., the stte where ll luxes nd luid phses exist). In our cse, the stte vector corresponds to the mx stte o our system. Nevertheless, the model structure chnges ccording to the discrete spects. For this reson, this section presents the hybrid dynmic spects o the model. Generl Aspects The objective is to study the HEX rector in stedy stte but lso in vrious trnsient sttes (strt up, shutdown, response to disturbnce, etc). In this context, hybrid dynmic model o the system hs to be estblished (Figure 5). In generl mnner, this pproch leds to mke discrete model S d interct with piecewise continuous or discontinuous model S c. In our cse, the hybrid model o the system is ormlized by Object Dierentil Petri Net (ODPN).  When hybrid dynmic system evolves, it psses through dierent conigurtions q, lso clled discrete stte x d (or modes) o the system. Ech conigurtion q is identiied by plce p q. I the continuous stte vribles X q must evolve in this conigurtion, then dierentil plce p q identiies this conigurtion q nd dierentil nd lgebric equtions (DAE) system F q is ssocited with it. In consequence, the evolution o the continuous vribles X c o the system is driven by piecewise continuous model while the discrete prt relizes the mngement o the legl sequences o switching between the continuous sub models. Ech continuous sub model is then speciic conigurtion o the Equtions (1) to (2). In order to detect chnge in system S, stte events or temporl events determine the crossing o ech trnsition t i. An event is usully mterilized by n lgebric eqution unction noted e i. It is monitored s soon s ll previous plces o the trnsition t i re mrked nd it is considered s n dditionl condition to the iring o the trnsition t i. When crossing the trnsition t i, ctions cn be executed. Action, clled j i, llows, or exmple, clculting the initil vlues o continuous sttes nd their time derivtives, in ccordnce with the ollowing conigurtion. Figure 6 illustrtes the evolution rules o this kind o Petri Net, pplied to system o dierentil nd lgebric equtions. From topologicl point o view, complex system such s btch or continuous processes must be decomposed hierrchiclly into severl entities (Figure 7). At the irst topologicl level, the control prt (the controller) nd the opertive prt (the process) re clerly distinguished. The controller is modelled by Petri net describing the recipe tht the process must ollow. This recipe is deined by continuous vlues (quntities o rectnts, operting conditions, etc.) nd genertes the events tht drive the simultion o the opertive prt. Figure 6. Exmple o Petri net representing the evolution o hybrid dynmic system.
7 Figure 7. Hierrchicl decomposition o the system. Concerning the modelling o the opertive prt, the ODPN cn be structured in dierent wys. It depends essentilly on the nture o the study to chieve nd the topologicl level considered. In this rmework, one or more Petri Nets re developed in order to represent the phenomenologicl evolution o the system. In prticulr, they include the het nd mss trnser mechnisms tht re speciic to the opertion steps, the evolution o the phse system s well s min nd secondry rection kinetics. In this rticle, we restrict the study to the strt up phse nd illing o the HEX rector. The Strt Up nd Filling Phse The strt up trnsient simultion is criticl spect in dynmic modelling, simultion nd control. The risk o incidents is higher during strt up thn in stedy stte opertion. Previous studies on control o HEX rectors or exothermic homogeneous systems highlight tht dynmic hybrid pproch llows robust control o strt up or sety purposes. [17,18] Two strtegies cn be implemented or the dynmic simultion o this phse. I ech vrible in the stte vector hs physicl mening, or ny stte o the system, then it is possible to build complete model rom the beginning o the simultion nd to reduce the impct o the hybrid etures. The second strtegy corresponds to progressive construction o the model. The importnce o this phse relies on the ct tht ll vribles re initilized. The more the system is complex the more diicult is the reserch o coherent initil stte. Hybrid dynmic modelling llows stged mngement o the trnsitions, including n initiliztion o ech sub model.  In this study, the second strtegy is dopted. According to the process, n empty rector is rector illed with n inert stgnnt luid phse, which is mostly gseous N 2 or ir. Filling recting plte with liquid phse is multiphse process nd the shpe o the trnsient luid luid interce evolves in complex mnner. A qulittive representtion o this evolution is the subject o computtionl luid dynmics studies, which require multidimensionl sptil grids nd imply higher computtionl burden. Such detiled description is not prt o the scope o this work. The one dimensionl model presented in the previous sections tkes into ccount low phenomen by integrting mcroscopic prmeters (xil dispersion coeicient, het nd mss trnser coeicients, explicit dispersed phse velocity/ volume rction equtions). The evolving interce is modelled by integrting void rction prmeter. For the simultion o the illing o the rector, severl hypotheses cn be considered: The illing in the sme time o ll the cells o the rector; The illing o cell only when the previous one is ull; And mixed pproch o the both previous cses. In this work, the second hypothesis is considered, since the conigurtion nd experimentl conditions o our rector justiy its use. Globl structure behviour In order to illustrte the principles o the progressive construction o the model, consider the illing step o recting chnnel, which is discretized into 4 unitry cells. Figure 8 shows the Petri net tht illustrtes the evolution o the model or this cse. Single rrows o cells represent convective luxes nd double rrows represent convective diusive luxes. At the initil stte, the recting chnnel is modelled s series o empty cells. The thermodynmic vribles tht deine the stte nd the properties o the rectnts do not hve physicl mening s long s the rector remins empty. Thereore, the het nd mss blnces re excluded rom the model beore the introduction o rectnts. The progressive model construction begins right ter the occurrence o n externl event, such s the opening o n inlet vlve, or the strt up o pump. The trnsition t M1 contins the necessry conditions to switch to the next model. When trnsition t M1 is ired, its ctions re perormed. For this prticulr trnsition, the inlet low o the recting chnnel is equl to F eed. The model equtions or cell A chnge by considering n inlet lux, nd new vribles re properly initilized. A token mrks the irst continuous plce M 1. Once cell A is illed, trnsition t M2 is ired nd the convective lux between the irst nd second cell is instntited. In the sme wy, the lux between the cells B nd C is set to convective when the trnsition t M3 is ired. Notice tht the convective dispersive lux rom cell B to cell C is set t t M4, ter the third cell is illed. The derivtive estimtion o
8 Tble 2. Cell sttes nd molr lux deinition o chemicl species k (Eqution (15)) Cell type Flux deinitions ¼ 0 M;ink;i ¼ 0 M;outk;i M;in k;i M;ink;i ¼ 0 M;outk;i M;in k;i M;outk;i ¼ F eed or the inlet cell V ¼ u i 1 e i 1 C i 1 x k;i 1 otherwise. ¼ u i 1 e i 1 C i 1 x k;i 1 ¼ u i e i C i x k;i M;ini;k M;outi;k M;ink;i M;outk;i ¼ u i 1 e i 1 C i 1 x k;i C x D k Þ x;i 1 ¼ u i e i C i x k;i C x k Þ x;i ¼ u i 1 e i 1 C i 1 x k;i 1 D x;i 1 ¼ u i e i C i x k;i C x k Þ i 1 i 1 Figure 8. Petri net o the strt up step. M;ink;i M;outk;i ¼ u i 1 e i 1 C i 1 x k;i 1 ¼ u i e i C i x k;i D C x k Þ i the dispersive lux term cn be done i the vribles or the irst three cells re deined. The discrete model evolves throughout the time spn. The inl structure is chieved when the recting chnnel reches trnsition t M5 nd the rector is completely illed. During the mrking o the continuous plce M 5, the simultion continues. The simultion ends when time is equl to t inl. This condition is the event o the trnsition t M6. Unitry Cell behviour Bsed on the globl structure behviour described in the previous sub section, this Petri net ormlism cn be used to describe the behviour o single cell. This cell cn tke six dierent sttes considering the current vlues o in nd out. The dierent cell types re denoted using the nomenclture presented in Tble 2. These cell types re described s ollows: Empty Cell: This type represents the empty stte o our cell. It is pplied i three conditions re veriied. The irst condition is the bsence o ny molr hold up (the cell is ctully empty). Secondly, there is no low or energy lux tht crosses the cell boundries. Thirdly, there re no sinks or sources considered. Consequently, it is unnecessry to pply the blnce equtions to the control volume, s ny o the luid vribles hve physicl mening. Filling Cell: A prtilly illed cell with only one inlet lux is modelled s single stirred tnk rector with one inlet low. This stte pplies exclusively during the illing phse. Axilly Dispersed Cell: The xilly dispersed cell represents control volume with two convective dispersive luxes crossing its boundries. The dispersive prt o lux cn be introduced once ll the cells needed to the pproximtion o the luxes re illed. This cell type corresponds to discretized cell o the xil dispersion model. The prtil molr blnce is given by Eqution (1). Plug Flow Cell: In order to be ble to pply the multiphse dispersion model (i.e., Eqution (1)), the cells used or the pproximtion o the dispersive lux term (e.g., cells A, B nd C or the pproximtion o the A B lux) need to be lredy illed. When it is not the cse, the dispersive term is neglected. The Plug Flow Cell represents this cse. Only the convective prt o the lux is considered. I this model is pplied to ech cell o the rector, the overll model would correspond to the plug low rector model. Opening Cell: The lux entering the cell is convective dispersive nd the lux leving is convective. The irst non illing cell o the series remins n opening cell i the recting plte is closed to dispersion t the entry. Closing Cell: This cell stte is chieved when the inlet lux is convective dispersive nd outlet is convective. It is used s trnsition cell stte between the plug low nd the xilly dispersed cell model nd trnsltes the discontinuity o the lux. Typiclly, the irst cell o plte or which the boundry t the outlet is closed to dispersion is closing cell. The lst two symmetric cell types closing nd opening re deined to implement the boundry conditions in Tble 1, s well s to model the strt up phse o the HEX rector or which the outlet boundry dvnces throughout the recting chnnel. The structurl chnges or ech elementry cell cn be extrcted rom Figure 8. They cn be represented by the cell Petri net in Figure 9.
9 Trnsition t 5 is ired or the second cell when the third cell is illed. Both in nd out re convective dispersive luxes. Trnsition t 6 is ired or the irst cell i the second cell is illed nd the outlet boundry is closed to dispersion. The outlet lux out is thus convective dispersive. Then, consider the illing o rector composed o 4 elementry cells (Figure 8). Ech elementry cell evolves thnks to the Petri net presented in Figure 9. Tble 3 represents the stte chnges o the our cells A, B, C nd D. Figure 9. Petri net o the elementry cell. The hybrid mthemticl model is then initilized s single illing cell. The cell trvels through dierent sttes s the rector structure evolves. In generl wy, the trnsition t 1 initilizes the cell model. I the previous cell o n empty cell is illed, convective lux is creted between the both o them. The irst cell remins n exception since the condition to ire t 1 depends on n event tht is independent o the rector stte vribles, (i.e., pump strt up). The inlet low o the irst cell is set equl to F eed. Trnsition t 2 is ired when the 3rd or higher illing cell is ull. Two ctions re perormed. Firstly, the inlet lux is then convective dispersive nd secondly the outlet lux is convective. Trnsition t 3 is ired when the irst or second cell is illed. The outlet lux is set to convective Trnsition t 4 is ired or closing cell, i the cell next to it is illed. To trnsorm the closing cell to n xilly dispersed cell, out is set to convective dispersive. SIMULATION EXAMPLES The complexity o the system requires progressive method or the model vlidtion. In this pper, the preliminry results o the HEX rector simultions re presented. A homogeneous system is studied. Six exmples re simulted. The irst one (cse 1) illustrtes the hybrid dynmic pproch presented in section 4. It concerns the simultion o the strt up nd the illing o the rector. The next results llow the vlidtion o the hydrodynmic nd therml prts o the model. The inite dierence method is used or these simultions (cses 2 to 6). The hydrodynmic prt o the model is vlidted thnks to the comprison o experimentl nd clculted residence time distributions (cse 2). Moreover, the response o the model is nlyzed when the composition o the inlet luid is chnged (cse 3). Cses 4, 5, nd 6 hve been simulted to test the therml prt o the model. The irst simultion (cse 4) concerns the heting o liquid with wll o constnt temperture. The second one (cse 5) represents the perorming o n exothermic rection in n dibtic rector. In the lst cse, exothermic rection is perormed with het exchnge with constnt wll temperture (cse 6). Cse 1: Strt up nd Filling o the HEX Rector For this section, let us consider recting plte with totl volume o m 3. The cse study prmeters re presented in Tble 3. Individul cell Petri nets Cell Cell petri net Trns. Relted ctions A t 1 Set the inlet low rte s F eed t 3 Set convective lux rom A to B Set convective dispersive lux t the outlet t 6 B t 1 Set convective lux rom A to B t 3 Set convective lux rom B to C Set convective dispersive lux rom B to C t 5 C t 1 Set convective dispersive lux rom B to C t 2 Set convective dispersive lux rom B to C Set convective lux rom C to D Set convective dispersive lux rom C to D t 4 D t 1 Set convective lux rom C to D t 2 Set convective dispersive lux rom C to D Set convective lux or the outlet
10 Tble 4. This rector is ed with wter t low rte, F eed o 10 kg h 1. The evolution o the illing rte or the irst our cells is shown in Figure 10. For N cells ¼ 4, the Petri net in Figure 8 describes the model evolution o the recting chnnel. The illing o the cell A strts when trnsition t 0 is ired. Once the irst cell is illed (b A ¼ 1), the trnsition t 1 is ired nd the illing o cell B strts. This behviour is repeted or the ollowing cells until the rector is completely illed. This mens tht the illing rtes o ll cells re equl to one. The chnnel mss hold up is shown in Figure 10. The xil dispersion coeicient is set to m 2 s 1. The mss hold up increses to the mximum vlue o kg (Figure 10b). It corresponds to the totl hold up o the rector. At t ¼ 4.2 s, the rector is illed. For N cells ¼ 30, the evolution o the illing rte long the rector or the complete set o cells is shown in Figure 10c. Cse 2: Residence Time Distribution Experimentl dt o residence time distributions or single phse systems on HEX rectors hve been obtined in the rmework o previous nd on going studies.  The ollowing experimentl dt hs been obtined by using Corning HEX rector under the operting conditions listed in Tble 5.  The model hs been used to reproduce the system response with Dirc type concentrtion disturbnce. Figure 11 shows experimentl nd simulted dt or the outlet concentrtion o trcer. The trcer is injected t t ¼ 7.2 s. The dispersion model is welldpted to represent the hydrodynmic behviour o the rector. A stisctory representtion is obtined with D x ¼ m 2 s 1 or Péclet number o 110 (P e ¼ ul/d x ). Tble 4. Geometric nd opertion prmeters or the illing o the rector Prmeters Rector length 3 m Chnnel height m Chnnel width m Inlet low rte 10 kg h 1 Density 995 kg m 3 Cse 3: Chnge o Inlet Fluid Composition A chnge o the composition o the eed is simulted: rom pure wter to pure ethnol. The sme rector s in cse 1 is considered (Tble 4). Tble 6 lists the operting conditions. For t < 2 s, the HEX rector is ed with wter t low rte o 5.5 kg h 1. At t ¼ 2 s, wter is completely chnged by ethnol t the sme temperture. The mss eed low rte is kept constnt. The composition chnge is not modelled by perect step signl. This step signl is smoothed to overcome the discontinuities o the model. Indeed, or DAE systems, discontinuities must be specilly studied.  Figure 12 shows the dynmic response o the system. Becuse o the chnge o luid in the rector inlet, ll system vribles chnge too. Then, or exmple the composition ront evolves ccording to the velocity o the luid nd long the rector (Figure 12). The eect o the xil dispersion cn be observed. Figure 12b shows tht the density hs the sme behviour thn those observed in Figure 12. The evolution o the velocity is illustrted in Figure 12c. The chnge o luid involves n increse o the velocity. At stedy stte, the velocity is still uniorm in ll cells. Figure 10. Filling phse o the rector. () Evolution o the illing rte o the irst our cells (b) Evolution o the mss hold up o the chnnel. (c) Evolution o the illing rte long the rector or N cells ¼ 30.
11 Tble 5. Geometric nd opertion prmeters o the HEX rector Prmeters Rector length 2.35 m Chnnel height m Chnnel width m Cross sectionl re m 2 Volume m 3 Flow rte 7 kg h 1 Temperture K Pressure 1.51 br(g) Tble 6. Operting conditions Chrcteristics o the eed Conditions or t < 2 s Conditions or t < 2 s x A ¼ Wter 1 0 x B ¼ ethnol 0 1 Density (kg m 3 ) Temperture (K) Mss low rte (kg h 1 ) luid temperture is due to the choice o closed to dispersion boundry condition t the inlet o the rector (see the second section). Stedy stte temperture proile is presented in Figure 14. The luid temperture reches the wll temperture t z/l ¼ This result is in greement with the experimentl behviour o HEX rector s reported Théron et l.  Cse 5: Exothermic Rection in n Adibtic Rector The rection o sodium thiosulte with hydrogen peroxide hs been considered. This rection, conducted in liquid solution, is very st nd strongly exothermic. 2N 2 S 2 O 3 þ 4H 2 O 2! N 2 S 3 O 6 þ N 2 SO 4 þ 4H 2 O ð17þ Figure 11. Residence time distribution o HEX rector. Cse 4: Heting With constnt Temperture Wll Tble 7 gives the geometric rector chrcteristics. The simultion hs been conducted with N cells ¼ 101. The xil dispersion coeicient is set to m 2 s 1 (Pe ¼ 100). At t ¼ 1 s trnsition t 1 is ired nd wll temperture is rtiicilly incresed to generte sptil grdients o the luid temperture. Figure 13 shows tht temperture proile evolves s soon s the wll temperture increses. The discontinuity between the irst node nd the inlet The thermokinetics prmeters hve been well studied, nd experimentl dt is vilble rom literture.  The rection is irst order in both rectnts. There could be sety problems relted to the evcution o het relesed by the rection. This rection hs Tble 7. Rector geometric dt Rector nd prmeters Chnnel length 7 m Cross sectionl re m 2 Speciic het exchnge re m 2 m 3 Het trnser coeicient 4.5 kw m 2 K 1 Figure 12. Filling phse o the rector. () Evolution o the illing rte o the irst our cells (b) Evolution o the mss hold up o the chnnel. (c) Evolution o the illing rte long the rector.
12 Tble 9. Inlet low dt or simultion o cses 5 nd 6 Inlet low properties Inlet low A Flow rte 3.3 L h 1 Composition (wt. %) 9 % N 2 S 2 O 3 91 % H 2 O Temperture 291 K Inlet low B Flow rte 1.7 L h 1 Composition (wt. %) 9 % H 2 O 2 91 % H 2 O Temperture 291 K Figure 13. Dynmic behviour o the luid temperture during the wll temperture chnge. been treted in sety studies s it constitutes st rection system tht cn be operted experimentlly in continuous rector presenting short residence time. [2,22] The rection rte is clculted s ollows: R ¼ k 0r exp E A r ½N 2 S 2 O 3 Š½H 2 O 2 Š: ð18þ ^RT Figure 14. Stedy stte temperture proile ter wll temperture chnge. Figure 15. Temperture proile evolution or the oxidtion o N 2 S 2 O 3 crried out in n dibtic xilly dispersed rector. Tble 8. Thermo kinetic dt. Rection kinetics E Ar kj mol 1 k or m 3 mol 1 s 1 DH r kj mol 1 o N 2 S 2 O 3 Thermo kinetic dt re presented in Tble 8. The chrcteristics o the rector re listed in Tble 7. Liquid densities nd speciic hets o the two inlet solutions re ssumed constnt nd equl to those o pure wter s the solutions re diluted (Tble 9). The simultion ws crried out with N cells ¼ 201. The xil dispersion coeicient is set to m 2 s 1 (Pe ¼ 100). The rector is initilized illed with solution o 9 % o sodium thiosulte. The inlet low properties re presented in Tble 9. At t 0 the inlet consists o inlet low rte A, t 3.3 L h 1. At t ¼ 1 s, inlet low B is injected t rte o 1.7 L h 1. Under these conditions the residence time is o 20 s. Figure 15 shows the temperture proile in the rector obtined in dibtic mode. The het relesed by the rection increses the luid temperture grdully rom the inlet temperture to orm the stte proile. The het generted by the rection is bsorbed by the luid. At t ¼ 35 s the rector reches stedy stte proile. The conversion chieved under these conditions, which is clculted in terms o the limiting rectnt (Sodium thiosulte), is 100 %. The luid temperture grdient between the inlet nd the outlet o the rector is C. Cse 6: Exothermic Rection With Het Exchnge With Constnt Wll Temperture An exothermic chemicl rection is conducted in the recting chnnel t constnt wll temperture (313 K). The simultion dt is given in Tbles 7, 8 nd 9. Petri net (Figure 16) represents the recipe o this simultion cse: At t ¼ 0 s, the rector is ull o the luid A. At t ¼ 1 s, trnsition t 1 is ired nd the wll temperture is incresed rom 292 K to 313 K. The stedy stte is reched t t ¼ 15 s (Figure 17). At t ¼ 20 s, trnsition t 2 is ired. The second inlet low cn be introduced nd the rection strts. At t ¼ 50 s, the simultion ends. The evolutions o the inlet low rte nd concentrtion re shown in Figure 17b c. Figure 18 represents the temperture proile between 0 s nd 20 s. At this stge, the trnsient temperture proiles, nd initil nd inl stedy sttes re presented in
13 Figure 16. Recipe Petri net o the simultion o the exothermic rection with het exchnge with constnt wll temperture. Figure 18. The luid temperture reches the wll temperture t z/l ¼ The luid temperture increses becuse o the het relesed by the rection. The stedy stte is reched t t ¼ 50 s. The mximum temperture o 315 K is obtined or z/l ¼ 0.09 (Figure 19). The outlet conversion t stedy stte is 79.1 %. The results re in greement with those presented by Théron et l.,  who obtined conversion o 82 % under these conditions. Figure 18. Trnsient temperture proiles between 0 s nd 20 s. CONCLUSION AND PERSPECTIVES In this pper, the dynmic modelling o HEX rector is discussed. The proposed model is bsed on the xilly dispersed plug low Figure 19. Trnsient temperture proiles between 20 s nd 50 s. rector model by tking into ccount the xil dispersion eects due to the non uniormity o the recting chnnel geometry. The chnnel is composed o series o independent cells, linked by luxes o mss nd het. This pproch diers rom the clssicl CSTR cscde, s it llows n independent choice o the dispersion on the hydrodynmic model nd the grid deinition. The overll rector representtion cn be clssiied s hybrid dynmic model. Petri Nets re used to represent it. Thnks to the Petri Net ormlism used, the model cn be simpliied nd its complexity cn be treted in stged mnner. The trnsitions between dierent models re esily mnged. Current work is willing to integrte this model within the hybrid dynmic simultion environment PrODHyS.  The min dvntges o the object oriented pproch rely on sotwre qulity (reusbility, mintinbility nd extensibility), s well s on modelling thnks to the bstrct hierrchicl description o rel systems. [16,25] The resulting simultion tool will be useul in the deinition o optiml operting conditions, to nlyze the process risks in order to conirm their inherently ser chrcteristics nd to cilitte the industriliztion process. In ddition to the simultion o norml opertion, or quntittive risk ssessment study, the inl model will lso llow to tke into ccount ilure mode mechnisms. Figure 17. Fluctutions o the inlet vribles. () Smooth step chnge o wll temperture (b) Evolution o the inlet mss low rte (c) Evolution o the inlet molr concentrtion. NOMENCLATURE Speciic het exchnge surce [m 2 m 3 ] C Molr concentrtion [mol m 3 ]
14 C p Het cpcity [Jmol 1 K 1 ] D x Axil dispersion coeicient [m 2 s 1 ] ~D z Dierentition mtrix or irst order derivtive pproximtion [m 1 ] E A Energy o ctivtion [Jmol 1 ] F Molr low rte [mol s 1 ] i 1/2 Molr lux [molm 2 s 1 ] H Enthlpy o the system [J] DH Enthlpy o rection [J mol 1 ] H M Term o molr production by chemicl rections [mol m 3 s 1 ] H T Term o het production by chemicl rections [J m 3 s 1 ] k o Pre exponentil ctor [m 3 mol 1 s 1 ] L Rector length M Rector model N cells Totl number o cells within plte N comp Totl number o chemicl species N p Totl number o recting pltes within the HEX rector N r Totl number o chemicl rections considered within phse N Totl number o phses within the rector p Petri net plce P Pressure [P] P Rector plte Pe Péclet number q Discrete stte Q M Term o mss trnser between phses [mol m 3 s 1 ] Q T Term o het trnser through the chnnel wll [J m 3 s 1 ] Q ðgþ k Mss trnser o component k rom phse g to phse [mol m 3 s 1 ] ^R Universl gs constnt R k,r Production/consumption o component k by rection r [mol m 3 s 1 ] s System output S System T Temperture [K] u Velocity [m s 1 ] u System input vrible U Input vribles stte spce t Time, independent vrible [s] t Trnsition between two plces o petri net. V Cell volume [m 3 ] x k Molr rction x Stte o system X Continuous vribles z Sptil vrible [m] Greek letters b Filling rte, or occupied volume rction o ech cell e Phse volume rction Undeined vrible l Eective xil therml conductivity [W m 1 K 1 ] y Stoechiometric coeicients V Chnnel cross section re [m 2 ] Indexes g Phse index Phse index c d eed i in k out u r Piecewise continuous or discontinuous system Discrete system Feed low property Discretized cell nd intercell lux index Entering the control volume Component index Leving the control volume Utility low property Rection index ACKNOWLEDGEMENTS This reserch work is supported by the French Ntionl Reserch Agency (ANR), in the rmework o the PolySe project (ANR 2012 CDII ). REFERENCES  Z. Anxionnz, M. Cbssud, C. Gourdon, P. Tochon, Chem. Eng. Process. Process Intensi. 2008, 47,  W. Beniss, S. Elgue, N. Gbs, M. Cbssud, D. Crson, M. Demissy, Int. J. Chem. Rect. Eng. 2008, 6, 1.  S. Hugwitz, P. Hgnder, T. Norén, Control Eng. Prct. 2007, 15, 779.  N. Niedblski, D. Johnson, S. S. Ptnik, D. Bnerjee, Int. J. Het Mss Trnser 2014, 70,  R. Bll, B. F. Gry, Environ. Prot. 2013, 91, 221.  S. Bhroun, S. Li, C. Jllut, C. Vlentin, F. D. Pnthou, J. Process Control 2010, 20, 664.  L. Despènes, S. Elgue, C. Gourdon, M. Cbssud, Chem. Eng. Process. Process Intensi. 2012, 52, 102.  M. Roudet, K. Loubiere, C. Gourdon, M. Cbssud, Chem. Eng. Sci. 2011, 66,  W. E. Schiesser, The numericl method o lines, Acdemic Press, Sn Diego  L. F. Shmpine, M. W. Reichelt, SIAM J. Sci. Comput. 1997, 18, 1.  L. F. Shmpine, M. W. Reichelt, J. A. Kierzenk, SIAM Rev. 1999, 41, 538.  A. Vnde Wouwer, P. Sucez, W. E. Schiesser, Ind. Eng. Chem. Res. 2004, 43,  W. E. Schiesser, G. W. Griiths, A Compendium o Prtil Dierentil Eqution Models: Method o Lines Anlysis with Mtlb, 1st edition, Cmbridge University Press, Cmbridge  A. Vnde Wouwer, Adptive Method o Lines, 1st ed., Chpmn nd Hll/CRC, Boc Rton, USA  Z. Anxionnz Minvielle, M. Cbssud, C. Gourdon, P. Tochon, Chem. Eng. Process. Process Intensi. 2013, 73, 67.  J. Perret, G. Hétreux, J. M. Le Lnn, Control Eng. Prct. 2004, 12,  S. Hugwitz, P. Hgnder, Anlysis nd design o strtup control o chemicl plte rector with uncertinties; hybrid pproch, IEEE Int. Con. Control Appl., 2007 p  S. A. Ky, S. A. Atti, J. Risch, A suboptiml control strtegy or the strt up o the Open Plte Rector, Proc. 10th IEEE Int. Con. Control Autom. Robot. Vis., Hnoi, Vietnm, 2008 p. 78.
15  J. Perret, G. Hétreux, J. M. Le Lnn, Rev. L électricité L électronique 2003, 71.  M. Moreu, N. Rimondi, S. Elgue, N. Le Suze, M. Cbssud, C. Gourdon, Comprison des comportements hydrodynmiques de récteurs échngeurs, Récents Prog. En Génie Procédés, SFGP, Lyon, Frnce  E. D. Lvric, P. Woehl, Chem. Tody 2009, 27, 45.  R. C. Vieir, E. C. Bisci Jr, Comput. Chem. Eng. 2001, 25,  F. Théron, Z. Anxionnz Minvielle, M. Cbssud, C. Gourdon, P. Tochon, Chem. Eng. Process. Process Intensi. 2014, 82, 30.  L. Prt, A. Devtine, P. Cognet, M. Cbssud, C. Gourdon, S. Elgue, F. Choprd, Chem. Eng. Technol. 2005, 28,  G. Hétreux, R. Théry, N. Olivier, J. M. Le Lnn, J. Eur. Systèmes Autom. 2007, 41, 585.