MATHEMATICAL MODELING FOR ADIPIC ACID CRYSTALLIZATION PROCESS. Caliane B. B. Costa, Aline C. Costa, Rubens Maciel Filho

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1 MATHEMATICAL MODELIG FOR ADIPIC ACID CRYSTALLIZATIO PROCESS Calae B. B. Costa Ale C. Costa Rubes Macel Flho Chemcal Egeeg School State Uvesty of Campas (UICAMP) CP 666 CEP; Phoe: Fax: Campas SP Bazl Abstact: Ths wok s pat of a study amg to mpove cystallzato pocesses. The batch cystallzato of adpc acd s chose as a case study a pocess whch the supesatuato ecessay fo the cystals to appea ad gow s geeated by the coolg of the soluto. Ths pat of the study volves the pocess modelg ad a aalyss of the fluece of the pocess vaables o the fal cystal sze dstbuto (CSD) ad o the quatty of solds. The esults have show that the developed model s a good epesetato of the pocess beg able to epoduce lteatue esults. Copyght 4 IFAC Keywods: Mathematcal models Solds pocessg Pocess models Dyamc Behavou Tempeatue pofles 1. ITRODUCTIO Cystallzato s a acet ut opeato ad s wdely used sce solds of hgh puty ca be obtaed. Batch cystallzato s specally used the poducto of hgh value added poducts maly because t offes flexble ad smple pocessg steps fo plats wth fequetly chagg ecpes ad poduct les (Zhag ad Roha 3; Rawlgs et al. 1993). Fo ths kd of mateal poduct puty ad the cystal sze dstbuto (CSD) ae of pme mpotace. Futhemoe the cystals poduced though a cystallzato pocess have a decsve fluece o the dowsteam pocessg ad theefoe the CSD should be epoducble each opeato ad egula as much as possble (Ma et al. ). Theefoe the vaables that affect the cystallzato pocess must be kow ad cotolled so as to be kept wth a acceptable age ode to satsfy the equemets coceg the fal poduct qualty ad the poducto pocess (Rawlgs et al. 1993). Ths wok s pat of a study amg to mpove the pefomace of batch cystallzato pocesses. The batch coolg cystallzato of adpc acd s chose as a case study. I a well-mxed batch cystallze the fal cystal poduct s detemed by the supesatuato pofle the tal seed mass ad the seed cystal sze dstbuto (Ma et al. ). The supesatuato evoluto dug batch cystallzato pocesses detemes the magtude of the may ketc pheomea that compse the pocess. I ode to pefom a successful study of the pocess the accuate modellg of the pocess s cucal. It tus possble the calculato of futhe o-le optmal opeatg polces as well as eal tme applcatos (Zhag ad Roha 3). I ode to aalyse the pefomace of sutable cotol stuctues ad to pefom cotol calculatos a accuate modellg s ecessay to be developed. Seedg s fequetly appled to avod a supesatuato peak at the begg of the pocess (Gulett et al. 1). Ths occus because the supesatuato values suffcet fo cystal gowth ae lowe tha the ecessay oes fo spotaeous ucleato. So f the system could be kept at a ego the metastable zoe wth ot so hgh supesatuato values ad wth addto of seeds just seeds gowth wll occu (Mull 1993; Rawlgs et al. 199). I ths wok the developed mathematcal modelg of the adpc acd cystallzato pocess s peseted. The ole of the seveal pocess vaables the fal poduct s evaluated as pat of the study of ways to mpove the pocess.. MATHEMATICAL MODELIG I ode to completely model a cystallzato pocess mass balace eegy balace ad a

2 descpto of the cystal sze dstbuto ae ecessay. Ths descpto s ecessay because the pocess poduce a mass of patcles of may szes whose descpto of sze dstbuto s cucal to ts chaactezato. The models used ths descpto ae called populato balace models. Accodg to Puel et al. (3) the populato balace modelg s fmly establshed as a basc theoetcal famewok fo all patcle pocesses. Two pheomea domate the cystallzato ketcs: ucleato ad cystal gowth. Both pheomea cosume the mass of the desed mateal dug the cystallzato pocess ad ae theefoe competg mechasms. They ae dffeet the mae they cosume mateal: ucleato volves the fomato of ew cystals whle cystal gowth the cystals become lage wth the deposto of mateal oto the exstg cystals (Lag et al. 1999). Apat fom ucleato ad gowth othe pheomea such as agglomeato ad beakage may occu dug the pocess..1 Populato Balace The Method of Classes. Populato Balace equato (PBE) s a hypebolc patal dffeetal equato whch volves all the cystallzato ketc pheomea. Beg stogly olea the PBE do ot possess a aalytcal soluto most cases equg the developmet ad adaptato of umecal techques. Accodg to Wulkow et al. (1) the peset state of PBE studes o stadad umecal method has bee establshed fo populato balace models ad oe of the pesetly avalable methods poduce a effcet ad accuate soluto fo a boad class of models. Dscetzato szg techques appea to be obust. Machal et al. (1988) developed the Method of Classes a method that tasfoms the patal dffeetal equato to a oday dffeetal equatos system by dscetzg the age of vaato of the vaable L elated to the cystal sze. The obtaed dffeetal equatos ae the o loge wtte wth populato desty fuctos but wth absolute umbes of cystals each class (allet et al. 1998). I a ecet wok Puel et al. (3) also used the Method of Classes to solve the populato balace equato but they exteded the method to cosde two chaactestc dmesos. I the Method of Classes the patcles szes ae defed as L L 1 L... L whee L s the sze of the ucle ad L s the sze of the lagest cystals. Ths szes deteme the exstece of gaulometc classes C whose wdths ae defed by C = L -L -1 ad the chaactestc sze of the class C s S = (L -1 L )/ (see Fgue 1). S S 1 L -1 L L 1 Class C Fg. 1. Dvso of Classes. Class C 1 The populato of cystals s descbed by the umbe desty fucto Ψ. Ψ(Lt)dL s the umbe of cystals of sze betwee L ad LdL pe ut volume at tme t (allet et al. 1998). (t) the umbe of cystals pe ut of volume the th class (C ) at tme t s gve by: L L (1) () t = Ψ( L) t dl The model assumes that the umbe desty fucto s costat at each class tug possble the tasfomato of the populato balace equato to a system of oday dffeetal equatos: d 1 dv Q Q GL ( ) GL ( ) = R R dt V dt V C C 1 1e susp s e A B susp susp 1 1 dvsusp s e e ( ) ( ) ( ) 1 susp susp 1 GL ( ) C R RB A 1 dvsusp Q s Q e e GL ( ) susp Vsusp C GL ( ) C R RB A d Q Q GL GL GL dt V dt V C C = d dt V dt = I the developmet of the modellg of the batch coolg cystallzato pocess the Method of Classes was chose because t s a effcet method wdely used lteatue cludg modellg of gowth-domated pocesses. Puel et al. (3) stess that ths umecal techque elablty s cofmed by thee dffeet papes dealg wth b-dmesoal balace equatos whee dscetzato methods wee used despte dffeeces the fomalsm ad the applcato aea. Howeve the umbe of classes cosdeed s the majo key to guaatee the accuate soluto as was poted out by Puel et al (3). Ths was also obseved the peset wok. Gowth Mechasm. The expesso fo the gowth ate developed by Machal et al. (1988) s based o the flm model ad may be wtte as: dl k a MM k c G = = η ( c c*) dt 3ρ k c v j () (3) The effectveess facto η whch expesses the dffusoal lmtatos cystal gowth causg a dstbuto of the gowth ates amog the classes s foud by the soluto of equato (4). The mass tasfe coeffcet k d s evaluated though the expesso fo Shewood (equato (5)). k k c d ( c c*) j η η 1 j = (4) 17 4 / 3 1/ ε Dam ν (5) k d L L Sh = = 47 D ν DamT D Fom the equato (5) t s easy to ote that each gaulometc class has a value fo the mass tasfe coeffcet what meas that the gowth ate s sze depedet. The data of solublty of adpc acd wate wee extacted fom Postkov ad alvako (1971).

3 ucleato Mechasm. Both pmay ad secoday ucleato ae cosdeed. The expessos ae espectvely gve by equatos (6) ad (7): 1 = Aexp l B ( HR) ( HR) * (6) k = k [( HR) ( HR) ] C (7) * Agglomeato Mechasm. Stll the wok of Machal et al. (1988) a model fo the agglomeato mechasm s poposed cosdeg oly the agglomeato of two patcles. The agglomeato s cosdeed as chemcal eacto of patcle m ad patcle leadg to the fomato of a cystal of sze q. Cosdeg that the pseudo-chemcal eacto descbg the agglomeato ca be schematcally epeseted by equato (9) ts stochometc coeffcet may be foud by ν est = (S 3 m S 3 )/S 3 q. The class q whee the agglomeate fts s foud by the elato: L q-1 <(S 3 m S 3 ) 1/3 L q. m ν q s (8) All possble agglomeatos betwee two patcles (m ad m) ca be aaged a sees assumg a ak l m that epesets ts posto ths sees. Fo gaulometc classes (1)/ dffeet bay agglomeatos ae peset ad the sees ae epeseted by (11) (1)... (1) () (3)... (m)... (). The posto l m of the agglomeato of a patcle m wth a patcle s foud by elato (9): m( m ) m = l ( m ) (9) Ths agglomeato wll affect class oly f s equal to m ( both cases ν est = -1) o q (wth ν est = (S m 3 S 3 )/S q 3 ). I ths way a oveall stochometc coeffcet of class wth espect to agglomeato of ak l ca be computed: ν S S δ δ δ 3 3 m est = l 3 q m S q ( ) whee δ j = 1 f = j ad δ j = f j. (1) The et ate of patcle poducto by agglomeato the th class s calculated by equato (11) whee (l) s the tsc ate of agglomeato of ak l beg a fucto of the umbe of collsos pe ut volume pe tme ad of the supesatuato. R ( 1) A = ν est l ( l) l = 1 (11) Davd et al. (1991) poposed a expesso fo the calculato of the tsc ate of agglomeato (equato 1) based o pheomeologcal ad flud mechacal cosdeatos takg to accout the cocetato of patcles the supesatuato the powe dsspato pe mass ut the cystallze sze ad of the cystals beg agglomeated. The agglomeato cotbuto the developed softwae s computed though equato (1). ( λ ) ( S Sm) λ S S = kasm 1 Dam f 1 kd ( c c*) Sm Sm e ( S δ) H S S m m e m (1) whee H(x) = 1 fo x ad H(x) = fo x <. The coeffcet k d s calculated by the same expesso fo k d (Eq. 5) but the sze of cystal cosdeed must be the sze of the agglomeate (S m S ). The f fucto epesets a elatve shape fucto of both cystals. Cosdeg both patcles as sphees: S 4 1 S Sm Sm S f = Sm S S 1 S S S 1 S S Sm m 3 S m Sm Sm Sm 3 3 (13) λ e s defed as Lagaga mcoscale whch s take as havg the same magtude as the Taylo mcoscale calculated by: 6ν λe = 3 π Dam 1ε 1/ (14) The calculato of futhe o-le optmal opeato polces takes a advatage wth the cluso of agglomeato the model compaed to most batch optmzato studes ecouteed lteatue. Most of them eglect agglomeato ad cosdes just ucleato ad gowth whch may be a poo cosdeato as s the case wth adpc acd a classc-kow substace that teds to agglomeate dug cystallzato pocesses.. Mass Balace Batch Cystallze. The mateal balace of the solute s made based o the fact that a chage the soluto cocetato esults a chage of the mass of cystals pe ut volume. Sce the smulated cystallze s a batch oe the solute peset the soluto the begg of the batch s the whole mass of adpc acd avalable fo cystallzato. The mass balace s made usg the dssocato costat of adpc acd cosdeed as a mooacd: the cocetato of sold the suspeso ca be coected wth the cocetato of potos the soluto (Machal et al. 1988): ( H ) K Vo VC o o = 1 Vo Cs K ( H ) MM 1 Cs ρ.3 Eegy Balace. (15) The eegy balace must take to accout the dffeeces ethalpy of the steams ad out the

4 heat of cystallzato ad the heat emoved by the coolg system. I the case study batch cystallzato the eegy balace equato s: dt p c c v susp c c dt ρ CV = H3ρ kv LGdL UA(T T) (16) Usg the Method of Classes the tegal peset the pevous equato s substtuted by a summato ove all gaulometc classes. The heat of cystallzato of adpc acd was extacted fom Postkov ad alvako (1971). The specfc heat of the sluy s cosdeed to be ot so dffeet fom the wate sce the pocess cosdeed s the cystallzato fom soluto whch s chaactezed to have ot so hgh cocetatos of solds. 3. MODEL VALIDATIO I ode to test the valdty of the costucted model esults fom the lteatue wee epoduced. The esults obtaed by Machal et al. (1988) ae compaed wth the oes geeated wth the developed model wth the same data. To llustate ths pot Fgue depcts a compaso betwee the esults fo evoluto of the elatve supesatuato dug the cystallzato. The esults geeated by the developed softwae ft petty well to the esults obtaed by Machal et al. (1988) both fo cystallzato cosdeg agglomeato o ot. Cosdeg the avalablty of data the model s valdated by the smlates of plots ot oly fo supesatuato evoluto but also fo solds cocetato evoluto CSD umbe ad mass at the ed of the batch ad effectveess facto fo thee gaulometc classes dug the batch. evaluated though aalyss of the effect of model paametes vaatos (±15%) o the fal espose though expemetal desg. The study showed that the model ucetates do ot affect the aalyss that ca be extacted fom the fluece of the pocess vaables: the same qualtatve esults llustated Fgues 3 5 ae obtaed. The fst esult s the fluece of the seedg the pocess. I ode to aalyze the fluece of seedg the plots o Fgue 3 ae peseted. It shows the esults of opeato tajectoes soluto tempeatue evoluto ad supesatuato evoluto fo two batches that dffe each othe oly the seedg: the fst oe (o-seeded) has sgfcat amout of seeds whle the secod oe has a cosdeable mass of cystals wokg as seeds. At these two smulatos a paabolc pofle of coolat tempeatue wth egatve secod devatve was employed. Cocetato (mol/m 3 of soluto) Soluto Tempeatue (K) o-seeded batch seeded batch sold le:equlbum cuve Tempeatue (K) o-seeded batch seeded batch Supesatuato (-) o agglomeato wth agglomeato Supesatuato (-) o agglomeato wth agglomeato Fg.: Compaso betwee Machal et al. (1988) esult (left) ad model pedctos (ght) 4. SIMULATIOS PROCESS VARIABLES IFLUECE Some smulatos wee made ode to evaluate the fluece of seveal pocess vaables. The ma esults ae peseted hee. They ae coceed to seedg coolg ate fluece of the global heat tasfe coeffcet ad of the powe dsspato. It should be poted that the effect of model ucetates (.e. model paametes) o the accuacy of the model pedctos should be aalysed. It was Supesatuato (-) o-seeded batch seeded batch Fg. 3: Opeato cuve evoluto of the soluto tempeatue ad evoluto of the supesatuato fo a o-seeded ad a seeded batch cystallzato The fst plot Fgue 3 llustates the opeato cuves (tajectoes) fo the two smulatos. The opeato tajectoy s fomed by the pas of values of soluto cocetato ad tempeatue followed dug the batch tme. The dffeece the path followed dug the pocess the two cases s extemely vsble. Whe o seeds ae added to the cystallzato pocess the soluto s cooled wth costat cocetato utl the metastable zoe lmt s pobably eached. A eomous quatty of

5 cystals s geeated by ucleato eleasg geat amout of heat of cystallzato (exothemc eacto) makg the soluto tempeatue to se (secod plot of Fg. 3) what meas that the coolg system was ot able to emove all the heat eleased. As a cosequece of the opeato cuve a geat peak of supesatuato s oted (last plot of Fg. 3). O the othe had f the cystallzato s coducted the same way except by the addto of seeds the coolg of the soluto happes softly whle the cocetato of adpc acd soluto deceases (fst plot of Fg. 3) pobably keepg the opeato fa fom the metastable zoe lmt. It s kept at a easoable dstace fom the equlbum le (Lag et al. 1999). ot so hgh values of supesatuato ae eached the pocess (last plot of Fg. 3) ad the ucleato s dsfavoued. Thee s o geat elease of heat of cystallzato ad so thee s o cease of soluto tempeatue (secod plot of Fg. 3). Apat fom the fluece of the seedg othe mpotat facto the cystallzato pefomace s the ate of coolg dug all the pocess. Accodg to Mull (1993) the use of atual coolg s ot the best choce. By atual coolg oe udestads that the coolat s passed though the exchage devce at costat both tempeatue ad flow ate. If atual coolg s used the tempeatue sde the cystallze deceases expoetally makg the supesatuato to cease vey fast at the fst momets of the pocess favoug ucleato. As a esult at the ed of the batch small cystals ad a lage CSD ae obtaed. But f the coolg pofle s chaactezed to have a soft decease at the ealy stages ad a moe poouced oe at the ed of the pocess the opeato s favoued (Mull 1993). Two seeded smulatos that dffe the coolat tempeatue pofle llustate appopately ths featue. I the fst oe atual coolg s smulated (coolat tempeatue ad flow ate ae both costat). I the ext smulato the same amout of seed s used but a paabolc pofle of coolat tempeatue wth egatve secod devatve s employed. Ths last coolg ate s chaactezed to have a soft decease at the ealy stages ad a moe poouced oe at the ed of the pocess. The esults of the two smulatos ca be obseved Fg. 4. Whe atual coolg s employed a geat umbe of fes s obtaed (last plot Fg. 4). A geat peak of supesatuato occus at the ealy stages of the cystallzato pocess what causes the appeaace of thousads of ucle. The cocetato of solds the suspeso ceases too fast the fst momets (fst plot of Fg. 4). Except fo the dsadvatage of the geat umbe of fes ths kd of coolg has the advatage of beg able of extactg a geat amout (mass) of solute of the soluto bgg them to cystals. Whe a egatve paabolc tempeatue pofle s employed much less umbe of fes s obtaed ad the sold cocetato evolves softly (see Fg. 4) wth seeds gowth ad ot so may fes beg geeated. Ths s a cosequece of the evoluto of the supesatuato dug the batch showg that t s desable to avod peak supesatuato ode to favou gowth. Ths cofms the esults fom lteatue (Lag et al 1999) dcatg that the optmum tempeatue pofle would poduce o peak supesatuato at all beg almost flat. Howeve the slow evoluto of the pocess whe the supesatuato s kept low mples less extacto of solute fom the soluto (less mass s obtaed compaed to atual coolg). Cs (mol/m 3 of suspeso) Supesatuato (-) umbe facto (%) atual coolg egatve paabolc coolat tempeatue pofle atual coolg egatve paabolc coolat 1 tempeatue pofle atual coolg egatve paabolc coolat tempeatue pofle Gaulometc class Fg. 4: Evolutos of sold cocetato ad supesatuato as well as the CSD umbe at the ed of the batch a seeded cystallzato wth atual coolg ad wth a egatve paabolc coolat tempeatue pofle. At the study of the fluece of the global heat tasfe coeffcet ts vaato would expess a vaato the coolat flow ate: the gate the value of the global heat tasfe coeffcet the geate the coolat flow ate. The study showed that a cease the flud flow ate would be advatageous oly cases whee the coolat tempeatue pofle deceases slowly ealy stages ad shape latte tme what meas to pass the coolat flud wth lowe tempeatue ad hghe flow ate as the batch tme passes. I ths way moe mass s obtaed the pocess. The ole of the powe dsspato s evaluated the peset wok ad ths vaable has show tself to be mpotat the dstbuto of cystals the vaous classes (Fg. 5). It s a expected esult sce the powe dsspato s peset the expesso of the

6 mass tasfe coeffcet whch flueces the gowth ate each class as well as the tsc ate of agglomeato. It has show to be ot so mpotat the amout (mass) of cystals obtaed. 7. ACKOWLEDGEMETS Facal suppot fom FAPESP s gatefully ackowledged ε =. ε = 1. REFERECES Mass facto (%) Gaulometc class Fg. 5: CSD mass at the ed of the batch two smulatos (ε =. ad ε = 1.). 5. BATCH OPTIMIZATIO STUDY I ode to cotol batch coolg cystallzato pocesses t s ecessay to have a off-le optmzed coolg pofle (set pot pofle) whch s calculated though optmal cotol theoy. The accuate model of the pocess s essetal obtag ths set pot pofle because t s obtaed though the soluto of a o-lea optmzato poblem. A geeal o-lea optmzato poblem subjected to equalty ad equalty costats h (x) ad g (x) espectvely may be defed as: M f (x) s. t.: h j (x) = j = 1... m (17) g k (x) k = 1... p I the batch coolg cystallzato optmzato the fucto to be mmzed s set as a fucto of the fal CSD ad the poblem costats ae the model equatos as well as physcal costats. O-le batch optmal opeatg polces ae calculated though the use of a model-based cotolle whch uses the pocess model to pedct futue aswes of the pocess ad take actos ode to obta the best poduct at the ed of the batch. The optmal opeatg pofles may be successfully obtaed by the poposed model togethe wth the optmal cotol theoy. 6. COCLUSIOS The developed model poved to be a good oe expessg epesetatve esults of the pocess. Lteatue esults wee satsfactoy epoduced fom the avalable fomato. The pedctos geeated though the model allow aalyzg the mpact of the pocess vaables the system behavou. The poposed model may be used a o-lea optmzato poblem ode to fd out the optmal coolg pofle. Davd R. P. Machal J.P. Kle ad J. Vllemaux (1991). Cystallzato ad Pecptato Egeeg III. A Dscete Fomulato of the Agglomeato Rate of Cystals a Cystallzato Pocess. Chemcal Egeeg Scece Gulett M. M.M. Seckle S. Deezo M.I. Ré ad E. Ceksk (1). Idustal Cystallzato ad Pecptato fom Solutos: State of the Techque. Bazla Joual of Chemcal Egeeg Lag Y.D. A.M. Cevates ad L.T. Begle (1999). Dyamc Optmzato of a Batch Coolg Cystallzato Pocess. Idustal &. Egeeg Chemsty Reseach Ma D.L. D.K. Taft ad R.D. Baatz (). Optmal cotol ad smulato of multdmesoal cystallzato pocesses. Computes ad Chemcal Egeeg Machal P. R. Davd J.P. Kle ad J. Vllemaux (1988). Cystallzato ad Pecptato Egeeg I. A Effcet Method fo Solvg Populato Balace Cystallzato wth Agglomeato. Chemcal Egeeg Scece Mull J.W. (1993). Cystallzato Buttewoth- Heema Oxfod. allet V. D. Mag ad J.P. Kle (1998). Model detfcato of batch pecptatos: applcato to salcylc acd. Computes ad Chemcal Egeeg S Postkov V.A. ad A.S. alvako (1971). Solublty ad Heat of Cystallsato of Adpc Acd. Russa Joual of Physcal Chemsty Puel F. G. Févotte ad J.P. Kle (3). Smulato ad aalyss of dustal cystallzato pocesses though multdmesoal populato balace equatos. Pat 1: a esoluto algothm based o the method of classes. Chemcal Egeeg Scece Rawlgs J.B. W.R. Wtkowsk ad J.W. Eato (199). Modelg ad cotol of cystallzes. Powde Techology Rawlgs J.B. S.M. Mlle ad W.R. Wtkowsk (1993). Model Idetfcato ad cotol of Soluto Cystallzato Pocesses: A Revew. Idustal &. Egeeg Chemsty Reseach Wulkow M. A. Gestlaue ad U. eke (1). Modelg ad smulato of cystallzato pocesses usg pasval. Chemcal Egeeg Scece Zhag G.P. ad S. Roha (3). O-le optmal cotol of a seeded batch coolg cystallze. Chemcal Egeeg. Scece