PRECIPITATION KINETIC PARAMETER ESTIMATION

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1 PRECIPITATIO KIETIC PARAMETER ESTIMATIO Kazs M Queensland Alumna Lmted, Gladstone, QLD, Austala Abstact The am of the Baye efney ectaton ccut s to maxmse oduct yeld whlst meetng oduct ualty constants, atculaly oduct sze. In ode to otmse the ectaton ccut t s motant to have a good undestandng of the ocesses of gowth, agglomeaton, and nucleaton; and to buld mathematcal elatonshs fo these ocesses nto ectaton ccut oulaton balance models. A mathematcal model has been develoed that s caable of estmatng the ectaton gowth (G), nucleaton (Bu), and agglomeaton (β ) knetc aametes fom batch exemental data. These knetc aametes ae eued fo solvng the oulaton balance euaton, whch s the conestone of ectaton ccut oulaton models. The method s maly based on wok conducted by Bamley et al., Ltste et al., and Wynn. Intoducton Gbbste ectaton s a comlcated system, wth the ocesses of gowth, agglomeaton, and nucleaton occung smultaneously. Investgatng the knetcs of gowth, agglomeaton, and nucleaton eues the ablty to extact knetc nfomaton fom ectaton eacton data n whch these ocesses ae occung smultaneously. Knetcs estmaton technues ae based on oulaton balance models. These models take the fom of an ntego-atal dffeental euaton, whch, fo a batch ocess, s gven below n euaton. En n( n( + G Bud( L L ) + B( D( t L whee n( s the numbe based sze dstbuton, t s tme and L s the chaactestc sze of atcles. The ate of atcle gowth s descbed by G, whch, as wtten n euaton, s the sze ndeendent lnea gowth ate and s defned as dl/dt. To account fo the movement of atcles nto the feld of vew of the atcle sze analyse a souce functon, B u, s ntoduced. B u s the ate of aeaance of atcles enteng the fst sze nteval, L, of the atcle sze analyse and s elated to the nucleaton ate. δ s the Dac delta functon. The tems B( and D( descbe the agglomeaton knetcs, and eesent the ate of aeaance and dsaeaance of atcles due to agglomeaton. Fo sze ndeendent agglomeaton they ae defned as En L βl n([ L λ ], n( λ, B( dλ ( L λ ) En D( βn( n( λ, dλ whee β s the agglomeaton kenel, o agglomeaton ate constant. The agglomeaton kenel s a measue of the feuency of collsons between cystals of szes L and λ that ae successful n oducng an agglomeate, and s defned as En 4 β β f ( λ) whee β s a functon of oeatng condtons such as luo comoston, temeatue and shea ate. The second facto, f ( λ), s some functon of sze and, n geneal, eflects the mechansm thought to have caused agglomeaton (e.g. gavtatonal settlng, shea etc.). Fo sze ndeendent agglomeaton systems f ( λ). Knetcs estmaton efes to the detemnaton of the gowth, nucleaton, and agglomeaton knetc aametes wthn the oulaton balance model (G, B u, and β esectvely). A sutable knetcs estmaton technue dentfed by Queensland Alumna Lmted (QAL) s that develoed by Bamley et al.. The uose of the wok conducted at QAL has been to aly the Bamley method to Baye gbbste ectaton, atculaly to lab-scale batch ectaton eactons. CSIRO have develoed a method to extact gowth (G), agglomeaton (β), and nucleaton (B u ) knetcs fom batch exements, whch was subseuently used n the AMIRA P5 sees of oects. Howeve, the oect sonsos do not have fee access to the knetcs estmaton ocedues develoed, as CSIRO clam that the technues wee develoed seaately to the AMIRA oects. Due to the cost of havng exemental data analysed by CSIRO, QAL decded to develo the technues n house. Exemental Batch ectaton exements wee conducted n QAL s laboatoy ectaton eactos usng synthetc luos and Alcoa C hydate as seed mateal. A total of 6 exements wee conducted unde the same condtons to allow develoment of the exemental ocedues. A lage batch of Alcoa C hydate was washed wth hot dstlled wate, and once dy, ffled nto kg dentcal batches fo use n the exementaton. Ths hydate was then used both fo makng synthetc luos and as a seed mateal n the ectaton exements. Synthetc luos wee made to the followng comoston: A/C.7 Caustc g/l C/S.8 The eued amounts of analytcal gade aoh ellets and Alcoa C hydate wee added to ~L of dstlled wate. Ths mxtue was heated and allowed to lghtly bol untl all of the hydate had dssolved (~.5 hous). The luo was then allowed to cool befoe beng vacuum flteed though a GF/C flte ae. Sodum cabonate soluton was made by dssolvng the eued amount of analytcal gade a CO owde nto ~.5L of dstlled wate. Once dssolved, the soluton was vacuum flteed though a GF/C flte ae. The two solutons wee then mxed togethe, and made u to 5.5L wth dstlled wate. Uon mxng the solutons, often t was obseved that vey fne nucle auto-ectated. When ths was 86 Poceedngs of the 8th Intenatonal Alumna Qualty Woksho 8

2 obseved the luo was e-flteed though a GF/C flte ae, befoe beng added to the ectaton eacto. All ectaton smulatons wee conducted n a lab-scale batch eacto. The man featues of ths eacto ae: A total volume of 5.5L Pecse temeatue contol (±. C) va oeaton of a wate acket Ovehead melle agtaton, ncludng a daft tube comlete wth baffles Batch ectaton eactons wee conducted n the lab-scale eactos at the followng condtons: Luo comoston As above Temeatue 8 C Seed Chage 45g/L Agtaton Rate 45 m Intal Volume 5L The necessay volume of synthetc luo was added to the ectaton eacto, and allowed to each the exementaton temeatue (~ hou) befoe a e-heated allotment of C hydate was chaged to begn the ectaton eacton. Samles wee taken fom the eacto at t, 5, 5,, 4, 5, 6, 75, 9 and mnutes. At each tme ont the followng samles wee taken: ml samle fo atcle sze analyss ml samle synge flteed fo ACS ttaton usng the Metohm automatc ttate ml samle fo solds concentaton The ml samle fo atcle szng was taken wth a 5mL Eendof ette and added to 4mL of mllq wate. Fom ths soluton two 5mL samles wee taken and each added to 95mL of soton (electolyte used fo coulte counte). These two solutons wee then analysed on the Coulte Counte Multsze III n volume mode, wth fve seaate ml samles analysed fom each soluton. These uns wee checked fo consstency aganst each othe befoe summng each atcle sze dstbuton (PSD) to gve the fnal sze dstbuton fo that tme ont n the ectaton exement. The ml samle fo solds concentaton s flteed though a mlloe fltaton set-u usng a.7μm flte ae befoe beng nsed thooughly wth ~ml of dstlled wate, ensung that no mateal s left on the funnel walls. The flte ae s then caefully emoved to ensue that no solds ae lost befoe beng laced n a 6 C oven to dy. Results and Dscusson Exemental Results Fgue below shows the evoluton ove tme of the PSD dung the batch eacton due to the ectaton henomena of gowth, agglomeaton, and nucleaton. Fom t to t, whle the suesatuaton s hgh, agglomeaton aeas to be the man contbuto to the change n the PSD shae, gowth then domnates fo the emande of the batch test. Tue nucleaton s not obseved, as any nucle oduced ae smalle than the mnmum measuable sze usng the 8μm ofce on the coulte counte (5.6μm). Fgue below shows the change n the total numbe of atcles wth tme dung the batch eacton. Intally the total numbe of atcles nceases (.e. fom t to t 5); t s beleved that ths s a esult of atcles smalle than the mnmum measuable sze agglomeatng and gowng nto the sze measuement egon. Evdence of ths s obseved n Fgue, wth the ncease n numbes fom t to t 5 occung ove a elatvely wde sze ange (5.6μm to ~μm). If the mnmum measuable sze wee to be μm, the data would show a sgnfcant ncease n numbes fom t 5 mn to t 5 mn; t s lkely that a smla henomenon occus below the cuent mnmum sze of 5.6μm. Afte ths ntal ncease the total numbe adly deceases as atcles agglomeate, the total numbes then becomes steady as the soluton suesatuaton and agglomeaton knetcs educe. o. e Inteval, (o./l) t t5 t5 t t4 t5 t6 t75 t9 t.e+8 9.E+7 8.E+7 7.E+7 6.E+7 5.E+7 4.E+7.E+7.E+7.E+7.E Sze (um) Fgue. PSD Evoluton Dung a Batch Pectaton Reacton Total umbe (o./ml) 4.E+6.5E+6.E+6.5E+6.E+6.5E+6.E+6 5.E Fgue. Total umbe vs Tme n a Batch Pectaton Reacton A key euement fo the knetcs estmaton ocedue s to detemne the ate of change of cystal mass wthn the system. As such the mass balance dung each batch eacton s montoed and checked fo consstency. Mass balances ae calculated fom the solds concentaton, luo ttaton, and atcle sze dstbuton at each samle tme (Fgue ). It s clea fom Fgue that thee s ageement between mass of solds detemned fom the ttaton and the measued solds concentaton. Howeve, the coulte calculated mass s consstently lowe than the othe measuements, whch s consstent wth an ssue wth samlng the lage atcles wthn the system (.e. the atcles contanng the mass). The data esented n Fgue s fom the thd batch ectaton test. It was notced dung sub-samlng fo the coulte measuement that some (only a vey small ecentage) of the atcles wee not beng susended (and hence not samled). Ensung that the melle was close enough to the bottom of the baffled beake to ovde suffcent agtaton ectfed ths. In the ntal two batch tests, the solds measuement was consstently hghe than that calculated fom the luo ttaton and also exhbted much geate vaaton. Ths was ectfed by: Poceedngs of the 8th Intenatonal Alumna Qualty Woksho 8 87

3 Ensung that a suffcently lage volume of wate was used to nse the solds and flte ae to emove any esdual luo, and Takng cae n emovng the flte ae such that no solds wee lost. Solds (g/l) Coulte Ttaton Ma -Ma 5-A -A 8-A A/C Solds (g/l) Fgue 5. Reoducblty of the Desuesatuaton Cuves fo the 5 Reeat Exements Fgue. Mass Balance Consstency Between Solds Content Calculated fom Solds, PSD, and Ttaton Data Ma -Ma 5-A -A 8-A Exemental Reoducblty To date a total of 6 batch exements have been comleted, and dung ths tme sgnfcant movements have been made n the exemental method to move exemental eoducblty. The ntal batch exement s excluded fom the eoducblty analyss due to the oo ualty of data obtaned. Fgue 4 below shows the exemental eoducblty fo the atcle sze obtaned afte mn of ectaton. The eoducblty n the desuesatuaton cuves s esented below n Fgue 5, showng vey lttle vaaton between the eeat exements. The caustc concentaton ove tme fo the 5 eeat exements s shown n Fgue 6. It s clea that the vaaton obseved between exements s due to the dffculty n achevng the am ntal caustc concentaton; afte t the caustc nceases smlaly fo each eeat exement. Fgue 7 shows the measued solds content wth tme fo 4 of the eeat exements (7 Ma data s excluded as ths was o to the moved method fo solds measuemen. It s clea that the solds concentaton s geneally consstent between the eeat batches and vaaton can be exlaned by eos n achevng the am stat solds concentaton and n oo washng of the solds and flte ae. o. e Inteval (o./l).e+8 9.E+7 8.E+7 7.E+7 6.E+7 5.E+7 4.E+7.E+7.E+7.E+7 7-Ma -Ma 5-A -A 8-A Sze (μm) Fgue 4. Reoducblty of PSD s Afte mn fo the 5 Reeat Exements Caustc (g/l) Fgue 6. Reoducblty of the Luo Caustc Concentaton fo the 5 Reeat Exements Solds (g/l) Ma 5-A -A 8-A Fgue 7. Reoducblty of the Measued Solds Content fo the 5 Reeat Exements Knetcs Estmaton Pocedue and Results The knetcs estmaton ocedue has been develoed usng a combnaton of the Bamley method fo knetcs estmaton and the wok conducted by Ltste et al. and Wynn on dscetsaton of the oulaton balance. The ocedue fo obtanng the knetcs estmates fom the exemental data wll be outlned n the followng sectons.. Patcle Sze Dstbuton Conveson The fst ste n the estmaton ocedue s to convet the sze dstbuton eoted by the Coulte Counte, whch can use ethe a lnea o logathmc sacng fo the sze ntevals, to a sze dstbuton that uses a geometc ogesson to detemne the sze ntevals,4. The ato of the geometc ogesson,, s detemned by: En 5 88 Poceedngs of the 8th Intenatonal Alumna Qualty Woksho 8

4 whee s an ntege; the lage, the fne the PSD dscetsaton. Fgue 8 below shows the aw coulte PSD as comaed to the conveted PSD, the sze nteval conveson smoothes most of the nose fom the aw coulte data. o. Concentaton (o./um/l).e+5.5e+5.e+5.5e+5.e+5 5.E+4 Coulte PSD Conveted PSD Sze (um) Fgue 8. Comason between the omalsed o. Concentaton fo Coulte and Conveted PSD Detemnaton of the Moments of the Dstbuton Afte the Coulte sze dstbutons have been conveted, as descbed above, the moments of the sze dstbutons ae calculated at each exemental tme ont. The k th moment fo a dscetsed sze dstbuton s defned by: En 6 m k L n k whee L s the mean atcle sze of the th sze nteval, and n s the numbe of atcles n the th sze nteval. The moments used n the knetcs estmaton ocedue ae: The zeoth moment, μ, whch s the total numbe of atcles The second moment, μ, whch s ootonal to the total aea of atcles The thd moment, μ, whch s ootonal to the total volume of atcles The othe oety of the conveted PSD used n the knetcs estmaton method s the numbe of atcles n the fst sze facton ( ), whch s smly ecoded fom the conveted sze dstbuton at each tme nteval.. Detemnng the Rate of Change of μ, μ, and The next ste n the ocedue s to calculate the ate of change of the zeoth moment, the thd moment and the numbe of atcles n the fst sze facton. Ths s the most ctcal ste n gettng easonable knetc aamete estmates. Thee ae two methods that have been used to detemne the ate of change: Ft a natual cubc slne functon to each set of data and evaluate the devatve of the functon at each tme ont Ft a model to the data usng a least suaes mnmsaton, and evaluate the devatve of the model functon at each tme ont In actce t s best to use a combnaton of the two methods as they both have the ostves and negatves. atual cubc slne Postves A natual cubc slne ogam can easly be coded nto a maco n vsual basc to ft the slne to the data, evaluate the devatve, and outut the devatve at the exemental tmes. egatves The slne has to ass though evey sngle exemental data ont, and as such, exemental eo n the data can cause sgnfcant eos n the calculaton of the devatve (see Fgue 9 below). When thee ae vey small changes n the data (e.g. th moment at t > 5mn n Fgue below), small exemental vaaton can cause swngs between ostve/negatve devatves, whch n tun ntoduces eo nto the knetc estmates. Model Fttng Postves Moe obust to andom exemental vaaton, as the model doesn t need to ass though evey data ont. egatves Moe comlex than a cubc slne. eed to detemne the ght fom of model euaton to ft the data. An euaton fom may not exst that fts the exemental data. Dffeentaton of the model euaton may be dffcult fo comlex model euaton foms. d Moment d Moment Slne d/dt d Moment 8.E+ 8.E+8 7.E+ 7.E+8 6.E+ 6.E+8 5.E+8 5.E+ 4.E+8 4.E+.E+8.E+.E+8.E+.E+8.E+.E+ -.E+8.E+ -.E Fgue 9. atual Cubc Slne Ft to Exemental d Moment Data th Moment 4.E+6.5E+6.E+6.5E+6.E+6.5E+6.E+6 5.E+5 th Moment Slne d/dt th Moment.5E+5.E+5 5.E+4-5.E+4 -.E Fgue. atual Cubc Slne Ft to Exemental th Moment Data. Detemnaton of Knetc Paametes Once the ate of change of each of μ, μ, and has been detemned at each samle tme, the knetc aametes (G, Bu, β ) can be found by solvng smultaneous euatons: En 7 μ β + B u En 8 μ G + Bu L En 9 + β G + B u d/dt d Moment d/dt th Moment Poceedngs of the 8th Intenatonal Alumna Qualty Woksho 8 89

5 The devaton of these euatons s detaled n Bamley et al., and,,, and ae detemned by the followng euatons and can be calculated fom the atcle sze dstbuton data, : En n e ( B + B + B + B4 D D ) En n e f ( L, ) L En ( + ) L En ( + Whee, En 4 B n e L + ) L S En 5 B f ( L S S En 6 B f ( L, L ) f ( L, L ) En 7 B 4 En 8 D + S + + S S + En 9 D n e S + f ( L f ( L, L ) f ( L, L ) En ln S Int ln +, L ), L ) / ( ) ( ) ( + ) L + ( ) ( ) n e s the total numbe of sze factons used to descbe the atcle sze dstbuton and f ( L, L ) s the sze deendency tem n the agglomeaton kenel (en 4 above) and s eual to fo sze ndeendent agglomeaton. Solvng the smultaneous euatons at each samle tme ont fo G, Bu, β and gves the gowth, nucleaton, and agglomeaton knetcs as a functon of tme dung the batch test (Fgues,, and below). Valdaton of the Estmates by Solvng the Poulaton Balance Euaton A batch ectaton model has been develoed whch uses a 4th ode Runge-Kutta technue to numecally solve the oulaton balance euaton fo n (. The model needs the followng nuts: + L fttng a model to the knetc estmates wth the exemental suesatuaton at each of the samle tmes) The model then uses the second moment of the PSD (ootonal to suface aea) and the gowth ate to detemne the amount of gbbste ectated dung the tme ste. The model can then calculate the suesatuaton at the end of the tme ste, whch s then used to calculate the knetc aametes at the new tme ont. The calculated atcle sze dstbutons (.e. n ( ) ae comaed aganst those obtaned exementally, and any sgnfcant devatons ndcate oo knetc estmates. Fgue 4 and Fgue 5 below show the model edcted atcle sze dstbutons (at t4mn and tmn) usng the t exemental data as ntal condtons and the knetc estmates detemned above vesus the exemental atcle sze dstbutons. Gowth Rate (um.h-) Fgue. Gowth Rate Estmates ucleaton Rate (ml-.h-).e+7.e+7 8.E+6 6.E+6 4.E+6.E+6.E Fgue. ucleaton Rate Estmates Agglomeaton Kenel (ml.h-).6e-6.4e-6.e-6.e-6 8.E-7 6.E-7 4.E-7.E Fgue. Agglomeaton Kenel Estmates An ntal atcle sze dstbuton and suesatuaton The gowth, nucleaton and agglomeaton knetc aametes as a functon of suesatuaton (ths s done by 9 Poceedngs of the 8th Intenatonal Alumna Qualty Woksho 8

6 Model Pedcted Exemental.6E+5.4E+5.E+5 Count (o./ml).e+5 8.E+4 6.E+4 4.E+4.E Sze (μm) Fgue 4. Model Pedcted Vs Exemental PSD fo t4 mn Model Pedcted Exemental.6E+5.4E+5.E+5 Count (o./ml).e+5 8.E+4 6.E+4 4.E+4.E Sze (μm) Fgue 5. Model Pedcted Vs Exemental PSD fo t mn Conclusons A mathematcal model has been develoed that s caable of estmatng the ectaton gowth (G), nucleaton (Bu), and agglomeaton (β ) knetc aametes fom batch exemental data. These knetc aametes ae eued fo solvng the oulaton balance euaton. The stes ae: Conduct a batch exement, collectng samles ove the duaton of the smulaton fo luo comoston, solds concentaton, and atcle numbe sze dstbuton Check fo mass balance consstency usng the analyses Convet the aw atcle sze dstbutons nto the coect fom fo analyss Calculate the zeoth and thd moments (μ and μ ) and the numbe of atcles n the fst sze nteval ( ) of the conveted PSD s Detemne the ate of change of μ, μ, and at each of the samle tmes Fom the PSD at each samle ont, calculate the values of the coeffcents,,, and Solve the system of smultaneous euatons fo the knetc aametes G, B u, and β Check the valdty of the estmates by solvng the oulaton balance euaton usng the values of G, B u, and β and comang the edcted PSD wth the exemental PSD Refeences. Bamley, A.S.; Hounslow, M.J.; Ryall, R.L. Aggegaton Dung Pectaton fom Soluton: A Method fo Extactng Rates fom Exemental Data ; J. Collod Inteface Scence 996, 8, Ltste, J.D.; Smt, D.J.; Hounslow, M.J. Adustable Dscetsed Poulaton Balance fo Gowth and Aggegaton ; AIChE J., 995, 4(), Wynn, E.J.W. Imoved Accuacy and Convegence of Dscetzed Poulaton Balance of Ltste et al., AIChE J., 996, 4(7), Hounslow, M.J.; Ryall, R.L.; Mashall, V.R. A Dscetsed Poulaton Balance fo ucleaton, Gowth, and Aggegaton, AIChE J., 988, 4(), 8 Poceedngs of the 8th Intenatonal Alumna Qualty Woksho 8 9