Lecture 18. MSMPR Crystallization Model
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1 ecture 18. MSMPR Crystalliati Mdel MSMPR Crystalliati Mdel Crystal-Ppulati alace - Number f crystals - Cumulative umber f crystals - Predmiat crystal sie - Grwth rate
2 MSMPR Crystalliati Mdel Mixed-suspesi, mixed-prduct-remval (MSMPR) mdel is useful fr the desig ad aalysis f draft-tube, baffled crystallier Assumptis (1) Ctiuus, steay-flw, steay-state perati (2) Perfect mixig f the magma () N classificati f crystals (4) Uifrm degree f supersaturati fr the magma (5) Crystal grwth rate idepedet f crystal sie (6) N crystals i the feed, but seeds are added iitially (7) N crystal breakage (8) Uifrm temperature (9) Mther liqur i prduct magma i equilibrium with the crystals (1) Nucleati rate is cstat, uifrm, ad due t secdary ucleati by crystal ctact (11) Crystal-sie distributi is uifrm i the crystallier ad equal t that i the magma (12) All crystals have the same shape
3 Crystal-Ppulati alace (1) The crystal-sie distributi ca be estimated as a fucti f the rpm f the draft-tube prpeller ad exteral circulati rate by a crystal-ppulati balace i the MSMPR mdel The umber f crystals per uit sie per uit vlume d( N V ) d Fr a cstat, crystal-sie grwth rate idepedet f crystal sie G d dt 1 V dn d : characteristic crystal sie (e.g. frm a scree aalysis) N : cumulative umber f crystals f sie ad smaller i the magma i the crystallier V : vlume f the mther liqur i the crystallier magma D GDt Gt D law f McCabe t : residece time i the magma i the crystallier fr crystals f sie
4 Crystal-Ppulati alace (2) Number f crystals i the sie rage d dn Vd Frm the perfect-mixig assumpti fr the magma, umber f crystals withdraw mther - liqur vlume withdraw umber f crystals i crystallier mther - liqur vlume i crystallier umber f crystals withdraw umber f crystals i crystallier mther - liqur vlume withdraw mther - liqur vlume i crystallier Dd D Q D t Q - - : vlumetric flw rate f mther liqur d V i the withdraw prduct magma
5 Crystal-Ppulati alace () D GDt D Q D t - V ü ý þ - D Q D d Q - GV d GV Takig the limit Reteti time f mther liqur i the crystallier, t V Q d - d Gt Itegrati exp( - G t ) Number f crystals per uit vlume f mther liqur belw sie N V ò d Number f crystals per uit vlume f mther liqur N V T ò d
6 Crystal-Ppulati alace (4) Cumulative umber f crystals f sie smaller tha, as a fucti f the ttal x ò ò - Gt e d 1 - exp( - Gt ) - Gt e d 1 - e - (/Gt) : dimesiless crystal sie Cumulative distributi (cumulative crystal ppulati) Differetial distributi dx d e - Fr a give value f, x is the fracti f crystals havig a smaller -value
7 Crystal-Ppulati alace (5) Mmet equati fr a relati f ( ) x ò k d k k ò d k : rder f the mmet Mmet Distributi basis Cumulative Differetial Zerth First Secd Third Number Sie r legth Area Vlume r mass x 1 - e - x 1- ( 1+ ) e - 2 x 1 - ( ) e - a 2 x 1 - ( ) e - m dx d e - dx d e - 2 dx d e - a dx d e - Predmiat crystal sie, pd i terms f the mass distributi: crrespdig t the peak f the differetial-mass distributi æ dx d m ö ç d è ø e e - \ pd Gt d 6 6 Gt m 2 6
8 Crystal-Ppulati alace (6) Grwth rate (G) depeds the supersaturati ad degree f agitati, ad residece time (t) depeds crystallier desig ad perati 1 dn 1 dn æ d ö ç V dt V d è dt ø 1 lim V 1 lim V dn dt d G dt dn d ü ï ý ï ï þ G exp( - G t ) Pwer-law fucti fr the effect f peratig cditis k G M N i j r N T exp( - G G t ) This equati ca be used t btai ucleati ad grwth rates
9 Crystal-Ppulati alace (7) Number f crystals per uit vlume f mther liqur c NT V Mass f crystals per uit vlume f mther liqur m c ò m d m p : mass f a particle, m p fv r p m 6 f r ( Gt ) p c v p 4 Number f crystals per uit mass f crystals c 1 m 6 f r ( Gt ) c v p Nucleati rate C ò d tg e d tg 9 G t 2 f r v p pd 9C ò - c mcv 2 f r C : mass rate f prducti f crystals v p V pd pd f v : vlume shape factr defied by v f D p v p i
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