Leture 16. Kineti and Ma Tranfer in Crytallization Crytallization Kineti Superaturation Nuleation - Primary nuleation - Seondary nuleation Crytal Growth - Diffuion-reation theory - Srew-diloation theory
Crytallization Kineti Crytallization i a omplex phenomenon involving three tep: (1) nuleation, (2) ma tranfer of olute to the rytal urfae, and (3) inorporation of olute into the rytal lattie Þ Crytallization kineti Superaturation i the driving fore for rytallization kineti A rytal ize dereae, olubility notieably inreae, making it poible to uperaturate a olution if it i ooled lowly without agitation The olubility of very mall rytal an fall in the metatable region Point a: the olution i underaturated; rytal of all ize diolve Point b: equilibrium between a aturated olution and rytal that an be een by naked eye
Superaturation Point : metatable region; rytal an grow but annot nuleate Point d: pontaneou nuleation of very mall rytal, that are inviible to the naked eye, our Limiting uperaturation, D limit m Superaturation, D Limit of uperaturation temperature differene Superaturation temperature differene Kelvin equation: a relationhip between olubility and rytal ize æ ö ln ç è ø 4v, L vrtd Relative uperaturation - p v : molar volume of the rytal,l : interfaial tenion v : number of ion/moleule of olute / : uperaturation ratio (S) In pratie, i uually le than 2% - 1 S - 1
Nuleation (1) To determine volume or reidene time for magma in a rytallizer, the rate of nuleation (birth) of rytal and their rate of growth mut be etablihed Relative rate of nuleation and growth are important beaue they determine rytal ize and ize ditribution Primary nuleation - Superaturated olution i free of rytalline urfae - Requre high uperaturation and i the prinipal mehanim in preipitation
Nuleation (2) - Homogeneou nuleation our with uperaturated olution in the abene of foreign matter, uh a dut moleule in the olution firt aoiate to form a luter, whih may dioiate or grow if a luter get large enough to take on the appearane of a lattie truture, it beome an embryo table rytalline nuleu rate of homogeneou nuleation B o é ê -16p v Aexp ê 2 3 ê 3v ( RT) éln ë ë N 2 3, L a ( ) B o : rate of homogeneou primary nuleation, number of nulei/m 3 A : frequeny fator N a : Avogadro number ù û 2 ù ú ú úû
Nuleation (3) Seondary nuleation - Superaturated olution ontain rytal - Key in ommerial rytallizer, where rytalline urfae are preent and large rytal are deired - Initiated by (1) fluid hear pat rytal urfae that weep away nulei, (2) olliion of rytal with eah other, and (3) olliion of rytal with metal urfae (rytallizer veel wall or agitator blade) - (2) and (3) mehanim are ontat nuleation: mot ommon ine they happen at the low value of relative uperatuation,, typial of indutrial appliation - Empirial power-law funtion B k M N N b j r T B : rate of eondary nuleation M T : ma of rytal per volume of magma N : agitation rate (e.g., rpm of an impeller) The ontant k N, b, j, and r are determined from experiment
Crytal Growth (1) Ma tranfer theory of rytal growth baed on equilibrium at the rytal olution interfae [Noye and Whitney, 1897] dm dt k A( - ) dm/dt : rate of ma depoited on the rytal urfae k : ma-tranfer oeffiient A : urfae area of the rytal : ma olute onentration in the bulk uperaturated olution : olute ma onentration in the olution at aturation Thin, tagnant film of olution adjaent to the rytal fae through whih olute moleular diffuion take plae [Nernt, 1904] k D d D : diffuivity d : film thikne Two-tep theory of rytal growth, diffuion-reation theory [Valeton, 1924] - Firt tep: ma tranfer of olute from the bulk of the olution to the rytal-olution interfae our dm dt k A( - ) i : uperaturated onentration at the interfae i
Crytal Growth (2) - Seond tep: a firt-order reation i aumed to our at the rytal-olution interfae, in whih olute moleule are integrated into the rytal-lattie truture (kineti tep) dm dt k A( - ) dm dt i i A( - ) 1 k + 1 k i - At low veloitie, growth rate may be ontrolled by the firt tep (ma tranfer) - The eond tep (kineti) i important when olution veloity pat the rytal urfa i high: k > k i - The ma-tranfer oeffiient, k, for the firt tep i independent of the rytallization proe - The kineti oeffiient, k i, i unique to the rytallization proe
Crytal Growth (3) - Srew-diloation theory a diloation i an imperfetion in the rytal truture predit a growth rate proprotional to ( i - ) 2 at low uperaturation and to ( i - ) at high uperaturation
Crytal Growth (4) - Although rytal do not grow a phere, an equation an be derived for the diameter of a pherial rytal dm dt K A( - ) K : overall ma-tranfer oeffiient ddp 2K ( - ) dt r p D A p D 2 p and m 6 If growth rate i ontrolled by k i, whih i aumed to be independent ( ) DDp 2ki D of D p Dt r Crytal-ize inreae i linear in time for a ontant uperaturation If growth rate i ontrolled by k at a low veloity K k 2D D D : olute diffuivity p ddp 4D( D) dt D r p 2K ( D) r 3 p r
Crytal Growth (5) Integrating from D po to D p 2 2 Dp - Dp 4D( D) o t 2 r If D po D p, D p æ 8D( D) t ö ç è r ø 1 2 In thi ae, the inreae in rytal diameter low with time At higher olution veloitie where k till ontrol, K k C D 1 2 1 p For thi ae, the inreae in rytal diameter alo low with time, but not a rapidly a predited in the previou ae