Dissolution of Solid Particles in Liquids: A Shrinking Core Model

Transcription

1 Wold Aademy of Siene, Engineeing and Tehnology 5 9 Dissolution of Solid Patiles in Liquids: A Shining oe Model Wei-Lun Hsu, Mon-Jyh Lin, and Jyh-Ping Hsu Astat The dissolution of spheial patiles in liquids is analyzed dynamially. Hee, we onside the ase the dissolution of solute yields a solute-fee solid phase in the oute potion of a patile. As dissolution poeeds, the intefae etween the undissolved solid phase and the solute-fee solid phase moves towads the ente of the patile. We assume that thee exist two esistanes fo the diffusion of solute moleules: the esistane due to the solute-fee potion of the patile and that due to a sufae laye nea solid-liquid intefae. In geneal, the equation govening the dynami ehavio of dissolution needs to e solved numeially. Howeve, analytial expessions fo the tempoal vaiation of the size of the undissoved potion of a patile and the vaiation of dissolution time an e otained in some speial ases. The pesent analysis taes the effet of vaiale ul solute onentation on dissolution into aount. Keywods dissolution of patiles, sufae laye, shining oe model, dissolution time. I. INTRODUTION HE dissolution of solid patiles in liquids an oughly e T desied y two poess in seies: the esape of solute moleules fom the solid sufae and the diffusion of these moleules towad the ul liquid phase. Depending on the opeating ondition, the ate of dissolution may e ontolled y one of these two steps [],[]. A model often used fo the desiption of the tanspot of solute moleules to the ul liquid phase is the film theoy poposed y Shewood et al []. This appoah povides a onvenient way of desiing the dynami phenomenon unde onsideation. Howeve, sine the onentation of solute in the ul liquid phase is assumed onstant, it is most appopiate fo the ase of low soluility. By assuming that the ul liquid phase is stagnant and of infinite size, hen and Wang [4] have deived an analytial expession fo the vaiation of dissolution time as a funtion of the onentation of solute in the ul liquid phase. Fo the dissolution of patiles ompise inogani and eletohemial mateials, the method of invaiant funtions is often adopted fo the desiption of its dissolution inetis [5]. In some ases, the soluility of only one of the omponents ontained in the solid phase is appeiale ompaed with the est omponents. In these ases, if the amount of the dissolving omponent is limited, the appeaane of the solid phase will W. L. Hsu is with the National Taiwan Univesity hemila Engineeing Depatment, Taipei, Taiwan 67, RO (phone: ; fax: ; @ ntu.edu.tw. emain essentially unhanged duing the ouse of dissolution. As dissolution poeeds, the undissolvale solid phase povides a esistane fo the diffusion of solute moleules towad the ul liquid phase. This an e advantageous in patial appliations. A well-nown example is the ontolled-elease of an ative agent in dug [6] and fetilize [7] whee the esistane due to the undissolvale solid is utilized to maintain the dissolution ate on a etain favoale level. The pupose of this epot is to examine the dynami ehavio of the dissolution of solid patiles in liquids. In patiula, we ae inteested in the ase the dissolution yields a solute-fee solid phase. An attempt is made to tae the effet of finite ul liquid on dissolution into aount. The mathematial epesentation thus otained us not limited to the magnitude of soluility. Fig. A shemati epesentation of the dissolution phenomenon unde onsideation II. ANALYSIS By efeing to Fig., we assume that thee exists a sufae laye nea the sufae of a spheial patile. Also, the dissolution of solute yields a solute-fee solid phase on the oute potion of the patile. As dissolution poeeds, the intefae etween the undissolved solid phase and the solute-fee solid phase moves towads the ente of the patile. The dissolution of solute onsists of thee steps in seies: (i Diffusion of solute moleules fom the solid-solid intefae though the solute-fee potion of the patile to the solid-liquid 9

2 Wold Aademy of Siene, Engineeing and Tehnology 5 9 intefae. (ii Diffusion of solute moleules fom the solid-liquid intefae though the sufae laye to the oute ounday of the sufae laye. (iii Diffusion of solute moleules fom the oute ounday of the sufae laye to ul liquid phase. Suppose that the onentation of solute on the solid-solid intefae is at satuation. On the asis of these assumptions, a solute alane ove the solute-fee potion of the patile and the sufae laye egion yields, espetively, D t D t (, < < (, < < whee denotes the onentation of solute, t is time, D and D ae espetively, the effetive diffusivity of solute in the solute-fee solid phase and the diffusivity of solute in the sufae laye, and ae, espetively, the adius of the patile and the adius of oe, δ is the thiness of the sufae laye, and is the distane measued fom the ente of the patile. If the diffusion ate of solute is muh geate than the ate of moving of the solid-solid intefae towads the ente of the patile, a pseudo-steady state an e assumed. In this ase, ( and ( edue to, espetively, D (, < < D (, < < δ (4 Integating these two equations, sujet to the onditions e at, s at, and, at δ, we otain e s ( s [ (, < <, δ ] δ < < δ whee e, s, and epesent the onentations of solute at dissolving font, at the solid-liquid intefae, and in the ul liquid phase, espetively. At the solid-liquid intefae, we have d d SD SD d d (7 whee S denotes the sufae aea of the patile. By efeing to (5 and (6, this expession eomes D e s s D (8 δ ( ( ( (5 (6 Solving this expession fo s, we otain s η [ η [ δ ] e ( δ ] ( whee η D D. Let m e the nume of moles of solute ontained in the solid patile. At pseudo-steady state, we have dm (9 d π D onstant ( d 4 Integating this equation with espet to, sujet to the ounday onditions e at and s at, yields dm ( ( 4π D ( e s ( Sustituting (9 into this expession and noting that m 4π ρε M, we have η( d [ ]( ( δ MDη ( e ρε[ ( δ ] ( whee M denotes the moleula weight of solute, ρ is the density of patile, and ε is the weight fation of solute. A. onstant Bul onentation If the volume of the ul liquid phase V is lage enough, the onentation of solute in the ul liquid phase is essentially onstant. Denote this onentation as. Fo onveniene, ( nondimensionlized as d [ ( P η η ]( η ( whee VM ( e 4 εn (a πρ δ δ ( P δ ( (d t πtd n V (e 4 whee n is the nume of patiles. Solving (, sujet to the initial ondition at t, we otain 94

3 Wold Aademy of Siene, Engineeing and Tehnology 5 9 t {[( ( P η ] [ η( ]} η (4 This equation desies impliitly the vaiation of the adius of a patile as a funtion of time. B. Vaiale Bul onentation If the volume of the ul liquid phase is finite, the vaiation of the ul liquid onentation is 4 δ πρε n( n( VM VM n( δ 4π d 4π d V 4 π d V (5 The vaiation of as a funtion of at pseudo-steady state an e otained y integating (5 and (6 with the onditions e at, s at, and at δ. The esults ae, < < (6 ' ', < < δ (7 whee ( e s ( (7a ( s e ( (7 ' δ ( s [ ] (7 ' δ [ s ( δ ] [ ] (7d Sustituting (6 and (7 into (5, we otain 4πρε n( VM 4πn{ ( ( ' ' [( δ ] [( δ ]} V (8 Sustituting s fom (9 into (7a though (7d and sustituting the esultant expessions into (8, we have whee 4πn[ ρε ( M f e ] V (9 4πnf V f f ( ( [( δ ] ( [( δ 6 ( ] 5 4 [( δ ] [( In these expession, ( a ( ( a δ ( a ( ( a ( 4 a [ 5 δ ] ( a [ 6 δ ] [ a ( δ ] [ 7 δ [ a ( δ ] [ 8 ] 8 δ whee a a η [ η [ η [ ( δ ] δ ] ( δ ] ( Sustituting (9 into (, gives [ ( P η η η[ whee e ( e ρε M f f d ] f e V ( f f ] ] ] 7 ( 95

4 Wold Aademy of Siene, Engineeing and Tehnology 5 9 f f V V 4πn Speial ases If δ <<, then (5 edues to n( 4 d V VM π VM 4πn{ ( ( If e << ρε M and δ <<, then ( t d {[( P η ]ln[ α ( α ] ( η α { ln[ α ( α] ln[ α (( α [tan ( tan α 4] (( α α ]}} η (6 D. Solid Phase Diffusion ontol If η >> (o D >> D, the diffusion esistane due to sufae laye is negligile. By following the simila poedue as that employed fo the deivation of (4 though (6, it an e shown that, fo the ase of onstant ul onentation, t [( ( ] (7 t d 6 (8 and VM ( Sustituting this expession into (, we otain, afte simplifiation, [ d ( P η η ] η ( ( Solving this equation, sujet to the initial ondition at t, we otain t {[( P η ]ln[( α ( α ] ( η α { ln[( α ( α] ln[(( tan [tan α ((( (( α whee α. α α 4 (( α α ]}} η α α 4] (4. Dissolution Time The dimensionless time equied to dissolve a patile, o the dimensionless dissolution time, t d, an e evaluated y letting in eithe (4 o (4. We have, fo the ase of onstant ul onentation t d [( P η η ] η (5 and fo the ase of vaiale ul onentation with e << ρε M and δ << t {ln[( ln[(( α tan ln[( t d α 4] ln[ α α ( α] (( α [tan α [tan ((( α ]}} η α ( α ] {lnα ( α (( α ( ln[ α ( α ] α tan 4 (( α α 4] (( α α α ]} α (9 ( fo the ase of vaiale ul onentation with e << ρε M and δ <<. E. Sufae Laye Diffusion ontol The definition of η suggests that if η << (o D << D, the diffusion esistane is mainly due to sufae laye. It an e shown that, fo the ase of onstant ul onentation, t ' P( ( t d ' P ( whee t ' 4πtD n V, and t d ' is the dimensionless dissolution time. Fo the ase of vaiale ul onentation with e << ρε M and δ <<, we have t ' P ln[( α ( α ] ( t d ' P ln[ α ( α ] (4 96

5 Wold Aademy of Siene, Engineeing and Tehnology 5 9 oated mateial. The analysis is essentially the same as that pesented in the pesent study. Fig. Simulated tempoal vaiation of the adius of the undissolved solid phase fo the ase δ., V, η.5,, and. Solid uve: onstant ul onentation, (4; long dash uve: vaiale ul onentation with e << ρε M and δ <<, (; shot dash uve: vaiale ul onentation, ( III. RESULTS AND DISUSSION Fig. shows the simulated vaiation of the adius of the undissolved solid phase as a funtion of time. As an e seen fom this figue, the dissolution is faste fo the ase of onstant ul liquid onentation than fo the ase of vaiale ul liquid onentation. This is due to the deease of onentation diving foe duing the ouse of dissolution fo the latte. In a study of the ontolled-elease of pogesteone fom mioapsules, Gupta and Spas [8] have deived a ineti expession simila to (4. In thei mathematial fomulation, the solute distiuted in the egion [, δ ] is negleted. As an e seen fom Fig., negleting the solute ontained in this egion may lead to an appeiale eo. The vaiation of dissolution time as a funtion of onentation diffeene is shown in Fig.. As suggested y (a, fo a fixed solid-liquid omination, ineases with the inease of V. Thus, is a measue of the elative signifiane of the vaiation of solute onentation in the ul liquid phase. The geate the, the less signifiant the vaiation of the solute onentation in the ul liquid phase. Figue eveals that if is on the ode of, the vaiation of the solute onentation in the ul liquid phase is negligile. In othe wods, if ( e - > 4π ρεn VM, the assumption of onstant ul onentation is appopiate in estimating the dissolution time of a patile. Note that the solute ontained in the egion [, δ ] have a signifiant effet on dissolution time fo small. The vaiation of dissolution time as a funtion of onentation diffeent values of fo the ase δ. is shown in Fig. 4. As suggested y the definition of η ( D D, if η is small, the dissolution is sufae laye diffusion ontolled-elease, a speifi mateial is often oated on the sufae of a patile to edue the ate of dissolution. Hee, the oated solid plays the ole of the sufae laye of this study and D is intepeted as the effetive diffusivity of solute in the Fig. Vaiation of dissolution time as a funtion of onentation diffeene. Solid uve: onstant ul onentation, (5; long dash uve: vaiale ul onentation with e << ρε M and δ <<, (6; shot dash uve: vaiale ul onentation with e << ρε M and δ << not satisfied Fig. 4 Vaiation of dissolution time as a funtion of onentation diffeene fo diffeent values of fo the ase δ., V, and. Solid uve: (6; long dash uve: (; shot dash uve: (4. (a: η.5; (: η ; (: η 5 If δ <<, then δ 4π d 4π Sine < e, we have APPENDIX d (A 4π d > 4π ed 4π e ( (A By efeing to (5 in the text, we have 97

6 Wold Aademy of Siene, Engineeing and Tehnology 5 9 > VM 4πn( V 4πn ρε ( e M e ( V (A Theefoe if e << M ρε, VM (A4 AKNOWLEDGMENT This wo is suppoted y the National Siene ounil of the Repuli of hina. REFERENES [] M. R. Riazi and A. Faghi, Solid dissolution with fist-ode hemial eation, hem. Eng. Si., vol. 4, pp. 6-6, 985. [] A. N. Bhasawa, On appliation of the method of ineti invaiants to the desiption of dissolution aompanied y a hemial eation, hem. Eng. ommun., vol. 7, pp. 5 4, 988. [] T. K. Shewood, R. L. Pigfod, and. R. Wile, Mass tansfe. New Yo: MGaw Hill, 975. [4] Y. W. hen and P. J. Wang, Dissolution of spheial solid patiles in a stagnant fluid: an analytile solution, an. J. hem. Eng., vol. 67, pp , 989. [5] E. M. Vidgohi and A. B. Shein, Mathematial modeling of ontinuous dissolution poesses. Leningad: Khimiye, 97. [6] K. A. Ovehoff, J. D. Stom, B. hen, B. D. Sheze, T. E. Milne, K. P. Johnston, and R. O. Williams, Novel ulta-apid feezing patile engineeing poess fo enhanement of dissolution ates of pooly wate-solule dugs, Eu. J. Pham.Biopham., vol. 65, pp , 7. [7] P. Loganathan, M. J. Hedley, M. R. Betheton, and J. S. Rowath, Aounting fo patile movement when assessing the dissolution of slow elease fetilizes in field soils, Nut. yl. Agoeosyst., vol. 7, pp , 4. [8] D. V. S. Gupta and R. E. Spas, ontolled elease of ioative mateials. New Yo: Aademi Pess, 98, pp