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Dissolution of solid particles in liquids
Dissolution of Solid Particles in Liquids J Y H - P I N G HSU 1 AND M O N - J Y H LIN Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10764, Republic of China
Received February 6, 1990;accepted May 18, 1990 The behavior of the dissolution of solid particles in liquids is modeled kinetically. It is assumed that a surface layer exists near the solid-liquid interface; the diffusion of solute moleculesthrough this layer is the rate controlling step. Analytical expressions for the description of the temporal variation of the size of a particle and for the dissolution time of a particle are derived. The present analysis takes the effect of the dissolved solute on the bulk liquid concentration into account, and, therefore, is applicable for both low solubility cases and for high solubility cases. The effects of polydispersed particles and variable surface layer thickness on dissolution kinetics are discussed. A criterion for neglectingthe effect of dissolved solute on bulk liquid concentration is suggested. © 1991AcademicPress,Inc. I. INTRODUCTION
phase/saturation concentration). For inorganic and electrochemical materials the method of invariant functions, a semiempirical approach, is often adopted for the description of the dissolution kinetics (6, 7). As pointed out by Bhaskarwar (2), however, this approach has under its surface some logical and empirical inconsistencies. The dissolution of hydroxyapatite (HA) in water results in the release of Ca 2+ and PO43-. It is found that these ions adhere on the surface of H A and form a surface layer; an equilibrium is reached between the solid surface and the bottom of the surface layer. The existence of this surface layer provides a resistance for the dissolution of HA (8). T h o m a n n (9) suggests that there exist two solute layers: a surface layer and a Nernst layer. The significance of the latter depends upon the operating condition. The more extensive the agitation is, the less significant it is. Previous investigations focus mainly on the case where the bulk liquid phase is of infinite size. In other words, the variation of solute concentration in this phase is negligible. This occurs if the solubility of particles is low. Since the solute concentration in the bulk liquid phase is constant, the driving force of dissolution due to the concentration difference be-
The dissolution of solid particles in liquids is a p h e n o m e n o n in which the solute molecules on solid surface leave the surface continuously at the expense o f the mass of the particle. The whole process can roughly be described by two steps in series: the escape of solute molecules from the solid surface and the diffusion of these molecules toward the bulk liquid phase. Depending on the operating conditions, the rate of dissolution may be controlled by one of these two steps. Riazi and Faghri ( 1 ) and Bhaskarwar (2), for example, assume that the mass transfer of dissolved solid away from the solid-solution interface is the rate-controlling step. A model often adopted to describe the transport of solute molecules to the bulk liquid phase is the film theory ( 3, 4). Since a pseudo-steady state assumption for the concentration profile of solute is made, this model is most appropriate for low solubility cases. Under the assumption that a film is of infinite thickness, Chen and Wang (5) have derived an analytical expression for the variation of dissolution time as a function of the ratio (concentration of solute in the bulk To whom correspondenceshould be addressed. 60 0021-9797/91 $3.00 Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved,
Journal of Colloid and Interface Science, Vol. 141, No. 1, January 1991
SOLID
PARTICLE
tween the solid surface and the bulk liquid phase remains essentially constant if the diffusion of solute molecules is the rate-controlling step. This assumption, although it simplifies the derivation of the governing equation for dissolution kinetics, is unrealistic in practical applications, where the bulk liquid phase is of finite size. The purpose o f this report is to derive a general kinetic model that takes the effect of dissolved solute on the bulk liquid concentration into account. The mathematical representation thus obtained is applicable for both the cases in which the solubility is low and the cases in which the solubility is high. The effects ofpolydispersed particles and variable surface layer thickness on the dissolution kinetics are also discussed. II. M O D E L I N G
We assume the following: ( 1 ) A solid particle is spherical. (2) A surface layer is formed near the solid-liquid interface. (3) The ratecontrolling step of the dissolution process is the diffusion of solute molecules through the surface layer. In other words, the concentration of solute on the solid surface is at saturation. On the basis of these assumptions, a mass balance on solute yields Ot
r 2 Or
r2 -~r
'
[ 1]
where C denotes the concentration of solute, t represents the time scale, De is the diffusivity of solute, and r is the distance measured from the center of a particle. Suppose that the surface layer is thin, and a pseudo-steady state is achieved. Equation [1] reduces to
d (r2dC 1 dr ] = O.
De-~r ~
[21
Integrating this equation with boundary conditions C = Ce at r = re and C = Cb, the bulk liquid concentration of solute, for r >/rc + 6 yields dC dr
rc(rc
+
6)(Ce - Cb) r26
[3]
61
DISSOLUTION
where re is the radius of a particle and 6 is the thickness of the surface layer. If n is the n u m ber of moles in the particle, then
dn
dC[
[4]
dt - SDe dr r=r~,
where S represents the surface area of the particle. Since n = 4zrr3p/3Mo and S = 4~rr~, this equation becomes dr~
DeMo(rc + 6)(C~ - Cb)
dt
pro6
,
[5]
where p and Mo are, respectively, the density of particle and the molecular weight of solute. 2.1. Constant B u l k L i q u i d Concentration
If the volume of the bulk liquid phase V is large enough, the concentration o f solute in the bulk liquid phase is essentially constant. Denote this concentration as Co. For convenience, [5] is recast as dr* dt*
-
C*(r* + 6")/r'6",
[6]
where r* = rc/ro
[6a]
6* = 6/ro
[6b]
t* - 4~rtDeroN
[6c]
3V ( C e -- Co)
C*
(47rpr3oN/3 V M o )
[6d]
In these expressions, ro is the initial radius of a particle, and N is the total number o f particles. Solving [ 6 ], subject to the initial condition r* = 1 at t* = 0, we obtain t* = { ( l - r * ) + 6*ln[(r* + 6 " ) / ( 1 + 6 " ) ] } 6 " / C * .
[7]
This equation describes implicitly the variation of the radius of a particle as a function o f time. Journal of Colloid and Interface Science, Vol. 141, No. 1, January 1991
62
HSU AND LIN cl = - 6 " 2 / ( a 3 - 6*3 )
2.2. Variable B u l k L i q u i d Concentration
I f the v o l u m e o f the bulk liquid phase is finite, the variation o f its solute concentration can be significant. In this case, the variation o f the bulk liquid concentration can be written
dl =
[12d]
a3/(0Z 3 -- ~.3)
[12e]
a2 = ( b l a 2 - cla + d l ) / 3 a 2
[12f]
b2 = ( 2 b l a 2 + cla - d l ) / 3 a 2
[12g]
c2 = ( - b ~ a 2 + c j a + 2 d , ) / 3 a .
[12h]
as
dCb dt
47rp { ~ _ ~
2dri\ N,.ri -77,/, at] V M o \ ~i=~
--
[8]
III. RESULT AND DISCUSSION
where Ni denotes the n u m b e r o f particles in class i, ri represents the radius o f a particle in class i, and M is the n u m b e r o f classes. The solution o f this equation, subject to the initial conditions Cb = Co a n d ri -- roi at t = 0, is
47rp
Cb = Co + ~
M
[~, N,.(r~i- r~)].
[9]
i=1
In the case in which all particles are of the same size, this e q u a t i o n reduces to Cb = Co +
47rp N t r 3 _ r3). 3VMo ~ o
[10]
The temporal variations o f the radius o f a particle for different values o f C* and for a fixed value o f 6" are illustrated in Fig. 1. As can be seen from this figure, the higher the value o f C*, the faster the dissolution. This is expected since the greater the C*, the greater the concentration driving force for dissolution, as suggested by [ 6 d ] . Figure 1 shows that the dissolution rate is faster for the case of constant bulk liquid concentration than for the case o f variable bulk liquid concentration. This is due to the decrease o f concentration driving force during the course o f dissolution for the latter. 3.1. Dissolution T i m e
Substituting this expression into [ 5 ], we obtain (r* + 6 " ) ( C * - 1 + r .3)
dr* -
[11]
dt*
6*r*
T h e time required to dissolve a particle, or the dissolution time t~, can be obtained by
T h e solution o f this equation, subject to the initial condition r* = 1 at t* = 0, is
0.006 "\
t* = -/~* { a l l n [ ( r * + 6 " ) / ( 1 + 6*)]
\
~,0.0o~
\ x
+ a21n[(r* + a ) / ( 1 + a ) ]
x
~;
\\
.~ 0.004
+ b2ln[((r* - a / 2 ) 2 + 3a2/4)/((1
\\
~
__ o~/2)2
"" o"--2.a
\\\\\
~o.oo3 o
+ 3a2/4)]/2
+ (b2a + 2c2)
"~ ~ 0,002
X [tan-l((2r * - a)/f3a)
"~ 0.001
-tan-'((2--a)/V3a)]/~a},
[12] o.ooo0.0
where a s= C*-
1
[12a]
al = - 6 * / ( a 3 - 6*3 )
[12b]
bl = 6 * / ( a 3 - 6*3 )
[12c]
Journal of Colloid and Interface Science, Vol. 141, No. 1, J anuary 1991
i
0.2
i
0.4
Dimensionless
t
0.6
i
0.8
Radius,
1.0
r*
FIG. 1. Temporal variation of the radius of a particle for different values of C* for the case 6" = 0.01. (--) Constant bulk liquid concentration; (---) variable bulk liquid concentration.
SOLID PARTICLE DISSOLUTION letting r* = 0 in [7] and [12] for the case of constant bulk liquid concentration and for the case of variable bulk liquid concentration, respectively. We have for the former t~ = 6"{1 + 6"1n[6"/(1 + 6 " ) ] } / C * , [13]
63
tion in the bulk liquid phase. Figure 2 reveals that if C* is on the order of 10, the variation of the solute concentration in the bulk liquid phase is negligible. In other words, [12] can reasonably be approximated by [7], if (Ce - Co) >1 407rpr3N/3 VMo.
and for the latter
3.2. Polydispersed Particles
t~ : - 6 * {alln[6*/(1 + 6*)] + a21n[a/(1 + a)] + bzln[a2/((1 -- a / 2 ) 2 + 3 a 2 / 4 ) ] / 2
M
dr*_ dt*
-- (baa + 2 c 2 ) [ t a n - ' ( 1 / f 3 ) + tan-'((2 - a)/V3a)]/V-3a}.
If the solid particles are polydispersed, and the variation of bulk liquid concentration is significant, [11] can be modified as
[14]
The variation of dissolution time as a function of C* for different values of 6" is illustrated in Fig. 2. The solid lines represent the results calculated for the case of constant bulk liquid concentration, and the dash lines denote the results for variable bulk liquid concentration. The definition of C* ( [ r d ] ) suggests that, for a fixed solid-liquid combination, the greater its value, the greater V is. In other words, C* is a measure of the relative significance of the variation of solute in the bulk liquid phase. The greater the C*, the less significant the variation of the solute concentra-
(r* + 6 " ) [ C * -
E£(r~ 3 j-1
-r*3)]/r*6 *
i = 1,2 . . . . , M ,
[15]
where r* = ri/ror
[15a]
r~j = roJro~
[15b]
t* -- 47rtDerorN 3V
[ 15c ]
( c e - c0) C* = (4wpr3orN/3VMo) f) = N J N
[15d] [15e]
M
N = Z NJ.
[15f]
\ \\ 0.1
In these expressions, ri is the radius of a particle in class i, for is a reference radius, r0j is the initial value of D, and Ni is the number o f particles in class i. The set of M equations expressed in [15] needs to be solved simultaneously.
\ N
"0.1
OiOO 1
o
x
k
x
3.3. Variable SurJace Layer Thickness
r~
The linear size of a particle decreases monotonically during the course of its disso0.0001 lution. If the linear size becomes very small, the thickness of the surface layer may reduce 10 accordingly, rather than remain constant. In Dimensionless Concentration, C* FIG. 2. Variation of dissolution time as a function of this case, an intuitive representation for the C* for differentvalues of 6". (--) Constant bulk liquid variation of the thickness of the surface layer concentration;(---) variablebulk liquid concentration. is Journal of Colloid and lnterfizce Science, Vol. 141, No. 1, January 1991
64
HSU AND LIN
6
f 6c, a constant, if
r~o ~ rc
+ 3 ~ 2 / 4 ) / ( ( 1 -- c~/2) 2 + 3c~214)]/2
1
r~¢ > re,
+ (b2a + 2 c 2 ) [ t a n - l ( ( 2 r * -
[ kro,
where k = 5~/r¢~, and rc~ is a certain critical radius. 3.3.1. Constant bulk liquid concentration. If the variation of the bulk liquid concentration is negligible, the variation of the size of a particle with time can be obtained by following the same procedure as that employed for the case of constant surface layer. We have, for the present case,
tan-i((2 - ~)/f3~)]/f3a}
-
+ k { l n [ ( r * + a ) / ( r * + a)] -
l n [ ( ( r *
t* =
+6")]}/C*,
-
oz]2)
2
+ 3e~2/4)/((r * - ~ / 2 ) 2 + 3c~2/4)]/2 -
~/3[tan-l((Zr * - ~ ) / f 3 ~ )
-
tan-l((2r * - ~)/V3a)]}/ 3c~(k+l),
I 6"{(1 - r*) + 5 * l n [ ( r * + 6*)/ (1
r*>~r*
~)/fJ~)
r*
[19]
where r* = r¢c/ro. The time required to dissolve a particle is
6*{(1-r*)+6*ln[(r*+6*)/ (1 + 6 " ) ] } / C * + k ( r .2 - r*2)/ 2(1 + k ) C * ,
[16]
r* < r * ,
t~ -- - 6 " { a l l n [ ( r * + 6 " ) / ( 1 + 6*)] + azln[(r* + 6 " ) / ( 1 + c~)] + b21n[((r* - c~/2) 2
where r* = r¢c/ro and 5" = 6~/ro. The time required to dissolve a particle is then
+ 3 a 2 / 4 ) / ( ( 1 - oz/2) 2 + 3 ~ 2 / 4 ) ] / 2 + (b2~ + 2 c z ) [ t a n - l ( ( 2 r * - a)/~/-3a)
t~ = [kr*2/2(1 + k ) C * ] + 6 " { ( 1 - r*) -
+6*ln[(r* +6")/(1+6")]}/C*.
[17]
3.3.2. Variable bulk liquid concentration. If the variation o f the bulk liquid concentration is appreciable, it can be shown that
+ k { l n [ a / ( r * + a)] - l n [ a Z / ( ( r * - or/2) 2 + 3 a 2 / 4 ) ] / 2 -
V3[tan-l( 1/V3) + t a n - l ( ( Z r * - a ) / V~a)]}/3a(k+l).
t* = -ac* { a l l n [ ( r * + 6 " ) / ( 1 + 6*)] + azln[(r* + ~ ) / ( 1 + ~)] + b21n[((r* - o~/2) 2 + 3a2/4)/((1 - a/2) 2 + 3a2/4)]/2 + (b2a + 2 c 2 ) [ t a n - l ( ( 2 r * - a ) / f 3 a ) - tan-l((2 - a)/~/Ja)]/f3c~}, r* >~ r*
tan-I((2 - a)/f3a)]/~/3~}
[18]
and t* = - 6 " { a l l n [ ( r * + 6 " ) / ( 1 + ~*)] + a21n[(r* + a ) / ( 1 + a)] + b21n[((r* - o~/2) 2 Journal of Colloid and Interface Science, Vol. 141, No. 1, January 1991
[20]
Figure 3 shows the temporal variation of the radius of a particle for different values of 6" for the case of variable bulk liquid concentration. The solid lines represent the results for the case where the thickness of surface layer is constant, and the dashed lines represent those for the case where the thickness of surface layer is variable. As can be seen from this figure, the dissolution rate of the former is slower than that of the latter. This is because the smaller the thickness of a surface layer, the less the resistance for solute molecules to diffuse toward the bulk liquid phase. For the latter case, the thickness of the surface layer decreases monotonically with the size of a par-
SOLID PARTICLE DISSOLUTION 0.12
d 0.09
~
~
~
~
-.
a2=o.7
,~ 0.06
\\\ 0.4
0
.~ 0.03
0.00
r0.2
0.0
~ 0.4
J
~ 0.6
i 0.8
1.0
Dimensionless Radius, r* FIG. 3. Temporal variation of the radius of a particle for different values of 6" for the case C* = 3. (--) The thickness of surface layer is constant; (---) the thickness of surface layer is variable with k = 1. ticle as the radius of the particle is smaller t h a n a critical value. Therefore, the d i s s o l u t i o n rate is faster t h a n if the thickness o f surface layer r e m a i n s c o n s t a n t t h r o u g h o u t the dissolution period. The variation of dissolution t i m e as a function of C* for different values o f 6* a n d for the case o f variable b u l k c o n c e n t r a t i o n is illustrated in Fig. 4. T h e solid lines represent
65
the results for the case where the thickness o f surface layer is c o n s t a n t , a n d the dashed lines represent those for the case where the thickness of surface layer is variable. As expected, the greater the value o f 6*, the slower the dissol u t i o n rate. The applicability o f the present k i n e t i c m o d e l is e x a m i n e d b y a n a l y z i n g the experim e n t a l data reported b y T h o m a n n et al. (9). I n their e x p e r i m e n t , 10 m g o f h u m a n e n a m e l powder ( H E P ) is dissolved i n 50 m l o f a n a q u e o u s s o l u t i o n c o n t a i n i n g 10-4 M CaC12, 7 X 10 -6 M KH2PO4, a n d 8 X 10 -2 m KC1 at various pH's. T h e s o l u t i o n is agitated at a stirring speed o f 1860 r p m . T h e H E P particle is roughly spherical with sizes r a n g i n g f r o m 165 to 200 ~ m a n d d e n s i t y 2.93 g / c m 3. T h e m o lecular weight o f H E P is 502 g. T h e i r experim e n t a l data are s h o w n i n Fig. 5. Since the volu m e o f the b u l k phase is limited, the v a r i a t i o n o f its solute c o n c e n t r a t i o n d u r i n g the course of dissolution c a n be appreciable. Therefore, [11 ] is a d o p t e d i n the data fitting procedure. T h e initial radius o f the H E P particle is ass u m e d to be 90 # m . T h e values predicted by 0.40
0.30 B
0 0.20
I 0.1
6:=0.9
"~ .~ ~ ...
0,10
0.00 ~
0,01
,
,
,
,
,
,
'110
Dimensionless Concentration, C* FIG. 4. Variation of dissolution time as a function C* for different values of 6*. (--) The thickness of surface layer is constant; (---) the thickness of surface layer is variable with k = 1.
4'o ' Time
do
12o
(min)
FIG. 5. The kinetic data of the dissolution of HEP particles in aqueous solution (9) along with the values predicted by [11]. (A) Dry HEP at pH 3.7, De = 3.86 X 10-7 cm2/s, C~- Co = 5.13 X 10 -3 M, 6 = 7.70 #m; (1~)dry HEP at pH 4.0, De = 4.39 X 10-7 cm2/s, Ce - Co = 1.06 X 10 -3 M, 6 = 3.98 ~m; (O) equilibrated HEP at pH 5.5, Dr = 4.63 X 10-7 cm2/s, Ce - Co - 1.18 X 10-4 M, 6 = 5.83 ~m. Journal of Colloid and lnt erfl~ce Science, Vol. 141, No. 1, January 1991
66
HSU AND LIN
[ 1 1] are s h o w n i n Fig. 5 as solid lines. It app e a r s t h a t t h e p r e s e n t k i n e t i c m o d e l is c a p a b l e o f s i m u l a t i n g t h e b e h a v i o r o f the dissolution o f t h e H E P particle. I n s u m m a r y , a n a l y t i c a l expressions for the d e s c r i p t i o n o f t h e v a r i a t i o n o f particle size as a f u n c t i o n o f t i m e a n d for the v a r i a t i o n o f t h e d i s s o l u t i o n t i m e o f a p a r t i c l e as a function o f its size are d e r i v e d . T h e p r e s e n t kinetic m o d e l t a k e s t h e effect o f t h e v a r i a t i o n o f b u l k l i q u i d c o n c e n t r a t i o n o n d i s s o l u t i o n kinetics into acc o u n t , a n d , therefore, is a p p l i c a b l e for b o t h low s o l u b i l i t y cases a n d high solubility cases. REFERENCES 1. Riazi, M. M., and Faghri, A., Chem. Eng. Sci. 40, 1601 (1985).
Journal of Colloid andlnterface Science, Vol. 141, No. 1, January 1991
2. Bhaskarwar, A. N., Chem. Eng. Commun. 72, 25 (1988). 3. Sherwood, T. K., Pigford, R. L., and Wilke, C. R., "Mass Transfer," p. 151. McGraw-Hill, New York, 1975. 4. Elenkov, D., Vlaev, S. V., Nikov, I., and Ruseva, M., Chem. Eng. J. 41, 75 (1989). 5. Chen, Y. W., and Wang, P. J., Canad. J. Chem. Eng. 67, 870 (1989). 6. Vidgorchik, E. M., andShein, A. B., "Mathematical Modeling of Continuous Dissolution Processes," p. 60. Khimiye, Leningrad, 197l. 7. Kurasova, L. Z., Volfkovieh, S. I., and Oleinikov, L I., DokL Akad. Nauk SSSR 252, 1192 (1980). 8. Gramain, Ph., Voegel, J. C., Gumpper, M., and Thomann, J. M., J. Colloid lnterface Sci. 118, 148 (1987). 9. Thomann, J. M., Voegel, J. C., Gumpper, M., and Gramain, Ph., J. Colloid Interface Sci. 132, 403 (1989).
Received February 6, 1990;accepted May 18, 1990 The behavior of the dissolution of solid particles in liquids is modeled kinetically. It is assumed that a surface layer exists near the solid-liquid interface; the diffusion of solute moleculesthrough this layer is the rate controlling step. Analytical expressions for the description of the temporal variation of the size of a particle and for the dissolution time of a particle are derived. The present analysis takes the effect of the dissolved solute on the bulk liquid concentration into account, and, therefore, is applicable for both low solubility cases and for high solubility cases. The effects of polydispersed particles and variable surface layer thickness on dissolution kinetics are discussed. A criterion for neglectingthe effect of dissolved solute on bulk liquid concentration is suggested. © 1991AcademicPress,Inc. I. INTRODUCTION
phase/saturation concentration). For inorganic and electrochemical materials the method of invariant functions, a semiempirical approach, is often adopted for the description of the dissolution kinetics (6, 7). As pointed out by Bhaskarwar (2), however, this approach has under its surface some logical and empirical inconsistencies. The dissolution of hydroxyapatite (HA) in water results in the release of Ca 2+ and PO43-. It is found that these ions adhere on the surface of H A and form a surface layer; an equilibrium is reached between the solid surface and the bottom of the surface layer. The existence of this surface layer provides a resistance for the dissolution of HA (8). T h o m a n n (9) suggests that there exist two solute layers: a surface layer and a Nernst layer. The significance of the latter depends upon the operating condition. The more extensive the agitation is, the less significant it is. Previous investigations focus mainly on the case where the bulk liquid phase is of infinite size. In other words, the variation of solute concentration in this phase is negligible. This occurs if the solubility of particles is low. Since the solute concentration in the bulk liquid phase is constant, the driving force of dissolution due to the concentration difference be-
The dissolution of solid particles in liquids is a p h e n o m e n o n in which the solute molecules on solid surface leave the surface continuously at the expense o f the mass of the particle. The whole process can roughly be described by two steps in series: the escape of solute molecules from the solid surface and the diffusion of these molecules toward the bulk liquid phase. Depending on the operating conditions, the rate of dissolution may be controlled by one of these two steps. Riazi and Faghri ( 1 ) and Bhaskarwar (2), for example, assume that the mass transfer of dissolved solid away from the solid-solution interface is the rate-controlling step. A model often adopted to describe the transport of solute molecules to the bulk liquid phase is the film theory ( 3, 4). Since a pseudo-steady state assumption for the concentration profile of solute is made, this model is most appropriate for low solubility cases. Under the assumption that a film is of infinite thickness, Chen and Wang (5) have derived an analytical expression for the variation of dissolution time as a function of the ratio (concentration of solute in the bulk To whom correspondenceshould be addressed. 60 0021-9797/91 $3.00 Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved,
Journal of Colloid and Interface Science, Vol. 141, No. 1, January 1991
SOLID
PARTICLE
tween the solid surface and the bulk liquid phase remains essentially constant if the diffusion of solute molecules is the rate-controlling step. This assumption, although it simplifies the derivation of the governing equation for dissolution kinetics, is unrealistic in practical applications, where the bulk liquid phase is of finite size. The purpose o f this report is to derive a general kinetic model that takes the effect of dissolved solute on the bulk liquid concentration into account. The mathematical representation thus obtained is applicable for both the cases in which the solubility is low and the cases in which the solubility is high. The effects ofpolydispersed particles and variable surface layer thickness on the dissolution kinetics are also discussed. II. M O D E L I N G
We assume the following: ( 1 ) A solid particle is spherical. (2) A surface layer is formed near the solid-liquid interface. (3) The ratecontrolling step of the dissolution process is the diffusion of solute molecules through the surface layer. In other words, the concentration of solute on the solid surface is at saturation. On the basis of these assumptions, a mass balance on solute yields Ot
r 2 Or
r2 -~r
'
[ 1]
where C denotes the concentration of solute, t represents the time scale, De is the diffusivity of solute, and r is the distance measured from the center of a particle. Suppose that the surface layer is thin, and a pseudo-steady state is achieved. Equation [1] reduces to
d (r2dC 1 dr ] = O.
De-~r ~
[21
Integrating this equation with boundary conditions C = Ce at r = re and C = Cb, the bulk liquid concentration of solute, for r >/rc + 6 yields dC dr
rc(rc
+
6)(Ce - Cb) r26
[3]
61
DISSOLUTION
where re is the radius of a particle and 6 is the thickness of the surface layer. If n is the n u m ber of moles in the particle, then
dn
dC[
[4]
dt - SDe dr r=r~,
where S represents the surface area of the particle. Since n = 4zrr3p/3Mo and S = 4~rr~, this equation becomes dr~
DeMo(rc + 6)(C~ - Cb)
dt
pro6
,
[5]
where p and Mo are, respectively, the density of particle and the molecular weight of solute. 2.1. Constant B u l k L i q u i d Concentration
If the volume of the bulk liquid phase V is large enough, the concentration o f solute in the bulk liquid phase is essentially constant. Denote this concentration as Co. For convenience, [5] is recast as dr* dt*
-
C*(r* + 6")/r'6",
[6]
where r* = rc/ro
[6a]
6* = 6/ro
[6b]
t* - 4~rtDeroN
[6c]
3V ( C e -- Co)
C*
(47rpr3oN/3 V M o )
[6d]
In these expressions, ro is the initial radius of a particle, and N is the total number o f particles. Solving [ 6 ], subject to the initial condition r* = 1 at t* = 0, we obtain t* = { ( l - r * ) + 6*ln[(r* + 6 " ) / ( 1 + 6 " ) ] } 6 " / C * .
[7]
This equation describes implicitly the variation of the radius of a particle as a function o f time. Journal of Colloid and Interface Science, Vol. 141, No. 1, January 1991
62
HSU AND LIN cl = - 6 " 2 / ( a 3 - 6*3 )
2.2. Variable B u l k L i q u i d Concentration
I f the v o l u m e o f the bulk liquid phase is finite, the variation o f its solute concentration can be significant. In this case, the variation o f the bulk liquid concentration can be written
dl =
[12d]
a3/(0Z 3 -- ~.3)
[12e]
a2 = ( b l a 2 - cla + d l ) / 3 a 2
[12f]
b2 = ( 2 b l a 2 + cla - d l ) / 3 a 2
[12g]
c2 = ( - b ~ a 2 + c j a + 2 d , ) / 3 a .
[12h]
as
dCb dt
47rp { ~ _ ~
2dri\ N,.ri -77,/, at] V M o \ ~i=~
--
[8]
III. RESULT AND DISCUSSION
where Ni denotes the n u m b e r o f particles in class i, ri represents the radius o f a particle in class i, and M is the n u m b e r o f classes. The solution o f this equation, subject to the initial conditions Cb = Co a n d ri -- roi at t = 0, is
47rp
Cb = Co + ~
M
[~, N,.(r~i- r~)].
[9]
i=1
In the case in which all particles are of the same size, this e q u a t i o n reduces to Cb = Co +
47rp N t r 3 _ r3). 3VMo ~ o
[10]
The temporal variations o f the radius o f a particle for different values o f C* and for a fixed value o f 6" are illustrated in Fig. 1. As can be seen from this figure, the higher the value o f C*, the faster the dissolution. This is expected since the greater the C*, the greater the concentration driving force for dissolution, as suggested by [ 6 d ] . Figure 1 shows that the dissolution rate is faster for the case of constant bulk liquid concentration than for the case o f variable bulk liquid concentration. This is due to the decrease o f concentration driving force during the course o f dissolution for the latter. 3.1. Dissolution T i m e
Substituting this expression into [ 5 ], we obtain (r* + 6 " ) ( C * - 1 + r .3)
dr* -
[11]
dt*
6*r*
T h e time required to dissolve a particle, or the dissolution time t~, can be obtained by
T h e solution o f this equation, subject to the initial condition r* = 1 at t* = 0, is
0.006 "\
t* = -/~* { a l l n [ ( r * + 6 " ) / ( 1 + 6*)]
\
~,0.0o~
\ x
+ a21n[(r* + a ) / ( 1 + a ) ]
x
~;
\\
.~ 0.004
+ b2ln[((r* - a / 2 ) 2 + 3a2/4)/((1
\\
~
__ o~/2)2
"" o"--2.a
\\\\\
~o.oo3 o
+ 3a2/4)]/2
+ (b2a + 2c2)
"~ ~ 0,002
X [tan-l((2r * - a)/f3a)
"~ 0.001
-tan-'((2--a)/V3a)]/~a},
[12] o.ooo0.0
where a s= C*-
1
[12a]
al = - 6 * / ( a 3 - 6*3 )
[12b]
bl = 6 * / ( a 3 - 6*3 )
[12c]
Journal of Colloid and Interface Science, Vol. 141, No. 1, J anuary 1991
i
0.2
i
0.4
Dimensionless
t
0.6
i
0.8
Radius,
1.0
r*
FIG. 1. Temporal variation of the radius of a particle for different values of C* for the case 6" = 0.01. (--) Constant bulk liquid concentration; (---) variable bulk liquid concentration.
SOLID PARTICLE DISSOLUTION letting r* = 0 in [7] and [12] for the case of constant bulk liquid concentration and for the case of variable bulk liquid concentration, respectively. We have for the former t~ = 6"{1 + 6"1n[6"/(1 + 6 " ) ] } / C * , [13]
63
tion in the bulk liquid phase. Figure 2 reveals that if C* is on the order of 10, the variation of the solute concentration in the bulk liquid phase is negligible. In other words, [12] can reasonably be approximated by [7], if (Ce - Co) >1 407rpr3N/3 VMo.
and for the latter
3.2. Polydispersed Particles
t~ : - 6 * {alln[6*/(1 + 6*)] + a21n[a/(1 + a)] + bzln[a2/((1 -- a / 2 ) 2 + 3 a 2 / 4 ) ] / 2
M
dr*_ dt*
-- (baa + 2 c 2 ) [ t a n - ' ( 1 / f 3 ) + tan-'((2 - a)/V3a)]/V-3a}.
If the solid particles are polydispersed, and the variation of bulk liquid concentration is significant, [11] can be modified as
[14]
The variation of dissolution time as a function of C* for different values of 6" is illustrated in Fig. 2. The solid lines represent the results calculated for the case of constant bulk liquid concentration, and the dash lines denote the results for variable bulk liquid concentration. The definition of C* ( [ r d ] ) suggests that, for a fixed solid-liquid combination, the greater its value, the greater V is. In other words, C* is a measure of the relative significance of the variation of solute in the bulk liquid phase. The greater the C*, the less significant the variation of the solute concentra-
(r* + 6 " ) [ C * -
E£(r~ 3 j-1
-r*3)]/r*6 *
i = 1,2 . . . . , M ,
[15]
where r* = ri/ror
[15a]
r~j = roJro~
[15b]
t* -- 47rtDerorN 3V
[ 15c ]
( c e - c0) C* = (4wpr3orN/3VMo) f) = N J N
[15d] [15e]
M
N = Z NJ.
[15f]
\ \\ 0.1
In these expressions, ri is the radius of a particle in class i, for is a reference radius, r0j is the initial value of D, and Ni is the number o f particles in class i. The set of M equations expressed in [15] needs to be solved simultaneously.
\ N
"0.1
OiOO 1
o
x
k
x
3.3. Variable SurJace Layer Thickness
r~
The linear size of a particle decreases monotonically during the course of its disso0.0001 lution. If the linear size becomes very small, the thickness of the surface layer may reduce 10 accordingly, rather than remain constant. In Dimensionless Concentration, C* FIG. 2. Variation of dissolution time as a function of this case, an intuitive representation for the C* for differentvalues of 6". (--) Constant bulk liquid variation of the thickness of the surface layer concentration;(---) variablebulk liquid concentration. is Journal of Colloid and lnterfizce Science, Vol. 141, No. 1, January 1991
64
HSU AND LIN
6
f 6c, a constant, if
r~o ~ rc
+ 3 ~ 2 / 4 ) / ( ( 1 -- c~/2) 2 + 3c~214)]/2
1
r~¢ > re,
+ (b2a + 2 c 2 ) [ t a n - l ( ( 2 r * -
[ kro,
where k = 5~/r¢~, and rc~ is a certain critical radius. 3.3.1. Constant bulk liquid concentration. If the variation of the bulk liquid concentration is negligible, the variation of the size of a particle with time can be obtained by following the same procedure as that employed for the case of constant surface layer. We have, for the present case,
tan-i((2 - ~)/f3~)]/f3a}
-
+ k { l n [ ( r * + a ) / ( r * + a)] -
l n [ ( ( r *
t* =
+6")]}/C*,
-
oz]2)
2
+ 3e~2/4)/((r * - ~ / 2 ) 2 + 3c~2/4)]/2 -
~/3[tan-l((Zr * - ~ ) / f 3 ~ )
-
tan-l((2r * - ~)/V3a)]}/ 3c~(k+l),
I 6"{(1 - r*) + 5 * l n [ ( r * + 6*)/ (1
r*>~r*
~)/fJ~)
r*
[19]
where r* = r¢c/ro. The time required to dissolve a particle is
6*{(1-r*)+6*ln[(r*+6*)/ (1 + 6 " ) ] } / C * + k ( r .2 - r*2)/ 2(1 + k ) C * ,
[16]
r* < r * ,
t~ -- - 6 " { a l l n [ ( r * + 6 " ) / ( 1 + 6*)] + azln[(r* + 6 " ) / ( 1 + c~)] + b21n[((r* - c~/2) 2
where r* = r¢c/ro and 5" = 6~/ro. The time required to dissolve a particle is then
+ 3 a 2 / 4 ) / ( ( 1 - oz/2) 2 + 3 ~ 2 / 4 ) ] / 2 + (b2~ + 2 c z ) [ t a n - l ( ( 2 r * - a)/~/-3a)
t~ = [kr*2/2(1 + k ) C * ] + 6 " { ( 1 - r*) -
+6*ln[(r* +6")/(1+6")]}/C*.
[17]
3.3.2. Variable bulk liquid concentration. If the variation o f the bulk liquid concentration is appreciable, it can be shown that
+ k { l n [ a / ( r * + a)] - l n [ a Z / ( ( r * - or/2) 2 + 3 a 2 / 4 ) ] / 2 -
V3[tan-l( 1/V3) + t a n - l ( ( Z r * - a ) / V~a)]}/3a(k+l).
t* = -ac* { a l l n [ ( r * + 6 " ) / ( 1 + 6*)] + azln[(r* + ~ ) / ( 1 + ~)] + b21n[((r* - o~/2) 2 + 3a2/4)/((1 - a/2) 2 + 3a2/4)]/2 + (b2a + 2 c 2 ) [ t a n - l ( ( 2 r * - a ) / f 3 a ) - tan-l((2 - a)/~/Ja)]/f3c~}, r* >~ r*
tan-I((2 - a)/f3a)]/~/3~}
[18]
and t* = - 6 " { a l l n [ ( r * + 6 " ) / ( 1 + ~*)] + a21n[(r* + a ) / ( 1 + a)] + b21n[((r* - o~/2) 2 Journal of Colloid and Interface Science, Vol. 141, No. 1, January 1991
[20]
Figure 3 shows the temporal variation of the radius of a particle for different values of 6" for the case of variable bulk liquid concentration. The solid lines represent the results for the case where the thickness of surface layer is constant, and the dashed lines represent those for the case where the thickness of surface layer is variable. As can be seen from this figure, the dissolution rate of the former is slower than that of the latter. This is because the smaller the thickness of a surface layer, the less the resistance for solute molecules to diffuse toward the bulk liquid phase. For the latter case, the thickness of the surface layer decreases monotonically with the size of a par-
SOLID PARTICLE DISSOLUTION 0.12
d 0.09
~
~
~
~
-.
a2=o.7
,~ 0.06
\\\ 0.4
0
.~ 0.03
0.00
r0.2
0.0
~ 0.4
J
~ 0.6
i 0.8
1.0
Dimensionless Radius, r* FIG. 3. Temporal variation of the radius of a particle for different values of 6" for the case C* = 3. (--) The thickness of surface layer is constant; (---) the thickness of surface layer is variable with k = 1. ticle as the radius of the particle is smaller t h a n a critical value. Therefore, the d i s s o l u t i o n rate is faster t h a n if the thickness o f surface layer r e m a i n s c o n s t a n t t h r o u g h o u t the dissolution period. The variation of dissolution t i m e as a function of C* for different values o f 6* a n d for the case o f variable b u l k c o n c e n t r a t i o n is illustrated in Fig. 4. T h e solid lines represent
65
the results for the case where the thickness o f surface layer is c o n s t a n t , a n d the dashed lines represent those for the case where the thickness of surface layer is variable. As expected, the greater the value o f 6*, the slower the dissol u t i o n rate. The applicability o f the present k i n e t i c m o d e l is e x a m i n e d b y a n a l y z i n g the experim e n t a l data reported b y T h o m a n n et al. (9). I n their e x p e r i m e n t , 10 m g o f h u m a n e n a m e l powder ( H E P ) is dissolved i n 50 m l o f a n a q u e o u s s o l u t i o n c o n t a i n i n g 10-4 M CaC12, 7 X 10 -6 M KH2PO4, a n d 8 X 10 -2 m KC1 at various pH's. T h e s o l u t i o n is agitated at a stirring speed o f 1860 r p m . T h e H E P particle is roughly spherical with sizes r a n g i n g f r o m 165 to 200 ~ m a n d d e n s i t y 2.93 g / c m 3. T h e m o lecular weight o f H E P is 502 g. T h e i r experim e n t a l data are s h o w n i n Fig. 5. Since the volu m e o f the b u l k phase is limited, the v a r i a t i o n o f its solute c o n c e n t r a t i o n d u r i n g the course of dissolution c a n be appreciable. Therefore, [11 ] is a d o p t e d i n the data fitting procedure. T h e initial radius o f the H E P particle is ass u m e d to be 90 # m . T h e values predicted by 0.40
0.30 B
0 0.20
I 0.1
6:=0.9
"~ .~ ~ ...
0,10
0.00 ~
0,01
,
,
,
,
,
,
'110
Dimensionless Concentration, C* FIG. 4. Variation of dissolution time as a function C* for different values of 6*. (--) The thickness of surface layer is constant; (---) the thickness of surface layer is variable with k = 1.
4'o ' Time
do
12o
(min)
FIG. 5. The kinetic data of the dissolution of HEP particles in aqueous solution (9) along with the values predicted by [11]. (A) Dry HEP at pH 3.7, De = 3.86 X 10-7 cm2/s, C~- Co = 5.13 X 10 -3 M, 6 = 7.70 #m; (1~)dry HEP at pH 4.0, De = 4.39 X 10-7 cm2/s, Ce - Co = 1.06 X 10 -3 M, 6 = 3.98 ~m; (O) equilibrated HEP at pH 5.5, Dr = 4.63 X 10-7 cm2/s, Ce - Co - 1.18 X 10-4 M, 6 = 5.83 ~m. Journal of Colloid and lnt erfl~ce Science, Vol. 141, No. 1, January 1991
66
HSU AND LIN
[ 1 1] are s h o w n i n Fig. 5 as solid lines. It app e a r s t h a t t h e p r e s e n t k i n e t i c m o d e l is c a p a b l e o f s i m u l a t i n g t h e b e h a v i o r o f the dissolution o f t h e H E P particle. I n s u m m a r y , a n a l y t i c a l expressions for the d e s c r i p t i o n o f t h e v a r i a t i o n o f particle size as a f u n c t i o n o f t i m e a n d for the v a r i a t i o n o f t h e d i s s o l u t i o n t i m e o f a p a r t i c l e as a function o f its size are d e r i v e d . T h e p r e s e n t kinetic m o d e l t a k e s t h e effect o f t h e v a r i a t i o n o f b u l k l i q u i d c o n c e n t r a t i o n o n d i s s o l u t i o n kinetics into acc o u n t , a n d , therefore, is a p p l i c a b l e for b o t h low s o l u b i l i t y cases a n d high solubility cases. REFERENCES 1. Riazi, M. M., and Faghri, A., Chem. Eng. Sci. 40, 1601 (1985).
Journal of Colloid andlnterface Science, Vol. 141, No. 1, January 1991
2. Bhaskarwar, A. N., Chem. Eng. Commun. 72, 25 (1988). 3. Sherwood, T. K., Pigford, R. L., and Wilke, C. R., "Mass Transfer," p. 151. McGraw-Hill, New York, 1975. 4. Elenkov, D., Vlaev, S. V., Nikov, I., and Ruseva, M., Chem. Eng. J. 41, 75 (1989). 5. Chen, Y. W., and Wang, P. J., Canad. J. Chem. Eng. 67, 870 (1989). 6. Vidgorchik, E. M., andShein, A. B., "Mathematical Modeling of Continuous Dissolution Processes," p. 60. Khimiye, Leningrad, 197l. 7. Kurasova, L. Z., Volfkovieh, S. I., and Oleinikov, L I., DokL Akad. Nauk SSSR 252, 1192 (1980). 8. Gramain, Ph., Voegel, J. C., Gumpper, M., and Thomann, J. M., J. Colloid lnterface Sci. 118, 148 (1987). 9. Thomann, J. M., Voegel, J. C., Gumpper, M., and Gramain, Ph., J. Colloid Interface Sci. 132, 403 (1989).