A population balance model for mass transfer in ... - Purdue Engineering
Chemical EngineeringScience,1973,Vol. 28, pp. 389-399.
Pcrgamon
Press.
Printed in Great Britain
A population balance model for mass transfer in lean liquid-liquid dispersions B. H. SHAH and D. RAMKRISHNA Department of Chemical Engineering, Indian Institute of Technology, Kanpur 16, India (Received24 February 1972; accepted 16 May 1972)
Abstract-A population balance model is considered for the description of mass transfer in liquidliquid dispersions in a mechanically agitated vessel operated continuously. The dispersions considered are of low dispersed phase fraction in which coalescence processes may be assumed to be negligible. The model incorporates the dynamic nature of breakage processes among the droplet population together with mass transfer processes at the droplet level into the analysis which is accomplished by the use of a trivariate number density using size, concentration of diffusing solute and age of the droplets. Mass transfer in single drops is assumed to be by diffusion. Droplet breakage is assumed to be binary and even. The results of the model are compared with predictions made from an alternative approach (reflective of methods used in the past) which ignores the fact that droplet size distributions are attained in time. Several interesting deviations are predicted in the concentration distributions for drops of definite sizes and in the overall mass transfer rates. It is concluded that the dynamics of droplet phenomena may play an important role in the analysis of mass transfer in dispersions. INTRODUCTION
transfer in liquid-liquid dispersions is a complex process involving droplet phenomena such as breakage and coalescence, and mass transfer from individual drops, both of which depend on a large number of a diverse variety of factors. These factors have been reviewed by Olney and Miller [ll. Various attempts in the past towards a quantitative description of mass transfer in liquid-liquid dispersions have been reviewed by Ramkrishna [2]. These attempts range from the coarse method of the overall volumetric mass-transfer coefficients to those which seek to incorporate mass transfer processes at the droplet level in the prediction of the overall mass transfer from the drop population. It is with this latter approach, in its application to liquid-liquid dispersions produced in a mechanically agitated vessel, that this paper will be concerned. Moreover, attention is focussed on lean liquid-liquid dispersions in which droplet coalescence is likely to be negligible so that droplets undergo only breakage. The necessity to incorporate the aspect of MASS
389
simultaneity of droplet processes (breakage and coalescence) and mass transfer processes (from single drops) has been pointed out [2,31. To accomplish this one must consider multivariate number densities in which must appear as variables the size of the droplet, concentration of the solute in the droplet, and all others pertaining to the drop, whose specification should be vital to the prediction of the instantaneous mass transfer rate from a given drop. The population balance equations which the multivariate number density must satisfy have been discussed in detail by Ramkrishna[21. In this paper we consider a specific model which employs a multivariate density including as variables, the size (radius), solute concentration and “age” of the droplet. This model is analyzed to demonstrate the interesting deviations of its predictions from those that arise from neglecting the dynamic nature of droplet phenomena. THE MODEL
The system under consideration consists of a mechanically agitated vessel which continually
B. H. SHAH and D. RAMKRISHNA
receives the continuous phase and the dispersed determine the instantaneous mass transfer rate. phase, the latter in the form of identically sized This follows from the assumption that the drop droplets such as those formed at the end of a has a uniform concentration at birth. hypodermic needle submerged in the continuous If we let W(Q,c, r; t) da dc dr represent the phase. Simultaneously, the dispersion is with- number of droplets per unit volume of the drawn at the same rate at which the two phases dispersion at time t with size between a and enter the mixer, which results in a constant hold- a +du, mean solute concentration between c up in the vessel. The dispersed phase is supposed and c+dc, and age between r and T+ dr, then to contain a solute which transfers to the contin- w must satisfy the population balance equation: uous phase entirely by the molecular process of diffusion within the drop with negligible resistance to mass transfer on the side of the continuous phase. The entering droplets possess identical concentrations of the solute. x qc- c&30) -;+ h(w, a, c, 7). (1) Each drop, when and if it breaks, is assumed to break into two equal halves,t Furthermore breakage, when it occurs, is taken to be an Note that w is independent of position in the instantaneous process. The daughter droplets vessel which is a consequence of perfect mixing. at birth are assumed to have a uniform solute The left hand side of equation (1) contains the concentration which will be equal to the average familiar “continuity operator” in the space of concentration in the parent droplet at the in- a, c and r while the right hand side lists the stant of breakage. Although prior to breakage source and sink terms for drops of state (a, c, 7) there would have existed in the parent a concen- within the vessel. The first term on the right hand tration profile consistent with that predicted side of (1) represents the input rate, the Dirac by the diffusion equation, it is assumed that there deltas indicating the unique state of the entering is sufficient internal mixing during breakage to droplets. The second term refers to the rate at ensure uniform concentration in the daughter which drops of state (a, c, 7) depart from the vessel. The third term h(w, a, c, T) is such that droplets at birth. Since mass transfer occurs by diffusion in the h(w, a, c, r) da dcdr represents the net rate of drop, the solute concentration is not spatially formation of drops of state (a, c, 7) by breakage; uniform in the drop. We choose to describe the it includes loss of drops by breakage and formaamount of solute in the drop by the average tion by breakage of other drops. We shall concentration. Specification of the drop size inquire into the form of h subsequently. The (which we shall take as drop radius) and average term C(u, c, T) in Eq. (1) represents the rate of concentration will not suffice to predict the change of concentration in a drop of state (a, c, 7) instantaneous mass transfer rate from the drop. by mass transfer. Since mass transfer occurs by However, incorporation of an “age” concept diffusion we have will overcome this difficulty. “Age” is defined as as the time elapsed after the droplet has assumed (2) its identity in the vessel by entering with the feed or by breakage of a parent droplet. A little where c’ is the local solute concentration in the thought will show that the average concentration, drop radius and the age of the drop will uniquely drop and 9 is the diffusion coefficient of the solute in the dispersed phase. The time t’ belongs anywhere in the interval (t - 7, t) where tSleicher[5] has observed that the most frequent form of breakage in turbulent pipe flow is into two approximately t denotes the absolute time and T the age of the equal parts. droplet. Equation (2) must be subject to the 390
A population balance model for mass transfer
initial condition c’(r,r-7)=ci
O=SrGa
(3)
and boundary conditions c’(a, t) = Kc” ad -_= ar
f-7
o
s t’s
t
r = 0.
(4) (5)
Spherical symmetry is implicit in Eqs. (2) through (5). The constant K in Eq. (4) is the equilibrium constant and c” is the concentration of the solute in the continuous phase assumed uniform in the vessel but varying with time. If it is assumed that the continuous phase feed to the mixer has no solute and that the dispersion is sufficiently lean that mass transfer does not materially alfect the concentration of the continuous phase then condition (4) becomes c’(a,t)=O
t-7s
t’s
t.
2 I,“”da’ I,” dc’ J; dT’I(u’, c’, T’)p(U, a’, C’, T’)W(U’,
C’, 7’; t)
c, 7;
da dc dT. (9)
The rate of drops of state (a, c, T) lostby breakage W-4 to other sizes is
This eliminates the need for writing the equation for the continuous phase, although such an equation may be readily written [21. The solution of Eq. (2) subject to (3), (4a) and (5) is obtainable in either of the forms:
E= 1+3T-6P2
However C can be expressed as a function of the initial concentration ct so that on specifying a, c and 7 one can obtain C by back-calculating cf from (6) or (7). We now return to consider the form of h(w, a, c, T) which represents the difference between the rate of drops of state (a, c, T)formed and lost by breakage. We denote the fiuctiond rate of breakage of droplets of state (a, c, 7) by I(u, c, 7) and the fraction of drops of state (a, c, 7) in the daughter droplets formed by breakage of parent droplets of state (a’, c’, T’) is denoted by p(u, C,T; u’, c’, 7’) da dc d7. Then the rate of formation of drops of state (a, c, T) by binary breakage is given by
I(u, c, T)W(U,
c, 7; t)
da dc dr.
(10)
If one assumes that the breakage of droplets has little to do with their solute concentration and age, and that breakage of drop of size a is into two even halves each of radius u’/2113then (9) may be replaced by t
,rr-1/2+2 2 ierfc ?I=, X
(7) where c is now the average concentration obtainable from the local concentration c’. The symbol T represents the dimensionless group (%/u2). The two forms (6) and (7) are presented because one. or the other permits an easier computation of C from a, c and 7 depending on the value of 7. It should be clear that C cannot be readily expressed as an explicit function of c since (8)
da’ dr’ da dc dT.
(9a)
In writing (9a) we have recognized that droplets have zero age at birth. We have resisted further simplification of (9a) since this form is more convenient to the use of the method of successive substitution for the solution of Eq. (1). The quantity h( w, u, c, T) may now be written as: tit is not inconceivable that breakage may depend on the solute concentration since any interfacial turbulence due to mass transfer [4] is likely to affect the deformation of the drop.
391
B. H. SHAH
and D. RAMKRISHNA
x wfu’, c, T’; t) da’ d+ - r(u)w(u, c, 7). (11) If we restrict ourselves to steady state operation then Eq. (1) may be written as
(12)
[tic] =-
[
;+r(u)
02
I
0
dr’
1
CC;T > 0 (13)
+()(a, C,T) = $~((a,c~,T)+ ,;
x
eat
+l+%(U
Equation (14) may also be derived mathematically by integrating Eq. (12) over the interval (E, 00)and comparing with the population balance equation written for the bivariate density in a and c, on letting c * 0.
Equation (13) is a linear first order partial differential equation which readily submits to the method of characteristics[61. The result when combined with the boundary condition (14) yields an integral equation of the Volterra type which can be solved by the method of successive substitution. Given a drop of state
-
s[0
r(a)+f+$
Equation (15) when combined gives the integral equation
r(d)6 a-$ $(a’,c,7’). (14) ( >
Solution
I
.
Since drops of zero age can appear only with the feed or from breakage of other droplets we may write
x da’
ct, 0)exp
#((a, c, 7) = flu,
Equation (12) may be rewritten with distinction for drops of non-zero and zero age as follows: %+$
(a, c, r) Eqs. (6) or (7) can be solved back for calculating the spatially uniform initial concentration of the drop denoted c~. The concentration of this drop at time r’ measured from the instant of its birth is expressible as a function of C~ and r’, where T’ varies from zero to its present age 7. This function in symbols is written as c(c*, 7’) remembering that the actual function is given by (6) or (7) and that c = c(ct, r), where the left hand side is the present specified concentration of the drop. (Note that the radius of the drop, a has been suppressed in the expression). The solution of equation (13) is now written as:
with
du’J; dT’K(u,
(14)
U', Ci,T)
k-i'@', f&T')
(16)
3 by 1 I c(C
7') 7')
dr’
(17) &U,U’,Ct,T)
=
2$4.2, Cf,T)r(U)a
The integral equation (16) is solved by successive substitution. The resulting solution is somewhat complex in form but reflects some characteristics of the process that are apparent through physical reasoning [2]. Thus consider a particular droplet state (a, c, T). The attaiment of this state is associable with an infmite number of mutually exclusive possibilities for the past history of the droplet. Hence, the number of 392
A population balance model for mass transfer
drops of state (a, c, T) must result from the cumulation of drops which have evolved to that state from different past histories. Such difference in past history arises not only in the number of generations in the background but also in the ‘life-spans” of the various generations. Remembering that the first generation drops are those that have entered the vessel with unique “birth states,” it should be clear that the variation of life spans of individual generations is not free of constraints. To present the mathematical form of the solution we rewrite Eqs. (6) and (7) by
c=F(a, 7).
This solution is more conveniently presented in the form of individual concentration probability density functions for each drop size. Of course for each drop size, the droplet state is described by a bivariate density in concentration and age. However we will integrate the bivariate densities with respect to age and present the resulting concentration distributions for each droplet size. The joint probability density of size and concentration denoted fAc(u, c) is clearly
(19)
ci
Given (a, c, T), if the drop is of the ti generation, then for a particular sequence of life spans , I 3 1) of the respective ancestors Tj 9 rj+l, - - - &(j of generationj, j+ 1, . . . (n - l), the maximum life span of the (j- 1)st generation denoted #T(jnlI is given by
f=jiQ.z, c, 7) d7
L&9 c) =
(24) J;duJ;dcl;d&(u,c,$
where the denominator is the total number density fi. The conditional probability density for concentration for a given droplet size f&u, c) may be readily obtained from
n-1 C=
GP(%~)
F(aj-l,
@I)
Ej
F(ak,T$
(25)
(20)
where aj = (~$2~3). Equation (20) is solved to where the density function fA(u) is the size obtain distribution. Combination of (24) and (25) (2 1) yields ?$!!I= $!I($, $+I,. . . , T&l). Further #$,is given by c = c&.z, 7)&Z,_,, 7$&).
The solution of Eq. ( 16) can now be written as
+i
2”6(u- a,)
n=1
I ; @(a, c, 7) dT WA@) *
(26)
Since the droplet sizes are discrete the conditional density f&c) = &lA(un,c) can be seen to be given by
4%
ii’r(ak) k=O
&A(& c) =
(22)
I 0 "7p
e-[r(an-1)+(1/@)kk-1
d7A_1
e-[r(a~)ff1/e)17;
. . .
where we have put
0
e-ma~+wemo~n)
F(u, ~2’)) d7; - C(uo, c, To’“‘) I*
(23) 393
(28)
B. H. SHAH and D. RAMKRISHNA
of a minimum droplet size which can be estimated by the Kolmogorov theory of local isotropy [&lo]. The case of unrestricted breakage is handled by equating a, to zero. Two other forms for T(a) were considered which postulate the existence of a droplet size amax above which the droplets would break instantaneously in the vessel. Thus I(a) would tend to infinity when a tends to amax. This concept is somewhat akin to that of Collins and
(29) (30)
(32) j=o,1,2,...
Knudsen who have considered a droplet size of a,., which breaks with probability one in a fixed arbitrarily chosen time interval[ 111. The forms denoted I,(a) and r,(a) are presented below.
The dimensionless grouping Rj may be written as -- aj219
g/g
Rj-l/lY(aj)+
8’
(33)
r2(a)=
The first of the terms in the right hand side of (33) provides a relative measure of the time scales of diffusion and droplet breakage. Since larger drops should break with higher probability, R. should be a conservative estimate of the importance of the dynamic nature of droplet breakage; a large value of R,, would mean that breakage occurs very rapidly although the R,‘s should progressively diminish with increasing j. A more adequate measure of the importance of breakage dynamics is referred to in the next section. RESULTS
OF COMPUTATION
Computations were performed on an IBM 7044 computer for selected forms of the breakage frequency function I(a). Valentas er al. 171have considered the following expression for the breakage frequency. a-a,
R
1’ ) k
I-,(u) =
L-&-u,
0
a,
C
OSa
Pcrgamon
Press.
Printed in Great Britain
A population balance model for mass transfer in lean liquid-liquid dispersions B. H. SHAH and D. RAMKRISHNA Department of Chemical Engineering, Indian Institute of Technology, Kanpur 16, India (Received24 February 1972; accepted 16 May 1972)
Abstract-A population balance model is considered for the description of mass transfer in liquidliquid dispersions in a mechanically agitated vessel operated continuously. The dispersions considered are of low dispersed phase fraction in which coalescence processes may be assumed to be negligible. The model incorporates the dynamic nature of breakage processes among the droplet population together with mass transfer processes at the droplet level into the analysis which is accomplished by the use of a trivariate number density using size, concentration of diffusing solute and age of the droplets. Mass transfer in single drops is assumed to be by diffusion. Droplet breakage is assumed to be binary and even. The results of the model are compared with predictions made from an alternative approach (reflective of methods used in the past) which ignores the fact that droplet size distributions are attained in time. Several interesting deviations are predicted in the concentration distributions for drops of definite sizes and in the overall mass transfer rates. It is concluded that the dynamics of droplet phenomena may play an important role in the analysis of mass transfer in dispersions. INTRODUCTION
transfer in liquid-liquid dispersions is a complex process involving droplet phenomena such as breakage and coalescence, and mass transfer from individual drops, both of which depend on a large number of a diverse variety of factors. These factors have been reviewed by Olney and Miller [ll. Various attempts in the past towards a quantitative description of mass transfer in liquid-liquid dispersions have been reviewed by Ramkrishna [2]. These attempts range from the coarse method of the overall volumetric mass-transfer coefficients to those which seek to incorporate mass transfer processes at the droplet level in the prediction of the overall mass transfer from the drop population. It is with this latter approach, in its application to liquid-liquid dispersions produced in a mechanically agitated vessel, that this paper will be concerned. Moreover, attention is focussed on lean liquid-liquid dispersions in which droplet coalescence is likely to be negligible so that droplets undergo only breakage. The necessity to incorporate the aspect of MASS
389
simultaneity of droplet processes (breakage and coalescence) and mass transfer processes (from single drops) has been pointed out [2,31. To accomplish this one must consider multivariate number densities in which must appear as variables the size of the droplet, concentration of the solute in the droplet, and all others pertaining to the drop, whose specification should be vital to the prediction of the instantaneous mass transfer rate from a given drop. The population balance equations which the multivariate number density must satisfy have been discussed in detail by Ramkrishna[21. In this paper we consider a specific model which employs a multivariate density including as variables, the size (radius), solute concentration and “age” of the droplet. This model is analyzed to demonstrate the interesting deviations of its predictions from those that arise from neglecting the dynamic nature of droplet phenomena. THE MODEL
The system under consideration consists of a mechanically agitated vessel which continually
B. H. SHAH and D. RAMKRISHNA
receives the continuous phase and the dispersed determine the instantaneous mass transfer rate. phase, the latter in the form of identically sized This follows from the assumption that the drop droplets such as those formed at the end of a has a uniform concentration at birth. hypodermic needle submerged in the continuous If we let W(Q,c, r; t) da dc dr represent the phase. Simultaneously, the dispersion is with- number of droplets per unit volume of the drawn at the same rate at which the two phases dispersion at time t with size between a and enter the mixer, which results in a constant hold- a +du, mean solute concentration between c up in the vessel. The dispersed phase is supposed and c+dc, and age between r and T+ dr, then to contain a solute which transfers to the contin- w must satisfy the population balance equation: uous phase entirely by the molecular process of diffusion within the drop with negligible resistance to mass transfer on the side of the continuous phase. The entering droplets possess identical concentrations of the solute. x qc- c&30) -;+ h(w, a, c, 7). (1) Each drop, when and if it breaks, is assumed to break into two equal halves,t Furthermore breakage, when it occurs, is taken to be an Note that w is independent of position in the instantaneous process. The daughter droplets vessel which is a consequence of perfect mixing. at birth are assumed to have a uniform solute The left hand side of equation (1) contains the concentration which will be equal to the average familiar “continuity operator” in the space of concentration in the parent droplet at the in- a, c and r while the right hand side lists the stant of breakage. Although prior to breakage source and sink terms for drops of state (a, c, 7) there would have existed in the parent a concen- within the vessel. The first term on the right hand tration profile consistent with that predicted side of (1) represents the input rate, the Dirac by the diffusion equation, it is assumed that there deltas indicating the unique state of the entering is sufficient internal mixing during breakage to droplets. The second term refers to the rate at ensure uniform concentration in the daughter which drops of state (a, c, 7) depart from the vessel. The third term h(w, a, c, T) is such that droplets at birth. Since mass transfer occurs by diffusion in the h(w, a, c, r) da dcdr represents the net rate of drop, the solute concentration is not spatially formation of drops of state (a, c, 7) by breakage; uniform in the drop. We choose to describe the it includes loss of drops by breakage and formaamount of solute in the drop by the average tion by breakage of other drops. We shall concentration. Specification of the drop size inquire into the form of h subsequently. The (which we shall take as drop radius) and average term C(u, c, T) in Eq. (1) represents the rate of concentration will not suffice to predict the change of concentration in a drop of state (a, c, 7) instantaneous mass transfer rate from the drop. by mass transfer. Since mass transfer occurs by However, incorporation of an “age” concept diffusion we have will overcome this difficulty. “Age” is defined as as the time elapsed after the droplet has assumed (2) its identity in the vessel by entering with the feed or by breakage of a parent droplet. A little where c’ is the local solute concentration in the thought will show that the average concentration, drop radius and the age of the drop will uniquely drop and 9 is the diffusion coefficient of the solute in the dispersed phase. The time t’ belongs anywhere in the interval (t - 7, t) where tSleicher[5] has observed that the most frequent form of breakage in turbulent pipe flow is into two approximately t denotes the absolute time and T the age of the equal parts. droplet. Equation (2) must be subject to the 390
A population balance model for mass transfer
initial condition c’(r,r-7)=ci
O=SrGa
(3)
and boundary conditions c’(a, t) = Kc” ad -_= ar
f-7
o
s t’s
t
r = 0.
(4) (5)
Spherical symmetry is implicit in Eqs. (2) through (5). The constant K in Eq. (4) is the equilibrium constant and c” is the concentration of the solute in the continuous phase assumed uniform in the vessel but varying with time. If it is assumed that the continuous phase feed to the mixer has no solute and that the dispersion is sufficiently lean that mass transfer does not materially alfect the concentration of the continuous phase then condition (4) becomes c’(a,t)=O
t-7s
t’s
t.
2 I,“”da’ I,” dc’ J; dT’I(u’, c’, T’)p(U, a’, C’, T’)W(U’,
C’, 7’; t)
c, 7;
da dc dT. (9)
The rate of drops of state (a, c, T) lostby breakage W-4 to other sizes is
This eliminates the need for writing the equation for the continuous phase, although such an equation may be readily written [21. The solution of Eq. (2) subject to (3), (4a) and (5) is obtainable in either of the forms:
E= 1+3T-6P2
However C can be expressed as a function of the initial concentration ct so that on specifying a, c and 7 one can obtain C by back-calculating cf from (6) or (7). We now return to consider the form of h(w, a, c, T) which represents the difference between the rate of drops of state (a, c, T)formed and lost by breakage. We denote the fiuctiond rate of breakage of droplets of state (a, c, 7) by I(u, c, 7) and the fraction of drops of state (a, c, 7) in the daughter droplets formed by breakage of parent droplets of state (a’, c’, T’) is denoted by p(u, C,T; u’, c’, 7’) da dc d7. Then the rate of formation of drops of state (a, c, T) by binary breakage is given by
I(u, c, T)W(U,
c, 7; t)
da dc dr.
(10)
If one assumes that the breakage of droplets has little to do with their solute concentration and age, and that breakage of drop of size a is into two even halves each of radius u’/2113then (9) may be replaced by t
,rr-1/2+2 2 ierfc ?I=, X
(7) where c is now the average concentration obtainable from the local concentration c’. The symbol T represents the dimensionless group (%/u2). The two forms (6) and (7) are presented because one. or the other permits an easier computation of C from a, c and 7 depending on the value of 7. It should be clear that C cannot be readily expressed as an explicit function of c since (8)
da’ dr’ da dc dT.
(9a)
In writing (9a) we have recognized that droplets have zero age at birth. We have resisted further simplification of (9a) since this form is more convenient to the use of the method of successive substitution for the solution of Eq. (1). The quantity h( w, u, c, T) may now be written as: tit is not inconceivable that breakage may depend on the solute concentration since any interfacial turbulence due to mass transfer [4] is likely to affect the deformation of the drop.
391
B. H. SHAH
and D. RAMKRISHNA
x wfu’, c, T’; t) da’ d+ - r(u)w(u, c, 7). (11) If we restrict ourselves to steady state operation then Eq. (1) may be written as
(12)
[tic] =-
[
;+r(u)
02
I
0
dr’
1
CC;T > 0 (13)
+()(a, C,T) = $~((a,c~,T)+ ,;
x
eat
+l+%(U
Equation (14) may also be derived mathematically by integrating Eq. (12) over the interval (E, 00)and comparing with the population balance equation written for the bivariate density in a and c, on letting c * 0.
Equation (13) is a linear first order partial differential equation which readily submits to the method of characteristics[61. The result when combined with the boundary condition (14) yields an integral equation of the Volterra type which can be solved by the method of successive substitution. Given a drop of state
-
s[0
r(a)+f+$
Equation (15) when combined gives the integral equation
r(d)6 a-$ $(a’,c,7’). (14) ( >
Solution
I
.
Since drops of zero age can appear only with the feed or from breakage of other droplets we may write
x da’
ct, 0)exp
#((a, c, 7) = flu,
Equation (12) may be rewritten with distinction for drops of non-zero and zero age as follows: %+$
(a, c, r) Eqs. (6) or (7) can be solved back for calculating the spatially uniform initial concentration of the drop denoted c~. The concentration of this drop at time r’ measured from the instant of its birth is expressible as a function of C~ and r’, where T’ varies from zero to its present age 7. This function in symbols is written as c(c*, 7’) remembering that the actual function is given by (6) or (7) and that c = c(ct, r), where the left hand side is the present specified concentration of the drop. (Note that the radius of the drop, a has been suppressed in the expression). The solution of equation (13) is now written as:
with
du’J; dT’K(u,
(14)
U', Ci,T)
k-i'@', f&T')
(16)
3 by 1 I c(C
7') 7')
dr’
(17) &U,U’,Ct,T)
=
2$4.2, Cf,T)r(U)a
The integral equation (16) is solved by successive substitution. The resulting solution is somewhat complex in form but reflects some characteristics of the process that are apparent through physical reasoning [2]. Thus consider a particular droplet state (a, c, T). The attaiment of this state is associable with an infmite number of mutually exclusive possibilities for the past history of the droplet. Hence, the number of 392
A population balance model for mass transfer
drops of state (a, c, T) must result from the cumulation of drops which have evolved to that state from different past histories. Such difference in past history arises not only in the number of generations in the background but also in the ‘life-spans” of the various generations. Remembering that the first generation drops are those that have entered the vessel with unique “birth states,” it should be clear that the variation of life spans of individual generations is not free of constraints. To present the mathematical form of the solution we rewrite Eqs. (6) and (7) by
c=F(a, 7).
This solution is more conveniently presented in the form of individual concentration probability density functions for each drop size. Of course for each drop size, the droplet state is described by a bivariate density in concentration and age. However we will integrate the bivariate densities with respect to age and present the resulting concentration distributions for each droplet size. The joint probability density of size and concentration denoted fAc(u, c) is clearly
(19)
ci
Given (a, c, T), if the drop is of the ti generation, then for a particular sequence of life spans , I 3 1) of the respective ancestors Tj 9 rj+l, - - - &(j of generationj, j+ 1, . . . (n - l), the maximum life span of the (j- 1)st generation denoted #T(jnlI is given by
f=jiQ.z, c, 7) d7
L&9 c) =
(24) J;duJ;dcl;d&(u,c,$
where the denominator is the total number density fi. The conditional probability density for concentration for a given droplet size f&u, c) may be readily obtained from
n-1 C=
GP(%~)
F(aj-l,
@I)
Ej
F(ak,T$
(25)
(20)
where aj = (~$2~3). Equation (20) is solved to where the density function fA(u) is the size obtain distribution. Combination of (24) and (25) (2 1) yields ?$!!I= $!I($, $+I,. . . , T&l). Further #$,is given by c = c&.z, 7)&Z,_,, 7$&).
The solution of Eq. ( 16) can now be written as
+i
2”6(u- a,)
n=1
I ; @(a, c, 7) dT WA@) *
(26)
Since the droplet sizes are discrete the conditional density f&c) = &lA(un,c) can be seen to be given by
4%
ii’r(ak) k=O
&A(& c) =
(22)
I 0 "7p
e-[r(an-1)+(1/@)kk-1
d7A_1
e-[r(a~)ff1/e)17;
. . .
where we have put
0
e-ma~+wemo~n)
F(u, ~2’)) d7; - C(uo, c, To’“‘) I*
(23) 393
(28)
B. H. SHAH and D. RAMKRISHNA
of a minimum droplet size which can be estimated by the Kolmogorov theory of local isotropy [&lo]. The case of unrestricted breakage is handled by equating a, to zero. Two other forms for T(a) were considered which postulate the existence of a droplet size amax above which the droplets would break instantaneously in the vessel. Thus I(a) would tend to infinity when a tends to amax. This concept is somewhat akin to that of Collins and
(29) (30)
(32) j=o,1,2,...
Knudsen who have considered a droplet size of a,., which breaks with probability one in a fixed arbitrarily chosen time interval[ 111. The forms denoted I,(a) and r,(a) are presented below.
The dimensionless grouping Rj may be written as -- aj219
g/g
Rj-l/lY(aj)+
8’
(33)
r2(a)=
The first of the terms in the right hand side of (33) provides a relative measure of the time scales of diffusion and droplet breakage. Since larger drops should break with higher probability, R. should be a conservative estimate of the importance of the dynamic nature of droplet breakage; a large value of R,, would mean that breakage occurs very rapidly although the R,‘s should progressively diminish with increasing j. A more adequate measure of the importance of breakage dynamics is referred to in the next section. RESULTS
OF COMPUTATION
Computations were performed on an IBM 7044 computer for selected forms of the breakage frequency function I(a). Valentas er al. 171have considered the following expression for the breakage frequency. a-a,
R
1’ ) k
I-,(u) =
L-&-u,
0
a,
C
OSa
