Ind. Eng. Chem. Res. 1995,34, 3223-3230
3223
Some Basic Aspects of Reaction Engineering of Precipitation
Processes
K. S. Gandhi
Department of Chemical Engineering, Indian Institute of Science, Bangalore, 560 012 India
R. Kumar*
Department of Chemical Engineering, Indian Institute of Science and Jawaharlal Nehru Centre for Advanced
Scientific Research, Bangalore, 560 012 India
Doraiswami Ramkrishna*
School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
Analysis of precipitation reactions is extremely important in the technology of production of
fine particles from the liquid phase. The control of composition and particle size in precipitation
processes requires careful analysis of the several reactions that comprise the precipitation system.
Since precipitation systems involve several, rapid ionic dissociation reactions among other slower
ones, the faster reactions may be assumed to be nearly at equilibrium. However, the elimination
of species, and the consequent reduction of the system of equations, is an aspect of analysis
fraught with the possibility of subtle errors related to the violation of conservation principles.
This paper shows how such errors may be avoided systematically by relying on the methods of
linear algebra. Applications are demonstrated by analyzing the reactions leading to the
precipitation of calcium carbonate in a stirred tank reactor as well as in a single emulsion drop.
Sample calculations show that supersaturation dynamics can assume forms that can lead to
subsequent dissolution of particles that have once been precipitated.
1. Introduction
The technology of modern ceramic materials has
provided a fresh breath t o the reaction engineering of
precipitation processes because of the unparalleled need
for precise control of composition of mixtures, size
distribution of fine particles down to the nanometer
range, and so on. The formation of precipitates of solid
particles follows the establishment of supersaturation,
a state characterized by concentrations of the appropriate ions allowing for the solubility product of the
reaction product to be exceeded. Particles,however, can
result only by the process of nucleation, an event
occurring more or less randomly but leading to the
formation of a nucleus which thereafter grows definitely
at the expense of the prevailing supersaturation. Subsequent nucleation must compete with the process of
growth for the available supersaturation so that the
formation of fine particles depends on circumstances in
which nucleation of new particles prevails over the
growth of those that already exist. While such circumstances can be investigated, their establishment and
maintenance require careful analysis of the reaction
engineering of precipitation processes which it is the
objective of this paper to address.
Chemical reaction engineering is a mature field and
it would seem that the foregoing issue leaves little that
is unknown, but this paper is inspired by our recognition
of some subtleties not often encountered with other
reaction systems. The issue concerns a mixture of
reversible reactions with such highly disparate rates
that some of them may be deemed to have reached
equilibrium while others are far from it so that the
number of species required to define the system is
correspondingly reduced. Sorensen and Stewart
* To whom correspondence may be addressed.
(1980a,b) have indeed been concerned with the basic
issues of how thermodynamic constraints are built into
the analysis of multicomponent systems in chemical
reactors. These papers are extremely important in the
formulation of the equations of multicomponent reacting
systems particularly for chemical reactors. Because of
the generality of the treatment of Sorensen and Stewart
(1980a,b),focus is, however, lost on the issue of specific
interest to this paper.
The elimination of reaction intermediatesfrom kinetic
expressions for overall reaction rates (by using steadystate hypotheses) constitutes an altogether familiar
exercise so that the aspect of accounting for equilibria
or pseudo-steady states in reactions systems should be
regarded as patently routine. Paradoxically, however,
the possibility of a pitfall lurks in the very routine
nature of this procedure when it is applied indiscriminately. To elucidate further, we briefly reminisce the
procedure below.
The analysis of a single reaction begins with a
sequence of elementary steps involving intermediates.
Mass action kinetics is used to express the rates of the
elementary steps in mass balances of each species
involved in the different steps. The rate of formation
of the product generally features the concentration of
reaction intermediates which are eliminated using the
algebraic relations resulting from the steady-state
hypotheses for the intermediates. The procedure is
straightforward,but this setting neither does (nor needs
to) acknowledge the possibility of a loss or gain of any
intermediate due to another reaction. If there were a
second reaction that shared a particular intermediate,
the analysis without proper accounting for this feature
could lead to serious errors. Proper accounting lies in
recognizing that equilibrium reactions featuring species
which are also involved in slower reactions cannot be
assumed to proceed at zero rates. In precipitation
0888-5885/95/2634-3223$09.QQIQ 0 1995 American Chemical Society
3224 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
reactions, one encounters ionic species (which are akin
to the intermediates above) that are shared among
several different reactions, compounding the possibility
of errors. Thus, sensitivity to this issue is particularly
important in the analysis of reactions in precipitation
systems.
Yet again the issue of interest to this paper is not
without parallel in the chemical engineering literature,
for it has arisen in a somewhat ad hoc way in the
simultaneous absorption of two gases in a liquid (Astarita and Savage, 1982)with chemical reaction. The
setting consists of two gases (say A1 and Az) absorbing
into the liquid, each undergoing fast reversible reactions
with a component B1 in the liquid t o form BOand B3,
respectively. A third reversible (also fast) reaction is
also entertained in the analysis between A2 and B2 to
form A1 and B3. The third reaction involving a shift in
concentrations of A1 and A2 forces the appearance of
reaction terms otherwise absent in the transport equation for the film. The analysis of Astarita and Savage
(1982)is particularly interesting in that it provides
contrasting combinations of physical and chemical driving forces for the absorption of the two gases.
In order to see the nature of the error that results
from improper application of equilibrium (or steadystate) assumptions, we briefly consider the following
simple chemical reaction system. (We are grateful to
the reviewer for suggesting discussion of the main issue
of this paper by relating to the simple reaction system
above.) Consider species A, B, and C in the consecutive
reaction scheme
the new set of reduced equations given by
da = -kla,
dt
-db
=
dt
Izia
+
(1 KJ
c =K,b
(1.3)
Equation 1.3 must clearly satisfythe total mass balance.
In order to obtain eq 1.2 from the above set, one must
require that K2 << 1 for which no rationale need exist.
It is the generalization of the process of deriving the
set of eq 1.3 for arbitrary reaction systems with which
this paper is concerned.
The objective of this paper, therefore, is to elucidate,
in chemical reaction systems, the proper transport
equations only featuring concentration variables that
are left over a€ter elimination of the others due to
assumption of equilibrium for some reactions. Use of
these equations will preserve the analysis from the
possibility of pitfalls. Since our focus is on precipitation
systems, we shall demonstrate the analysis for the
specific example of reactions involved in the precipitation of calcium carbonate in a well-stirred system as well
as in a single liquid drop. It will also become clear from
our analysis how unusual combinations of physical
transport and chemical reactions can arise in systems
of this kind.
2. Reaction System
We shall assume n chemical species (AI, A2, ..., &}
undergoing m reversible chemical reactions where n L
m. For convenience of notation, we let S, represent the
set of integers from 1 to m. The reactions are represented by
n
Ca&=O, i ES,
j=l
The mass balance for a well-stirred batch reactor leads
to the differential equations
da-- -k,a,
dt
db = k,a - k2b
dt
+ k-+,
dc =
-
dt
kib - k-+
(1.1)
where the lower case letters denote concentrations. Now
suppose that the forward and backward reactions of B
to C occur at a considerably faster rate than the reaction
of A to B. We assume that the second reaction is a t
equilibrium so the first instinct is to write the differential equations
Indeed eq 1.1 does not conserve the total mass of
reactants because the differential equation for B does
not account for its disappearance through its reaction
to C. The methodology of this paper would instead call
for eliminating the equilibrium reactions by adding the
two equations for species B and C in eq 1.1 and writing
da - -kla,
dt
db + dc = k,a,
dt
c = K,b
dt
Differentiating the equilibrium relationship with respect
to time and substituting K2 dbldt for dcldt, we arrive at
where, in the ith forward reaction, the stoichiometric
coefficient ac is positive if 4 is a product, negative if Aj
is a reactant, and zero if Aj does not participate. We
shall denote the molar concentration of 4 by uj and
represent the concentration vector by a = [al, u2, ...,a,].
We let the ith intrinsic reaction rate (which is the net
rate in the forward reaction) be ri ri (a),and then the
total rate of generation per unit volume of A,, denoted
Rj, is given by
Rj = x $ j r i
(2.1)
ZCS,
We now assume that there are m - r reactions very
fast relative to the other r reactions. These reactions
will be very nearly a t equilibrium over a time scale
small compared with that of evolution of the others. In
symbols, we let zi be the time scale of evolution of the
ith reaction (relative to that of observation) and introduce the set Zm-r = {i E S,: zi *: l}, the subscript m r signifying the number of reactions satisfying the
characterization of the set
We now make the
assumption that
riri M 0, i E 2m-r
(2.2)
which implies that, over the time scale of observation,
the reactions belonging to the set Ern-? are nearly a t
equilibrium. In particular, note that eq 2.2 does not
imply that the above reaction rates, as they appear in
a balance equation for each chemical species, are
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3225
themselves zero. Next, we postulate that the m - r
equilibrium reactions are linearly independent and that
eq 2.2 can be solved uniquely for the last m - r
concentration variables in terms of the first n - m r
concentration variables so that we may write
Multiplying eq 3.1 by bkj and summing overj E
we obtain in view of eq 3.2
Sa,
+
uj = 4jli(ul,u2,...,u,-m+r),
j E Sm-r=
{n - m r 1,n - m r
+ +
+ + 2, ...,n} (2.3)
We tacitly assume that none of the above equations can
be manipulated to obtain a relationship purely among
the concentration variables al, a2, ..., an-,+?. The
relationships expressed by eq 2.3 among the different
concentrations are as though the reactions concerned
were exactly at equilibrium. We thus need only n - m
r concentration variables for a description of this
reaction system. Let the vector of this select set of
concentrations be denoted by I = [ul, u2, ..., an-m+rlT,
the superscript T referring to the transpose of the row
vector. We may now restate our objective here as one
of identifying the transport equations for the above
select variables entirely free from the eliminated variables. The continuity equation for the species Aj is wellknown (Bird et al., 1960).
+
3
+ V - N =~ R ~ j, E S,
at
3. Analysis of the Reaction System
We rewrite eq 2.4 as
%+
cqr +
qjri, j
J z
E
S,
(3.1)
= {i E S,: i P Xm-r) identifies the
where the set
slower reactions far away from steady or equilibrium
values. Of particular interest is the second term on the
extreme right-hand side of eq 3.1. We wish to eliminate
this term by finding n constants {bj} such that
c q j b j= 0, i E
(3.2)
jeS,
Alternatively, we seek homogeneous solutions of the
algebraic equations (3.2) involving the matrix {ug;i E
X,-,, j E S,} whose rank is m - r; it is then possible by
standard arguments of linear algebra (see, for example,
Amundson, 1966)to fmd n - m r linearly independent
vectors that satisfy the homogeneous equation (3.2).The
actual methods t o determine them are well-known. We
K E Sn-m+r},
shall assume that they are given by {b(k);
each being an n-dimensional vector. In what follows
we let be be the j t h component of the vector bck).
+
Equation 3.3 basically represents the required transport
equation. The second term on the left-hand side of eq
3.3 consists of items related to the variables to be
eliminated. Their complete elimination is facilitated by
invoking eq 2.3. For example, the partial time derivatives are dealt with as follows.
(2.4)
where Rj is as in eq 2.1 so that it includes the rates of
all the reactions including those close to equilibrium;
Nj represents the total molar flux of component Aj. It
is also opportune t o recall the earlier warning that it
would be erroneous to simply eliminate the reactions
close to equilibrium in Rj. A systematic method to
arrive a t the proper transport equations in the leftover
variables is elaborated below.
- V * NJ .= RJ. =
at
icz
Splitting the sums above
Similarly, one may evaluate the convective and diffusion
terms in detail by using eq 2.3 on the appropriate
constitutive equations for transport. A demonstration
will take us far afield, but the essential result which
concerned the identification of transport equations in
the leftover variables is already contained in eq 3.3
when supplemented by calculations of the type (3.4).
Since the rank R of the matrix {%; i E X m - r , j E S,} is
m - r, the number of equations in eq 3.3 equals the
number n - m r of concentration variables. We have
thus obtained the description of the reaction system in
terms of the reduced concentration vector I as required.
Since many of the reactions of interest to this paper
are ionic in nature, the issue of whether or not electroneutrality considerations must be included in the analysis above arises. Electroneutrality considerations are,
of course, an integral part of the analysis of transport
of ionic species. However, our focus here has been on
chemical reaction rates at every point in the medium
which, because of stoichiometry, automatically conserve
component masses and satisfy charge balances everywhere, at least with respect t o the generation rates of
opposite charges. Thus, the linear combination of
chemical species in eq 3.3 is consistent with the generation of ionic species satisfying electroneutrality. The
constraints imposed by electroneutrality on the transport of different ionic species must therefore impact the
subsequent treatment of eq 3.3 (rather than that prior
to it).
3.1. Boundary and Initial Conditions: Equation
3.3 must indeed be considered together with boundary
and initial conditions. These conditions are, of course,
t o be obtained from those which apply to the original
system (3.1). The strategy for deriving the proper
boundary conditions for the reduced set is as follows.
We form the same n - m r linear combinations of
the boundary conditions for the different species as we
do the differential equations (3.1). These boundary
conditions will involve the concentrations and/or fluxes
at the boundary of all chemical species present including
+
+
3226 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
those in the reduced set as well as the others. We
assume that the near-equilibrium reactions are the
same at the boundary as in the bulk. In this case, the
concentration of each species not belonging to the
reduced set can be expressed in terms of those in the
reduced set via eq 2.3, while their fluxes (involving
gradients of concentration) can be computed using eq
3.4 in which the time derivatives are replaced by spatial
derivatives. This strategy automatically accounts for
transport of any of the species across the system
boundary and their consequent effect on the reduced set
species.
The initial conditions, on the other hand, are somewhat more subtle. Assume for the present that the
spatial distributions of all the species are specified in
the original problem (3.1). Denote the original initial
concentration of the j t h species by aj,owhich is a function
of spatial coordinates. The problem at hand is to specify
for the reduced set the initial concentration of thejth
species 0' E Sn-m+r),denoted aj(0,x) where x represents
spatial coordinates. From eq 2.3 we must have
following reactions are all found to be important in this
system (Butler, 1964).
ki
(1)
Ca(OH),(aq)
kdKi
Ca2++ 20H-
k3
(3)
C02+H20
HC03-+Hf
kdK3
k4
(4)
Ca2+
+ ~ 0 ,kdK4
~ -~ a ~ ~ , ( a q )
(7)
+
which represent m - r equations relating n - m r
unknowns aj(0,x) 0' E SmJ where it is implicit that n
- m r > m - r or n > 2(m - r). Thus, we need n 2(m - r ) more equations in order to identify the initial
conditions aj(0,x) 0' E SLJ.
These additional equations can be arrived at as follows:
(1)For species participating in the slower reactions
the initial concentrations will be those originally specified. Thus
+
aj(O,x)= aj,o
(2) For species that are involved in the fast reactions,
we basically seek redistribution z of the fured total
amount of each species implied by the original initial
conditions subject t o local mass balance constraints for
each of the constituent atomic species and equilibrium
relationship (2.3). For the well-stirred case the specification of the initial conditions is relatively straightforward.
We now consider an example below.
4. Precipitation of Calcium Carbonate
As an illustration, we first consider a reaction scheme
involving calcium hydroxide and dissolved carbon dioxide to form calcium carbonate. The overall reaction
is given by
Ca(OH),
+ CO,
-
CaCO,
+H20
The precipitation occurs when the calcium ions and
carbonate ions are present in proportions such that the
solubility product of calcium carbonate is exceeded. The
The above 9 reactions cover 12 species: Ca2+,co32-,
HC03-, OH-, H+,
CaHC03+, C02, CaOH+, CaCOdaq),
HzCOs(aq), Ca(OH)n(aq),H2O. We assume that reactions (1)and (4)-(9) are always at equilibrium. We
have thus 7 equilibrium reactions so that only 5 species
(out of 12) need be considered for a description of the
system. In terms of the notation of the previous section,
we have n = 12, m = 9, and r = 2 so that m - r = 7;
thus, the number of concentration variables for the
system is given by n - m r = 5. We let these be A1
A4 = Ca(OH)2 (aq), & =
COS,A2 Cos2-, & = H+,
H2O. (We remind the reader that the choice of these
five variables has been made so that no relationship
exists among the variables selected as a result of
equilibria. There are several other ways of doing it.
However, if water were to be replaced by, say Ca2+ion
concentration, it would have been possible to use the
equilibrium relationships for reactions (1)and (9) to
establish a relationship between the concentrations of
Ca2+,Ca(OH)2, and H+. This choice is therefore to be
avoided.) The 7 variables to be eliminated are As =
HC03-, A7 = HzCOs(aq),As OH-, A9 = CaCOa(aq1,
A10 E CaHC03+, A11 = CaOH+, and A12 = Ca2+. The
equilibrium relationships are given by:
+
K9
a6= K5-'a2a3, a7= K , - ' K , - ~ u ~ , ~ ,a8 = -,
a3
a, =
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3227
Using formulas of the type (3.3), we get from (4.1)
,"
j=l
By defining a matrix fi
{6kj}, where 6 k j
[bkj
x j f 6 bki &#~i(A)/&zjI,
the above equation becomes
+
Substituting the values of the different quantities on
either side of the equation above, one obtains
0 0
0 0
b(4)T =
b"5)T
0 0
0 0
0
0
-10
0 0
0 0 1 0 - 1
-1001-1
[O 0 0 2 1 0 0 1 0 0 1 0 1
=
[O -1 0 -1 -1 -1 -1 -1 0 0 0 11
The above analysis is valid for calculation of supersaturation prior to the birth of the first nucleus. Subsequent
to that the growth of the particle phase a t the expense
of the solutes in the liquid phase has to be accounted
for in the mass balance of species. Of course, the
method of analysis of the reactions will be unaltered.
4.1. Precipitation in a Well-Stirred Batch Reactor. Suppose that we are concerned with reactions in
a well-stirred reactor under batch conditions with
spatially uniform concentrations of all species. Under
these circumstances, the appropriate macroscopic form
of eq 3.2 becomes
where the matrix &A) is given by Chart 1. The inverse
of the matrix fi can be calculated analytically so that
eq 4.2 can be solved explicitly for the rates of change of
concentrations of each of the selected out species entirely
in terms of their concentrations. (If, for some realizations of the concentration vector, the matrix fi(a) is
singular, eq 4.2 is ill-defined. Under these circumstances, however, the rate of change of the reduced
concentration vector dudt is either nonexistent or
nonunique, from which it must be inferred that the
approximation in question is inadmissible for such
realizations.) The prime purpose of our pursuit is now
accomplished. Equation 4.2 should be solved subject to
initial conditions stipulated only for the concentration
vector A = [al, a2, ...,a51Tas specified in section 3.1.
Our focus in the foregoing analysis has been on
obtaining a reduction in the number of equations to be
solved for a precipitation process. This reduction is
strictly that connected with the reactions that are
assumed t o be a t equilibrium.
For the purposes of demonstration we present numerical calculations for precipitation in a batch system.
(We are grateful to Rajdip Bandyopadhyay for the
calculations of this paper.) Instead of the well-stirred
batch reactor, we shall consider an emulsion drop such
as, for example, in the experiments of Kandori et al.
(19881, in which carbon dioxide was initially bubbled
into a water-in-oil emulsion stabilized with the surfactant CaOT. The aqueous phase contains Ca(OH12
through its addition in the form of solid particles; the
total calcium, however, is dependent also on CaOT.
Precipitation of calcium carbonate is considered subsequent to termination of the bubbling of carbon dioxide
at saturation. The calculations are made for two
different values of the parameter R, defined as the
molar ratio of water to CaOT added at the beginning of
the experiment. Since our purpose here is to demonstrate the dynamics of supersaturation with respect to
the calcium carbonate precipitate prior to the appearance of precipitate, we shall not account for precipitation
itself. Thus, the need for population balance for the
calcium carbonate particles is obviated in this ap-
3228 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995
Chart 1
0
0
0
K1K4
Kt
2
-1
.Iza3
K1K7
3
K5KtJZu3
KlK,
2+
K,”3
1
1
15
j
;
i
CO, = ,0389m/l
10
0
I
.
0
.02 .04 .OB .08
.1
0-
Pu
cc
5
15 -
lo+--0
’
’
I
’
’
5
0
,-
10
15
time ( 8 . )
Figure 1. Supersaturation dynamics for a “batch” emulsion drop
initially saturated with carbon dioxide and with stated calcium
hydroxide concentration. The inset shows small time behavior.
proximation. The supersaturation with respect to calcium carbonate, denoted Acaco3,is defined by
where K,,is the solubility product of calcium carbonate
and the expression on the extreme right in the equation
above results from exploiting other equilibrium relationships. The values for the various equilibrium
constants and the solubility product were taken from
Butler (1964) and Danckwerts (1970).
The initial concentrations for the five species are
obtained from the value of R, total calcium balance,
equilibrium reactions (11, (81, and (91, and finally a
charge balance. The calculations for two different
values of the ratio R are shown in Figures 1 and 2.
Particularly interesting is Figure 2 in which the supersaturation, after rising to very high values at short
times because of the very fast ionic reactions, eventually
starts to drop again and reaches levels below that of
saturation. Under these circumstances the smaller
particles that have been formed by precipitation during
supersaturation will tend to dissolve rapidly, again
changing the concentration of the ions in the liquid
solution. Of course, more appropriate calculations must
account for nucleation and growth of particles because
the reaction equations must be coupled to the population
0
5
10
15
20
time ( 6 . )
Figure 2. Supersaturation dynamics for a “batch” emulsion drop
initially saturated with carbon dioxide and with stated calcium
hydroxide concentration. Note how supersaturation drops below
the saturation level for calcium carbonate (dashed line). The inset
shows small time behavior.
balance equation since particle growth must change the
concentration of species in the liquid phase.
If the precipitation process had been carried out in
an open system (such as a continuously stirred tank
reactor or an emulsion droplet), eq 4.1 should be
replaced by another macroscopic balance in which
allowance is made for transport of the different species
across the system boundaries. In view of the importance of precipitation in small emulsion droplets, we
shall stop to consider briefly an application of the
techniques of this paper to this case.
4.2. Precipitation of Calcium Carbonate in a
Drop. Consider a drop exposed to a continuous phase
through which carbon dioxide is continuously bubbled
t o maintain a fmed concentration in that phase (e.g., at
its solubility). Assume transport of all components
occurs purely by diffusion within the drop with a
multicomponent diffusion coefficient matrix, say D,
which depends on concentrations of all the diffusion
species. The continuity equation may then be written
as:
After calculations similar to those used in the previous
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3229
40 1
204
I
/
in which several of the ionic dissociation reactions are
so rapid (relative to other reactions in the system such
as those generating the dissociating salts) that they may
be effectively assumed to be at equilibrium. Thus, the
number of species to be included in the analysis of the
process is considerably reduced, resulting in a correspondingly small number of equations representing
balances of the selected-out species. Such reduced
balances must include the contributions of equilibrium
reactions to the rates of change of the different species.
This paper shows that the contributions of equilibrium
reactions can be systematically calculated by the methods of linear algebra. A complete analysis of the
precipitation process must couple the analysis in this
paper to the population balance equation for the precipitated particles.
Acknowledgment
0
5
15
10
20
time (8.1
Figure 3. Supersaturation dynamics for an emulsion drop
exposed to carbon dioxide (initial concentration = 0)saturated in
the surrounding organic phase for different initial calcium hydroxide concentrations as shown. Supersaturation drops slightly
below the saturation level for calcium carbonate.
example, the equation in P is seen to be
The authors gratefully acknowledgethe support of the
Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India, and the Department of Science
and Technology, India, for the support of this research.
D.R. also thanks the International Programs Division
of the National Science Foundation for a travel grant
in the summer of 1994 to collaborate with K.S.G. and
R.K.
Notation
12
12
Equation 4.4 shows that the effect of equilibrium
reactions when used to select out species is to modify
their diffusion coefficients as contained within the
parentheses on the right-hand side of the equation. The
modification of the diffision coefficients depends very
much on the equilibrium relationships, consequently
implying indefinite signs for the diffision coefficients.
The contrasting combinations of physical and chemical
driving forces clearly arise from this source.
Calculations were also made for an open system
comprising a drop sufficiently small in which diffusional
rates are rapid enough t o allow the assumption of wellmixedness. Carbon dioxide was continuously bubbled
during the process with the initial concentration of
dissolved C02 in the drop at zero. Thus, the equations
solved were a macroscopic version of eq 4.4obtained by
integrating it over the drop volume and supplying the
boundary fluxes by incorporating mass transfer coefficients where necessary. The rather straightforward
equations are not included here. In this case, Figure 3
shows how the supersaturation in the drop varies.
Again for the smaller value of R , the supersaturation
drops to subsaturation levels, leading to dissolution of
the calcium carbonate particles. As pointed out earlier,
the calculations neglect the coupling between the population balance equation and the reaction equations. A
proper analysis of the precipitation process must therefore entertain the full equations including those in this
paper for the solution phase variables as well as the
population balance equation for the particles.
5. Conclusions
This paper presents a compact methodology based on
the methods of linear algebra for dealing with reactions
AJ =jth chemical species
a, = concentration of jth species
a = concentration vector of all the species present
I = concentration vector of selected-out species
bkJ =jth component of vector b(k)
iE
b(k)= kth vector in the null space of the matrx {q;
& - r , j E Sal
6, = kjth coefficient of the matrix B
$ = matrix defined above eq 4.2
k, = forward rate constant for the ith reaction
K, = equilibrium constant for the ith reaction
m = total number of chemical reactions in the system
n = total number of species in the system
r = number of selected-out species for system description
rL= rate of the ith reaction
R = molar ratio of water to surfactant (CaOT)
S, = set of n integers from 1to n
K-r = set of n - r integers from r 1to n
+
Greek Symbols
a, = stoichiometric coefficient of the jth species in the ith
reaction; also the zjth coefficient of the stoichiometric
matrix
Xm-r = set of m - r integers from the set {1,2, ...,m} which
correspond to reactions that are at equilibrium
= set of r integers from the set (1,2, ..., m } not in the
set E,-,, corresponding to reactions far removed from
equilibrium
4 = function defined by eq 2.3
Literature Cited
Amundson, N. R. Mathematical Methods in Chemical Engineering.
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Received for review November 2, 1994
Revised manuscript received July 17, 1995
Accepted August 7, 1995@
IE940639+
@
Abstract published in Advance ACS Abstracts, September
15, 1995.