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 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. Matrices and Their Applications; Prentice-Hall: Englewood Cliffs, NJ, 1966.
3230 Ind. Eng. Chem. Res., Vol. 34,No.10, 1995 kstarita, G.; Savage, D. W. Simultaneous ansorption with reversible instantaneous chemical reaction. Chem. E M - . Sci. 1982, 37,677-686. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wilev & Sons: New York. 1960. Butler, J. N. Ionic Equilibrium: A Mathematical Approach; Addison Wesley: Reading, MA, 1964. Danckwerts, P. V. Gas-Liquid Reactions; McGraw-Hill: New York, 1970. Kandori, K; Kon-No, K.; Kitahara, A. Formation of Ionic w/o Microemulsions and Their Application in the Preparation of CaC03 Particles. J . Colloid Interface Sci. 1988,122, 78. Sorensen, J. P.; Stewart, W. E. Structural Analysis of Multicomponent Reaction Models, Part I. Systematic Editing of Kinetic and Thermodynamic Values. MChE J . 1980a,26,98-104.
Sorensen, J. P.; Stewart, W. E. Structural Analysis of Multicomponent Reaction Models, Part 11. Formulation of Mass Balances and Thermodynamic Constraints. AZChE J . 1980b,26, 104111.
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.
