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Mixing Effects on Particle Precipitation Giridhar Madras* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
Benjamin J. McCoy Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803
A common but complex multiphase chemical reactor process is the mixing of two chemicals to precipitate particles. We build on previous work [Madras, G.; McCoy, B. J. AIChE J. 2004, 50, 835] to study how turbulent mixing influences precipitation reactions. A fragmentationcoalescence model for turbulent dispersal of a limiting reactant A into a homogeneous batch of dissolved B shows how mass transfer controls the reaction rate. The reaction product Q undergoes either hetero- or homogeneous nucleation to form stable nuclei of precipitate that then grow until the supersaturation of Q is exhausted. Nucleation and growth are mathematically represented by a particle size distribution model that shows how the average precipitate particle size and numbers change with time. The model predicts, in agreement with experimental observations, that increased turbulence reduces the average precipitate particle size. Introduction Mixing liquids to precipitate solid particles is a common multiphase chemical process that comprises several complex phenomena.1-3 In its simplest form, a soluble reactant is mixed by dispersion into a batch of a second soluble reactant. When the reaction occurs between the two soluble reactants, a product is formed that is initially also soluble, but in a supersaturated, metastable state relative to its equilibrium solubility product. Either heterogeneous or homogeneous nucleation produces stable nuclei that grow into precipitate particles, causing the supersaturation to decline toward its equilibrium value. As supersaturation decreases and the nucleation rate decreases, the stable nucleus size increases according to the Gibbs-Thomson effect, which governs the solubility of a curved interface. If the system is allowed to age, the increasing stable nucleus size leads to Ostwald ripening, coarsening of the particulate size distribution, by dissolution of the smallest particles and transfer of their mass to the larger particles.4 If the particles are only slightly soluble, then an extremely long time may be required for observable ripening. Aggregation and breakage, in addition to reversible growth and dissolution, can influence the particle size distribution.5-7 Attrition may cause secondary nucleation; i.e., fragmentation can produce small particulates that act as heterogeneous nuclei.5,8 In previous work, we have applied general methods of distribution kinetics (population balance modeling) to describe dispersive mixing of a tracer or reactant into a bulk fluid.9 The cascade of decreasing fluid elements in turbulent mixing causes increased interfacial areas for transfer of the dispersed reactant into the bulk, and allows quantitative description of competing chemical reaction rates.10,11 Several of our papers were also devoted to investigating how cluster distribution kinetics mathematically portrays the liquid-solid-phase transition dynamics of nucleation, growth, and ripening.12 We wish now to * To whom correspondence should be addressed. Tel.: 91080-2293-2321. Fax: 91-080-2360-0683. E-mail: giridhar@ chemeng.iisc.ernet.in.
combine these processes of mixing, reaction, and phase transition to study the effect of turbulent mixing on precipitation. The objective is to show how the concepts of distribution kinetics and dynamics of turbulent mixing of two reactants dissolved in liquids can be used to determine the production rate of particulate precipitate. The reactant A is mixed into a batch vessel containing B so that a reaction occurs at the interfaces of fluid elements to form a product Q:
A+BfQ This very fast reaction is controlled by the rate of mass transfer between the dispersed and continuous liquids, and thus depends on the area for the fluid elements13 of A dispersed in B. We assume that the dispersed fluid elements are fragmented in a cascade of increasingly smaller sizes and larger interfacial area.14 The model, with variables reinterpreted, can be applied to mixing of jets, given that fragmentation and dispersal are the key processes. The engulfment model of Baldyga and Bourne similarly postulates deforming slabs of dispersed fluid with diffusion occurring at the interfaces.1 In contrast to the engulfment model, we describe the reversible fragmentation-coalescence by a population dynamics equation that has an exact self-similar solution for the size distribution as a function of time. The instantaneous dissolved concentration of Q can be expressed as a supersaturation ratio S, which is the concentration of Q divided by its saturation value. The product Q will nucleate according to the classical homogeneous nucleation theory expressed in terms of S. The nuclei grow (and possibly ripen) as S approaches its equilibrium value. The equation for the concentration of Q, or its supersaturation, is coupled with the equations of nucleation and growth. If S is large because the mixing is good, then many nuclei will be formed and will grow until equilibrium is reached. The numerous particles can each grow only until the mass of Q (and thus of the limiting reactant A) is consumed, and so the particles will be uniformly small. On the other hand if
10.1021/ie049217i CCC: $30.25 © 2005 American Chemical Society Published on Web 02/02/2005
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S is small because of poor mixing, the number of nuclei will be fewer and the smaller number of particles can grow to form a broadly dispersed (nonuniform) particle size distribution. The model is based on assumptions that allow us to focus on the essential features of mixing with precipitation. Temperature and heat effects are considered negligible, although these could be added.15 Soluble A in liquid is added to a batch of homogeneous liquid solution of B. The reaction, A + B f Q, is so fast relative to mixing that reactant A is instantaneously converted to product Q when it comes into contact with reactant B.10 This assumption is quite reasonable for precipitation reactions.2,3 Precipitation of the reaction product Q occurs in the continuous liquid, which is assumed homogeneous. Classical nucleation theory and the GibbsThomson equations are applied to ascertain the influence of supersaturation, liquid-solid interfacial energy, and particle curvature on nucleation and growth. An aim of this study is to propose a simple (but not too simple!) and readily implemented approach to mixing with precipitation. The central principles are common to chemical engineering science, namely, population balance modeling, mass transfer controlled chemical reactions, and cluster kinetics of particulate nucleation and growth. It may seem surprising that a system of straightforward ordinary differential equations governs the complex problem; this is a consequence of applying the moment method to the population balance equations. The model employs parameters related to properties and rates of actual materials and processes. Any model of turbulent mixing, including those models involving computational fluid dynamics, will incorporate parameters whose values are determined by experimental measurements. The nucleation and growth processes also require parameters that have their basis in classical nucleation theory. The illustrative computational results presented here show what parameters must be estimated so that experimental data can be represented.
dispersed fluid elements per vessel volume at time t in the mass range (y, y + dy). It follows that yp(y, t)dy is the mass of such elements per volume. The moments consequently are defined by
Dispersive Mixing Kinetics
To represent the interfacial area among the dispersed fluid elements and the bulk fluid,10 we consider that the dispersed fluid volume per vessel volume is R. If the elements are spherical of radius r, then the fluidelement area/volume is 3/r ) 3(3y/4πF)-1/3 ) gy-1/3, in terms of g ) (36πF)1/3. The total dispersed-fluid area per vessel volume for elements in the range (y, y + dy) is Rgy-1/3p(y, t)dy. For nonspherical fluid elements, the coefficients of yν will have a different form. For example, if fragmentation occurs by stretching cylindrical fluid elements of radius r(t) and then severing them into two random segments of area/volume ) 2/r, the cylinder mass in terms of its length z is y ) πr2zF, from which the total element area per vessel volume is Rgy-1/2, where g ) 2(πzF)1/2. Finally, if fragmentation occurs by stretching flat slabs and then severing them into two slabs, and if the ribbonlike slabs, or striations, are thin of thickness h and surface 2S, then edge areas can be neglected and the element area per element volume is 2/h. In terms of the element mass y ) ShF, the element area per vessel volume is proportional to Rgy-1 with g ) 2SF. In this case, the moment with ν ) -1 in eq 5 diverges, but we can compute the moment for ν arbitrarily close to -1. We see that ν ) -1/3, -1/2, and near -1 apply for the spherical, cylindrical, and slab geometries, respectively, suggesting that a low-order negative moment of p(y, t) governs the kinetics of a reactive mixing system. This is particularly significant
The reactant A is dispersed into the bulk liquid containing reactant B. As in our earlier work10 we consider that fluid elements P(y) of mass y undergo shearing and merging in the turbulent stirred tank. As diffusion between the dispersed and continuous liquids is much slower than turbulent dispersion, it is feasible to assume the dispersed phase has distinct boundaries that define the interfacial area. This assumption may be less reasonable for mixed gaseous reactants where diffusion is orders of magnitude more rapid. Given the distinction between the dispersed and continuous fluid, if fragmentation and coalescence of the dispersed fluid are uncorrelated, the simultaneous binary processes can be represented as kb(y)
P(y) {\ } P(y′) + P(y - y′) k a
(1)
Here, kb(x) and ka represent the rate coefficients for fragmentation (breakage) and coalescence (aggregation), respectively, both depending on the turbulent state through the Reynolds number. All parameters are considered independent of position in the vessel; postulating interconnected compartments positioned within the vessel could add the spatial influence. The distribution is defined so that p(y, t)dy is the number of
p(ν)(t) )
∫0∞p(y, t)yν dy
(2)
The number of fluid elements per vessel volume is p(0)(t). The dispersed-fluid volume fraction, R, is related to p(1)(t) ) p0(1), the constant mass per vessel volume, and its mass density F as R ) p0(1)/F. The variables can be made dimensionless:
ξ ) y/p0avg P(ξ, θ)dξ ) p(y, t)dy/p0(0) P(ξ, θ) ) p(y, t)p0avg/p0(0) P(ν)(θ) ) p(ν)(t)/p0(0)(p0avg)ν (3) For the scaled distribution P(ξ, θ), the reaction rate is determined by the νth moment,10 defined as
P(ν)(t) )
∫0∞ ξνP(ξ, t)dξ
(4)
McCoy and Madras16 have shown that when kb ()κbx) is proportional to the mass x and ka is constant, any initial distribution evolves to the exact self-similar exponential solution whose moments are simply
P(ν)(t) ) [Pavg(t)]ν-1Γ(ν + 1)
(5)
The time dependence is determined solely by the average element size, which has the quite simple asymptotic behavior if fragmentation dominates over coalescence:16
Pavg(θ) ∼ 1/(1 + κbt)
(6)
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considering that the negative moments increase strongly when fragmentation dominates the mixing process.10
that produced by the reaction A + B f Q, evolves according to
Reaction Kinetics
∂q(x, t)/∂t ) -q(x, t)
For a batch operation with A as the limiting reactant, and when the reaction is mass transfer controlled, the interfacial area between the continuous and dispersed phases determines the reaction rate. The surface area depends on the degree of mixing, that is, on the relative effects of fragmentation and coalescence. The reaction of A and B can be represented as an integral over the distribution of fluid elements, p(y, t). When A is the limiting reactant, the reaction ceases when A is completely consumed and thus the total moles of A reacted must be considered. When A is depleted the concentration of B stops changing and remains constant. If R and 1 - R are the dispersed- and continuous-fluid volume fractions, respectively, then
RdcA/dt ) (1 - R)dcB/dt
-dcB/dt ) km cB(t)gRp(ν)(t)
∫x∞kd(x′)c(x′, t)δ(x - xm)dx′ - Iδ(x - x*)x*/xm + kmcB(t)gRp(ν)(t) (12) where we have incorporated both the production of Q by chemical reaction and loss by nucleation. Nucleation of critical nuclei of mass x* at the rate I uses a number of monomers equal to x*/xm. The distribution changes according to eq 11, which becomes, when the integrations over the Dirac distributions are performed, the finite-difference differential equation
∂c(x, t)/∂t ) -kg(x)c(x, t)q(0) + kg(x - xm)c(x - xm, t)q(0) - kd(x)c(x, t) + kd(x + xm)c(x + xm, t) + Iδ(x - x*) (13)
(7)
For the continuous phase concentration, the rate expression is
(8)
where the moment p(ν)(t) is defined by eq 2. The reactants are initially segregated, so initial conditions are cA(t ) 0) ) cA0, cB(t ) 0) ) cB0, and cQ(t ) 0) ) 0. The rate of Q produced by the reaction in the homogeneous continuous bulk fluid is -dcB/dt.
The governing equations show that c(x, t) increases by addition and decreases by the loss of mass xm. Equation 5 can be expanded for xm , x to convert the differences into differentials and obtain a partial differential equation in x and t. Microscopic reversibility for the growth process implies
kd(x) ) qeq(0)kg(x)
kg(x) ) γxλ
The size distribution of precipitated particles is defined by c(x, t)dx, representing the concentration of precipitate clusters at time t in the differential mass range (x, x + dx). Moments are defined as integrals over the mass
∫0∞c(x, t)xn dx
(9)
The zeroth moment c(0)(t) and the first moment c(1)(t) are the time-dependent number and mass concentration of precipitate clusters, respectively. The ratio of the zeroth and first moments is the average cluster mass, cavg ) c(1)/c(0). Reversible addition of the monomer molecules Q which have molecular weight xm, to a cluster, C(x), with rate coefficients, kg(x) and kd(x), for growth and dissolution, respectively, is represented by kg(x)
} C(x + xm) C(x) + Q(xm) {\ k (x)
(10)
d
The distribution of the precipitate clusters, c(x, t), is governed by the kinetic equation (population balance)15
∂c(x, t)/∂t ) -kg(x)c(x, t)
∫0 kg(x - x′)c(x - x′, t)q(x′, t)dx′ - kd(x)c(x, t) + ∫x∞ kd(x′)c(x′, t)δ(x - (x′ - xm))dx′ + Iδ(x - x*) (11) The distribution of monomers, q(x, t) ) q(0)(t)δ(x - xm), for a closed system with no input of monomer, except
(15)
Thus, the mass dependences of the growth and dissolution rate coefficients are identical with λ ) 0, 1/3, 2/3 representing various deposition rates. The power is λ ) 1/3 for diffusion-controlled and λ ) 2/3 for surfacecontrolled processes. Heterogeneous nucleation can be simply incorporated by seed particles represented as an initial particle size distribution, c(x, t ) 0). Homogeneous nucleation, however, is a more difficult matter. For spherical clusters the nucleus energy is the sum of interfacial and volume terms
W(r) ) 4πr2σ - (4/3)πr3
(16)
where ) (F/xm)kBT ln S is expressed in terms of mass density F, monomer mass xm, the Boltzmann constant kB, and absolute temperature T, and σ is the interfacial energy between the condensed-phase and the free monomers. The term -kBT ln S is the chemical potential difference between the two phases in terms of supersaturation S. The maximum energy occurs at r* ) 2σ/, or in terms of the spherical-particle mass, x* ) (4/3)πr*3F, thus
∫0∞q(x′, t)dx′ +
x
(14)
and we assume the growth rate increases as a power of mass
Particle Nucleation and Growth
c(n)(t) )
∫0∞kg(x′)c(x′, t)dx′ +
ξ* ) x*/xm ) (ω/ln S)3
(17)
where
ω ) (σ/kBT)(32xm2π/3F2)1/3 Classical nucleation theory accounts for cluster growth by means of the cluster energy W. For a supersaturated
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Figure 1. Effect of the breakage parameter β on the time evolution of (a) concentration CB, (b) concentration of Q (supersaturation) S, (c) number concentration C(0), and (d) average mass Cavg. The parameters are CA0 ) 10, CB0 ) 100; S0 ) 100, κm ) 0.1, ν ) -0.333, R ) 0.5, Jo ) 0.001, and ω ) 5.
(metastable) system, the energy of a cluster of radius r, W(r), reaches a maximum value, W*, at the critical cluster radius, r*, given by eq 17, and equivalent to the Gibbs-Thomson equation
W3* ) (16/3)πσ3/2
(18)
For r < r* (or in terms of mass, x < x*) the energy W increases with size, implying that the cluster is unstable and subject to fluctuations that dissociate most clusters to monomer. For r > r* the energy decreases with size, and the particle can grow or dissolve by diffusive or other reversible kinetic mechanisms. The classical expression for the nucleation rate (mols of nuclei/vol‚ time) is the flux over the maximum energy barrier (at r ) r*)
Inuc ) knuc exp(-W*/kBT)
(19)
with a prefactor knuc that varies with the square of monomer concentration, and thus with the square of the supersaturation, S. We will consider the prefactor as a parameter for the computations described below. Dimensionless quantities12 are defined as
ξ ) x/xm θ ) tγq∞(0)xmλ S ) q(0)/q∞(0) C ) cxm/q∞(0) C(n) ) c(n)/q∞(0)xmn J ) I/γq∞(0)2xmλ (20) where ξ is the number of monomers in a cluster. The dimensionless time θ, cluster distribution C, nucleation rate J, and monomer concentration S are scaled by the
monomer concentration q∞(0) in equilibrium with an uncurved surface. The dimensionless monomer equation is
dS(θ)/dθ ) [-S(θ) + eΩa]C(λ) - Jξ* + κmRCBP(ν)(θ) (21) where
κm ) km gp0(0)(p0avg)ν/(γq∞(0)xmλ)
(22)
The Damkohler parameter κm is the ratio of mass transfer (reaction) rate to precipitate growth rate and incorporates both kinetics and geometry of the fluid element. This reveals how mixing alters the reaction rates by including the effects of turbulence in κm. We can assume km ) D/δ2 for a diffusion-controlled rate over a characteristic distance δ. Increased mixing intensity will decrease δ, so κm depends on the hydrodynamic state, and in particular on the Reynolds number. As in other reactive mixing models, for example, the engulfment model,1,17,18 the Damkohler number can be related to the energy dissipation rate, and hence to power input and geometry of the mixer. Because eq 21 is a moment equation, Ωa is evaluated for the average-sized cluster, Ωa ) ω/(Cavg)1/3. Exact for an infinitely narrow distribution, this approximation has been shown to be reasonable in previous computations for crystal growth and Ostwald ripening.12,15
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Figure 2. Effect of Damkohler parameter κm on the time evolution of (a) concentration CB, (b) concentration of Q (supersaturation) S, (c) number concentration C(0), and (d) average mass Cavg. The other parameters are the same as those in Figure 1.
where the asymptotic result in eq 6 is substituted, and we have defined the breakage parameter
β ) κb/(γq∞(0)xmλ)
(26)
The initial conditions are
CA(θ ) 0) ) CA0 CB(θ ) 0) ) CB0 S(θ ) 0) ) 0 (27) Integrating eqs 24 and 25 yields
CA(θ) ) CA0 + (CB(θ) - CB0)(1 - R)/R Figure 3. Variation of the average mass Cavg with κm at various times. The other parameters are the same as those in Figure 1. The five points are experimental data.19
We can write the dimensionless mass balances in terms of concentrations scaled by the equilibrium solubility of a flat surface, q∞(0), of Q
CA ) cA/q∞(0) CB ) cB/q∞(0) S ) cQ/q∞(0)
and
CB(θ) ) CB0 exp[-κmR(2 - ν)-1Γ(ν + 1)[(1 + βθ)2-ν - 1]/β] (28b)
(23)
The dimensionless equations corresponding to eqs 3-5 are
RdCA/dθ ) (1 - R)dCB/dθ
(28a)
(24)
-dCB/dθ ) κmRCB(θ)P(ν)(θ) ≈ κmRCB(θ)Γ(ν + 1)(1 + βθ)1-ν (25)
Because A is the limiting reactant, we require that initial moles of B are greater than or equal to the initial moles of A, thus CB0 g R/(1 - R)CA0. The mass balance on reaction product and precipitate mass is
d(S + C(1))/dθ ) κmRCB(θ)P(ν)(θ)
(29)
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Figure 4. Effect of ν on the time evolution of (a) concentration CB, (b) concentration of Q (supersaturation) S, (c) number concentration C(0), and (d) average mass Cavg. The other parameters are the same as those in Figure 1.
which can be integrated to
S(θ) ) C0(1) - C(1)(θ) + κmRΓ(ν + 1)
∫0θCB(θ)(1 + βθ)1-ν dθ
(30)
Upon substitution of eq 28b, the integral over θ can be computed analytically, thus
κm R Γ(ν + 1)
∫0θ CB(θ)(1 + β θ)1-ν dθ )
CB0 [exp{-κm R (2 - ν)-1 Γ(ν + 1)/β} exp{-κm R (2 - ν)-1 Γ(ν + 1) (1 + β θ)2-ν/β}] (31) The resulting expression for S(θ) can then be substituted into eq 13 made into a dimensionless equation
∂C(ξ, θ)/∂θ ) S(θ)[-ξλ C(ξ, θ) + (ξ - 1)λC(ξ - 1, θ)] ξλ exp(ωξ-1/3)C(ξ, θ) + (ξ + 1)λ exp(ω(ξ + 1)-1/3)C(ξ + 1, θ) + Jδ(ξ - ξ*) (32) This nonlinear difference-differential equation for C(ξ, θ) requires a numerical solution.12 Ripening changes the nucleation J into denucleation, which cannot be expressed analytically and thus also entails numerical computation.12,15 Particles smaller than the critical nucleus size at any value of S vanish instantaneously, giving up their mass to solution and subsequent deposition on larger particles.
When the process is terminated before Ostwald ripening can begin and the growth rate is independent of particle mass (λ ) 0), a moment solution for the particulates is practicable and is applied in the current work. The nonlinear ordinary differential equations for n ) 0 and 1 moments are as follows:
dC(0)/dθ ) J(θ)
(33)
dC(1)/dθ ) [S(θ) - exp{ω/(Cavg)1/3}]C(0) + J(θ)ξ*(θ) (34) where Cavg ) C(1)/C(0) is a function of θ. Although this method does not reveal the details of the entire particle size distribution, it does provide crucial information in the form of mass and number of particles and thus their average size. Mixing affects both nucleation and growth through the time dependence of the supersaturation, S(θ). Equation 30 combined with eq 31 shows how S(θ) depends on the mixing parameters, κm, ν, and β, and on the initial conditions, R, CA0, and CB0. Results and Discussion As expected for a process as complex as mixing with precipitation, the number of parameters must be larger than for simpler processes. An advantage of our model, however, is that the parameters are physically meaningful and the differential equations (eqs 28a,b, 30, 33, and 34) are readily solved. The generic temporal behavior of supersaturation, S(θ), and Gibbs-Thomson factor, exp [ω/(Cavg)1/3] in eq 34, is such that both
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Figure 5. Effect of dispersed volume fraction R on the time evolution of (a) concentration CB, (b) concentration of Q (supersaturation) S, (c) number concentration C(0), and (d) average mass Cavg. The other parameters are the same as those in Figure 1.
approach unity at long times with their difference being the driving force. The effects of entropy and energy driving forces in nucleation are governed by the prefactor, JoS2, and the scaled exponential, exp[-ω3/ 2(ln S)2], respectively, which are in turn related to Jo, ω, and S. The parameters ω and Jo influence the nucleation and could also be varied. We are primarily concerned with how the four factors, κm, ν, R, and β, affect the time evolution of the moments of the cluster size distribution and the supersaturation. Figures 1-4 show how these parameters influence the time evolution of the concentration of B, concentration of Q, particle number concentration, and average size. The initial conditions CB0 and CA0 and the dispersed volume fraction R will influence the results. When the limiting reactant A is depleted, the concentration of the coreactant B must become constant and equal to CB0 CA0R/(1 - R). If initially the moles of A equal the moles of B, or RCA0 ) (1 - R)CB0, then A and B are depleted at the same time. Figure 1a shows that increasing the dispersed-fluid breakage parameter β by an order of magnitude speeds the decrease of the concentration of B. The initial conditions cause CB to decline from 100 to 90, at which time CA is zero. Thus, increased turbulence enhances reaction rate, causing a larger supersaturation (Figure 1b) and increased nucleation rate. The concentration of reaction product Q (represented by the supersaturation S) rises to a maximum before the precipitation process causes its decline. Figure 1c,d shows that the particle number concentration and average size attain a constant value that depends on β. As β increases, the
average size is reduced because the number concentration increases due to increased nucleation. Decreased turbulence thus leads to fewer, larger particles. The influence of the Damkohler number κm, representing the ratio of mass transfer to precipitate growth rates, is more complex (Figures 2 and 3). Smaller κm slows the decrease of CB (Figure 2a) and increases the time when product Q, or supersaturation S, reaches its maximum (Figure 2b). Figure 2c,d shows that the particle number concentration and average size attain a constant value dependent on κm. The evolution of particle number concentration and average size is not monotonic but reaches a maximum value dependent on κm (Figure 3). If the Damkohler number κm is proportional to Reynolds number and the impeller/stirring speed, this observation is consistent with experimental data19 for the variation of the mean particle size with power input. The data are shown to be in agreement with eqs 33 and 34 when the dimensionless quantities are appropriately scaled. The specific power input19 is multiplied by 10 and the mean particle size is multiplied by 0.0007. Then only the dimensionless time is free for fitting, and we find that θ ) 0.5 fits the data quite well (Figure 3). Plots (Figure 4) for the effect of ν indicate that the eddy geometry influences the reaction and subsequent precipitation. Reactant concentration CB, supersaturation S, and average precipitate mass Cavg (Figure 4a,b,d, respectively) are reduced as the geometry passes from sphere (ν ) -1/3) to cylinder (ν ) -2/3) to slab (ν ) -1). A slight maximum near ν ) -2/3 (Figure 4c) is found for the number of precipitated particles, C(0).
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area for transfer of the dispersed reactant into the bulk is the basic phenomena represented by the Damkohler number κm and the dispersed-fluid breakage parameter β. In principle these parameters can be measured by dispersing an inert tracer into the vessel in the same turbulent state (same Reynolds number). A similar strategy was proposed8 to perform nonreactive dispersion (blending) experiments and to use the results to predict reactive experiments. Literature Cited
Figure 6. Variation of the average mass Cavg with R/(1 - R) at various times. The other parameters are the same as those in Figure 1.
Figure 5 shows the monotonic effects of dispersed volume fraction R. Decreasing R slows the decrease of the concentration of reactant B as shown in Figure 5a, where the dotted line indicates when A is depleted. Figure 5b shows that increasing R lowers the supersaturation and shifts the maximum S to later times. Decreasing R decreases the number of particles (Figure 5c) and increases the average particle mass (Figure 5d). Figure 6 plots the average particle size versus the ratio of dispersed to bulk volume, R/(1 - R). In agreement with experimental observations,1 the aged particles generally increase with increased ratio.1 The results of the model computations are in general agreement with observations of the effects of mixing on precipitation.1 Increased mixing (larger values of β and κm) ultimately decreases particle size by increasing the supersaturation and thus the nucleation rate. The classical homogeneous nucleation theory applied here ensures that the nucleation rate is quite sensitive to supersaturation. This can also explain the influence of the feed point for dispersed A; for example, if A is injected near the impeller, turbulence is increased and particles are smaller. Conclusion It may seem surprising that this easily constructed and readily implemented model of turbulent mixing with precipitation describes the complex phenomena. The model apparently captures the principal behavior, while omitting nonessential detailed dynamics of turbulent fluctuations. The key to good liquid mixing is to disperse one liquid into another, thus increasing the interfacial contact between the dispersed and bulk liquids. We have argued that if diffusion between the two liquids is slow, then the Damkohler number will be small, less than unity. The dispersed fluid contact area is then more important than precise turbulent statistics, because the dispersion-diffusion process controls the kinetics. No matter what model is used to represent turbulent mixing, some parameters must be estimated through experimentation. In the current work we have proposed that dispersive mixing of a tracer or reactant into a bulk fluid is a result of the fragmentation-coalescence of fluid elements causing a cascade of increasingly smaller fluid elements in turbulent mixing. The increase in interfacial
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Received for review August 26, 2004 Revised manuscript received November 18, 2004 Accepted November 24, 2004 IE049217I
