Heterogeneous Modeling of Gas Absorption in Emulsions - American

specific rate of interphase transport. The behavior of the mass-transfer rate with respect to the various parameters, such as microdispersed-phase hol...
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Ind. Eng. Chem. Res. 1999, 38, 2460-2468

Heterogeneous Modeling of Gas Absorption in Emulsions Anurag Mehra† Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Bombay 400 076, India

A heterogeneous, multilayer, mass-transfer model is proposed for explaining the effect of dispersed microparticles/microdroplets, which act as solubilizing and/or reactive agents, in enhancing the specific rate of interphase transport. The behavior of the mass-transfer rate with respect to the various parameters, such as microdispersed-phase holdup, partition coefficient of the solute between the microparticles and continuous phase, microparticle size, various rate constants, etc., has been modeled. Specific issues, such as the effect of multiple microparticle layers near the interphase boundary, the location of the first layer of microparticles closest to this boundary, and the incorporation of rate-retarding effects arising out of high apparent viscosities of the overall dispersion, have been addressed, for the first time. It is shown that the heterogeneous approach should be preferred when the dispersed-phase sizes are on the order of (but still less than) the diffusional lengths of the rate-limiting diffusant. The proposed model has been validated by analyzing experimental data from the literature, and it has been shown that it works well for the cases of high microphase holdups or large microparticle sizes. Introduction The rates of interphase mass transfer in multiphase systems (e.g., absorption, extraction, and solid dissolution) are known to be enhanced in the presence of fine particles or droplets which can “interact” with the diffusional gradients of the rate-limiting solute. The enhancement in the mass-transfer flux occurs because of the physical or chemical “consumption” of the diffusing species by these fine constituents, which are also known as microphases because these usually lie in the micron size range. The overall process of rate enhancement is therefore often termed microphase catalysis. Figure 1 shows a schematic picture at the gas-liquid interface across which interphase mass transfer of the gaseous solute A occurs. The receiving phase is a liquid medium into which A dissolves and diffuses; it may be accompanied by chemical reaction with another liquidphase reagent B. As the diffusion proceeds, the dissolved solute is taken up by the microphase particles/droplets, because of plain physical adsorption/solubilization and/ or surface/internal reaction. This consumption of the rate-controlling diffusant by the microphase results in steeper concentration gradients near the gas-liquid interface, leading to an enhanced specific rate of absorption of A. The solute taken up by the microparticles may ultimately be delivered to the bulk phase, which is lean in A, or may be transformed chemically within/on the particle. The basic conditions, for the microphase action to be effective in enhancing the interphase flux, are that the diameter of the microconstituents be much less than the thickness of the solute penetration depth, at the gas-liquid interface, and that the microparticles show a marked physical or chemical affinity for the diffusant A. (The gas phase may well be replaced by a solid or even by a liquid phase immiscible with the receiving liquid phase. Also, throughout this paper, “droplet” and “particle” have been used interchangeably.) † Tel.: +91-22-576 7217. Fax: +91-22-579 6895. E-mail: [email protected].

Figure 1. Schematic drawing of gas-liquid mass transfer in the presence of a microdispersed phase (top) and its heterogeneous idealization (bottom).

A large number of experimental and theoretical studies are available in the literature which deal with a variety of microphases, such as solid catalysts, adsorbents and reactants in the form of fine particles, small droplets of a dispersed/emulsified liquid (immiscible with the original continuous liquid phase) as physical solubilizing agents, micelles and microemulsion constituents as solubilizing and reaction sites, and even fine gas bubbles. The nature of the multiphase systems studied includes gas absorption, liquid-liquid extraction, and solid dissolution, with and without superposed

10.1021/ie980653c CCC: $18.00 © 1999 American Chemical Society Published on Web 05/11/1999

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2461

chemical reaction. Typical examples of industrial relevance are the absorption of olefinic gases into emulsions of organic solvents in aqueous solutions of sulfuric acid, the use of fine catalyst/adsorbent particles to enhance the absorption of hydrocarbon gases, and the utilization of dispersed perfluorocarbon phases as oxygen vectors in aerobic fermentations as well as artificial blood. Details of references to these and other studies may be found elsewhere.1-4 The modeling of the rate behavior of these systems has largely been attempted by a pseudohomogeneous approach. In this framework, an additional reaction or accumulation type of term is incorporated into the diffusion equation which is still written for a singlephase continuum, as is done in the extended models of mass transfer with chemical reaction. A treatment of this type “pretends” that the original continuous phase, in which diffusion occurs, has been replaced by another homogeneous phase having a different effective reactivity or solution capacity. Because the system is not really homogeneous, such a diffusion medium is termed as pseudohomogeneous, in which the effects of the microconstituents are essentially smeared out uniformly throughout the system volume. The theories proposed by Mehra,5 and more recently by Nagy and Moser,4 are typical representatives of this class of models. Pseudohomogeneous models are subject to several constraints, the most notable being that these are valid only for low holdups of the microdispersed phases. At high holdups, internal diffusion through the dispersed phase cannot be neglected. Moreover, the local gradients, of the species being consumed by the microphase, around an individual particle/droplet tend to interfere with those of its nearest neighbors. The other significant constraint on these models is related to the notion of “fine” particles. Typically, a microparticle may be considered to be fine if its characteristic dimension (say, diameter) is much smaller than the length scale of the diffusional gradients that it interacts with. In other words, the microparticle should be small enough to “fit into” the gradient. As the particle size increases, and especially in the range where it is on the order of the diffusional lengths, the discontinuous nature of the fluid-microphase interface cannot be ignored. The microconstituents can no longer be treated as “negligibly small points” on the diffusion gradient because each particle interacts with a large part of the concentration profile of the diffusant. In these limits, of high holdup of the microdispersed phase or when the particles are “large”, the pseudohomogeneous models therefore break down. Some of the more recent studies in the literature have focused on heterogeneous approaches to modeling of mass transfer into a microheterogeneous media. Junker et al.6 examined the role of fine droplets of perfluorocarbons in enhancing the rates of oxygen transfer to aerobic biochemical systems. They proposed an unsteadystate, penetration theory model for the case when the perfluorocarbon droplet size is comparable to the diffusion film thickness, and the droplet may even partially jut out of the interfacial zone into the bulk liquid. In this scenario, only the first layer of droplets is active because all of the subsequent layers anyway reside outside the gas-liquid diffusion zone, in the bulk liquid. Karve and Juvekar7 developed a cell model to predict the enhancement of gas absorption rates into slurries

containing fine catalyst particles. This model accounts for interparticle interactions and essentially views the interfacial region to consist of small cylindrical cells, of equal size, each enclosing a single particle. A computation-intensive, finite element technique was used for solving the model equations in view of the curved boundaries that were present. In the context of reactant slurries made up of fine particles, Yagi and Hikita8 have reported a two-dimensional model for gas absorption into such slurries. These authors have criticized the use of a mass-transfer coefficient around the reactant microparticles and have suggested that, instead, a parameter representing the average spacing between particles be used. Yet another approach has been suggested by Van Ede et al.,3 who have proposed the “film varying interfacial holdup” (FVIH) model. Here, the investigators first show that, for a large particle touching the gas-liquid interface, the cumulative holdup profile of the dispersed phase in the diffusion film gradually increases from the interface and finally becomes equal to that in the bulk. Even though the argument begins from large particles, they ultimately propose that it might be reasonable to assume this to be true even for small particles. Vinke et al.9 have reported a film theory based, heterogeneous model which attempts to quantify the effect of fine catalyst/adsorbent particles that have a tendency to “adhere” on the gas-liquid interface, on the absorption rates. Such particles show a much greater concentration at the interface than in the bulk which may actually be quite lean in terms of particle loading. Here, the interaction of the first layer of particles, “covering” the interface, with the absorbed diffusant is sought to be modeled. Nagy10 has recently proposed a more generalized, film-penetration model, where only a single layer of particles is present near the gas-liquid interface. This model allows for the particles to be located at an arbitrary distance from the gas-liquid interface, and only the first layer is assumed to play a role in enhancing the interphase mass-transfer flux; the contributions of successive layers of particles/droplets is not accounted for. It was therefore thought desirable to develop a multilayer, heterogeneous model for microphase catalysis which endeavors to overcome the basic constraints placed on the pseudohomogeneous models and also makes it possible to examine the various issues mentioned above. The proposed model explicitly allows for diffusion, with simultaneous solubilization and/or reaction, in the microdispersed phase. The scenarios arising out of high holdup of the microphase and for large particles (though not large enough to reside partly in the bulk phase) are specifically examined. The effect of increased overall viscosity of the dispersion (continuous plus microphase) in lowering the hydrodynamic surface renewal frequency is incorporated. Various other issues, such as the location of the first layer of microparticles and its dominant influence and the incremental contribution of subsequent layers, are also addressed. The model is used for the quantitative prediction of rate enhancements observed experimentally by various workers, for systems pertaining to physically solubilizing (nonreactive) microphases. Basic Model Development The heterogeneous model developed here is for the case of physically solubilizing microsinks incorporating

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within these the possibility of a pseudo-first-order reaction. A similar reaction may also take place in the continuous liquid phase between the dissolved gaseous solute A and a dissolved liquid-phase reagent B. Only microphases that do not show any preference for adhering to the gas-liquid interface are considered here, so that all of the microparticles may be taken to be uniformly dispersed throughout the volume of the continuous liquid. A heterogeneous idealization of the physical scenario is also shown in Figure 1. A microconstituent is taken to be represented by a cube whose volume is that of an actual particle. All of the microparticles are assumed to be of the same size. The microparticle cubes are located in the liquid phase, in the interfacial penetration zone (where the diffusional gradients of A exist), in the form of a regular “lattice”. The entire lattice is taken to be located at some distance from the gas-liquid interface, which can vary from zero to some upper limit; this is discussed later. One-dimensional diffusion into the depth of the microheterogeneous system, comprising the particle lattice immersed in the continuous phase, perpendicular to the plane of the gas-liquid interface, is assumed; lateral diffusion, in the other two dimensions parallel to the interface, is neglected. This description amounts to proposing that the solute moves in straight, one-dimensional channels. When it encounters a microparticle in its path, it begins diffusing through it and then emerges from the opposite end of the microparticle. Such a geometric arrangement has also been assumed in some previous studies.6,10 The idealization proposed above thus allows for the concentration gradients around adjacent particles to “interfere” with each other. The liquid-phase concentration profile of the diffusant starts from the releasing edge of one particle and continues up to the receiving edge of the one in the next layer; it is thus a link between two neighboring particles. In contrast, in the pseudohomogeneous picture, each microparticle only picks up the solute through its entire surface and never releases anything into the diffusion zone; the residual solute within the continuous phase then diffuses forward to be picked up by particles at locations more remote from the gas-liquid interface.5 Consider the lattice arrangement shown in Figure 1. Each microcube may be taken to be present within a cubic liquid cell whose length is given by

δc )

(dp3/l0)1/3

(1)

which is also the center-to-center distance between two microparticles located in successive layers. Here, dp is the cube length and l0 is the volumetric fractional holdup of the microphase material. The thickness of the liquid layer sandwiched between two successive microphase layers, i.e., the edge-to-edge distance between two adjacent microparticles, is given by

δl ) δc - dp

(2)

The thickness of the liquid between the gas-liquid interface and the nearest edge of a microparticle located in the first microphase layer is denoted by δ0. Now, if N particles are located, spaced δl apart, within the depth of the penetration zone of the solute A, then the liquid gap between the edge of the Nth microparticle facing

the bulk and the boundary of the penetration zone is given by

δN ) δp - δ0 - dpN - δl(N - 1)

(3)

where δp is the maximum penetration depth of A in the absence of the microphase and with no reaction anywhere. From the surface renewal model of mass transfer, as proposed by Danckwerts,11 this may be written as

x

DA,c DA,c )4 ) 4δ S kL

δp ) 4

(4)

where S is the surface renewal frequency of the liquid at the gas-liquid interface, DA,c is the diffusivity of A in the continuous liquid phase, kL is the physical, gasliquid (liquid-side) mass-transfer coefficient (without the microphase), and δ is a characteristic diffusion length. N is also the number of microparticle layers that are present in the interfacial, diffusion zone. Danckwerts’ surface renewal model essentially states that mass transfer of A from the gas into the liquid phase proceeds by penetration/diffusion of the solute into liquid elements located at the gas-liquid interface, which circulate back and forth between the bulk liquid and the interfacial zone. The fraction of liquid elements having a residence time between t and t + dt at the interface is given by S exp(-St) dt, where t can range from 0 to ∞. This implies that

kL ) xDA,cS

(5)

The specific form of Danckwerts’ age distribution enables very convenient use of Laplace transforms to compute surface-averaged properties of the system, such as absorption rates, without having to invert back to the time domain. This becomes possible because of the similarity between the definition of the Laplace transform and the surface age distribution function used. The pictures depicted in Figure 1 are for a typical, interfacial penetration element. The unsteady-state species balance for A in the continuous liquid is given by

DA,c

∂2CA,c ∂x

2

)

∂CA,c + k1,cCA,c ∂t

(6)

whereas in a microparticle it is written as

DA,d

∂2CA,d 2

∂x

)

∂CA,d + k1,dCA,d ∂t

(7)

The initial condition for the entire diffusion zone may be stated as

t ) 0 4δ > x > 0

CA,c ) CbA ) 0

(8)

CA,d ) CbAm ) 0

(9)

The subscripts c and d denote variables pertaining to the continuous and dispersed phases, respectively; m is the distribution coefficient of the A partitioning

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between the microphase and the continuous phase. The relevant boundary conditions are given by

with the conditions

across the first liquid layer of thickness, δ0 t>0

x)0

CA,c ) C/A

(10)

t>0

x ) δ0

CA,c ) C1,i A,c

(11)

t>0

CA,c ) Cj,b A,c

x ) δ0 + dpj + δl(j - 1) x ) δ0 + dpj + δlj

CA,c )

Cj+1,i A,c

(12)

(21)

x ) xb

C ) Cb

(22)

sinh[xk/D(xb - x)] C - Cb ) Ci - Cb sinh[xk/D(x - x )] b

-D

() dC dx

)

x)xi

(

Ci xkD (xb - xi) tanh[xk/D(xb - xi)] Cb

x ) δ0 + dpN + δl(N - 1) CN,b A,c

(14) -D

x ) δ0 + dpN + δl(N - 1) + δN CA,c )

CbA

) 0 (15)

() dC dx

)

x)xb

(

Ci xkD (xb - xi) sinh[xk/D(xb - xi)] Cb

x ) δ0 + dp(j - 1) + δl(j - 1) CA,d ) Cj,i A,cm (16)

t>0

x ) δ0 + dpj + δl(j - 1)

CA,d ) Cj,b A,cm (17)

Here, the above equations also state the condition of partitioning equilibrium that prevails between the liquid and microphase concentration of A at the microparticle-continuous liquid interface. The numbering scheme labels the concentrations at the two opposite j,b faces of the jth microparticle as Cj,i A,c and CA,c, respecj,i j,b tively. The concentrations CA,c and CA,c for j ) 1 to N not only are functions of x and t but are also the variables to be solved for. Thus, there are 2N variables, and values of these may be obtained by solving the equations that result from applying the condition of flux continuity at the microparticle-liquid interfaces of every microparticle. A total of N microparticles, each with i and b faces, will therefore yield 2N equations. The flux continuity conditions, for the jth particle, may be written as

t>0

x ) δ0 + dp(j - 1) + δl(j - 1) ∂CA,c ∂CA,d ) -DA,d (18) -DA,c ∂x ∂x

t>0

x ) δ0 + dpj + δl(j - 1) ∂CA,d ∂CA,c -DA,d ) -DA,c (19) ∂x ∂x

It can be easily shown that a differential equation of the type 2

D

dC ) kC dx2

)

(25)

tanh[xk/D(xb - xi)]

across dp for the jth microparticle t>0

(24)

as well as

CA,c ) t>0

)

sinh[xk/D(xb - xi)]

across the last liquid layer of thickness, δN t>0

(23)

i

and also gives

(13)

where, i and b in the superscripts denote concentrations at the microparticle surfaces facing the interface and bulk, respectively.

C ) Ci

results in

across δl, between the jth and (j + 1)th particles (j ) 1 to N - 1) t>0

x ) xi

(20)

Transforming eqs 6-19 with respect to t, into the Laplace domain, and applying the solutions indicated by eqs 24 and 25 to the (transformed) flux continuity equations (18) and (19), after considerable algebraic manipulation, results in the following tridiagonal system of linear algebraic equations

[

Qm0 Qd 0 0 0 0 . 0 0 0 0

Qd Qm Qc 0 0 0 . 0 0 0 0

0 Qc Qm Qd 0 0 . 0 0 0 0

0 0 Qd Qm Qc 0 . 0 0 0 0

0 0 0 Qc Qm Qd . 0 0 0 0

0 0 0 0 Qd Qm . 0 0 0 0

. . . . . . . . . . .

0 0 0 0 0 0 . Qm Qd 0 0

0 0 0 0 0 0 . Qd Qm Qc 0

0 0 0 0 0 0 . 0 Qc Qm Qd

]

0 0 0 0 0 × 0 . 0 0 Qd QmN

[ ][ ] Q0C/A C h 1,i A,c S C h 1,b A,c 0 2,i C h A,c 0 C h 2,b A,c 0 C h 3,i 0 A,c 3,b ) 0 C h A,c . . N-1,i 0 C h A,c 0 C h N-1,b A,c 0 C h N,i A,c QNCbA C h N,b A,c S

(26)

2464 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999

where

Qc )

Qd )

xDA,c(S + k1,c) sinh(x(S + k1,c)/DA,cδl) mxDA,d(S + k1,d) sinh(x(S + k1,d)/DA,ddp)

-Qm )

xDA,c(S + k1,c) tanh(x(S + k1,c)/DA,cδl)

+

mxDA,d(S + k1,d) tanh(x(S + k1,d)/DA,ddp)

Q0 )

xDA,c(S + k1,c) sinh(x(S + k1,c)/DA,cδ0)

QN )

xDA,c(S + k1,c) sinh(x(S + k1,c)/DA,cδN)

-Qm0 ) mxDA,d(S + k1,d) xDA,c(S + k1,c) + tanh(x(S + k1,c)/DA,cδ0) tanh(x(S + k1,d)/DA,ddp) -QmN ) mxDA,d(S + k1,d) xDA,c(S + k1,c) + tanh(x(S + k1,c)/DA,cδN) tanh(x(S + k1,d)/DA,ddp) where the overlined concentrations are the transformed concentrations of A at the various microparticlecontinuous phase interfaces in the Laplace domain and S is the Laplace domain variable. The surface-average value of the specific rate of absorption of A, in the framework of Danckwerts’ model, is given by11

Rdef A )

( )

∫0∞-DA,c

∂CA,c ∂x

Figure 2. Enhancement factor, Eu, versus thickness of liquid layer between the gas-liquid interface and the first layer of microparticles, δ0, at varying microphase holdups, l0. Inset: Eu versus microparticle size, dp, at varying l0. Data used: kL ) 1 × 10-5 m/s, DA,c ) DA,d ) 1 × 10-9 m2/s, k1,c ) k1,d ) 0 s-1, and m ) 100.

to a limiting value which may be computed by applying the definition stated in eq 24 to the first particle layer (i )1). Similarly, the average specific rate of absorption into a clear, microparticle-free channel is simply given by / Rba A ) CAxDA,c(S + k1,c)

so that the total specific rate of absorption into the microheterogeneous media (continuous liquid plus microphase) becomes

RuA )

()

( ( ))

dp 2 m,u dp RA + 1 δc δc

2

2/3 m,u Rba A ) l0 RA +

S exp(-St) dt ) -SDA,c

( )

x)0

∂CA,c ∂x

x)0

(27)

Therefore, multiplication of the instantaneous flux in the Laplace domain, as obtained from the solution of the tridiagonal system of equations given above, by S yields the desired average values. When this definition is applied to obtain the surfaceaveraged specific rate of absorption of A into the channel made up of alternating liquid and microparticle slabs, it results in

(

Rm,u A ) xDA,c(S + k1,c)

C/A

tanh(x(S + k1,c)/DA,cδ0) 〈C1,i A,c〉

-

)

sinh(x(S + k1,c)/DA,cδ0)

(28)

This is essentially the average mass-transfer flux of A into the continuous liquid layer, of thickness δ0, that sits between the gas-liquid interface and the first layer of microparticles. When δ0 f 0, the value of Rm,u A tends

(29)

(1 - l02/3)Rba A (30)

and the enhancement factor can be defined as the rate in the presence of the microphase to that in its absence, i.e.,

Eu ) RuA/Rba A

(31)

where superscript m is for the channel in which the microphase is present, superscript u denotes at a fixed value of δ0, and Rba A is the flux into a channel made up purely of the continuous liquid phase and is also the base rate in the absence of the microphase. The above absorption rates are for a fixed value of δ0 and therefore need to be appropriately averaged over all possible values of this parameter. This exercise is done later. Behavior of the Basic Model We now examine the absorption rate behavior of the system as a function of the thickness of the initial liquid layer, in the absence of any chemical reaction. Figure 2 shows that Eu declines as δ0 increases. The enhancements increase with dp, for a fixed value of δ0, and this is shown in the inset to the figure. An increase in δ0

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2465

reason for this is that the number of layers contributing to the enhancement effect is lower for greater m. Thus, even though physically the same amount of microphase is present, the diffusant penetrates up to a lesser number of layers in the case of larger m. Initial Spacing-Averaged, Viscosity-Corrected Model

Figure 3. Enhancement achieved, Eu,n, versus the number of microparticle layers taken to be present, n, at various microphase holdups, l0, and rate constants, k1,c. Inset: Eu,n/m versus n, at different partition coefficients, m. Data used: kL ) 1 × 10-5 m/s, DA,c ) DA,d ) 1 × 10-9 m2/s, k1,d ) 0 s-1, dp ) 1 µm, and δ0 ) 0 µm.

increases the resistance to mass transfer, because the liquid phase shows a much lower solubility for A compared to the microphase. The trend, with respect to dp, can be explained on the basis that the solute now gets to diffuse in the high-solubility microparticle for a longer distance in larger microparticles; the diffusant sees more of the microphase, with increasing dp, instead of the continuous liquid. The extent of enhancement achieved, Eu,n, if only the initial n layers contribute, is shown in Figure 3. At low l0 (0.1), fewer layers contribute to the rate enhancement effect as compared to high l0 (0.6). Because the microphase layers are more closely packed in the latter case, more of these have access to the diffusant than in the former scenario where the layers are spaced further apart. Thus, for the conditions used in this figure, the enhancements become constant at about 15 microphase layers for l0 ) 0.1 but continue to be influenced by more than 50 layers for l0 ) 0.6. Similarly, in the presence of a continuous phase reaction, the number of layers that effectively participate in solubilizing A (and enhancing the absorption rate) is less than that in the absence of this reaction, with other conditions being the same. This happens because of the consumption of A by reaction and consequently less solute reaching the microparticles more remote from the gas-liquid interface. The inset shows Eu,n, scaled with respect to m, versus the nth layer. The number of participating layers is least for the highest m value. The microparticles closest to the gas-liquid interface solubilize the largest fraction of A and leave very little for the subsequent layers. Therefore, only about 10 layers are active for m ) 1000, but the diffusing solute remains available to more than 50 layers for m ) 10. In general, layers other than the first may also contribute quite substantially to solute uptake and absorption rate enhancement. The magnitude of Eu,n/m with respect to m indicates that the enhancement is not linearly proportional to m. Whereas the absolute values of the enhancements are indeed higher for greater m, the scaled values are lower; i.e., the efficiency of utilization of the solubilizing power of the microphase becomes poorer with increasing m. The most evident

The model, up to now, merely assumes the microphase lattice to be located at a fixed distance δ0 from the gas-liquid interface. This distance itself may be expected to be a function of, at least, the particle size and the holdup of the microphase. We now propose here the limits between which δ0 may vary. One limit is given by the scenario that the first microparticle layer commences immediately from the gas-liquid interface (δ0 ) 0 ). The other limit may be taken to be on the order of δl/2; the physical meaning of this limit is that if δ0 ) δ0max is set exactly equal to δl/2, it implies that the liquid cell containing a first layer particle starts from the gasliquid interface. For microparticles that show no preference for adhering to the interface, the probability of δ0 having any value within the range 0 to δ0max is equal, so that the specific rate of absorption, averaged over all possible (equally probable) lattice positions, is given by

RA )

∫0δ

0

max

RuA(δ0) dδ0/(δ0max)

(32)

and the corresponding enhancement factor becomes

E ) RA/Rba A

(33)

When the microphase volumetric holdup exceeds a few percent, the viscosity of the suspension/emulsion will increase as compared to the plain continuous phase, and for very high concentrations of the microphase material, the increase in the apparent viscosity may be drastic. This implies that the surface renewal rates of the fluid elements, composed of the continuous plus microphase, will be lower than that computed using the physical mass-transfer coefficient measured for the continuous phase alone. Therefore, the surface renewal frequency, based on the continuous phase alone, needs to be corrected for the influence that the microphase exerts via an increase in the apparent viscosity of the system. In conventional mass-transfer theory it is common to use the Stokes-Einstein relation12

DAµ/T ) constant

(34)

in order to deduce the effect of viscosity on the diffusion coefficient in a homogeneous medium, so that, under isothermal conditions, when the viscosity is increased from µ1 to µ2, the diffusivity reduces from DA,1 to DA,2, i.e.

DA,1µ1 ) DA,2µ2

(35)

Consequently, using eq 5, the mass-transfer coefficients are given by

kL,2 ) kL,1

x

µ1 µ2

(36)

for a given surface renewal frequency. The above arguments may be adapted to a microheterogeneous

2466 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999

medium by applying these to the renewal frequency, S, instead of the diffusion coefficient, DA. Because we are not using the concept of an effective diffusion coefficient in our model, given its heterogeneous nature, the reduction in the mass-transfer coefficient upon an increase in the suspension viscosity may be taken to be produced by a lowering of the surface renewal frequency (instead of a decreased value of the diffusion coefficient), so that

Sdispµdisp ) Sµc

(37)

which is consistent with

kL,disp ) kL

x

µc µdisp

(38)

The above formulation thus represents a way of estimating the extent to which the eddy circulation rate, between the gas-liquid interface and the bulk, is “slowed down” because every eddy has to “rush past” a crowd of fine particles. Therefore, if the viscosity ratio is known, kL,disp or Sdisp may be computed and used in eqs 26-28. Because the viscosity ratio may not be readily available along with the experimental data on the mass-transfer rates into such systems, and this is indeed true of the data reported in the prior literature, it is desirable to locate suitable generic relations for the estimation of viscosities in suspensions/emulsions. Such relations will also provide an a priori predictive power to the theory. The literature on the estimation of the apparent viscosities of suspensions/emulsions is vast, and a variety of approaches have been applied to this endeavor. Usually, the viscosity of such systems has been strongly correlated with the fractional volumetric holdup of the microdispersed phase and also with the individual viscosities of the continuous and dispersed phases. It is also known that the overall viscosity depends upon the mean size as well as the microparticle size distribution in addition to other factors such as the shape, interparticle interactions, etc.13 A critical review of this literature is beyond the scope of this paper. For the purpose of demonstration and use in the subsequent analysis of experimentally reported rate enhancements, the following semiempirical equation was chosen14

(

2 µd + µc µdisp 5 (l + l05/3 + l011/3) ) exp 2.5 µc (µd + µc) 0

(

)

)

(39)

This relation, which is essentially a modified version of Taylor’s equation, has been proposed for oil-in-water systems. It has the advantage of having a sound conceptual basis and has also been validated for the dispersed phase occupying up to about 40-50% of the system volume. As is evident, this equation includes the effect of internal circulation within the dispersed microdroplets and is therefore of direct applicability to emulsions where relatively large, deformable liquid droplets are present. Equation 39 may now be used to compute values of Sdisp or kL,disp from eq 37 or eq 38, respectively. The modification of the enhancement factor values arising out of the use of eq 39 has been labeled as the deformable droplet correction, in the following text, and is denoted by Ed. This represents the final,

Figure 4. Comparison of model predictions with experimental data. Enhancement factors: experimental, Eve, Eh, Ed (δ0max ) δl/ 2), and Ed (δ0max ) δl/4), as a function of microphase holdup, l0. Inset: E, Ed, and Er versus l0, for δ0max ) δl/2. Data used: kL ) 8 × 10-5 m/s, DA,c ) 2 × 10-9 m2/s, DA,d ) 3.6 × 10-9 m2/s, k1,c ) k1,d ) 0 s-1, m ) 18, dp,s ) 22 µm, dp ) 17.7 µm, µc ) 8.5 × 10-4 kg/ m‚s, and µd ) 5 × 10-4 kg/m‚s. Experimental data, on the absorption of oxygen into aqueous dispersions of octene, and parameter values taken from Van Ede et al.3

theoretical value which may be compared with experimental measurements. Analysis of Experimental Data and Discussion Experimental rate enhancement data pertaining only to gas-liquid systems, and for which operating as well as physicochemical parameter values have been given, have been analyzed in this study. The experimental enhancement factors for the absorption of oxygen in aqueous dispersions of octene are shown in Figure 4 and have been taken from Van Ede et al.3 The various theoretical enhancement factors are also shown here. The Ed ( δ0max ) δl/2) values are reasonably close to the experimental points though consistently slightly lower, whereas the Ed predictions for the case δ0max ) δl/4 seem to provide an even better match with the experimental data. The theoretical plot reported by Van Ede et al., from their own model, also agrees with the experiments but is essentially linear (marked Eve) and does not capture the “levelling off” trend in the experiments that is visible (with the exception of the last data point at l0 ) 0.5). The flattening trend was noted by these investigators themselves, and they suggested that a dispersion viscosity related correction to the enhancements may be needed, in order to capture the leveling off. Indeed, it is important to provide the viscosity corrections to the E values, which become much larger than the experimental values at increasing values of l0. The difference between E and Ed is shown in the inset to the figure. As the holdup increases, the E values change very steeply for small increases in l0 and the extent of this steepness continues to increase as l0 is raised further. In contrast, the pseudohomogeneous model suggests that Eh ∝ l01/2.5,15 The Eh values are much lower than the experimental values. Evidently, for such large droplets used in these experiments (dp,s ) 22 µm and δ ) 25 µm) the heterogeneity of the system cannot be ignored. This observa-

Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 2467

Conclusions

Figure 5. Comparison of model predictions with experimental data. Enhancement factors: experimental, Ebr, Eh, Ed (δ0max ) δl/ 2), and Ed (δ0max ) δl/4), as a function of microphase holdup, l0. Data used: kL ) 8.5 × 10-5 m/s, DA,c ) 2 × 10-9 m2/s, DA,d ) 2 × 10-9 m2/s, k1,c ) k1,d ) 0 s-1, m ) 11.6, dp,s ) 10 µm, dp ) 8 µm, µc ) 8.5 × 10-4 kg/m‚s, and µd ) 8.5 × 10-4 kg/m‚s. Experimental data, on the absorption of oxygen into aqueous dispersions of hexadecane, and parameter values taken from Bruining et al.18

tion holds for all values of l0 and, importantly, applies in the range of low l0 too. At the higher l0 values, the pseudohomogeneous model should not be applied in any case because it breaks down on conceptual grounds. Some investigators have reported that there occurs a maximum in the enhancement factor with respect to l0,16,17 which cannot be predicted by any of the prior models in the literature. The proposed model is capable of producing such a trend depending upon the magnitude of the viscosity corrections, such as when the viscosity of the dispersed phase is very high (say, because of the presence of surfactants). For instance, the computed enhancement, in the limit of µd f ∞ (i.e., a rigid drop), is shown in the inset to Figure 4 and denoted by Er. It can be observed here that Er is always lower than Ed and, whereas the values of Er start decreasing around l0 ) 0.3, the Ed trend only becomes flat by about l0 ) 0.5. The observations made in relation to the analysis of the experimental data reported by Van Ede et al., with regard to microdroplet size, can be further buttressed by examining the experimental data of Bruining et al.18 for the absorption of oxygen into hexadecane emulsions in water. This is shown in Figure 5 and is limited to l0 ) 0.08. The value of dp/δ here is about 0.4 so that the particles may be considered to be large. Even at these low holdups, the experimental points are closer to the Ed curves than to Eh. Yet again, the Ed values for the case δ0max ) δl/4 seem to give a somewhat closer match with the experimental points, as compared to δ0max ) δl/2. (The theoretical enhancement used by Bruining et al. in their work, marked Ebr, is the maximum enhancement that may be obtained from the pseudohomogeneous model.) Therefore, the rate enhancement effects due to large particles should be modeled by the use of the proposed heterogeneous models rather than pseudohomogeneous ones, even at low holdups of the microphase.

An approach for interphase mass transfer, in the presence of a microphase, through the use of a heterogeneous, multilayer model, based on Danckwerts’ surface renewal theory, has been presented in this work. The effect of an irreversible first-order reaction has also been included. It has been shown that the heterogeneous approach must be used when the microphase holdup is large or when the microparticle size is not much smaller than the thickness of the interfacial, diffusion zone. One of the most important model parameters is the thickness of the first liquid layer, δ0, sandwiched between the gas-liquid interface and the first layer of microparticles. It has been demonstrated that the specific rate of mass transfer should be computed by averaging over all possible values of δ0. The experimental data examined in this study indicates that δ0 may lie in the range of 0 to δl/2. Corrections to the values of the rate enhancements computed from the heterogeneous models, because of the large viscosities of the dispersion constituted by the microphase plus continuous phase, relative to the pure continuous phase, have been applied in order to deduce the effect of lowered surface renewal rates at the gasliquid interface. These corrections have been shown to be especially relevant at large microphase holdups. The proposed model has been validated by analyzing experimental data from the literature, and it has been shown that it works well for the cases of high microphase holdups or large microparticle sizes. The heterogeneous approach proposed in this work may be further explored and improved. For instance, the effect of neglecting lateral diffusion needs to be examined by building two- or even three-dimensional diffusion models. Because microphases are usually not used in a monodispersed state and there always exists a size distribution, it might also be instructive to look at how microparticles of varying sizes can be accommodated in the heterogeneous models. Similar arguments may be applied to the case of nonspherical particles, especially those with large aspect ratios, and these may be modeled by idealizations involving rectangular blocks rather than cubes. Also, there still remain operating zones where experimental data are required to be generated, such as, for example, high l0 with small dp, variation of dp and m systematically for a given system at different l0, and so on. Subsequent efforts directed toward parametric sensitivity studies and reconciliation of the heterogeneous approach with the earlier pseudohomogeneous models (in regions where the latter are applicable) would be desirable. Nomenclature CA ) concentration of A, kmol/m3 C/A ) solubility of A in the continuous phase, at the gasliquid interface, kmol/m3 b CA ) bulk concentration of A in the continuous phase, kmol/m3 DA ) diffusivity of A, m2/s dp ) microdroplet cube length, m E ) enhancement factor in the specific rate of absorption (ratio of rate in the presence to that in the absence of the microphase) k1 ) first-order rate constant for consumption of A, s-1

2468 Ind. Eng. Chem. Res., Vol. 38, No. 6, 1999 kL ) liquid-side mass-transfer coefficient for gas-liquid contact in the absence of the microphase, m/s l0 ) volumetric fractional holdup of the microphase m ) distribution coefficient of solute A between microdroplets and the continuous phase, [kmol/m3(d)]/[kmol/m3(c)] N ) number of microparticle layers that fit in penetration depth δp Qc ) coefficient defined in eq 26, m/s QN ) coefficient defined in eq 26, m/s Qd ) coefficient defined in eq 26, m/s Qm ) coefficient defined in eq 26, m/s Qm0 ) coefficient defined in eq 26, m/s QmN ) coefficient defined in eq 26, m/s Q0 ) coefficient defined in eq 26, m/s RA ) specific rate of absorption, kmol/m2‚s S ) surface renewal frequency of the gas-liquid interface, s-1 T ) temperature, °C t ) time on element contact time scale, s x ) distance from the gas-liquid interface within the penetration element, m Greek Symbols δ ) characteristic diffusion length defined in eq 4, m δc ) center-to-center distance between two microparticles located in adjacent layers (also liquid cubic cell length), m δl ) thickness of the liquid layer between adjacent microparticle layers (microphase interlayer spacing), m δN ) thickness of liquid between the last microphase layer in the diffusion zone and the bulk phase, m δ0 ) thickness of liquid between the gas-liquid interface and the first microparticle layer, m δp ) penetration depth of solute in the absence of the microphase and chemical reaction (thickness of the diffusion zone), m µ ) viscosity, kg‚m/s Superscripts b ) pertaining to cube side facing the bulk phase ba ) pertaining to the base case (absence of the microphase) d ) pertaining to deformable droplet correction def ) generic definition h ) pertaining to the pseudohomogeneous model i ) pertaining to the cube side facing the interface j ) pertaining to the jth particle m ) pertaining to the channel containing the microparticle max ) maximum possible value n ) pertaining to the presence of only n microparticle layers r ) pertaining to rigid droplet correction u ) unaveraged with respect to the thickness of the initial liquid layer, i.e., at a fixed value of δ0 Subscripts c ) pertaining to the continuous phase d ) pertaining to the microdispersed phase

disp ) pertaining to the entire dispersion (applies to µ, kL, and S) s ) diameter of an equivalent sphere [(π/6)dp,s3 ) dp3], m Special Characters C h ) Laplace domain variable 〈 〉 ) surface-averaged quantity

Literature Cited (1) Mehra, A. Intensification of Multiphase Reactions by Using Microphases. In Handbook of Heat and Mass Transfer; Cheremisinoff, N. P., Ed.; Gulf Publishing: Houston, TX, 1990; Vol. 4. (2) Beenackers, A. A. C. M.; van Swaaij, W. P. M. Mass Transfer in Gas-Liquid Slurry Reactors. Chem. Eng. Sci. 1993, 48, 3109. (3) Van Ede, C. J.; Van Houten, R.; Beenackers, A. A. C. M. Enhancement of Gas to Water Mass Transfer Rates by a Dispersed Organic Phase. Chem. Eng. Sci. 1995, 50, 2911. (4) Nagy, E.; Moser, A. Three-phase Mass Transfer: Improved Pseudo-homogeneous Model. AIChE J. 1995, 41, 23. (5) Mehra, A. Intensification of Multiphase Reactions Through the Use of a MicrophasesI. Theoretical. Chem. Eng. Sci. 1988, 43, 899. (6) Junker, H. B.; Wang, D. I. C.; Hatton, T. A. Oxygen Transfer Enhancement in Aqueous/Perfluorocarbon Fermentation Systems: II. Theoretical Analysis. Biotechnol. Bioeng. 1990, 35, 586. (7) Karve, S.; Juvekar, V. A. Gas Absorption into Slurries Containing Fine Catalyst Particles. Chem. Eng. Sci. 1990, 45, 587. (8) Yagi, H.; Hikita, H. Gas Absorption into a Slurry Accompanied by Chemical Reaction with Solute from Sparingly Soluble Particles. Chem. Eng. J. 1987, 36, 169. (9) Vinke, H.; Hamersma, P. J.; Fortuin, J. M. H. Enhancement of the Gas-absorption Rate in Agitated Slurry Reactors by Gasabsorbing Particles Adhering to Gas Bubbles. Chem. Eng. Sci. 1993, 48, 2197. (10) Nagy, E. Three Phase Mass Transfer: One Dimensional Heterogeneous Model. Chem. Eng. Sci. 1995, 50, 827. (11) Danckwerts, P. V. Gas-Liquid Reactions; McGraw-Hill: London, 1970. (12) Doraiswamy, L. K.; Sharma, M. M. Heterogeneous Reactions; John Wiley & Sons: New York, 1984; Vol. 2. (13) Sherman, P. Rheological Properties of Emulsions. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1983; Vol. 1. (14) Sherman, P. Industrial Rheology; Academic Press: London, 1970. (15) Mehra, A.; Pandit, A.; Sharma, M. M. Intensification of Multiphase Reactions Through the Use of a MicrophasesII. Experimental. Chem. Eng. Sci. 1988, 43, 913. (16) Ju, L.-K.; Lee, J. F.; Armiger, W. B. Enhancing Oxygen Transfer in Bioreactors by Perfluorocarbon Emulsions. Biotechnol. Prog. 1991, 7, 323. (17) Van der Meer, A. B.; Beenackers, A. A. C. M.; Burghard, R.; Mulder, N. H.; Fok, J. J. Gas/liquid Mass Transfer in a Fourphase Stirred Tank Fermentor: Effects of Organic Phase Holdup and Surfactant Concentration. Chem. Eng. Sci. 1992, 47, 2369. (18) Bruining, W. J.; Joosten, G. E. H.; Beenackers, A. A. C. M.; Hofman, H. Enhancement of Gas-Liquid Mass Transfer by a Dispersed Second Liquid Phase. Chem. Eng. Sci. 1986, 41, 1873.

Received for review October 13, 1998 Revised manuscript received March 3, 1999 Accepted March 7, 1999 IE980653C