Three-Phase Mass Transfer: Effect of the Size Distribution

Ind. Eng. Chem. Res. 2003, 42, 5363-5372

5363

Three-Phase Mass Transfer: Effect of the Size Distribution Endre Nagy* and Pe´ ter Hadik Kaposvar University, Research Institute of Chemical and Process Engineering, P.O. Box 125, 8201 Veszpre´ m, Hungary

The absorption of oxygen into the aqueous phase was measured in the presence of a dispersed silicone oil phase. In the most experiments, a thin, permeable membrane layer resulting from interface polymerization encapsulated the organic droplets. The size distribution and average droplet size, obtained at different stirrer speeds, were measured. Using a heterogeneous, multilayer mass-transfer model, enhancement was predicted as a function of mass-transfer coefficient without particles and average particle size. Enhancements measured and predicted were compared to each other as a function of the dispersed-phase hold-up and the average particle size. It was concluded that the particle size distribution can significantly alter the absorption rate; thus, its effect should take into consideration in the prediction of absorption rate enhancement. Introduction The absorption rate of oxygen in the presence of a second, dispersed, organic phase can be significantly increased, because of the higher solubility and diffusivity of oxygen in the organic phase. The use of an organic phase in a fermentation broth can cause some negative effects on the cell growth and productivity.1 Encapsulation of the organic droplets within an ultrathin, oxygenpermeable membrane eliminates direct contact between the cells and the organic phase; thus, the toxicity problems related to direct contact can be avoided.2,3 The aim of this work is to show enhancement of the absorption rate of oxygen in the presence of encapsulated silicon oil in a perfectly mixed laboratory tank reactor. Mass transport into fine solid particles4-6 or liquid drops7-12 can essentially alter the concentration gradient in the liquid boundary layer at the gas-liquid interface and, consequently, the absorption rate.13 For example, the oxygen absorption rate has been increased by a factor up to about 4 (in the dispersion of octene droplets,11 H ) 18, Dr ) 0.56, ) 0.05-0.5), by a factor up to 1.4 (in the presence of hexadecane dispersion,7 H ) 11.6, Dr ) 1, ) 0.01-0.08), and by a factor of 4.5 (in the presence of n-dodecane,8 H ) 7.9, Dr ) 1, ) 0.050.25) in agitated vessels. Littel14 measured the enhancement of carbon dioxide absorption, obtaining an enhancement of up to 2.5 in the dispersion of toluene droplets (H ) 2.87, Dr ) 0.5, ) 0-0.4). The presence of an organic phase can alter both the mass-transfer coefficient and the interfacial area.12 To describe the mass-transfer rate theoretically, both effects have to be taken into account. Several mathematical models have been developed for mass transport in the boundary layer in the presence of a third, dispersed phase. Brilman et al.15 presented an excellent review of the models. In principle, heterogeneous mass-transfer models are suitable for predicting the three-phase absorption rate for larger particles, i.e., where size of the dispersed droplets is the same order of magnitude as the thickness of the laminar boundary layer.5,9-11,15-22 * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: 36 88 421614. Fax: 36 88 424424.

All of these models apply known mass-transfer theories, such as film theory,16,17 penetration theory,20,22 or filmpenetration theory,18,19 for the hydrodynamic conditions in the boundary layer. The latter model involves the film and surface renewal theories as limiting cases. The spherical, uniformly sized droplets are modeled as cubic particles of identical volume5,9,10,16,18,21 or are regarded as spheres.11,15,17,19,20,22 In practice, all of these models divide the boundary layer into equal-sized cubic or cylindrical cells within one particle. The size of the cells depends on the particle size and dispersed-phase holdup. An important unknown parameter is the distance of the first droplets from the gas-liquid interface because its value strongly affects the absorption rate. The assumption that the droplets can be located in positions between 0 and δp - R with equal probability15,18 seems to be reasonable. The three-dimensional model of Brilman et al.15 should be the most accurate. The authors found that droplets (or particles) influence local mass-transfer rates over an area that greatly exceeds the projection of the droplets onto the gas-liquid interface. This effect can only partly be involved in the two-dimensional model,22 and it is not taken into account in the one-dimensional models.18-20 The question is how this lateral diffusion alters the absorption rate and under what conditions, in terms of particle capacity, diffusivity, and particle size, it can be neglected. Nagy1 compares the absorption rates obtained by a one-dimensional model to those obtained by a two-dimensional model.22 According to these results, the difference between the modeled rates is less than 20% when you use a common organic phase with H up to 20-22. From a practical point of view, this difference can be neglected. Brilman et al.15 also stated that the one-dimensional model is sufficiently accurate for small particles with low capacities. Because multidimensional models can only be solved numerically, so that their use in simulating the effect of the particle size is very complicated, we used Nagy’s one-dimensional model19 in this paper. Little attention has been paid to the effect of the mean particle size and the size distribution on the masstransfer rate in the literature. Alper and Deckwer4 found that particles larger than the film thickness do

10.1021/ie030110p CCC: $25.00 © 2003 American Chemical Society Published on Web 09/10/2003

5364 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003

Figure 1. Boundary layer with particles in the diffusion path and the important notations used.

not enhance the absorption rate. Van der Meer et al.24 indirectly investigated the effect of particle size on the absorption rate. They varied the revolution rate in a stirred vessel between 5 and 17 s-1 to generate organic droplets of decreasing size. They found that, in the revolution rate regime investigated, the enhancement factor increases by a factor of 1.5-1.8 with increasing stirrer speed, that is, with decreasing droplet size. The size distribution as a function of the stirring rate was not measured, although it can also be important. In the case when the mean particle diameter is large, a significant enhancement of the gas absorption rate might be observed as a result of the presence of smaller particles.15 This was confirmed experimentally by Tinge and Drinkenburg,25 who added fine particles to a slurry that already contained larger ones. A detailed analysis of the real effect of the size and size distribution on the absorption rate has not yet been reported in the literature. This paper will present some theoretical and experimental results concerning this question.

taken into account; the lateral diffusion is neglected. Consider the differential interface element T∆ (Figure 1, T∆ ) 2R*π∆R*), at the projection of a differential volume element (∆V ) T∆δ) on the gas-liquid interface. For this element, the different diffusion distances, including that between the gas-liquid interface and the first particle, δ0 + κ; that between the particle interfaces, δp + 2κ; and that inside the particles, d, are constants. The boundary layer can be divided into 2N + 1 segments [δ0 + κ + Nd + N(δp + 2κ)] perpendicular to the gas-liquid interface, with D (in the continuousphase segments) and Dd (inside particle segments) diffusion coefficients in them. Assuming that the surface renewal theory is valid for this heterogeneous system, the Fick II differential mass balance equation can be written for each segment.18,19 We must define boundary conditions for the internal interfaces between the continuous and droplet phases (of which there are 2N - 1) and for x ) 0 and x ) ∞ (or δ). For the internal interfaces, we use the known boundary conditions,19,21 namely, (1) the fluxes over each internal boundary between each section must be equal, i.e., D(∂A/∂x) ) Dd(∂Ad/∂x), and (2) the concentrations at the interfaces are in equilibrium, i.e., Ad ) HA. The 2N + 1 differential mass balance equations were analytically solved by Laplace transform.18,19 Using the above conditions, we obtained 2(2N + 1) algebraic equations to determine the same number of parameters upon solution of the 2N + 1 differential equations. This algebraic equation system was solved by traditional mathematical methods. The solution also required some intuitive deductions because of the arbitrary value of N. The time-averaged value of the local absorption rate for surface renewal theory27 can be expressed as follows

j ) β(A* - A°)

(1)

where N nN 2 - n1 1 β ) β° 2 cosh λ0 nN

(2)

0

Theoretical Background two-22

three-dimensional15

The and the models for mass transfer do not discuss in detail the effect of the particle size and size distribution on the absorption rate. The local enhancement in the particle size range of 2-24 µm was given by Brilman et al.,15 whereas Lin, Zhou, and Xu22 simulated the enhancement in the range of 5-40 µm at different locations of the first particle in the boundary layer. In his more detailed analysis,1 Nagy used a one-dimensional model.19 It was shown that the one-dimensional model could be applied with good accuracy for estimations of the mass-transfer rate in the presence of conventional organic phases (H up to ∼20). A mathematical mass-transfer model was developed according to the physical model given schematically in Figure 1 for a given particle size. It was assumed in this model that both the particle size distribution and the residence time of the particles in the boundary layer are the same as in the bulk continuous phase. The distance between the droplets, δp, can be calculated from the hold-up of the dispersed phase for spherical particles,19 that is, δp ) dp(0.806/1/3 - 1). The value of δp is assumed to be constant during our calculations. Only diffusion perpendicular to the gas-liquid interface is

The values of nN t can be obtained by the following N equations, with the nN t and mt values calculated from i ) 1 to i ) N in series (1 e i e N)

nit ) ni-1 t (1 + p tanh λp tanh λd) + 1 mi-1 tanh λp + tanh λd t p

(

)

t ) 0, 1, 2

mit ) ni-1 t (tanh λp + p tanh λd) + 1 1 + tanh λp tanh λd mi-1 t p

(

)

t ) 0, 1, 2

where

p)H

x

Dd , D

λ0 )

x

sδ02 , D

λp )

x

sδp2 , D λd )

x

sd2 Dd

For the case of i ) N, the values of nit and mit have to be modified in the following manner because there is

Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5365

no particle behind the Nth one, so that λp ) 0

κ)

If i ) N, then tanh λp ) 1 nit

mit

and for the For the calculation of the values of case of i ) 1 (the values of n1t and m1t ), the values of n0t and m0t have to be known. They are

n00

) tanh λ0,

m00

n02 ) -m02 ) exp λ0 From eq 2, the mass-transfer rate of j can easily be obtained for any value of N. Because of its importance, the mass-transfer rate for the case of N ) 1 with infinite penetration depth behind it, that is, with tanh λp ) 1, is given as follows

tanh λ0(tanh λd + p) + p(1 + p tanh λd) β ) β° tanh λd + p + p tanh λ0(1 + p tanh λd)

(2a)

The only unknown parameter, the distance of the first particle from the gas-liquid interface, δ0, should be estimated. It should be noted that the value of δ0 has a significant effect on the absorption rate, which is why knowledge of its true value is very important for the estimation of the absorption rate. Up to now, no generally accepted method has been available to measure or predict its value. The first droplets in the boundary layer can be taken to be located within a cell with a length of dp + δp. We assume that δ0 can randomly vary between 0 and δp. [The rising bubbles divide the continuous liquid phase between the neighboring droplets into two streams that are flowing on the two sides of bubbles (Figure 7). This can decrease the δ0 value, as will be discussed in the last section in detail.] The number of droplets behind each other perpendicular to the gas-liquid interface in the boundary layer, N, can be calculated from the values of δp and the thickness of the laminar boundary layer or from the penetration depth.19 The above model gives the absorption rate for cubic particles, or for a differential volume element of spherical particles, perpendicular to the gas-liquid interface, where the values of δ0, δp, and d can be regarded as constant. This volume element is a tube with a wall thickness of ∆R*, with infinite length for penetration theory (or δ in case of a given penetration depth), and with a radius of R*. For a spherical particle, the δ0(R*), δp(R*), and d(R*) values vary continuously as a function of R*. The function of d(R*) vs dp can easily be given by geometric considerations. According to Figure 1, we obtain

d(R*) ) dpx1 - R*2/R2 From this expression, the values of δ0(R*) and δp(R*) can easily be obtained as

δ0(R*) ) δ0 + κ and δp(R*) ) δp + 2κ where

The specific absorption rate, jave, averaged over the values of R′* and δ0, can be obtained by means of double integration, as follows (with R′* ) R*/R19)

jave )

)1

n01 ) -m01 ) -exp(-λ0)

dp - d(R*) 2

1 δp

∫0δ ∫012R′*j(R′*,δ0) dR′* dδ0 p

(3)

With eq 3, one can obtain the absorption rate of the heterogeneous part of the gas-liquid interface (which is the projection of the droplets onto the gas-liquid interface) for a given particle size, dp. The absorption rate for the total gas-liquid interface for a given particle size, Jhet, can be given as19

Jhet ) 1.209jave2/3 + J(1 - 1.2092/3)

(4)

Calculating the effect of the size distribution of the dispersed phase on the absorption rate is rather complicated. Assuming a differential volume element of the dispersed phase, dVi(dp,i), in which the particle size, dp,i, is constant, the effect of the size distribution on the absorption rate can be obtained by integration over the whole particle size range. This can be done both for the absorption rate of the heterogeneous portion of the interface, jave (models B and C), and for the absorption rate related to the total interface, Jhet (model A). Detailed discussions of these models are provided in the Appendix. Most of the simulations presented here were carried out using model C because it seems to be the most accurate and the closest to reality. Model C takes into account the fact that droplets are perfectly mixed in the liquid independently of their size and that the smaller droplets can be located between larger ones, decreasing their effect on the absorption rate. Model A seems to be the simplest. Enhancements calculated by the different models are compared in Figure A2, in the Appendix. According to model C, averaging the value of jave, expressed in eq 3, over the whole particle size range, should be carried out as follows (the value of jave(dp,i) is equal to jave at particle size dp,i) ave jdisp

)

∫dd

p,max

p,min

jave(dp) τi(dp) ddp 2/3

M

)

jave(dp,i) τi(dp,i) ∑ i)1 2/3

(5)

with i

τi(dp,i) )

∑ k)1

i-1

[∆Vk(dp,k)]2/3 -

[∆Vk(dp,k)]2/3 ∑ k)1

(6)

The particle size range from dp,min to dp,max was divided into M uniform-size sections, with the ∆Vi volume fraction corresponding to the ∆dp,i size fraction, andintegration over the particle size range was carried out numerically using the expression on the right-hand side of eq 5. The value τi(dp,i) represents the current value of the heterogeneous interface corresponding to the dp,j particle size fraction. The second term on the right-hand side of eq 6 gives the area of the heterogeneous interface


occupied by particles smaller than dp,i, taking into account in the calculation all steps from k ) 1 to k ) i - 1. The heterogeneous interfacial area corresponding to particles of size dp,i is reduced by this value because it has already been taken into account in the former calculation steps. From eq 6, one can easily see that the following equation is fulfilled M

τi(dp,i) ) 2/3 ∑ i)1

(7)

The values of Jhet can be calculated in this case by eq ave instead of jave. Obviously, you must know 4 using jdisp the value of dVi(dp,i) as a function of dp,i. The size distribution of droplets was measured before the absorption measurements were performed. This measured distribution was used for our calculations, as will be shown later. Experimental Section Oxygen absorption experiments were carried out in a stirred vessel with 0.6 dm3 liquid volume and with a moderate stirrer speed of 250 rpm. The gas-liquid interfacial area was higher than the geometrical area because of rippling of the liquid interface and the small hold-up of gas bubbles generated by the stirrer. Its value was assumed to be independent of the dispersed-phase hold-up and droplet size. Oxygen was transferred from the gas phase (which contained 70 vol % oxygen and 30 vol % nitrogen) to a sulfite solution,7,11 where it reacted with sulfite to yield sulfate. The oxygen concentration in the liquid was kept at zero by adjusting the reaction rate with the Co2+ catalyst concentration. The oxygen concentration of the continuous gas stream essentially did not decrease during absorption because of the high volumetric gas flow rate: 200 dm3/h at 30 °C. A schematic diagram of the experimental setup is shown in Figure 2. The stirred cell, suggested by Danckwerts,27 was made of glass. The internal diameter of the vessel with two symmetrically located baffles is 0.1 m. The bottom section of this cell, containing the liquid phase, has six-bladed and four-bladed impellers in the upper and lower parts of this section with diameters of 0.05 and 0.06 m, respectively. The gas phase, with air added to the oxygen stream to adjust its oxygen content, was introduced into the upper part of the cell through an external bubble column to presaturate it with water. A four-bladed impeller with a diameter of 0.048 m was located in the gas phase of the cell. Further details concerning the experimental conditions and chemicals are provided in Table 1. To determine suitable experimental conditions, we measured the oxygen absorption rate as a function of the catalyst concentration at a stirring speed of 4.17 s-1 and pH ) 8.5. The cobalt ion concentration in these experiments ranged7,11 from 10-7 to 10-1 mol/m3. A double logarithmic plot of the oxygen absorption rate vs the cobalt concentration (not shown) indicated that no enhancement of the absorption rate occurred in the range of catalyst concentrations from 1 × 10-7 to 5 × 10-6 mol/m3. These results are in agreement with the data of van Ede et al.11 and Bruining et al.7 taking into account the sensitivity of the reaction rate to the impurities in the sulfite solutions. Thus, the cobalt ion concentration was chosen in the range of (1-2) × 10-7 mol/m3 in our three-phase experiments.

Figure 2. Experimental setup with dimensions of stirrers (dimensions are in millimeters). Table 1. Chemicals and Operating Conditions Chemicals for Enhancement Measurements deionized water, silicone oil (silicone oil, 350; Reanal) Na2SO3 (puriss, Reanal), CoSO4‚7H2O (pro-analysis, Reanal) H2SO4 (pro analysis, Reanal) oxygen gas; compressed air Chemicals for Microcapsulation Na2CO3 (puriss, Reanal), deionized water Dow Corning 190 fluid (Dow Corning) sebacoyl chloride (Sigma) hexane diamine (Sigma) polyethylenimine (Sigma) silicone oil HCl (pro-analysis, Reanal) Experimental Conditions 0.8-1 M Na2SO3, pH adjusted 8.5 with H2SO4 T 303 K 0-0.3 stirring speed for experiments 4.17 1/s stirring speed for emulsification up to 5000 1/s oxygen concentration in liquid 0

The sulfite method allows for the measurement of the physical mass-transfer coefficient (and the interfacial area) in the fast reaction regime.27 The absorption rate of oxygen was measured at different values of the kinetic constant. The line obtained by plotting the quantity (J/A*)2 as a function of the overall reaction kinetic constant28 permits the simultaneous determination of the interfacial area and the mass-transfer coefficient in the liquid phase from the slope and the ordinate at the origin, respectively. The kinetic data of Laurent29 were used to obtain the values of the secondorder reaction rate at different catalyst concentrations. Under the operating conditions of this study, the partial orders of the oxidation reaction with respect to oxygen, sodium sulfite, and cobalt sulfate are 2, 0, and 1, respectively.11,30 The liquid mass-transfer coefficient measured was 1 × 10-4 m/s, and the interfacial area obtained was higher by a factor 5, at a stirrer speed of


4.2 s-1, than the geometrical area of the cell because of the rippling of the liquid interface and the gas bubbles generated. The interfacial area was regarded as a constant value under the experimental conditions investigated (dispersed-phase hold-up, , was varied in the range of 0-0.3, while the average droplet size was up to 310 µm). The values of the enhancement measured give the ratio of mass-transfer coefficients with and without dispersed phase, β/β°. Microencapsulation of the silicon oil yielded very stable, rigid particles. The dispersed silicone oil droplets were encapsulated in polyamide nylon membranes formed by an interfacial polymerization reaction.2,31 In this reaction, the monomers of the nylon are an oilsoluble sebacoyl chloride and a water-soluble hexanediamine. After the oil droplets are mixed into the aqueous solution, both phases contain a reactive compound; the molecules react at the interface to form a quasi-peptide bond and, thus, a cohesive ultrathin membrane. Crosslinking with polyethyleneimine, which was added to the aqueous hexanediamine solution,2 enhanced the strength of the nylon membrane. Munaretto31 gives the exact recipe of the encapsulation process. The membrane thickness was about 1-3 µm, according to our measurements with a light microscope. The size distribution of the particles was affected by the stirrer speed of the turbine impeller in the range of 400-5000 rpm. A laserdiffraction analytical instrument, Malvern 2600c, was used to measure the size distribution of the silicon emulsion. Its average values varied in the range of 17310 µm.

Table 2. Parameter Values Used for Simulation and Verification of the Experimental Data

a

parameter

value

β° D D/Dd H Na

1 × 10-4 m/s 2.3 × 10-9 m2/s 1 21 0.05-0.3 30 for dp < 25 µm 15 for 25 µm e dp < 100 µm 5 for dp g 100 µm

For β° ) 1 × 10-4 m/s.

Results and Discussion

Figure 3. Enhancement as a function of the physical masstransfer coefficient (without droplets in the liquid phase), β° ( ) 0.1; other parameter values reported in Table 2).

It is well-known that both the physical mass-transfer coefficient (that without dispersed phase) and the particle size have strong effects on the mass-transfer rate. First, we show some simulated results concerning enhancement of the mass-transfer rate depending on the physical mass-transfer coefficient. Then, the experimental results are discussed and evaluated as a function of the dispersed-phase hold-up and average particle size. The measured enhancement data is verified taking into account the measured size distribution of the dispersed organic phase. The dispersed silicone oil used was coated by a thin, permeable, polymeric membrane layer in most experiments. A. Simulation Results: Effect of the Physical Mass-Transfer Coefficient, β°. The role of β° seems to be very significant, essentially affecting the enhancement, and consequently, knowing its real values during our experiments is also important for the verification of the experimental data. Brilman et al.15,20 investigated the effect of the contact time on the enhancement at different distances of the first particle from the gasliquid interface for a given particle size. Recently, Nagy1 showed how enhancement can change as a function of physical mass-transfer coefficient, β°, at different number of particles behind each other at dp ) 10 µm. The first particle was located at the gas-liquid interface, i.e., at δ0 ) 0. As was shown, the penetration depth strongly determines the role of particles behind each other. When the value of β°, that is, increasing the penetration depth, is decreased, the number of particles behind each other in the boundary layer, which has an effect on the absorption rate, is also increased. It follows from this statement that the particle size, which enhances the absorption rate, increases with decreasing

value of β°. (The numbers of particles behind each other used for our simulations are given in Table 2 for the case of β° ) 1 × 10-4 m/s. Decreasing the β° value, even more droplets can influence the absorption rate. This was taken into account at our simulations.) How this effect acts, in reality, is illustrated in Figure 3 for heterodisperse (solid lines) and monodisperse (dashed lines) third phases. Enhancement was simulated by two measured size distributions with average particle sizes of 39 and 92 µm as a function of β°. The parameter values used for the calculation are given in Table 2. The physicochemical parameters used are characteristic of the silicone oil that was used for the experiments as the dispersed phase. The mass-transfer coefficient ranged between 0.01 × 10-4 and 10 × 10-4 m/s. The curves clearly show the strong effect of β° on the enhancement. The curves tend to a limiting value with decreasing mass-transfer coefficient. The maximum value of the enhancement can even be reached with particles 215 µm in diameter, when the β° value is low enough. This enhancement behavior is related to the penetration depth, as was mentioned earlier. In the particle size range corresponding to the plateaus in the enhancement curves, the penetration depth is greater than δ0 + dp. At larger values of β°, the enhancement gradually decreases and tends to zero. The penetration depth decreases monotonically. In the limiting case, the penetration depth is less than the value of δ0; thus, particles at the interface are not able yet to alter the absorption rate. The curves also show that the enhancements for monodisperse particles are lower than those for heterodisperse particles (the size distribution of the third phase with an average particle size of 39 µm is


Figure 4. Typical measured particle size distribution curves of the dispersed phase (+, d h p ) 39 µm; *, d h p ) 92 µm).

given in Figure 4). This effect is discussed in the next section. B. Experimental Results and Their Verification. The absorption of oxygen into Na2SO3 solutions was measured in the presence of emulsified silicon oil as the dispersed phase. The particle size distribution was measured in all cases when the operating conditions of the dispersed phase (that is, the mixing rate during emulsification) was changed. The effect of the dispersedphase hold-up on the enhancement, , was measured in the presence of a dispersed phase with a given size distribution centered at d h p ) 39 µm. The particle size distribution of the dispersed phase in these experiments is plotted in Figure 4 (+). Figure 4 also shows another size distribution of the dispersed phase, as an example, with d h p ) 92 µm, which was also used to predict the enhancement as a function of the average particle size (*). The two size distributions are similar to each other. This similarity of the size distributions is also fulfilled for higher or lower average particle sizes. In the legend of Figure 5, the absorption rate without a third, dispersed phase is also given, J ) 1.03 × 10-3 mmol/(s m2). From this value and from the enhancement, the absorption rate in the presence of the dispersed phase can be calculated. The deviation of the average value of repeated experiments was less than 10%. The measured enhancements (*, 0) and the predicted ones (continuous lines) as a function of the dispersedphase hold-up are shown in Figure 5. (The dashed line in this figure is analyzed in the following section.) During the absorption experiments (which took 2-3 h), we measured the sulfite concentration 5-8 times as a function of time. From the slope of the function of the sulfite concentration vs time, the absorption rate of oxygen was calculated. The evaluation of the measured distribution data (Figure 4, d h p ) 39 µm) was carried out using model C. All of the models mentioned in this paper, and thus model C as well, involve three numerical integrations, namely, integration over R′* and δ0 (using eq 3), as well as over dp (in eq 5). Experiments were partly carried out with a dispersed phase without coating of the organic droplets by a polymeric film (*) and partly with encapsulated droplets (0). Both types of dispersed phases had the same particle size distribution and average particle size. This allows for a comparison of the enhancement data of the two cases to determine whether the thin polymeric film alters the

Figure 5. Effect of dispersed-phase hold-up on enhancement: points, experimental data (0) with encapsulated dispersed droplets and (*) without encapsulation; lines, data simulated using model C with (s) d h p ) 39 µm or with (- - -) δ0 ) 0-δp, such that, if dp < 14 µm, then δ0 ) 0-δp/2, and if dp g 14 µm, then δ0 ) 10 µm. The parameters for the simulations are in Table 2; the absorption rate without particles is J ) 1.03 × 10-3 mmol/(s m2).

absorption rate. According to these results, the effect of the diffusion resistance of this thin polymeric layer can be neglected. From a comparison of the experimental and theoretical results, it can be stated that certain differences exist between them. The theoretical enhancement, obtained using the parameter values in Table 2, exhibits essentially linear functionality with increasing dispersed-phase hold-up, whereas the experimental data show a slight curvature. This can especially be seen at ) 0.05 and 0.1, where the experimental data are somewhat higher than the simulated values. The repeated experimental data show rather high differences from each other at ) 0.1. The straight line of the predicted data is caused by the assumption that the value of δ0 can vary between 0 and δp with uniform probability. How the average particle size alters the enhancement is illustrated in Figure 6. As was mentioned earlier, the average particle size of the dispersed phase was varied by changing the rate of stirring during the emulsification of the organic phase. The size distribution and the average particle size were measured for every experiment. The points (×) represent experimental data, and the curves represent simulated data. For curves 1 and 2, the measured size distribution functions were used in the calculation of every point. (Curve 1 was simulated using model C with the δ0 value averaged between 0 and δp; curve 2 is explained in the next section.) As can be seen, the simulated data of curve 1 are about 2040% lower than the experimental data. This difference is even larger at lower values of the enhancement. The cause of this difference cannot really be determined from these experiments. We look for an explanation in the following section. As can be seen in Figure 6, the measured experimental data tend to a limiting value with decreasing particle size. A similar effect was obtained as a function of β° in Figure 3. The limiting value seems to be about 1.8. Looking for this limiting value at very low particle sizes, we recalculated the enhancement with the homogeneous models of Bruining et al.7 (dashed line 3) and Nagy and Moser23 (line 4). According to Bruining’s model, the enhancement is independent of the particle size. The


Figure 6. Effect of average particle size on enhancement: points, experimental data; lines, data simulated using model C with ) 0.1. Curve 1, δ0 ) 0-δp; curve 2, if dp < 14 µm, then δ0 ) 0-δp/2, and if dp g 14 µm, then δ0 ) 10 µm; curve 3, homogeneous model of Bruining et al.;7 curve 4, model of Nagy and Moser.23 The parameters for the simulations are in Table 2, ) 0.1; the absorption rate without particles is J ) 1.03 × 10-3 mmol/(s m2).

enhancement was determined to be 1.73 by their model. Nagy and Moser also take into account the internal mass transport in their model. Consequently, the enhancement decreases as a function of particle size in their model. In the size range of 3-20 µm, the enhancement changes from 1.82 to 1.49. For this latter simulation, we used the surface renewal theory as a limiting case of the film-penetration theory used in Nagy and Moser’s paper.23 It can be stated that, with decreasing particle size, the measured enhancement tends toward the calculated values obtained by homogeneous models that fall between about 1.73 and 1.82. The measured enhancement does not exceed the value of 1.8. C. Additional Effects on the Absorption Rate. The differences between the measured and predicted enhancements might arise for various reasons, as will be briefly discussed in this section. The model presented takes into account the diffusional transport perpendicular to the gas-liquid interface only. The lateral diffusion into the droplets in the boundary layer can also be important, especially in the case of a high solubility (partition) coefficient. In our case (H ) 21, ) 0.050.3), the increase in the enhancement should be1 less than 10-15%. The physical mass-transfer coefficient might also change because of the presence of the dispersed phase. As shown in Figure 3, the physical mass-transfer coefficient has a significant effect on the absorption rate in the presence of a dispersed phase. The question arises as to how the third phase can alter the value of the physical mass-transfer coefficient. Mehra21 suggested a mathematical method for predicting the effect of the dispersed phase on the masstransfer coefficient. According to this method, the holdup of the dispersed phase should increase the enhancement. This is also confirmed by the paper of Cents et al.,32 who investigated the effect of the dispersed-phase hold-up on the values of β° and on the interfacial area. According to Peeva’s measurements,33 the value of β° essentially does not increase as a function of the dispersion hold-up. Thus, the increase in enhancement due to the dispersed phase should be less than 10% under the conditions investigated.

Figure 7. Schematic diagram of a moving bubble and droplets around it in the stirred tank. (The figure is a scale drawing regarding the sizes of and distances between droplets.)

An important effect on the absorption rate can be caused by the small rising bubbles colliding with the moving silicon oil particles. According to our measurements, the interfacial area was about 5 times higher than the geometrical interfacial area. During the stirring of the liquid, many small bubbles were generated. The diameter of these bubbles was estimated to be 0.002 m according to our visual observations. Using the known real interfacial area, the hold-up of the bubble phase was determined to be about 0.04. The rising velocity of the bubbles was estimated to be 0.2 m/s, using the Stokes law.34 Figure 7 (which is a scale drawing) illustrates this situation in a special case, namely, for monodisperse particles with dp ) 100 µm at ) 0.1. A bubble with a size of 2000 µm rises in a liquid phase with Re ≈ 408, in the perpendicular direction, where it meets with droplets at a distance of about 75 µm. Many droplets cross the path of the large bubble. The moving bubble can capture many droplets, as is known to occur in the case of rising bubbles and solid particles, as a result of the van der Waals and electrostatic forces.38 Small particles, following streamlines around the bubble, can get very close to it. A number of droplets could hit each other, and a number of droplets could hit the rising bubble. As a result of these collisions, some droplets approach as close as possible to the bubble. At Re ≈ 408 for bubbles, flow separation can occur at the rear of bubble, and the resulting oscillating wake that forms there gradually becomes turbulent.35 This turbulence promotes collisions between the drops and the bubble.36 The bubbles can move in another flow direction as well, because the stirrer causes the continuous liquid to circulate in the vertical direction. The velocity of the circulation should be higher than the rising velocity, which causes turbulent conditions. These streams also cause collisions between droplets and bubbles with large probability. (The dashed line represents the resulting direction of the moving bubble.) The question arises as to how close the drops can get to the moving bubbles. Let us assume that the larger rigid droplets, coated with a thin polymer film layer, have a viscous sublayer in a turbulent continuous phase. The thickness of this layer can be estimated to be 10 µm.37 According to this estimation, we can assume that the larger droplets, dp >14 µm (at this size, the value of δp is simply equal to 10 µm for ) 0.1; the viscous sublayer can involve droplets smaller than 10 µm), can be located as close


as δ0 ) 10 µm to the bubble interface. This assumption could also be true for the plain gas-liquid interface to which the rising bubbles push the droplets. For smaller droplets, dp< 14 µm, we assumed that the δ0 value can randomly vary between 0 and δp/2 [the rising bubbles divide the continuous liquid phase between the neighboring droplets into two streams that are flowing on the two sides of bubbles (Figure 7)]. Using these assumptions, we recalculated the experimental results (only the value of δ0 was changed; all other parameter were the same as earlier). The predicted results are plotted as curve 2 in Figure 6. As can be seen, we obtained higher enhancements in this way. Thus, the measured and predicted results are in good agreement. Similarly, the results as a function of hold-up were reevaluated (Figure 5, dashed line). The shape of this curve differs somewhat from that of the continuous line. The recalculated enhancements are again in agreement with the measured values. Does this mean that the above assumption is true? The δ0 value used for the simulation is rather speculative. Experimental results are needed that can be used to determine the true value of δ0 on the moving bubbles and that make the mass transport into rising bubbles in three-phase, turbulent systems more clear.

i ) index of particles in the diffusion path of the boundary layer or index of particle size fraction in the size distribution j ) mass-transfer rate for the heterogeneous part of interface, i.e., portion of the gas-liquid interface where there are particles in the diffusion path, mol/(m2 s) jave ) average value of j obtained by eq 3 for uniform particles, mol/(m2 s) jave disp ) average value of j obtained for a dispersed phase (eq 5) with a distribution of particle sizes, mol/(m2 s) J ) mass-transfer rate without a dispersed phase, mol/ m2s Jhet ) absorption rate related to the total interface obtained by the heterogeneous model, mol/(m2 s) N ) total number of particles in the boundary layer perpendicular to the gas-liquid interface p ) see eq 2 R ) particle radius, m R* ) radius as a space coordinate (see Figure 1), m R′* ) R*/R Re ) Reynolds number of the rising bubble s ) surface renewal frequency, s-1 Vi ) ith volume fraction of particles, m3 ∆Vi(dp,i) ) dimensionless value of the volume fraction corresponding to particles of size dp,i

Conclusion

Greek Letters

The particle size distribution of a dispersed phase can have a significant effect on the absorption rate. The onedimensional multilayer heterogeneous model is able to predict the absorption rate in the presence of nonuniformly sized droplets in the dispersed phase. With increasing standard deviation, or spread, of the particle size distribution function, the volume portion of the smaller particles in the dispersed phase increases. The small particle fraction lowers the decrease of the enhancement with increasing average particle size. An emulsified dispersed phase made simply by mixing has a wide range of particle sizes. Thus, its effect on the absorption rate should be considered.

β° ) physical mass-transfer coefficient () xDs), m/s β ) mass-transfer coefficient in the presence of particles as given by eq 2, m/s δ ) thickness of the boundary layer or penetration depth, m δ0 ) distance of the first particle from the gas-liquid interface, m δp ) distance between particles, m ) hold-up of the dispersed phase ψ ) standard deviation of the size distribution τi ) heterogeneous interfacial area for the ith size fraction of the dispersed phase, m2/m3

Acknowledgment This work was supported by the Hungarian Research Foundation under Grants OTKA T 15864 and T 29272. Notation ai ) heterogeneous interfacial area of the dp,i particle size fraction, m2/m3 A° ) concentration of the absorbed component in the bulk phase, mol/m3 A* ) concentration of the absorbed component at the gasliquid interface at x ) 0, mol/m3 d ) diffusion distance within particles, m dp ) particle size, m dp,i ) diameter of the ith particle size fraction, m d h p ) average particle size, m dp,min ) diameter of the smallest particle of the dispersed phase, m dp,max ) diameter of the largest particle in the dispersed phase, m d(R*) ) dpx1-(*R*2/R2) D ) diffusion coefficient in the continuous phase, m2/s Dd ) diffusion coefficient in the dispersed phase, m2/s Dr ) D/Dd E ) enhancement () β/β°) H ) solubility (partition) coefficient of the absorbed component between the dispersed and continuous phases () Ad/A)

xsδ02/D λp ) xsδp2/D λd ) xsd2/Dd

λ0 )

Appendix Depending on the particle size distribution, the volume portion of particles with a particular size can strongly vary with size. Particles with different sizes in the gas-liquid boundary layer act simultaneously on the absorption rate. It is complicated task to take into account these size-dependent effects in mathematical equations. Here, three slightly different methods are discussed as possibilities for predicting the above effect. Model A. Model A is the simplest method shown here, but it also might be to the most approximate one. According to this method, the enhancement, Jhet,i(dp,i), calculated for a differential size range, ∆dp,i, is weighted by the differential volume element, ∆Vi(dp,i), corresponding to this size range. Thus, the average value of the enhancement, Jhet, can be obtained as

Jhet )

∫dd

p,max

p,min

Jhet(dp) dV(dp) ≡

∑i Jhet,i(dp,i) ∆Vi(dp,i)

(A1)

The mathematical function of ∆Vi(dp,i) vs dp,i must be known for the value of Jhet to be calculated, and eqs 1-4 must also be used. The physical representation of this


Figure A1. Illustrations of the physical representations of (a) model A and (b) model C.

mathematical model is illustrated in Figure A1a. The gas-liquid interface is divided into a large number of sections such that particles with a size of dp,i, corresponding to the differential volume element ∆Vi(dp,i), belongs to the ith section. These sections affect the absorption rate according to eq 4 independently of each other. The hold-up of the dispersed phase is the same for each section, that is, it is , and each section has uniformly sized particles. Model B. The same physical model is used for calculations in model B as in model A. Here, the absorption rate of the heterogeneous part of the total interface, obtained for particle size dp,i, is weighted with the differential heterogeneous interfacial area, ai. This area is proportional to the volume element ∆Vi(dp,i) according to the relation ∆ai(dp,i) ∝ ∆Vi2/3(dp,i). The equations used for these calculations are as follows

E)

Jhet jave ) + 1 - 1.2092/3 J J

Figure A2. Comparison of the different models (models A-C) at ψ ) 0.5 for a logarithmic normal distribution with the parameters listed in Table 2 ( ) 0.1).

affected by smaller particles are marked by dashed lines starting from the large particle. The absorption rates for these portions of the heterogeneous interface will be much higher than those for the other portions, where there are no small particles in the diffusion path to the large particle. One can take these additional effects into account by using the following calculation steps: Let M be the number of sections of differential particle size elements. In the Mth step of the calculation, going from dp,min, to dp,max (i.e., from i ) 1 to i ) M), the value of ave jdisp can be obtained as M

(A2) ave jdisp )

with

jave )

2/3 ∑i jave i (dp,i) ∆Vi (dp,i)

∑i ∆Vi

jave ∑ i (dp,i) τi(dp,i) i)1 M

(A4)

τi(dp,i) ∑ i)1 (A3)

2/3

(dp,i)

Model C. The shortcoming of the above models is that they do not take into account the interactions of particles of different sizes. Brilman et al.15 investigated this effect using a two-dimensional model with two- and three-particle configurations. They defined an interaction factor as the ratio of the actual enhancement to the sum of the contributions of the individual particles. In our one-dimensional model, we want to take into account the possibility that several small particles might be located between larger particles, which decreases the distance between the particles, as well as the distance of the first particle from the interface. This is illustrated in Figure A1b. The distance between particles, δp, strongly depends on the particle size [δp ) dp(0.806/1/3 - 1)]. As can be seen in this figure, several small particles can be located in front of a large particle. The absorbing component has to diffuse through these smaller particles to reach the larger one. Thus, the small particles actually decrease the distance between the larger particles, as well as the distance of the large particle and the interface; consequently, they increase the absorption rate. In Figure A1b, the regions of the heterogeneous interface where the absorption rate is

where i

τi(dp,i) )

∑ k)1

i-1

[∆Vk(dp,k)]2/3 -

[∆Vk(dp,k)]2/3 ∑ k)1

(A5)

The quantity τi(dp,i) represents the current area of the heterogeneous interface corresponding to the dp,i particle size fraction. The second term on the right-hand side of eq A5 gives the heterogeneous interfacial area occupied by particles smaller than dp,i, taking into account the calculation steps from k ) 1 to k ) i - 1. With this sum, the heterogeneous interfacial area corresponding to particles of size dp,i is reduced by the area that has already been taken into account in the earlier calculation steps. It follows from eq A5 that M

τi(dp,i) ) 2/3 ∑ i)1

(A6)

ave Given the value of jdisp , the enhancement can be calculated using eq A2. The simulated enhancements obtained using the above three models are compared in Figure A2 for a logarithmic normal distribution26 of particle sizes given by eq A7 at a value of ψ ) 0.5. The parameter values


used for the calculation are given in Table 2.

∆Vi(dp,i) )

∆dp,i ψdp,ix2π

(

exp -

)

(log dp,i - log d h p)2 2ψ2

(A7)

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Received for review February 3, 2003 Revised manuscript received July 31, 2003 Accepted August 5, 2003 IE030110P

Three-Phase Mass Transfer: Effect of the Size Distribution

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