Reactive Dissolution of Particle Clusters - ACS Publications

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Ind. Eng. Chem. Res. 2001, 40, 4050-4057

Reactive Dissolution of Particle Clusters Anurag Mehra,* P. Basu, and A. K. Suresh Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Bombay 400 076, India

The reactive (hydrolysis) dissolution of a solid ester suspended in alkaline, aqueous solutions has been investigated with respect to the major operating variables, such as the agitation intensity, alkali concentrations, and, most significantly, the solid reactant’s loading. The main objective of the study was to explore the effects of solid clustering on the conversion behavior of the system. It has been shown that the experimentally obtained conversions of the solid reactant can be explained by invoking the notion of a declining area for solid-liquid mass transfer due to the suspended particles forming clusters or agglomerates. A semiempirical framework, for analyzing the experimental data, which utilizes a power law type of rate expression for the loss of interfacial area due to clustering, is used to provide quantitative support for the proposed hypothesis. Further qualitative evidence of cluster formation has been given in images of the suspended particles as seen under an optical microscope. Introduction A variety of chemical processes involve the use of slurries of a solid reactant which reacts with another dissolved species, usually via a dissolution step. Typical examples include the alkaline hydrolysis of solid esters (aromatic benzoates), reactions between dissolved bases and solid acids (benzoic and salicylic acids), and the dissolution of silver halides in thiosulfate solutions (photography). More complex systems are encountered in gas-slurry processes such as the reaction between calcium hydroxide suspensions and carbon dioxide (manufacture of calcium carbonate) and sulfur dioxide (cleanup of sulfurous gases).1 Depending on the relative rates of the dissolution and reaction steps, the overall rate may lie in an operational regime where the solidliquid interfacial area, the mass-transfer coefficient, and the reaction rate constant are all important in determining its value. The motivation for using solid reactants usually arises from the fact that they are sparingly soluble in the reactive medium. In many instances, where an aqueous medium is used, it is likely that the particles will form clusters, driven by their hydrophobic interaction with water. The formation of such structures may be expected to lead to a reduction in the solid-liquid interfacial area and to a lesser extent in the mass-transfer coefficient as well, thus affecting the solid conversion-time trajectory. These effects on the reactor performance, arising out of reactant particle clustering, have not been reported in the literature. The role of particle clustering, agglomeration, or aggregate formation in multiphase reactor design has not been examined. Reactor size or batch time estimates based on some fixed, intrinsic value of the solid-liquid interfacial area can therefore be severely underpredicted. A model system, where the reaction course is dependent on the solid-liquid interfacial area and other transport parameters, was chosen to explore and probe such effects in reactive dissolution. The effects of stirring intensity and solid hold-up on the solid conversion were sought to be examined, incorporating the specific aspects associated with the clustering tendency of the solid reactant. Particle clusters may * Corresponding author. E-mail: [email protected].

form for a variety of reasons, such as, for instance, surface bonding between contacting solid interfaces or the trapping of smaller particles in the interstices formed by larger particles. The strength with which such particles are held together depends on the forces responsible for their “bonding”: weakly held clusters are usually called aggregates, while the strongly held ones are labeled as agglomerates. In this study, the reasons for cluster formation or the nature of the clusters formed have not been investigated. A survey of the literature reveals that there have been some efforts to study the effects of the particle size distribution on the dissolution rates.2,3 Bhaskarwar4 has discussed the extension of Smoluchowski’s theory to the agglomeration of dissolving (nonreactive) solid particles, but the focus has largely been on the agglomeration aspects such as collision rates, radius of coagulation, and so on. There are no studies available that relate the effect of particle agglomeration to the solid conversion or dissolution rates. The alkaline hydrolysis of solid esters has often been used to study various features of solid-liquid reactions.1 In the current context, preliminary experiments involving the hydrolysis of powdered phenyl benzoate suspended in an aqueous, alkaline medium at different solid loadings suggested that the solid-liquid area available per unit solid loading declined as the solids concentration was increased. Figure 1 shows an optical image of phenyl benzoate particles in water under stagnant conditions; the particles can be seen to form clusters. Visual inspection of mildly stirred suspensions also indicated that particle clustering persisted despite the stirring. It was therefore thought worthwhile to examine the reactive dissolution of phenyl benzoate in aqueous solutions of potassium hydroxide. Experimental Section Materials. Phenyl benzoate was prepared by the Schotten-Baumann reaction.5 In this method, phenol is reacted with benzoyl chloride in the presence of sodium hydroxide. The acid chloride was added in parts to a mixture of phenol and 5% (w/w) sodium hydroxide, with constant shaking. The mixture was further stirred,

10.1021/ie0007769 CCC: $20.00 © 2001 American Chemical Society Published on Web 08/23/2001

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4051

Figure 1. Optical micrograph of phenyl benzoate powder suspended in alkaline, aqueous solutions. Agglomerates can be seen clearly.

Figure 2. Optical micrograph of phenyl benzoate powder suspended in paraffin oil. Agglomerate formation is reduced very significantly.

after complete addition of the reactant, for periods exceeding 600 s, until the odor of benzoyl chloride could not be detected. The solid product obtained was filtered, washed with water, and recrystallized in ethanol. The other chemicals, including potassium hydroxide, were obtained from firms of repute. Distilled and deionized water was used for preparing all aqueous solutions. Procedure. A mechanically agitated contactor of 67 mm internal diameter, made of glass, was used to carry out the solid-liquid reaction. The contactor was provided with four baffles, and a six-bladed, glass turbine impeller of 23 mm diameter was used for agitation. The total liquid volume was kept constant at 150 cm3 for all of the runs, and the reactor was operated at three different agitation speeds, namely, 10, 20 and 30 rps. All of the experiments were carried out at a room temperature of 27 °C. The phenyl benzoate loadings used were 0.5, 0.667, 6.67, 40, and 100 kg/m3, whereas the initial concentrations of the aqueous potassium hydroxide solutions employed were 0.25, 0.5, 1.0, and 2.0 kmol/m3. The amount of the solid reacted up to any given point in time, in the above reactor, was estimated by the loss of weight method. At the end of the specified batch time, the reaction mixture was filtered and the solid so obtained was washed, dried, and weighed to assess the amount of solid hydrolyzed. The efficacy of this simple method was checked by measuring the quantity of phenol (product) formed up to the specified batch time using gas-liquid chromatography. The chromatographic measurements involve acidification of the sample and subsequent extraction of the phenol into tributyl phosphate. More details may be found elsewhere.5,6 A comparison between the phenyl benzoate conversions estimated by the weight loss and the chromatographic methods showed that the difference in the measurements was within a few percent. Some of the experiments were repeated to assess reproducibility, which was found to be within 10%. Physicochemical Properties. Table 1 lists the relevant physicochemical properties of the system.

The solubility of phenyl benzoate in the aqueous phase was measured by saturating a large volume of water with the ester. The loss of weight in the initial solid added was carefully estimated. A number of such experiments involving different solid loadings, saturation times, and water volumes were carried out. The reproducibility of these experiments was found to be excellent, and the solubility value given in Table 1 matches well with that reported earlier by Pandit and Sharma,7 i.e., 5.1 × 10-4 kmol/m3. The reaction rate for the alkaline hydrolysis of phenyl benzoate is known to be first order in the alkali as well as the dissolved ester concentrations and thus to be a second-order reaction overall.7 This second-order rate constant was determined by carrying out the reaction in a system of known solid-liquid interfacial area and the measured value is given in Table 1. The details pertaining to these measurements are given in the Appendix. Results and Analysis Figure 1 shows an optical micrograph of the powdered phenyl benzoate ester suspended in aqueous solutions of potassium hydroxide. It can be seen that the individual particles, of irregular shape but broadly cylindrical, form clusters in abundance. In contrast, Figure 2 shows that clusters do not form when these particles are suspended in an organic solvent like paraffin oil. Table 2 (columns 2, 5, 8, and 11) presents the experimentally obtained fractional conversions of phenyl benzoate due to alkaline hydrolysis, at a fixed time of 300 s, for various alkali concentrations, solid loadings of phenyl benzoate, and stirring speeds. It can be seen that the conversions go up with increasing alkali concentration. A rise in the conversion is also observed with increasing stirring speeds, whereas the conversions show a decrease with higher values of the solid ester loading. Figure 3 (symbols) shows the fractional conversions at different times, for some selected operating conditions. With increasing batch times, more of the phenyl benzoate gets converted. All of these observations are qualitatively consistent with what may be

Table 1. Values of Physicochemical Properties diffusion coefficient of phenyl benzoate, DA aqueous solubility of phenyl benzoate, C/A second-order rate constant for alkaline hydrolysis, k2

6.4 × 10-10 m2/s 5.0 × 10-4 kmol/m3 1.0 × 101 m3/kmol‚s

estimated by Wilke and Chang13 measured measured

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Table 2. Fractional Conversions of Phenyl Benzoate Due to Alkaline Hydrolysis [Experimental (expt), Computed (comp), and Those Computed without Including the Effect of Cluster/Agglomerate Formation (nocl)] at t ) 300 s, for Various Alkali Concentrations, Solid Ester Loadings, and Stirring Speedsa initial KOH concentration (kmol/m3) phenyl benzoate loading ls (kg/m3)

a

0.25 comp

expt

nocl

expt

0.50 comp

nocl

expt

1.00 comp

nocl

expt

2.00 comp

nocl

0.245 0.210 0.166 0.151 0.125

0.221 0.218 0.184 0.145 0.121

0.252 0.252 0.251 0.247 0.240

0.271 0.230 0.182 0.164 0.136

0.254 0.251 0.212 0.168 0.142

0.289 0.289 0.288 0.286 0.283

0.5 0.67 6.67 40 100

0.212 0.165 0.143 0.131 0.094

0.192 0.189 0.158 0.118 0.093

0.218 0.218 0.214 0.196 0.169

Stirring Speed ) 10 rps 0.232 0.202 0.230 0.186 0.199 0.230 0.157 0.168 0.228 0.139 0.130 0.219 0.108 0.106 0.205

0.5 0.67 6.67 40 100

0.280 0.250 0.157 0.149 0.132

0.255 0.251 0.203 0.145 0.108

0.296 0.296 0.289 0.253 0.201


0.301 0.280 0.178 0.164 0.155

0.275 0.271 0.223 0.170 0.139

0.320 0.320 0.318 0.309 0.295

0.320 0.299 0.193 0.181 0.165

0.300 0.295 0.244 0.188 0.156

0.347 0.347 0.346 0.342 0.336

0.5 0.67 6.67 40 100

0.341 0.270 0.171 0.170 0.152

0.301 0.296 0.236 0.163 0.118

0.355 0.354 0.344 0.292 0.220


0.371 0.320 0.184 0.190 0.173

0.317 0.312 0.253 0.189 0.152

0.373 0.373 0.371 0.358 0.337

0.380 0.330 0.202 0.197 0.183

0.337 0.331 0.270 0.204 0.168

0.395 0.395 0.394 0.388 0.378

Input data for computations taken from Tables 1 and 4.

expected intuitively. A quantitative assessment of the observed trends requires material balances combined with the theory of mass transfer with chemical reaction. The mass transport, in this problem, becomes important because the phenyl benzoate has to dissolve and diffuse into the reactive, alkaline liquid and then undergo the hydrolysis reaction. For a given set of monodisperse individual, nonclustering particles, it can be easily shown that the total particle area and volume (per unit volume of clear liquid) are related by

ap

) 0

ap

() Vp

V0p

R

Now, the rate of disappearance of the solid is described by the balance

(

(2)

where RA is the molar flux of the phenyl benzoate into the liquid phase through the available area, ap. The corresponding differential balance for the rate of disappearance of the solid-liquid interfacial area may be obtained by differentiating eq 1 with respect to time, and this results in

(1)

where R ) 2/3 for spheres and cubes and R ) 1/2 for long cylindrical particles. The superscript 0 denotes the values of the area and volume at the initial (batch) time, t ) 0.

)

Fp dVp ) RAap Mpw dt

a0p dVp dap ) 0 RrVR-1 p dt dt (V )

(3)

p

Substituting for dVp/dt in the above equation, from eq 2, gives

-

dap R Aa p a0p ) 0 RrVR-1 p dt Fp/Mpw (V )

(4)

p

This equation represents the rate of disappearance of the solid surface area due to the dissolution of the particles. The specific rate of dissolution, accompanied by chemical reaction, is described by the Danckwerts model as1

(

RA ) kslx1 + M C*A -

CbA 1+M

)

(5)

where M is the square of the Hatta number, defined as

M) Figure 3. Fractional conversions of phenyl benzoate due to alkaline hydrolysis: experimental (symbols), computed (lines) versus time for various alkali concentrations, solid ester loadings, and stirring speeds. Input data for computations taken from Tables 1 and 4. Frames A-C show the effects of stirring speed, initial concentration of the liquid-phase reactant, and solid loading, respectively.

DAk2CB ksl2

(6)

Here, DA is the diffusion coefficient of phenyl benzoate, CbA and CB are the aqueous concentrations of unreacted phenyl benzoate and potassium hydroxide, respectively, k2 is the second-order rate constant for the hydrolysis reaction and ksl is the solid-liquid mass-transfer coef-

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4053 Table 3. Regressed Values of the Mass-Transfer Coefficient, Initial (Total) Solid-Liquid Interfacial Area, and Initial (Specific) Interfacial Area for Each Combination of Stirring Speed and Solid Loadinga stirring speed, N 10 rps phenyl benzoate loading ls (kg/m3)

a

(m2/kg)

34.00 43.00 281.87 1328.40 3177.00

68.01 64.45 42.26 33.21 31.77

ksl × (m/s)

0.5 0.67 6.67 40 100

1.11 0.89 1.20 1.47 1.18

20 rps

atot p (m2)

104

as

ksl × (m/s)

104

1.53 1.36 1.32 1.88

30 rps

atot p (m2/s)

asp (m2/kg)

33.41 44.89 281.87 1220.40

66.82 67.30 42.26 30.51

ksl × (m/s)

104

1.85 1.38 1.52

atot p (m2)

asp (m2/kg)

35.77 50.67 267.07

71.53 75.97 40.04

Values for the lower, right side of the table are not uniquely determined by the regression technique.

ficient. Generally, the mass-transfer coefficient may be expected to be dependent on the particle size. However, Treybal8 has suggested, after a critical evaluation of the literature, that under well-agitated conditions this dependence is somewhat weak for particle sizes smaller than 2 mm; it can be shown that for the recommended correlation ksl ∝ dp-0.17. Therefore, in this analysis we assume ksl to be independent of the particle size. Some experimental evidence to support this assumption is given later in this paper. The expression for RA, given by eq 5, holds when xM , CB/zC*A, where z is the stoichiometric factor in

C6H5COOC6H5 + zKOH f C6H5OH + C6H5COOK and is equal to 1 for this reaction. This condition is valid for all of the experiments carried out in this study. The material balances for the concentrations CbA and CB are described by

(

)

CbA dCbA aps2 C*A [1 ) dt K s s + k2CB exp(-Khp)] -

apsk2hpCbA s + k2CB - k2CBCbA

(7)

and

-

dCB ) zRAap dt

(8)

respectively, with the accompanying conditions that at t ) 0, CbA ) 0, and CB ) CB0. Here, s ) ksl2/DA, which is the surface renewal frequency from Danckwerts’ masstransfer model, K ) x(s+k2CB)/DA and hp ) 5DA/ksl, which denotes the average penetration depth into a (liquid phase) surface element located at the solidliquid interface. Equation 7 essentially denotes a material balance for the exchange of solute A between fluid elements located at the interface and the bulk liquid and involves the integration of the time-averaged concentration profile of A in a penetration element. More details about deriving eq 7 may be found elsewhere.9 Equations 2, 4, 7, and 8 may now be solved simultaneously to obtain Vp, ap, CbA, and CB as functions of batch time t. The solid conversion is computed from

XA ) (1 - Vp/V0p)

(9)

In the above formulation, the values of the parameters a0p (reported as asp ) a0p/ls) and ksl are not known and may be obtained by “fitting” the above equations

to the experimental data given in Table 2. This exercise is an integral version of the well-known Danckwerts method for estimating the interfacial area and masstransfer coefficient in gas-liquid systems.1 A Levenberg-Marquardt search algorithm10 was used for the purpose of finding the “best fit” values of the above two parameters. The above-mentioned differential equations were solved simultaneously using the solver LSODE.11 Because the particles are of irregular shape and many different shapes exist simultaneously, R [the shape factor in eq 1] was set to 2/3 (sphere) for the sake of simplicity. To assess the sensitivity of the computations with respect to this shape factor, the value of R was set to 1/2 in the equations given above and the regression exercise carried out again; the estimated values of the regressed parameters are within 2% of those reported for the spherical shape in Table 3. This table shows the estimated values of the mass-transfer coefficients and the effective interfacial areas available for mass transport, as regressed from the experimental data for each specific combination of stirring speed and solid loading. From Table 3, it can be observed that the estimated mass-transfer coefficients show no particular trend with the solid loading of the ester and lie within a small range of values in a given column. This observation also suggests that these coefficients do not seem to depend on the solid loading and therefore by implication on the particle size, because the particle sizes may be expected to be larger at higher solid hold-ups due to greater clustering. This also provides support to the assumption stated earlier that the mass-transfer coefficients may be taken to be independent of the particle size for the analysis in this paper. The mass-transfer coefficients increase, as expected, with rising speeds of stirring. The initial interfacial areas available increase with higher ester hold-ups and show a remarkable consistency in value, across different stirring speeds. However, the important observation in this table relates to the decline in the initial area available per unit mass of the solid reactant with increasing loading of the solid. For nonclustering particles, this quantity may be expected to remain constant and independent of solid loading. Therefore, the fall in solid conversion with increasing solid hold-up, shown in Table 2, possibly arises out of a declining availability of area (per unit mass of the ester) for mass transfer and provides evidence of some kind of cluster formation leading to “blocking” of some of the solid surfaces. The observations made from the analysis, so far, only indicate that the initial area available per unit mass of the solid reactant seems to decline with the absolute value of the solid hold-up. However, the question remains whether this initial area is sufficient to explain

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Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 Table 4. Parameter Values Estimated by Fitting the Model to All of the Experi-mental Data parameter

value

asp n Asl g Acl h

78.43 1.369 2.760 × 10-5 0.537 1.375 × 10-4 0.256

units m2/kg m‚s-1/rpsg (m2/m3)1-n‚s-1/rpsh

To analyze all of the experimental data, across the entire spectrum of operating conditions, the masstransfer coefficient ksl can be decomposed as

ksl ) AslNg

(11)

This is a commonly used form in the literature. Treybal,8 for instance, recommends that ksl ∝ N0.62 for solid-liquid mass transfer for particles of size lesser than 2 mm. Similarly, the dependence of the cluster formation rate coefficient, kcl, on the stirring speed, N, can be written as Figure 4. Fractional conversions of phenyl benzoate due to alkaline hydrolysis: experimental (symbols), conventional theory without cluster formation (lines) versus time for some selected runs. Input data for computations taken from Tables 1 and 3.

the behavior of the system at higher times (beyond 300 s). The transient experimental data of Figure 3 may now be analyzed by using the parameter values of Table 3 in the equations given above. The results of this exercise are shown in Figure 4, for two typical data sets. At both of the ester loadings under consideration, the experimental conversions are much smaller than the computed values. This analysis seems to suggest that, for a given hold-up of the solid reactant, the solid-liquid interfacial area decreases continuously throughout the course of the experiment because of cluster formation. It is thus clear from the above discussion that a formulation which does not take into account the clustering of the solid reactant cannot describe the reaction behavior of this multiphase system. In view of the irregular shape of the particles as well as the illdefined nature of the particle clusters, it was thought desirable that, as a first approximation, we simply track the evolution of the total area available with the objective of quantifying the proposed hypothesis rather than develop detailed population balance based models. Now, the rate of loss of the available interfacial area (∝dp2) due to cluster formation is proportional to the projected area (also ∝dp2) for collisions, which suggests that in some manner the rate of area loss may be taken to be proportional to the existing value of the area itself. (A high value of the interfacial area would imply the presence of small, unclustered particles ready to agglomerate at high rates; conversely, a small value of the area would indicate a system where agglomeration has already occurred, thus having low agglomeration rates.) Generalizing this argument, eq 4 may therefore be modified to become

-

dap RAap a0p + kclapn ) 0 RRVR-1 p dt Fp/Mpw (V )

(10)

p

where the second term on the right-hand side denotes a power law type of expression for the reduction in the solid surface area due to cluster formation.

kcl ) AcINh

(12)

based on the typical forms used in the coalescence/ agglomeration literature.12 The mathematical formulation is now complete. With the objective of assessing whether the proposed description of clustering and related area loss are viable or not, all of the experimental data on solid conversion reached under different operating conditions were analyzed using the above equations (eqs 2, 7, 8, and 10). As before, a Levenberg-Marquardt search algorithm was used to regress the best estimates for the parameters asp (the intrinsic, unclustered surface area of the solid reactant per unit mass of solid), n (power law index for cluster formation), Asl and g (for the mass-transfer coefficient, ksl), and Acl and h (for the cluster formation rate coefficient, kcl). The differential equations were solved simultaneously using the solver LSODE. The value of the shape factor, R, was retained as 2/3. Table 4 gives the “best” values of the regressed parameters, as estimated from 105 experimental data points. The conversions, computed from the above equations using the values of parameters listed in Table 4, are shown in Table 2 (columns 3, 6, 9, and 12; for data points at 300 s) and Figure 3 (lines: selected data points including those at 300 s). It can be seen that the proposed approach, to account for loss of area available for interphase mass transport, is able to interpret the experimental measurements very well (with the exception of a few points). In contrast, the conversions predicted without including loss of area due to clustering, using the relevant parameters from Table 4, are much higher than their experimental counterparts. See Table 2 (columns 4, 7, 10, and 13) as well as Figure 4. The estimated extent of loss of interfacial area, as a fraction of the initial value, computed with and without including effects due to cluster/agglomerate formation is shown in Table 6 and Figure 5, for the various operating situations. The area lost for clustering particles is obviously much higher than that in the absence of such a phenomenon. Table 5 shows the estimated values of ksl and kcl at different speeds of stirring. The values of ksl reported in Table 5 and estimated by using all of the experimen-

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4055 Table 5. Values of the Mass-Transfer Coefficient and the Cluster-Forming Rate Constant at Different Operating Conditions N (rps)

ksl × 104 (m/s)

kcl × 104 [(m2/m3)1-n‚s-1]

10 20 30

0.95 1.38 1.71

2.48 2.96 3.29 23

Table 6. Solid-Liquid Interfacial Area Lost, as a Fraction of the Initial Area, Computed Assuming Only Reactive Dissolution (nocl) and Computed after Including the Effect of Cluster Formation (cl), at t ) 300 s, for Various Alkali Concentrations, Solid Ester Loadings, and Stirring Speedsa initial KOH concentration (kmol/m3)

phenyl benzoate loading ls (kg/m3)

nocl

0.5 0.67 6.67 40 100

0.151 0.151 0.149 0.135 0.116

0.5 0.67 6.67 40 100 0.5 0.67 6.67 40 100 a

0.25

0.50 cl

nocl

1.00 cl

nocl

2.00 cl

nocl

cl

Stirring Speed ) 10 rps 0.349 0.160 0.356 0.176 0.367 0.160 0.373 0.176 0.553 0.159 0.558 0.175 0.724 0.152 0.728 0.172 0.804 0.142 0.807 0.167

0.367 0.385 0.565 0.733 0.811

0.203 0.203 0.203 0.201 0.199

0.387 0.404 0.578 0.740 0.816

0.209 0.209 0.203 0.176 0.139


0.432 0.450 0.630 0.786 0.853

0.247 0.247 0.247 0.244 0.239

0.446 0.464 0.639 0.791 0.857

0.253 0.253 0.245 0.205 0.153


0.477 0.495 0.671 0.815 0.876

0.285 0.285 0.284 0.280 0.272

0.488 0.506 0.678 0.819 0.879

Input data for computations taken from Tables 1 and 4.

rising stirring speeds are as expected. Interestingly, the values of kcl indicate that the clustering rate goes up with the stirring speed possibly because more collisions, leading to greater agglomeration, occur per unit time, and the likely increase in breakage rates with increased stirring does not seem to compensate or overtake the increased agglomeration rates. Over the entire operating range, the largest value of M is 1.1, which indicates that for some operating situations there is significant reaction near the solidliquid interface as well as in the bulk liquid. The smallest value of about 0.2, on the other hand, indicates that in some cases all of the hydrolysis proceeds in the bulk liquid. The largest bulk concentration of the ester, CbA, is estimated to be around 10% of the solubility of the ester in the aqueous phase but is practically negligible for a majority of the experiments. The highest alkali conversion reached was around 25%. Conclusions The case of a solid-liquid reaction, where the solid has a tendency to form clusters or agglomerate, has been examined. It has been shown that such behavior can severely restrict the area available for diffusion and reaction to occur. Not accounting for clustering and resultant area loss may lead to great overprediction of the reactant conversions or equivalently an underdesigned reactor or batch times. The proposed analysis also suggests a way of using Danckwerts’ method for estimating mass-transfer coefficients and interfacial areas for systems containing solids with clustering tendencies. There are a few studies in the literature that deal with the effect of a reactant particle distribution on the dissolution rates in the presence and absence of chemical reactions. However, the effect of agglomeration or clustering on the reactant conversion has not been studied so far. This work uses a semiempirical approach to account for changes in the interfacial area, due to particle clustering or agglomeration, in the reaction behavior of the system. Appendix: Determination of the Second-Order Rate Constant for the Alkaline Hydrolysis of Phenyl Benzoate

Figure 5. Solid-liquid interfacial area lost, as a fraction of the initial area, computed assuming only reactive dissolution (thin lines) and that computed after including the effect of cluster formation (thick lines), versus time, for various alkali concentrations, solid ester loadings, and stirring speeds. Input data for computations taken from Tables 1 and 4. Frames A-C show the effect of stirring speed, initial concentration of the liquid-phase reactant, and solid loading, respectively.

tal data match well with those reported in Table 3 which have been computed from the solid conversion data at different specific combinations of solid hold-up and stirring speeds. Also, the fitted value of asp, mentioned in Table 4, seems to be reasonable when compared to the values of this quantity reported in Table 3 for the low solid loading zone. The increasing values of ksl with

Molten phenyl benzoate was poured into a glass beaker and allowed to solidify at the base of the vessel in the form of a circular solid plate having a crosssectional area identical with that of the glass beaker. The exposed surface of this solid plate was polished with sand paper to smoothen it and make it completely flat. Aqueous solutions of potassium hydroxide, of a known initial strength, were then introduced into this glass vessel at room temperature, and the liquid phase was stirred for a period of 600 s. The total amount of product, phenol, formed up to this time was measured by chromatography. The initial potassium hydroxide concentrations used here were 0.5, 1.0, and 2.0 kmol/m3, and the contactor was operated at two stirring speeds, namely, 25 and 41.7 rps. The volume of the liquid solution was kept at 100 cm3 for all of the experiments, and the solid-liquid contact area was taken to be essentially the cross-sectional area of the reaction vessel which was of 55 mm internal diameter, with the area thus being 23.7 cm2.

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The rate of formation of phenol is described by the equation1

dCP/dt ) kflslC*AxM(a/VL)

(A1)

with the conditions t ) 0 and CP ) 0 and where xM ) fl fl the physical xDAk2CB/ksl is the Hatta number, ksl is mass-transfer coefficient, and DA and C*A are the diffusion coefficient and the solubility of phenyl benzoate, in the liquid phase, respectively. CB is the prevailing concentration of the alkali in the aqueous phase, and a/VL is the solid-liquid contact area per unit volume of the liquid phase. The amount of phenol formed in 600 s was found to be small so that the consumption of the hydroxide on this account, as compared to its initial concentration, is negligible. Therefore, CB in the Hatta number may be replaced by its initial concentration CB0 and eq A1 may be integrated very simply, followed by rearrangement, to give

k2 )

(CfP)2 (C*A)2DA(a/VL)2tf2

(A2)

where the subscript f denotes the end time of the batch run. The mass-transfer coefficient does not enter the calculation of k2, but the condition of validity of eq Al) states that

xDAk2 > 3 kflsl

(A3)

The mean value of k2 determined from the above expression, using six experiments (2 stirring speeds × 3 KOH concentrations), was found to be 10 m3/kmol‚s, with the maximum difference between the lowest and the highest estimates being about 7%. The mass-transfer coefficients, at the two agitation speeds used for the flat interface contacting, were found by replacing the phenyl benzoate plate with an identical plate of benzoic acid and carrying out its physical dissolution into pure water. This method of measurement is necessitated because of the very low solubility of phenyl benzoate in water so that the aqueous phase concentrations of the benzoate cannot be measured conveniently. In contrast, the solubility of benzoic acid is substantial enough for concentrations of this order of magnitude to be estimated by simple titration procedures. A straight line plot of -ln(1 - C/C*) versus t gives kflsl(a/VL) as the slope. Here, C is the concentration of the benzoic acid measured at different times and C* is the solubility of benzoic acid in water. The kflsl values, obtained from the above procedure, after correction for the different diffusion coefficients for the solutes, phenyl benzoate and benzoic acid, are 1.5 × 10-5 and 2.1 × 10-5 m/s for the agitation speeds of 25 and 41.7 rps, respectively. Substituting these values of kflsl and the value of k2 mentioned above, the condition given by eq A3 is found to be valid for all of the experiments done in the current context. Nomenclature Acl ) constant defined in eq 12, (m2/m3)l-n‚s-1/rpsh Asl ) constant defined in eq 11, ms-1/rpsg a ) cross-sectional area of a flat-interface cell, m2

ap ) solid-liquid interfacial area per unit volume of the aqueous phase, m2/m3 a0p ) initial solid-liquid interfacial area per unit volume of the aqueous phase, m2/m3 s ap ) intrinsic solid surface area per unit mass of solid reactant, m2/kg 2 atot p ) total solid-liquid interfacial area, m b CA ) bulk aqueous phase concentration of A, kmol/m3 C/A ) solubility of A in the liquid phase, kmol/m3 CB ) liquid-phase concentration of B, kmol/m3 CB0 ) initial concentration of B in the liquid phase, kmol/ m3 CP ) concentration of phenol in a flat-interface cell, kmol/ m3 f CP ) final concentration of phenol in a flat-interface cell, kmol/m3 DA ) diffusivity of A in the liquid phase, m2/s dp ) reactant particle diameter, m g ) power index in eq 11 h ) power index in eq 12 hp ) average penetration depth into a penetration element, m k2 ) second-order rate constant in the aqueous phase for reaction between A and B, m3/kmol‚s K ) constant defined in eq 7, m-1 kcl ) cluster/agglomerate formation rate constant, (m2/ m3)1-n‚s-1 ksl ) mass-transfer coefficient for solid dissolution, m/s kflsl ) mass-transfer coefficient for solid dissolution in a flat-interface cell, m/s ls ) solid reactant loading per unit volume of the liquid phase, kg/m3 M ) Hatta number squared, defined by eq 6 Mpw ) molecular weight of the solid reactant, kg/kmol N ) speed of agitation, rps n ) power index in eq 10 q ) CB/zC/A; stoichiometric ratio RA ) specific rate of dissolution, kmol/m2‚s s ) surface renewal frequency from Danckwerts’ model, s-1 t ) batch time, s tf ) end time for a flat-interface run, s VL ) volume of the aqueous phase, m3 Vp ) volume of suspended solid per unit volume of the aqueous phase V0p ) initial volume of suspended solid per unit volume of the aqueous phase XA ) fractional conversion of A Xarea ) fractional area of solid A lost A z ) stoichiometric factor for reaction A + zB f products Greek Symbols R ) shape factor in eq 1 Fp ) density of the solid reactant, kg/m3

Literature Cited (1) Doraiswamy, L. K.; Sharma, M. M. Heterogeneous Reactions; Wiley: New York, 1984; Vol. 2. (2) LeBlanc, S. E.; Fogler, H. S. Population balance modeling of the dissolution of polydisperse solids: rate-limiting regimes. AIChE J. 1987, 33, 54-63. (3) Dabral, M. M.; Roy, S.; Bhaskarwar, A. N. General kinetic invariant model of dissolution of large polydisperse particles. Chem. Eng. J. 1996, 61, 161-170. (4) Bhaskarwar, A. N. Extension of Smoluchowski’s theory to the agglomeration of dissolving solid particles. ACHsModels Chem. 1994, 131, 809-817. (5) Vogel, A. I. Practical Organic Chemistry; Longman’s Green: New York, 1962.

Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4057 (6) Gunter, Z.; Sherma, J. CRC Handbook of Chromatography, General Data Principles; CRC Press: Boca Raton, FL, 1984. (7) Pandit, A.; Sharma, M. M. Intensification of heterogeneous reaction through hydrotropy: Alkaline hydrolysis of esters; oximation of cyclodode-canone. Chem. Eng. Sci. 1987, 42, 2517-2523. (8) Treybal, R. E. Mass-Transfer Operations; McGraw-Hill: Singapore, 1981; pp 602-603. (9) Mehra, A. Intensification of multiphase reactions by using microphases. In Handbook of Heat; Mass Transfer, Advances in Reactor Design; Combustion Science; Cheremisinoff, N. P., Ed.; Gulf Publishing: Houston, TX, 1990; Vol. 4, Chapter 16. (10) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C; Cambridge University Press: New Delhi, 1992.

(11) Radhakrishan, K.; Hindmarsh, A. C. Description: use of LSODE; Reference publication NASA RP-1327; NASA: Washington, DC, 1993. (12) Wright, H.; Ramkrishna, D. Factors affecting coalescence frequency of droplets in a stirred liquid-liquid dispersion. AIChE J. 1994, 40, 767-776. (13) Wilke, C. R.; Chang, P. Correlation of diffusion coefficients in dilute solutions. AIChE J. 1955, 1, 264-270.

Received for review August 24, 2000 Revised manuscript received May 10, 2001 Accepted May 15, 2001 IE0007769

Reactive Dissolution of Particle Clusters - ACS Publications

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