Iran. J. Chem. Chem. Eng. Vol. 24, No.4, 2005

Transcription

1 Iran. J. Chem. Chem. Eng. Vol. 4, No.4, 5 The Contribution of Molecular Diffusion in Silica Coating an Chemical Reaction in the Overall Rate of Reaction of Aluminum Hyroxie with Fluosilicic Aci Bayat, Mahmou * + an Taeb, Abbas Faculty of Chemical Engineering, Iran University of Science an Technology, P.O.Box , Tehran, I.R.IRAN Rastegar, Saee Faculty of Polymer, Amirkabir University of Technology, P.O.Box , Tehran, I.R.IRAN ABSTRACT: The kinetic of the heterogeneous chemical reaction of aluminum hyroxie an fluosilicic aci was stuie. It was foun that the iffusion of the reactants through the porous silica coating to the aluminum hyroxie surface an the interfacial chemical reaction between the iffusing reactant an aluminum hyroxie platelets control the overall reaction rate. These two phenomena were stuie an their contributions to the overall reaction rate were erive using experimental ata. By combining these terms a relation for the overall reaction rate was obtaine. The activation energy of the chemical reaction was calculate to be kcal/mol an the activation energy of the iffusion into the silica coating was foun as 8 kcal/mol. A numerical proceure was ajuste to etermine the variation of the specific surface area of un-reacte core, its average particle size an the specific surface area for mass transfer. KEY WORDS: Heterogeneous reaction, Aluminum hyroxie, Fluosilicic aci, Aluminum fluorie, Silica, Effective iffusion coefficient, Aggregation, Kinetics. INTRODUCTION The reaction of fluosilicic aci with aluminum hyroxie is a heterogeneous one in which an aqueous solution of fluosilicic aci reacts with aluminum hyroxie power an leas to a solution of aluminum fluorie an silica precipitates. H SiF 6 + Al (OH) Al (SiF 6 ) + 6H O () Al (SiF 6 ) + 6H O AlF + SiO + HF () Accoring to a mechanism, propose by Skrylev [], * To whom corresponence shoul be aresse. + mmahmoo_in@ yahoo.com -9986/5/4/5 /$/. at first the water-soluble aluminum fluosilicate is forme, which is then hyrolyze to aluminum fluorie, silica precipitates an fluoriric aci. Grobelny [] stuie the kinetics of this reaction an propose an empirical first-orer rate equation(eq.) for the overall reaction, in which the iniviual contributions of the above mentione phenomena, was not taken into account. r = k C L () 5

2 Iran. J. Chem. Chem. Eng. Bayat, M., et al. Vol. 4, No.4, 5 log k = 9. 5/T - log(a) (4) In the authors previous paper [] have presente the filtration rate of the silica precipitates as relate to their morphology. It has been foun that in the course of reaction the prouce silica particles asorb on aluminum hyroxie platelets in an aggregate state. The morphology of the aggregates, which has a profoun influence on the filtration rate of the silica precipitates, is influence by the reaction conitions. On the other han, at every stage of the reaction, the overall reaction rate is influence by the morphology of the silica coating. In the other wors, the reaction conitions ictate the morphology of the silica coating an this factor influences the overall reaction rate in turn. In this paper, the results of the experimental stuy of the overall reaction kinetics are presente. This reaction is a complex one in which, two phenomena, the chemical reaction an the molecular iffusion of the mobile reactant in the silica coating are controlling the overall reaction rate. The contribution of the above-mentione phenomena in the overall reaction rate has been consiere. The overall reaction rate is necessary for reactor esign which is comprise of etermining the reactor size, the type an size of propeller, rotational spee of agitator an operating conitions of reaction. EXPERIMENTAL Chemicals Fluosilicic aci was prepare by the reaction of the analytical grae silica gel an the technical grae fluoriric aci. Aluminum hyroxie was of technical grae having a purity of 99% an an average particle size of 5 microns. All of the reactants were use as supplie an without any further purification. Apparatus Accoring to Fig. the 5 ml reaction mixture was mixe using a igitally-controlle mechanical stirrer in a ml reactor. The reactor was heate by a hot water bath equippe with an accurate thermoregulation system. A platinum crucible was use for the alkali fusion reactions. The electron micrographs were taken using a Topcon SR5 scanning electron microscope. Particle size micro analyzer A Fritch has been use for size istribution measuring of Al(OH) power. 6 rpm C C Fig. : Experimental Setup. ) Digitally-controlle mechanical Stirrer (heiolph RZR), ) Digitally-Controlle Hot Water Bath (Falk Sb5), ) Reactor (Diameter = cm & Volume = ml), 4) Teflon Propeller (Diameter = 6 cm & Type = Four turbine Blae), 5) Digitally Thermometer. Experimental Proceure In each run, the first sample was taken after secons an the next ones in minutes intervals. The volume of each sample was nearly ml. Each sample was filtere an the filtrate was analyze for aluminum concentration. In orer to etermine the concentration of aluminum in the filtrate, a ml sample was rawn from the filtrate an evaporate in a platinum crucible uner gentle heating for minutes. The aluminum fluorie rie completely an was expose to alkali fusion. The aluminum content was then etermine by reverse titration using EDTA [4]. Final prepare cakes of several experiments have been use for SEM stuies. Each experiment repeate three times an the experimental error was calculate to be less than %. RESULTS AND DISCUSSION The reactions were performe at 6, 7 an 8 C. The initial concentration of fluosilicic aci was selecte.76 an.8 mol/l. The initial concentrations of aluminum hyroxie were an 8 g/l. The stirrer spee was set at various spee 4, 9, 6 an rpm. In all experimental runs, reactants were use in the stoichiometric ratio an only in some runs one of them (fluosilicic aci of Aluminum hyroxie) was 6% excess. The reaction time was epene on the reaction temperature an varie 5 to 6 minutes. Figs. an show the SEM results of the aluminum hyroxie platelets before the reaction an after 6

3 Iran. J. Chem. Chem. Eng. The Contribution of Molecular Vol. 4, No.4, 5 Fig. : SEM micrograph of aluminum hyroxie particles. Fig. : SEM micrograph of Coate particles of aluminum hyroxie after minutes of reaction onset. minutes of the reaction. A silica coating is forme on the aluminum hyroxie platelets after the start of the reaction (shown in Fig.). The reaction of aluminum hyroxie with fluosilicic aci procees to form the unstable aluminum fluosilicate intermeiate, which is immeiately hyrolyze to aluminum fluorie, silica an fluoriric aci. Because the intermeiate prouct is unstable, it has not enough time to iffuse backwar to the outer surface of the silica coating, an hence the reaction procees at the surface of the aluminum hyroxie core. The latter phenomena is similar to the unreactecore moel theory [5]. The mass- transfer rate in the particle bounary layer, using equations (5) an (6) [6], has been calculate to be.68 mol/l.s [7](with emphasis on C = C L ), while by using the experimental ata, the overall reaction rate has been measure about - mol/l.s. Sh = +.5 Re.5 Sc / (5) r b.l = k m a C (6) Therefore the mass-transfer rate in the particle bounary layer is about 4 times greater than the overall reaction rate. It may be euce that the molecular iffusion of the reactant in the silica coating an the interfacial chemical reaction control the overall reaction rate. To etermine the contribution of the above-mentione phenomena in the overall reaction rate, it may be propose that the overall reaction rate is the resultant of the rates of the iniviual phenomena. The rate of the reaction at every stage was etermine by fitting to the experimental ata. Rate of the Chemical Reaction The rate of the chemical reaction can be measure at the start of the reaction only, when no silica coating has been forme on the aluminum hyroxie particles. Sampling through the first -secon of reaction startup may help to etermine the accurate slope of the rate curve at the initiation of reaction. The rate equation of the chemical reaction was consiere to be as follows: r ch = k exp(-ea/rt) C S α a β (7) The coefficients of this equation can be etermine from the exp. values of the reaction rate at the start of the reaction. To etermine the exponents of Eq. (7), we stuie the variation of the reaction rate by varying one of the parameters each time only. Fig. 4 shows the variation of the concentration of aluminum fluorie versus time at 6, 7 an 8 C. Base on these exp. results the activation energy of the reaction has been calculate to be kcal/mol while Grobelny [] suggeste a value of 6 kcal/mol. This value correspons to the activation energy of the reaction obtaine from the mi points of the curves in Fig. 4. In other wors, Grobelny [] was reporte an overall activation energy, which is not a reliable value for the chemical reaction. Fig. 5 shows the variation of aluminum fluorie concentration versus time at two ifferent fluosilicic aci concentrations. Again base on the slope of the curves at t =, the exponent α was calculate to be.864. For the sake of simplicity the reaction orer was suppose to be (as in Grobelny s equation (Eq.)). 7

4 Iran. J. Chem. Chem. Eng. Bayat, M., et al. Vol. 4, No.4, 5 Fig. 6 shows the variation of the concentration of aluminum fluorie versus time for two ifferent aluminum hyroxie concentrations. For the etermine the exponent β, the slope of the curves at t = is necessary. The problem in this case is to fin the specific surface area of the aluminum hyroxie at the course of reaction. We have use a numerical moel to relate the variation of the specific surface area to the concentration of aluminum hyroxie [7]. The algorithm of the calculations is presente in Appenix A. since the specific surface area has been obtaine as a function of aluminum hyroxie concentration, it is efine by surface area of the particles in unit volume of the slurry (m /m ). The correlation equations for two initial concentrations of an 8 g/l are given below: a =.5 exp(.7 C (8) C Al(OH) = g/l Al(OH) a = 648.exp(.758 C (9) C Al(OH) = 8 g/l Al(OH) Regaring the values, the exponent β was etermine to be.5 an the rate constant (k ) to be,. Therefor, the rate equation of the chemical reaction is obtaine as follows: r ch = 4 exp(-64/t) C S a.5 () Effective Diffusion Coefficient in Silica Coating The silica coating forme uring the chemical reaction is compose of submicron silica particles. So, this coating can be consiere as a porous layer. The porosity of the latter epens on the chemical reaction parameters an hyroynamic conitions. In other wors, as reporte in a previous stuy [] the forme silica particles may stick to each other an form layers of ifferent compactness. Therefore the iffusion in the silica coating may obey the equations of the iffusion in a porous soli. In a porous soli, the soli itself is completely enriche by the liqui an the molecular iffusion in the trappe liqui (through the pores) shoul be the mechanism of the mass transfer in the pores of the soli [8]. Because the iffusion surface an iffusion path length can not be exactly etermine, an equivalent iffusion surface an path length of iffusion are efine, where an effective iffusion coefficient is obtainable [8]. It is normally to suppose the outer surface of the soli as C AlF (mol/l).5.5 T = 8 C T = 7 C T = 6 C t (s) Fig. 4: Effect of temperature on the reaction rate of aluminum hyroxie with fluosilicic aci [H SiF 6 ] =.76 mol/l, [Al(OH) ] = g/l, N = rpm. C AlF (mol/l) [H SiF 6 ] =.8 mol / L [H SiF 6 ] =.76 mol / L 5 5 t (s) Fig. 5: Effect of Fluosilicic aci concentration on the reaction rate of aluminum hyroxie with fluosilicic aci [Al(OH) ] = 8 g/l, T= 7 C, N=6 rpm..6 [Al(OH) ] = 8 g / L.4 [Al(OH) ] = g / L. C AlF (mol/l) t (s) Fig. 6: Effect of aluminum hyroxie concentration on the reaction rate of aluminum hyroxie with fluosilicic aci [H SiF 6 ] =.76 mol/l, T = 7 C, N = 6 rpm. 8

5 Iran. J. Chem. Chem. Eng. The Contribution of Molecular Vol. 4, No.4, 5 the equivalent iffusion surface an the porous film thickness as the equivalent iffusion path length [8]. In theas the mass transfer rate in the silica coating may be efine by Eq.() [8]. Deff rm.t = a (CL CS ) () R P rc To etermine the values of the effective iffusion coefficient from the experimental ata, by supposing the steay state conitions, the chemical reaction an mass transfer rates for any thickness of silica coating may be equalize. r = r ch = r m.t () Deff r = a(cl CS) = exp ( )CS a () R r T P C By eliminating the C S, the overall reaction rate is given by Eq.(4). C L r = (4) (R P rc ) + 64 D 4.5 eff a exp ( ) a T Using the experimental ata of the overall reaction rate, the value of the effective iffusion coefficient was calculate. The values of the r c, a an a were obtaine using a numerical moel given in Appenix A. In the latter the best-fit equation to the obtaine values is presente for the initial concentrations of aluminum hyroxie of an 8 g/l. R P is the initial average particle size of Al(OH) power. Accoring to the unreacte core moel, R P was assume to be constant uring the course of reaction. The values of the effective iffusion coefficient may be obtaine using the experimental reaction rate (r) values as follows exp ( ) a r D = T eff (5) a exp ( ) a CL r R r T P C Normally, the effective iffusion coefficient is the same at all points of the porous layer. However in this case it has ifferent values at ifferent positions along the film. This can be attribute to the in-homogeneity of the ensity of porous film along its thickness. in-turn, the in-homogeneity in ensity is cause by the inhomogeneity in the fractal imension of the silica aggregates of which the porous layer is compose. This is because the aggregates forme uring the chemical reaction are forme in ifferent chemical conitions, since the fluosilicic aci is consume uring the reaction an the ph value of the solution rises too. At the start of the reaction the concentration of the fluosilicic aci is high an therefore the ph of the reacting liqui is very low (ph =.5). During the reaction the aci is consume an the ph value rises to.-.5. Comparing these values with the iso-electric point of silica an aluminum hyroxie surfaces (. an 9. respectively) it can euce that at the start of the reaction both silica an aluminum hyroxie surfaces are positively charge. During the chemical reaction as the ph value increases, there will be a transition point (ph=.), at which the net surface charge of the silica particles changes from positive to negative.as can be seen in Fig. 7, the variation of the effective iffusion coefficient (calculate by Eq. (5)) is not so sharp at ph =.. Therefore it can be conclue that the aggregation of the particles at ph < is of the same character of the aggregation at ph >. In other wors the aggregation at ph < is a heteroaggregation inuce by ifference between charge ensity of the aluminum hyroxie an silica particles an the aggregation at ph > is a hetero-aggregation inuce by ifference in the net charge of the surfaces [9]. Fig. 7 shows that the effective iffusion coefficient or in other wors, the permeability of silica coating is increase in the course of reaction by increasing ph. From this curve an the above iscussion one can conclue that the aggregation of the particles has a more RLCA nature at the start of the reaction an it changes graually to a more DLCA nature at the en of the reaction. In general, the aggregation may cause to the formation of aggregates having two extreme fractal imensions,.7 for DLCA an. for RLCA []. Xiao- Yan an Logan [], Gmachowski [] an Romm [] have stuie the permeability of the fractal objects. As a general conclusion a power law relationship was foun between the ensity or packing factor of the fractal object an the ratio of the size of the primary particles an the aggregates. The exponent of the relationship epens on the fractal imension. As the fractal imension ecreases (a more DLCA nature), the permeability of the aggregates increases. In Fig. 7 we observe a continuous increase in 9

6 Iran. J. Chem. Chem. Eng. Bayat, M., et al. Vol. 4, No.4, 5 the effective iffusion coefficient as the reaction procees. In other wors, the silica coating is a film the ensity of which ecreases as the istance to the aluminum hyroxie surface ecreases. The next step is to erive a general equation for the effective iffusion coefficient which may show the epenency of this parameter on the temperature, fluosilicic aci concentration (ph), the number of particles in the slurry (the initial concentration of aluminum hyroxie), an the imensionless stirrer spee (the ratio of the stirrer spee to the critical stirrer spee for complete suspension). This equation may be given by: D eff = D exp (-Ea/RT) C L δ n t γ (N/Nc) η (6) The experimental ata presente in Figs. 4,5,6 an 8 within the aluminum fluorie concentration range of. to.9 mol/l were use to etermine the exponents of Eq. (6). This is because at concentrations less than. mol/l the silica coating forme is not enough thick an it has not any consierable effect on the overall reaction rate. At concentrations greater than.9 mol/l, the high film thickness of the silica coating may cause the film to crack. In orer to etermine the exponents of Eq. (6), the variation of the effective iffusion coefficient was consiere by varying one of the parameters while other parameters were fixe. Effect of Temperature As may be observe from Fig.4, one can fin a profoun effect of temperature on the effective iffusion coefficient. Accoring to Table the average activation energy of iffusion has been obtaine to be 8 kcal/mol, which is more than times of the activation energy of the chemical reaction. This is the outcome of both increasing the iffusion coefficient of the reactants in the meia an to a much more extent, increasing of the porosity of silica coating ue to higher reaction rate. When temperature an hence the reaction rate increases, the rate of prouction of silica particles increases an so the letter have little time to experience ifferent positions for sticking to each other. On the other han, the increase rate of the chemical reaction can cause a faster ph change an so the sticking probability increases. This can cause the formation of the aggregates having a more porous structure (more DLCA nature) an so a higher effective iffusion coefficient. Table : The effective iffusion coefficient use for Calculation of activation energy at ifferent temperatures (C L=.76 mol/l, C Al(OH)= g/l, N=6 rpm). D eff (m /s) C AlF mol/l T ( C) T ( C) D eff (m /s) D eff (m /s) Ea kcal/mol E-.E- 8.E- 6.E- 4.E-.E-.E [H SiF 6 ] (mol/l) Fig. 7: Variation of effective iffusion coefficient (calculate by eq. (5)) versus fluosilicic aci concentration. C alf (mol/l) N = 9 rpm N = 6 rpm t (s) Fig. 8: Effect of stirrer spee on the reaction rate of fluosilicic aci wit h aluminum hyroxie. [H SiF 6 ] =.76 mol/l, [AL(OH) ] = g/l, T = 7 C.

7 Iran. J. Chem. Chem. Eng. The Contribution of Molecular Vol. 4, No.4, 5 C AlF (mol/l) (Experimental Data) C AlF (mol/l) (Calculate) C AlF (mol/l) (Experimental Data) C AlF (mol/l) (Calculate) Run7: T=6 C, [H SiF 6 ]=.8 mol/l, Al(OH) =8 g/l, N=6 rpm Run8: T=7 C, [H SiF 6 ]=.8 mol/l, Al(OH) =8 g/l, N=6 rpm Run9: T=8 C, [H SiF 6 ]=.76 mol/l, Al(OH) =8 g/l, N=9 rpm Run: T=6 C, [H SiF 6 ]=.8 mol/l, Al(OH) =8 g/l, N=9 rpm Run: T=8 C, [H SiF 6 ]=.8 mol/l, Al(OH) =8 g/l, N=9 rpm Run: T=7 C, [H SiF 6 ]=.8 mol/l, Al(OH) =8 g/l, N=9 rpm Fig. 9: Exp. vs. calculate of concentration of AlF for runs 7,8,9. Effect of Fluosilicic Aci Concentration (ph) As mentione earlier, the concentration of the fluosilicic aci has a pronounce effect on the porosity of the silica coating. Regaring the Exp. ata presente in Fig. 5 the mean value for the exponent of the fluosilicic aci concentration term (δ) is. (Eq. (6)). Effect of the Number of Particles in the Slurry (Aluminum Hyroxie Concentration) Regaring the Exp. ata presente in Fig. 6 it can be conclue that by increasing the concentration of the aluminum hyroxie in the slurry, the effective iffusion coefficient ecreases. Eq. (7) relates the number of the isperse particles to their concentration. C n t = π p ρs 6 Al(OH) (7) As a outcome of increasing the concentration of the particles in the slurry, the number of inter-particle impacts increases. This may cause the silica coating to rearrange an become more compact. An increase in the compactness of the silica coating leas to a ecrease in the effective iffusion coefficient. Regaring the Exp. ata presente in Fig. 6, the exponent γ was etermine to be. (Eq. (6)). Fig. : Exp. vs. calculate of concentration of AlF for runs,,. Effect of Stirrer Spee To generalize the stirrer spee term in Eq. (4), a imensionless term, which is the ratio of the stirrer spee to the critical stirrer spee was efine [4]. The latter has the benefit of taking into account the effect of the istance of the propeller to the bottom of the reactor an the ratio of the iameter of the propeller to the iameter of the container. Otherwise by using the Reynols number it will loss this benefit. Regaring the Exp. ata presente in Fig. 8, it may be seen that an increase in the stirrer spee, causes a ecrease in the effective iffusion coefficient. Because the stirrer spee oes not influence the rate of chemical reaction, the negative effect of this parameter on the mass transfer rate is obvious in the overall reaction rate. Regaring the Exp. ata presente in Fig. 8, the exponent η is etermine to be.. From the experimental ata presente in Figs. 4-6 an 8, one can obtain the equation constant D as 4.7. Regaring to the abovementione values, Eq. (6) becomes: D eff = 4.7 exp (-4/T) C L -. n t - (N/Nc) -. (8) To check the valiity of the relations obtaine using the previous ata, a number of experimental runs were performe, from which the results of 6 experimental runs are presente in Figs. 9 an. Using Eq. (4) an the

8 Iran. J. Chem. Chem. Eng. Bayat, M., et al. Vol. 4, No.4, 5 initial conitions of the 6 experimental runs, the overall reaction rate was calculate an its value was use to preict the concentration of aluminum fluorie versus time. The obtaine values were compare with the experimental ata. where the erive relation fits the experimental ata satisfactorily. The small eviations from the moel at the en of the reaction may be attribute to the formation of cracks in the silica coating. CONCLUSIONS A physical moel for the heterogeneous reaction of aluminum hyroxie an fluosilicic aci was propose. Regaring the moel, the controlling mechanism at the early stages (first secons) of the reaction is the chemical reaction itself. After that porous silica coating starts to form an the iffusion of the fluosilicic aci into the coating becomes important. The porosity of the silica coating increases as the chemical reaction procees an the ph value increases. In other wors, the nature of the aggregation is varie from RLCA-like at the start of the reaction to DLCA-like at the en of the reaction. The effective iffusion coefficient is influence by the temperature an inversely by the stirrer spee, fluosilicic aci an aluminum hyroxie concentrations. Acknowlegement The authors thank so much from Research & Technology Company of National Iranian Petrochemical Company for financial support of the project. Also the authors thank Miss Anvary for her help in the experimental work. APPENDIX (A) The moel use to calculate the variation of specific surface area of aluminum hyroxie particles uring the chemical reaction In orer to calculate the variation of specific surface area of aluminum hyroxie particles uring the chemical reaction the whole particle size range was ivie into 5 subgroups. The ivision is the same as the output of the particle size analyzer LMS. Using an iterative algorithm, the iameter of the largest (5 r.) particle was ecrease by. microns stepwise. A reactivity ratio was efine which is: V i = 5 V A A i 5 (A.) This ratio confirms that the amount of ecrease in the volume of the other 5 particles is proportional to the ratio of their surface area to the surface area of the reference particle. By rearranging equation (A.), equation (A.) is obtaine: (i) (5) (i) (5) = i 5 (A.) The following algorithm has been use for calculating specific surface area(a), specific surface area for mass transfer (a ) an mean iameter of Al(OH) power (p): The number of each of 5 groups of particles is calculate by Eq. (A.). for I= to 5 n i o Al(OH) * x i s * π * (i) 6*C = (A.) ρ In each stage. micrometer is subtracte from the biggest particle (particle group 5) an then by use of Eqs. (A.5) an (A.6), the new iameter of other 5 groups of particles an the new specific surface area have been calculate. 6 (5) (5) * for I= to 5 = (A.4) = (A.5) i / (i) ((i) ( ) *((5) (5) )) 5 5 a = n * π * (A.6) i= i (i) To calculate the specific surface area for mass transfer, the unshrinking particle moel was applie, in which the raius of the particle is consiere to be constant uring the reaction ue to the formation of the silica coating. But it shoul be note that it`s just applicable until the un-reacte core has not been consume completely. a = n * π * (A.7) i i: (i) (i) Also the new concentration of aluminum hyroxie is calculate by Eq.(A.8)

9 Iran. J. Chem. Chem. Eng. The Contribution of Molecular Vol. 4, No.4, 5 5 Al(OH) = n i * ρs * π * (i) /6 i= C (A.8) The new concentration of aluminum hyroxie has been calculate by this metho for any new iameter of particles,. The proceure continue until the iameter of the biggest particle was equal to zero. The above-mentione calculation have been one for two initial concentrations an 8 g/l of Al(OH). The calculate values of a,a an p in any stages have been rawn versus C Al(OH) an are fitte to equations presente as follow: for C Al((OH) = g/l a =.58 exp (.7 C ) (A.9) a p 6 C Al(OH) Al(OH) = 4.58 (.997) (C ) (A.) Al(OH) C Al(OH) +.79 (C Al(OH) ) = (A.) for C Al((OH) = 8g/l a = exp (.7585 C ) (A.) a Al(OH) 6 CAl(OH).4449 p.587 (.99799) (CAl(OH) ) = (A.) C Al(OH) +.7 (C Al(OH) ) = (A.4) Notations a Surface area of aluminum hyroxie cores per unit volume of the slurry, (m /m ) a Surface area of coate aluminum hyroxie particles per unit volume of the slurry, (m /m ) A i Surface area of ith particle, (m ) A 5 Surface area of 5r particle, (m ) C Al(OH) Aluminum hyroxie concentration, (g/l) C Al(OH) Initial concentration of aluminum hyroxie, (g/l) C L Initial concentration of fluosilicic aci C L Fluosilicic aci concentration in the bulk of solution, (mol/l) C S Fluosilicic aci concentration at the surface of aluminum hyroxie particle, (mol/l) C Different concentrations of fluosilicic aci over bounary layer of particles(mol/l) (i) (i) (5) (i) (5) i Initial iameter of ith particle Diameter of ith particle of aluminum hyroxie at every time, (m) Diameter of 5r particle of aluminum hyroxie at every time, (m) Diameter of ith particle of aluminum hyroxie after t, (m) Diameter of 5r particle of aluminum hyroxie after t, (m) Mean iameter of ith particle, (m) 5 Mean iameter of 5r particle, (m) p Mean iameter of aluminum hyroxie particles, (m) D Diffusion coefficient, (m /s) D a Propeller iameter, (m) D Effective iffusion coefficient pre-factor, m /s.(mol/l)..(particle/l) D eff Effective iffusion coefficient, (m /s) D t Reactor iameter, (m) Ea Activation energy, (cal/mol) k Reaction rate coefficient, (/s) k Constant of chemical reaction rate, (m.5 /s) k m Mass transfer coefficient, (m/s) n i Number of ith group of particles, (particles/m ) n t Number of particles in unit volume of the slurry, (particles/l) N Stirrer spee, (rpm) Nc Critical stirrer spee, (rpm) r Overall reaction rate, (mol/l.s) r b.l Mass transfer rate in bounary layer of particles, (mol/l.s) r C Average raius of unreacte cores of aluminum hyroxie, (m) r ch Chemical reaction rate, (mol/l.s) r m.t Mass transfer rate, (mol/l.s) R Gas constant, (.987 cal/mol. K) Re /4 Reynols Number, (ρ ε /µµ) R P Average raius of coate aluminum hyroxie particles, (m) Sc Schmit Number, (µ/ρd) Sh Sherwoo Number, ( km p/d T Temperature, (K) V i Volume change of the ith particle, (m ) V 5 Volume change of the 5 r particle, (m ) p

10 Iran. J. Chem. Chem. Eng. Bayat, M., et al. Vol. 4, No.4, 5 Greek Symbols α Exponent of fluosilicic aci concentration in equation of chemical reaction rate β Exponent of specific surface area in equation of chemical reaction rate δ Exponent of fluosilicic aci concentration in equation of effective iffusion coefficient, Eq.(6) γ Exponent of particles rampancy in equation of effective iffusion coefficient, Eq.(6) η Exponent of stirrer spee in equation of effective iffusion coefficient,eq.(6) ρ Flui Density, (kg/m ) ρ S Density of aluminum hyroxie, (kg/m ) µ Viscosity, (kg/m.s) ε Abbreviation AFT DLCA RLCA Agitation power, (N/6) D a 5 /D t, (w/kg) Aluminum Fluorie Trihyrate Diffusion Limite Colloi Aggregation Reaction Limite Colloi Aggregation [8] Satterfiel, C. N., Mass transfer in heterogeneous catalysis, M.I.T. Press, Cambrige, Mass., (97). [9] Kim, A.Y., an Berg, J.C., Fractal heteroaggregation of oppositely charge collois, Journal of colloi an interface Science, 9, 67, (). [] Martin, J. E., Wilcoxon, J. P., Schaefer, D., an Oinek, J., Fast aggregation of colloial silica, Physics Review A, 4, 479, (99). [] Xiao-Yan, L. an Logan, B. E., Permeability of fractal aggregates, Water Research, 5, 7,( ). [] Gmachowski, L., Hyroynamics of aggregate meia, Journal of Colloi an Interface Science, 78, 8, (996). [] Romm, F., Moeling of percolation/permeability in multiphase systems, Journal of Colloi an Interface Science,,, (). [4] Murugesen, T., Critical impeller spee for soli suspension in mechanically agitate contactors, Journal of Chemical Engineering of Japan, 4(), 4, (). Receive : th October ; Accepte : 8 th April 5 REFERENCES [] Skrylev, L. D., Reprocessing of fluorosilicic aci from superphosphate plants on aluminum fluorie an active silicon oxie, Zhurnal Priklanoi Khimii, 4(),, (968). [] Bayat, M., Taeb, A. an Rastegar, S., Investigation of the filtration rate of silica in aluminum fluorie prouction from silicic aci, Chemical Engineering Science, 57, 879, (). [] Grobelny, M., Stuy of the reaction of fluorosilicic aci with various aluminum substrates, Przemysl Chemiczny, 57(), 65, (978). [4] Vogel, A. T., Textbook of quantitative inorganic analysis, 4 th e., Longman, Lonon, p. 9,(978). [5] Levenspiel,O., Chemical Reaction Engineering, n e., John Wiley & Sons, New York, p , (97). [6] Armenante, P. M. an Kirwan, D.J., Mass transfer to microparticles in agitate systems, Chemical Engineering Science, 44, 78, (989). [7] Bayat, M. an Taeb, A., Kinetics of synthesis an crystallization of aluminum Fluorie, Ph.D. Thesis, Iran University of Science an Technology, (). 4