Modelling the adsorption of fluoride onto activated alumina in the presence of other ions

Modelling the adsorption of fluoride onto activated alumina in the presence of other ions

Naga Samrat MVV Gandhi K S Kesava Rao K Department of Chemical Engineering, Indian Institute of Science, Bengaluru 560012, India

Most of the studies on the adsorption of \ceF- are experimental, and have been done with synthetic solutions. Such studies rarely mimic the field situation. Therefore, selection of an adsorbent that can remove \ceF- from any kind of feed requires models that can predict the adsorption behavior for any given set of input conditions. From our observations and as also reported by many authors, the adsorption of \ceF- is affected by the presence of many ions. When modelling the adsorption of \ceF-, it is usually taken as a single entity getting adsorbed on the adsorbent. As this is not a proper assumption, a model was developed which takes into account all the speciation reactions that take place during adsorption, and all the species like \ceH+, \ceOH-, \ceNa+, \ceCl-, and \ceNO3- present in the solution along with \ceF-. As an electrolyte system is involved, the Nernst-Planck equations were used to obtain the flux of each species. Using the model, the equilibrium constants and rate constants for the reactions were obtained. For one initial concentration of \ceF-, a reasonable fit was obtained to the batch adsorption data, except at short times. Because of an uncertainty in the amount of impurity present in the commercial adsorbent used, there was a significant discrepancy between predictions and data at higher initial concentrations of \ceF-. The present model can be applied to any charged adsorbent.

Activated alumina; batch adsorption; defluoridation; Nernst-Planck equations;
journal: arXiv

1 Introduction

Most of the studies on adsorption of \ceF- are experimental, and have been done with synthetic solutions. Such studies rarely mimic the field situation. Therefore, selection of an adsorbent that can remove \ceF- from any kind of feed requires models that can predict the adsorption behavior for any given set of input conditions. Data on most of the adsorbents used for the adsorption of \ceF- are observed to agree well with Langmuir or Freundlich isotherms (Mondal and George, 2015). In the context of kinetics, they are fitted either to Langmuir kinetics, pseudo-first-order or pseudo-second-order kinetics. With the use of an analytical or numerical solution, we can predict the performance of the adsorbent without the need for laborious experiments.

In the light of the above observations, the motivation for the present work is as follows. Many papers show that the presence of an another ion along with \ceF- influences the adsorption of \ceF- (Nigussie et al., 2007; Goswami and Purkait, 2012; Chatterjee and De, 2014). However, with respect to modelling the adsorption of \ceF- , there are about 2 to 3 other ions always present in water. These ions often affect the structural properties of the adsorbent (Okamoto and Imanaka, 1988; Su and Suarez, 1997; Al-Abadleh and Grassian, 2003) and very few theoretical studies exist on the combined effect of multiple ions on the adsorption of \ceF-. Most of the available models do not account for the presence of other ions. Therefore, here a model has been developed to predict the adsorption behaviour of \ceF- in the presence of \ceH+, \ceOH-, \ceCl-, \ceNO3-, and \ceNa+.

In the papers of Fletcher et al. (2006) and Tang et al. (2009), the concentration of \ceH+ is taken as an independent variable. Hao and Huang (1986) assume that the surface concentration of \ceH+ depends on the electric potential of the surface, as the alumina surface becomes charged in solution. It is also assumed that the equilibrium constants depend on the pH of the solution (Hao and Huang, 1986; Fletcher et al., 2006). In the present work, the pH is permitted to change with time, and the complexity arising because of the surface potential is neglected.

2 Materials and methods

2.1 Experimental setup

Adsorption was studied in the batch mode. Two setups were used. In one setup, a conical flask containing 200 mL of solution and 1 g of activated alumina adsorbent was used. The details of the adsorbent are given in Table 1.

Property Quantity
Manufacturer Experimental
bulk density (kg/m) 880 786
surface area (m/g) 200 170
porosity 0.2 0.33
loss on attrition (%) 0.10
\ceAl2O3 (weight %) 93.74
\ceFe2O3 (weight %) 0.05
\ceSiO2 (weight %) 0.20
\ceNa2O (weight %) 0.35
loss on ignition (at ) (weight %) 5.66
Table 1: Characteristics of AA pellets of grade OAS37, as provided by the manufacturer Oxide (India) Catalysts Pvt. Ltd., Durgapur, India and experimentally measured.

The flask was placed in a rotary shaker which was rotated at a speed of 100 rpm. In these experiments, it is expected that shaking will provide good mixing and hence a uniform distribution of the solution and the adsorbent. However, the adsorbent accumulated as a stationary heap near the center of the flask even with a rotation speed of 100 rpm. Thus the relative velocity between the fluid and particles is uncertain. When the shaker speed was increased beyond 100 rpm, there was fluidization of the adsorbent pellets, but after some time of operation, attrition of the pellets occurred, leading to a reduction in their size.

In order to prevent the attrition and also to get an estimate of the relative velocity of the fluid with respect to adsorbent, a different experimental setup was used. The need to estimate arises because of the dependence of the external mass transfer coefficient on . Therefore, the data were acquired using a differential bed adsorber (DBA). The DBA experiments were done using a glass column of inner diameter 18 mm and the flow was controlled using a peristaltic pump (Fig. 1).

Figure 1: Schematic view of the differential bed adsorber used for the batch experiments.

The flow rate of the liquid through the DBA was maintained at 1.5 mL/s and could not be increased beyond this value owing to limitations of the available peristaltic pump. The adsorbent bed of height 1 cm was sandwiched between 5 cm beds of glass beads to ensure a uniform distribution of the fluid, especially when flow was from the top to the bottom. The lower bed was supported on a steel mesh, and the contents were loaded into the column after filling it with the solution of interest. This prevented the formation of air gaps in the bed, which could cause channelling of the fluid. The adsorbent height of 1 cm corresponds to a weight of 2 g and the total volume of the circulating solution was taken such that the solids concentration or bulk density was 5 g/L.

The bed porosity was determined by noting the change in the level of water immediately after adding the pellets. It was found that . Using a microscope, the diameters of about 10 particles were measured. The masses of the group of particles was measured and hence the averaged particle density could be calculated. It was found that kg/m.

Before loading into the column, the adsorbent was soaked in deionised water for 24 h. This was performed to remove any unwanted impurities that were present on the adsorbent, as the used AA was a commercial grade alumina. Adsorption of single ions onto AA was studied at first to determine the adsorption efficiency of AA towards these ions. The concentration of ions were varied from 40 mg/L to 5 mg/L for \ceF-, 500 mg/L to 20 mg/L for \ceSO4^2-, and 1000 mg/L to 10 mg/L for \ceHCO3-. After these experiments, binary combinations of the ions (i) \ceF- and \ceCl-, (ii) \ceF- and \ceNO3- were used. The change in the concentrations of the ions in the solution due to adsorption was measured by collecting samples at regular intervals of time. All the experiments were conducted at room temperature (26 - 30 C).

2.2 Chemicals and chemical analysis

All the chemicals used in the preparation of the solutions were of analytical grade, and were used without any further purification. Deionised water having a conductivity of S/cm was obtained from a Millipore unit.

Analysis of the samples was done using an ion chromatograph (Metrohm 883 Basic IC plus) for anions and cations. The 95% confidence limits were calculated using the data for the standards and the t-distribution (Snedecor and Cochran, 1968). The error bars shown in the figures correspond to these limits. In order to operate the chromatographic column below the maximum number of exchange sites available in the column, it was recommended to dilute the samples. During the analysis, the samples of ROR, SRW, and SNW were diluted 5 or 10 times and samples of SFW were used without dilution. As the bicarbonate concentration cannot be measured using this method, it was calculated using titration (Eaton et al., 2005, p. 2.27). The concentration of total carbonate comprising \ceCO3^2-, \ceHCO3-, and \ceCO2(aq) was calculated from the pH of solution, the pK values of carbonic acid, and the bicarbonate concentration obtained by titration. The quality of the analysis based on the ion chromatograph was checked using international standards like ION 915, ION 96.4, and MISSIPPI - 03. It was found that the concentrations of all ions quantified were within 5% of the certified values of these standards.

2.3 Surface studies

The determination of the possible reactions that can occur on the adsorbent can be determined by a study of its surface. It is known that the surface of AA changes with time upon soaking in water (Wijnja and Schulthess, 1999; Lefevre et al., 2002; Ryazanov and Dudkin, 2004; Carrier et al., 2007). This is mainly attributed to the hydration of the alumina surface, which was observed to change to bayerite or gibbsite based on the pH of the solution (Carrier et al., 2007). As the hydration reactions involve the use of \ceH+ ions, the use of titration can give us an insight into the equilibrium constants of these reactions.

Potentiometric titration experiments were done using a Mettler Toledo auto titrator (DL50 Rondolino). This was equipped with an automatic microburet and the pH of the solution after the subsequent addition of the titrant was measured by a Mettler Toledo pH electrode (DG111 - SC) containing 3 M \ceKCl solution. The pH meter was calibrated using Merck standard solutions of pH 4.0 and 7.0. For all the experiments, 0.02 N \ceHCl solution was used as the titrant. It was calibrated using 5 mL of 0.02 N tris-hydroxy methyl amino methan (TRIZMA) solution. The solution to be titrated consisted of 0.25 g of the adsorbent suspended in 10 mL of de-ionized water. This solution was made alkaline by the addition of 1 mL of 1 N \ceNaOH, and the pH was 12.0.

3 A model for titration and batch adsorption

A good model is needed to predict the adsorption behaviour for various concentrations of the ions. Isotherms like Langmuir, Freundlich, and Sips have been used to predict the equilibrium behaviour for the adsorption of \ceF- (Mondal and George, 2015), but it is assumed that there is only one species in the solution. It is also assumed that the adsorbent consists of only occupied and vacant sites. In practice, there is more than one species in the solution and there is a competition between the species for adsorption. Further, the adsorbent consists of different kinds of sites which can form varied complex species. Therefore, it is essential to include the concentrations of the other species in the model as well as different products formed by different sites. With these assumptions one can predict the equilibrium composition on the adsorbent and of the solution for any given initial condition.

The papers which show a decreased uptake of \ceF- in the presence of other ions do not mention or model how the concentrations of these ions change with time (Islam and Patel, 2007; Tang et al., 2009; Dou et al., 2011; Chatterjee and De, 2014). During the adsorption process, the following reactions have been proposed (Hao and Huang, 1986)


where \ce\bond3 denotes a surface group. Here the reaction

is not considered because it can be obtained by the addition of reaction (1), (3), and the water reaction.

In the most commonly used modelling approaches (Nigussie et al., 2007; Mondal and George, 2015) only (3) has been used. From (1) - (3) it can be seen that along with the adsorbed species \ce\bond3AlF, there are also \ce\bond3AlOH, \ce\bond3AlOH2+, and \ce\bond3AlO- present on the adsorbent. So there is need to consider these species also. The groundwater usually contains many other species like \ceHCO3-, \ceSO4^2-, etc. along with \ceF-. If there is competitive adsorption, then a few more species will also be produced (Wijnja and Schulthess, 1999; Appel et al., 2013).


In the present section, the assumptions and governing equations for the multicomponent adsorption are discussed. First, it is assumed that the charge neutrality is separately obeyed by the adsorbent and the solution. In existing models, the surface of the adsorbent is assumed to be charged, and the charge is balanced by an electrical double layer of ions in the solution (Bockris et al., 2006) (Fig. 2).

Figure 2: Schematic view of the possible distribution of the ions at the surface of a charged solid. Here \ceC+ and \ceA- denote cations and anions, respectively, and S denotes a surface site.

The surface concentration and bulk concentration of an ion at equilibrium, i.e. when there is no mass tranfer, are related by


where is the surface potential, or potential at the outer Helmholtz plane, is Faraday constant, is universal gas constant, and is the absolute temperature of the system (Hao and Huang, 1986). However, accounting for the diffuse double layer makes models of continuous contacting unwieldy. Further (6) is not strictly valid when the rate of adsorption is non-zero. The assumption of charge neutrality for the solid phase is equivalent to assuming a sharp double layer equivalent to a group of capacitors in parallel. Hence the concentrations at the surface and the bulk are assumed to be equal.

The charged species formed on the solid surface are assumed to be neutralised by non-reactive ions present in the solution. In the present study, chloride, nitrate and sodium were found to not complex with the adsorbent i.e., they were non-reactive. Similarly Nagashima and Blum (1999) found that \ceNa+ did not affect the adsorption of \ceH+ onto alumina. Thus, it is assumed that \ce\bond3AlOH2+ is neutralized by either \ceCl- or \ceNO3-. In the present case, the basis for taking \ceCl- in a solution made from \ceNaF and deionized (DI) water is that the pH of the DI water was about 6.0, but the water without any ions should have a pH of 7.0. Therefore, upon analysis of this DI water for anions using an ion chromatograph, it was also observed that there was a minute peak for \ceCl- which is below the limit of detection (Fig. 3).

Figure 3: Chromatogram from an ion chromatograph for anions in deionized water. Here the x-axis represents the time and the y-axis represents the conductivity.

Similarly, for \ce\bond3AlO-, \ceNa+ is the counter ion, as it is added along with \ceF-. Hence (1) - (3) are replaced by


Equilibrium constants for (7) and (8) were obtained by titration, and for (9) by batch adsorption.

3.1 Titration

The reactions that occur during titration are (7) and (8) and the equilibrium constants for these reactions are given by


where - , and are the electrochemical activities (Benjamin, 2002, pg. 598) of \ce\bond3AlOH2+Cl-, \ce\bond3AlOH, \ce\bond3AlO^-Na+, and \ceH+, respectively. As the double layer is collapsed onto a single plane in our model, . Hence, the electrochemical activities may be replaced by the chemical activites. Considering the adsorbed phase to be an ideal solution, the activities may be replaced by the mole fractions in the adsorbed phase. Similarly, considering a dilute solution where mol/L is a reference concentration. Therefore, with these assumptions (10) and (11) can be rewritten as


where is the concentration of adsorbed species (moles of per unit mass of the adsorbent), is the total number of sites on the adsorbent (moles per unit mass of the adsorbent), and is the concentration of solute species (moles of per unit volume of the solution).

For (7) and (8), with varying from 1-3 corresponds to the species \ce\bond3AlOH2+Cl-, \ce\bond3AlOH, \ce\bond3AlO^-Na+, respectively, and corresponds to the concentration of \ceH+ in the solution. The expressions for the equilibrium equations (12) and (13) are consistent with simple mass action kinetics for the special case where the charge-neutralising counter ions do not participate in the adsorption or complexing process. Neutralisation occurs just based on Coulombic attraction. Similar mass action kinetics were found suitable by Winkler and Thodos (1971) for the removal of phosphate by activated alumina activated with \ceHNO3, where the measured rate was independent of the concentration of nitrate in the solution.

In the titration experiment, the titrant was \ceHCl and the solution to be titrated consisted of AA in \ceNaOH. The mass balances for \ceCl- and \ceNa+ are given by


where and are the concentrations of \ceCl- and \ceNa+, respectively, is the concentration of the titrant (acid), is the initial concentration of the base, is the volume of the base taken, is the volume of the acid added, is the mass of the adsorbent taken, and is the total concentration of impurity present in the solution after addition of the acid. As mentioned earlier, the AA used in the experiments was a commercial adsorbent which contained some impurities. In view of the leaching of \ceNO3- from the fresh adsorbent when it is soaked in deionized water, it was assumed that the impurity was \ceNO3-. The term represents an ad hoc attempt to include the leaching of nitrate into the solution during titration. More satisfactory alternatives will be explored in the future. As shown later, there was no adsorption of \ceCl- and \ceNO3- on the adsorbent. Also, becasue of the same charge of these ions, \ceCl- and \ceNO3- are treated interchangeably in modelling. In addition, the solution must be electrically neutral, or


Here is the concentration of \ceH+ and is ionic product of water. In (16), gives the concentration of \ceOH- in solution. There can be a loss of aluminium in the form of soluble complexes such as \ceAlF_n(H2O)_6-n^(3-n), \ceAlF_n^3-n, and \ceAlOH_n^3-n (Nordin et al., 1999; George et al., 2010; Jin et al., 2010). This factor has been ignored in the present work.

The mass balance for the adsorbates is given by


where are the concentrations of species on the adsorbent. There are six equations (12) - (17) and seven unknowns , , , , , , and . Therefore, in order to obtain a unique solution, we need to specify one of the seven unknowns. In the titration experiment, the pH of the solution is measured as a function of the volume of the titrant added . Equations (12) to (17) can be solved to obtain in terms of as


The above equation can be rewritten in terms of the pH as


Equation (19) can be used to estimate the values of , , , and by fitting it to the data of pH vs. .

3.2 Batch adsorption: equilibrium

The above methodology can be followed for relating the variables involved in the batch adsorption of \ceF-. An equation for the mass balance of \ceF- has to be added, and the other mass balances have to be modified as titrant is not added to the solution. Equations (14) - (15) are modified to


where and are the concentrations of \ceF- and \ce\bond3AlF, respectively, and is the mass of adsorbent pellets per unit volume of the solution in the bed and the stirred vessel (Fig. 4). Here the mass balance for \ceCl- is changed because during the batch experiments, the adsorbent was taken after soaking in DI water for 24 h and the solution which contained the leachate was discarded, whereas the titration experiments were done along with the leachate. Here is the initial concentration of the impurity/leachate present on the adsorbent. The equilibrium constant for the reaction (9) is defined as


The charge neutrality equation for the solution is


Therefore, there are eight unknowns , , , , , , , and eight equations (12, 13, 20 - 24).

3.3 Batch adsorption: kinetics

Till now, the equations governing equilibrium were discussed. Modelling batch adsorption or continuous processing in a column requires rate data, in addition to equilibrium data. The setup shown in Fig. 1 was used to obtain data on adsorption kinetics, and the notation for the concentrations is shown in Fig. 4.

Figure 4: Sketch of the differential adsorber used for the batch experiments.

The volume of the adsorber bed is very small compared to the volume of the stirred vessel.

For convenience, the species in the solution are numbered such that 1 denotes \ceH+, denotes \ceOH-, and denotes any species whose concentration is eliminated using the electroneutrality condition. For example, for a 5-species system containing \ceH+, \ceCl-, \ceOH-, \ceF-, and \ceNa+, we have 1 = \ceH+, 2 = \ceF-, 3 = \ceCl-, 4 = \ceOH-, and 5 = \ceNa+. One of the important variables is the pH. As it affects the adsorption of \ceF-, special effort was made to account for the changes in pH, by treating the reaction between \ceH+ and \ceOH- as instantaneous.

The mass balance for species in a well-mixed stirred vessel is given by


where is the volume of the liquid in the vessel, and are the inlet and outlet concentrations, respectively, is the volumetric flowrate of the liquid, and is the molar rate of production of per unit volume by chemical reactions. We have


where and are the rate constants for the reaction


It is assumed that (28) is always close to equilibrium. Hence and


where is the activity of species and is the concentration of \ceH+ in mol/L. Hence, . To eliminate , we subtract the mass balance for from the balance for to obtain


In (30), is replaced by .

The charge neutrality condition is used to eliminate the concentration of the species. Thus


where is the charge of species , expressed in multiples of the charge of the electron. Using (29) and noting that , and , (31) reduces to


Therefore, it is necessary to solve only mass balances, and can be obtained from (32). Hence the mass balances for species in the stirred vessel are given by


Let us now consider the model for the differential packed bed. The external mass transfer rate to each pellet is calculated using a film model with the assumption that a liquid film of thickness surrounds the pellet. As is customary, convection and accumulation are neglected in the film. The dissociation of water is instantaneous and is permitted to occur in the film. The other reactions need adsorbent and occur only in the pellets. As before, the solution is assumed to remain electrically neutral. The film thickness is assumed to be small compared to the particle size, and the curvature of the film is neglected. Let denote the molar flux of in the x-direction (negative radial direction) (Fig. 5).

Figure 5: Sketch of the different fluxes into the pellet and the liquid film surrounding it. Here is the radius of the pellet, and n is the unit outward normal to the external surface of the pellet.

The mass balance for species in the bulk liquid is given by


where is the molar rate of production of per unit volume of the bed, is the porosity of the bed, and is the external surface area of the pellets per unit volume of the bed. As in the case of the stirred vessel, in the bulk liquid, we have


Proceeding as in the case of the stirred vessel, we obtain


For isothermal diffusion, the fluxes are related to the driving forces such as the gradient of the chemical potential by the generalized Maxwell-Stefan equations (Krishna and Wesselingh, 1997)


where is the driving force acting on per unit volume of the mixture, is the body force acting per unit mass of , is the molar flux of , is the mole fraction, is the volume fraction of species , is the mass fraction of , and is the chemical potential of and is the gradient of the chemical potential at a constant temperature and pressure . Here , and are the total concentration, the gas constant, and the Maxwell-Stefan diffusivity, respectively and is the total number of species. As discussed in Bird et al. (2002, pp. 765-768), the first of (41) arises naturally in the expression for the entropy production rate in a multicomponent fluid mixture. The term is called a “driving force”, even though it does not have the dimensions of a force. However, may be regarded as a force per unit volume. The expression for contains terms of the form , where is the mass flux of relative to the mass average velocity and is the density of . Assuming that each flux is a linear function of all the driving forces , the flux relations can be inverted to obtain the second of (41) (Curtiss and Bird, 1999). Predictions of (41) coupled with the mass balances agree fairly well with data obtained from many systems (Krishna and Wesselingh, 1997).

For isobaric diffusion in a dilute electrolyte solution, (41) can be simplified to obtain (Krishna, 1987)


where is the velocity vector of the solution, is the Faraday constant, and is the electrostatic potential generated because of the ions. For a dilute system, the diffusivity in (41) may be replaced by i.e. the diffusivity of species with respect to water. This quantity is defined as in (42). Equation (42) is called the Nernst-Planck equation and has been used by many authors to model ion-exchange onto resins (Frey, 1986; Jia and Foutch, 2004; Bachet et al., 2014). In the context of defluoridation with an oxide adsorbent, this approach has not been used earlier. During adsorption, it is assumed that the adsorbent and solution are electrically neutral separately.

The current density is given by


or, using (42),


As the system is electrically neutral, (31) and (44) imply that


As no current passes through the system . Hence (45) reduces to


Using standard electrochemical engineering terminology (Gu et al., 1997; Newman and Thomas-Alyea, 2012), the conductivity number and transport numbers are defined by




Substituting (46) and (49) in (42), we obtain





and is the Kronecker delta.

It may be noted that the fluxes are coupled to the driving forces of all the species due to enforcement of charge neutrality. Ignoring the axial diffusion and axial dispersion in the bulk liquid,


where is the z-component of the molar flux and is the interstitial velocity.

If we ignore convection in the liquid film, and if diffusion occurs only in the x-direction (Fig. 5), (50) reduces to


Ignoring the accumulation in the liquid film, and if the only reaction in the film is the dissociation of water, the mass balances are given by


Following the procedure used earlier, equations (55) are combined to eliminate the rate of the decomposition of water and may be rewritten as

The above equation can be integrated to obtain


Assuming that the concentration varies linearly across the film, and using (53) in (56) we obtain


where is the concentration in the bulk liquid, is the concentration at the surface of the pellet, and is the thickness of the film (Fig. 5). Note that the ’s are functions of the concentrations, but have been treated as constants while deriving (57) from (53). Similarly,


The expressions derived for fluxes to the surface of the pellet can be substituted into the balances (38) and (39) for the bed. These reduce to


Let us now consider the liquid inside the pellets. The mass balance are given by


where is the porosity of the pellet, is the molar concentration of per unit volume of the fluid, is the molar flux of species in the pellet, is the molar rate of production of per unit mass of the pellet, is the density of the pellet, and is the rate of production of due to the water reaction (28). Thus


Subtracting the mass balance equation for from that of , we obtain


To simplify the analysis, (64) and (65) are integrated over the the volume of the pellet. Thus


where overbars denote volume-averaged values, is the external surface area of the pellet, is the unit outward normal, and is the radius of the pellet (Fig. 5).

The pellets that are normally used for adsorption are porous materials, and hence the diffusion process may occur by bulk, Knudsen, and surface diffusion (Ruthven, 1984). Therefore, inside the pellets there will be an effective diffusion. Ignoring convection within the pellets, the flux of a species in dilute electrolyte solution is given by an equation analogous to (50), i.e.


where denotes an effective diffusion coefficient. The above equation is similar to the dusty gas model for a dilute mixture (Krishna and Wesselingh, 1997), but with the incorporation of an electrical force. Therefore


where is an effective ion-water bulk diffusion coefficient, and is an effective Knudsen diffusion coefficient of species in the porous medium. In the present model, the Knudsen diffusion is neglected. For the porous medium, (Tjaden et al., 2016). The radial component of the flux is given by


In the spirit of the linear driving force model (Gleuckauf, 1955; Sircar and Hufton, 2000; Moreira et al., 2006; Tefera et al., 2014), we approximate by


where is the volume averaged concentration of in the pellet, is the value of at the external surface of the pellet, and is a constant. Hence the fluxes are given by


Using (72) and (71), (66) and (67) can be rewritten as


The concentration in the above equations may be eliminated by ensuring the continuity of fluxes at the surface of the pellet. Thus


Substituting the fluxes we obtain


Equations (79) and (80) involve unknowns . However, as discussed earlier, the final form of the expressions for the fluxes in the bulk and the pellet can be obtained by eliminating the species and , and can be written as (see A)


where and are effective mass transfer coefficients for transfer from the bulk solution to the surface of the pellet, and from the surface of the pellet to the interior, respectively. They are given by