Modeling of Ion Exchange Performance in a Fixed Radial Flow

Ind. Eng. Chem. Res. 1997, 36, 2359-2367

2359

Modeling of Ion Exchange Performance in a Fixed Radial Flow Annular Bed Yan Tsaur and David C. Shallcross* Department of Chemical Engineering, The University of Melbourne, Parkville, Victoria 3052, Australia

A model is developed to simulate ion exchange performance within a fixed, annular ion exchange bed. In the bed the injected solution flows radially from the inlet located at the annular bed’s axis to the outlet located at the bed’s periphery. The model considers radial dispersion to be a function of the solution velocity and assumes instantaneous pointwise equilibrium exists throughout the bed between the ions in the solution and the exchanger phases. Experiments to validate the model are conducted using a wedge-shaped radial ion exchange bed having 30° of arc and containing commercially-available ion exchange resin. The performance of the radial bed is studied for both the exhaustion and regeneration cycles for the Na+-H+ and Ca2+-H+ cation binary systems. The predictions made by the radial ion exchange performance model are found to be in close agreement with the experimental observations for the systems studied. Introduction Over the past decades the ion exchange process has found increased applications in a range of diverse fields. Most of these new applications however still use cylindrical column ion exchange beds in which the electrolyte solution to be treated is injected in one end and the effluent is collected at the other. The resulting process involves ion exchange in essentially one-dimensional, linear flow. This paper describes a radial ion exchange process in which a solution is injected into an annular ion exchange bed at its center, and the effluent is collected from the bed’s periphery. The conventional one-dimensional, linear ion exchange column has been well investigated, and many models have been developed to predict ion exchange performance in vertical flow (Helfferich, 1962; Dorfner, 1991; Slater, 1991). However, most models assume no axial dispersion and either concentration independence of the rate constant or instantaneous pointwise ion exchange equilibrium with ideal behavior in both the solution and exchanger phases. These same assumptions are also made in the few models developed for radial flow chromatography (Rachinskii, 1968; Rice, 1982). In modeling ion exchange performance in an annular bed, Rachinskii made two simplifications. He assumed ideal ion exchange with a linear equilibrium relationship, and he further assumed that there was no dispersion in the radial flow direction. Although some radial models do consider radial dispersion, the dispersion coefficient is assumed to be constant, regardless of the fact that the radial dispersion is a function of the radial velocity, which in turn varies with distance from the bed’s axis (Rice and Heft, 1991; Hang et al., 1988). For systems other than those dealing with dilute solutions and plug flow, the existing models lack accuracy in the prediction of breakthrough, which is fairly sensitive to the nonideal behavior of the exchange equilibria and the flow of the solution through the porous bed. Several workers have studied simple dispersion within porous media in radial flow (Tan and Homsy, 1987; Zimmerman and Homsy, 1991; Yortsos, 1988). In these radial dispersion studies, the values of the dispersion coefficients are either assumed or calculated from molecular diffusivity and radial pore velocity prior to * Author to whom all correspondence should be addressed. E-mail: David Shallcross.Chem Eng@ muwaye.UNIMELB.EDU.AU. S0888-5885(96)00622-7 CCC: $14.00

incorporation into the model. No experiments have been performed to date to test the simple radial dispersion models in porous media. Dispersion reflects the back mixing which occurs with the flow. It is a function of the fluid velocity. Lack of flow symmetry about the axis of a 360° radial flow bed will also result in a very strong dispersive effect. In the absence of flow symmetry, breakthrough will occur at different times, resulting in increased effective dispersion. Also, depending on the system and solution concentration, the exchanging ions can exhibit nonideal behavior in the solution and exchanger phases. In this study, we have developed an ion exchange model which predicts ion exchange performance in radial flow through a packed bed of ion exchange resin. Our model takes account of both dispersion and the nonideal behavior of the exchanging ions in both the solution and the exchanger (i.e., resin) phases. The radial ion exchange model predictions are tested by comparing them with the observations made during experiments in a radial, wedge-shaped bed. The radial ion exchange model is based upon a simple material balance. It incorporates dispersion and uses a modification to the Mehablia (1994) equilibrium model (Mehablia et al., 1994, 1996) to predict ion exchange equilibrium. While the form of the model presented here considers binary cation exchange only, the fundamental concepts behind the model are not limited to such simple systems. Radial Ion Exchange Model Consider the annular resin bed shown in Figure 1. The fixed and homogeneous annular bed is made up of randomly-packed spherical resin beads. The horizontal bed has a constant thickness, H, and a uniform porosity, . The radius at the bed inlet is RI, and the radius at the bed outlet is RO. For the purposes of applying the model equations, it shall be assumed that RI < RO, i.e., the fluid flows outward away from the bed’s axis. The annular cross-sectional area, AR, is a function of the radius. Consider the binary system involving exchange between ions A and B. Initially the resin bed is in the B-form (i.e. all the exchanger sites of the resin are occupied by B ions) and the solution phase within the pores of the resin bed is free of A cations. At some time, τ ) 0, a solution containing A ions is injected into the © 1997 American Chemical Society

2360 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

resin bed through the face at radius RI. The solution is injected at a volumetric flow rate of FL, which is constant with time and uniform across the inlet of the bed. The concentration of A ions in the injected solution is known and constant with time. At the same time a solution is produced from the peripheral outlet of the resin bed at a rate uniform around the entire circumference of the bed. For some time after injection begins, no A ions are present in the effluent. As the resin gradually takes up A ions, the concentration of A ions in the effluent rises from zero, eventually reaching the concentration of the injected solution. Plotting the concentration of A ions in the effluent against the volume of effluent produced yields the breakthrough curve. When the solution is in contact with the resin bed, A ions in the solution exchange onto the resin and B ions on the resin exchange into the solution:

bAa + aBb H bAa + aBb

(1)

where a and b are the valencies of the ionic species A and B, respectively, and the underline indicates that the particular ion is on the resin rather than in the solution phase. Now consider a thin but finite annular section of the bed at position R. The annular section has a thickness ∆R. At some time, τ > 0, CA will be the volumetric concentration of A ions in the solution phase, and CRA will be the concentration of A ions in the resin phase. JA is the dispersive flux of A ions through the elemental annular section. A material balance for A ions over the annular section in cylindrical coordinates is

dCA dCRA + (1 - )(V|R+∆R - V|R) ) (V|R+∆R - V|R) dτ dτ (A|RJA|R - A|R+∆RJA|R+∆R) + FL(CA|R - CA|R+∆R)

Figure 1. Representation of a thin annular fixed resin bed showing the annular element.

In this equation the terms on the left hand side arise from the accumulation of A ions in the solution phase and in the resin phase. The first term on the right hand side arises from the dispersion terms while the second term results from the convective terms. Fick’s Law may be used to relate the dispersion flux to the concentration gradient in cylindrical coordinates:

JA ) -DLr

DLr ) f(URP)

URP )

V|R+∆R ) π[(R + ∆R)2 - RI2]H

(4)

A|R ) 2πRH

(5)

A|R+∆R ) 2π(R + ∆R)H

Substitution of eq 7 for VR+∆R - VR into eq 2 yields

dCA dCRA + (1 - )AR∆R ) dτ dτ (ARJA-R - AR+∆RJA-R+∆R) + FL(CA-R - CA-R+∆R)

AR∆R

(8) Dividing by (AR∆R) and taking limits as ∆R f 0,

AR+∆R ≈ AR

(9)

and

FL ∂CA ∂CA ∂ 1 -  ∂CRA + ) - (JA) (10) ∂τ  ∂τ ∂R AR ∂R

(

)

(

)

∂CA ∂ 1 ∂ (J ) ) RDLr ) ∂R A R ∂R ∂R

[

(6)

(7)

FL UR ) ) f(R)  A R

(13)

In cylindrical coordinates, the derivative of the radial dispersion flux is

- DLr

and neglecting the second-order terms of ∆R

VR+∆R - VR ) AR∆R

(12)

UR is the superficial velocity at the position R, and URP is the pore velocity at the position R. Since the solution is injected at a constant flow rate and the resin bed has a uniform porosity, we have

where

(3)

(11)

where the dispersion coefficient DLr is assumed to be independent of CA but is a function of the solution velocity through the porous resin bed:

(2) V|R ) π(R2 - RI2)H

∂CA ∂R

2

∂ CA ∂R

2

]

DLr ∂CA ∂DLr ∂CA + (14) R ∂R ∂R ∂R

+

Substitution of this expression for the derivative into eq 10 yields

∂2CA DLr ∂CA ∂CA 1 -  ∂CRA + ) DLr + + ∂τ  ∂τ R ∂R ∂R2 ∂DLr ∂CA FL ∂CA (15) ∂R ∂R AR ∂R

(

)

The concentration of A ions in the solid resin phase, CRA, may be written in terms of the resin phase mole fraction, y. This is defined as the fraction of ions on the resin that are ion species A. Hence,

( )

CRA ) y

QFR zA

(16)

where y is the mole fraction of A ions in the resin phase,

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2361

Q is the ion exchange capacity of the resin on a mass basis, FR is the resin density, and zA is the valence of A ions. Similarly, the concentration of A ions in the solution phase, CA, may be written in terms of the solution phase mole fraction, x. This is defined as the mole fraction of the ions in the solution that are ion species A. Hence,

()

CA ) x

C0 zA

( ) () ()

(18)

∂CA C0 ∂x ) ∂τ zA ∂τ

(19)

∂CA C0 ∂x ) ∂R zA ∂R

(20)

and

∂R

)

()

C0 ∂2x zA ∂R2

(21)

Thus, upon substitution, eq 15 becomes, after division by (C0/zA),

1 -  QFR ∂y ∂2x DLr ∂x ∂DLr ∂x ∂x + ) DLr 2 + + ∂τ  C0 ∂τ R ∂R ∂R ∂R ∂R FL ∂x (22) AR ∂R

(

)

By expressing this equation in a dimensionless form, the problem may be more easily solved using numerical techniques. Define

R RO

(23)

τFL τFL ) ARORO 2πROHRO

(24)

r)

t)

FL QFR ∂y DLr ∂2x 1- ∂x + ) 2 2+  2πR2 H C0 ∂t 2πR2 H ∂t R ∂r

(

O

)

O

O

DLr ∂x FL 1 ∂DLr ∂x 1 ∂x + (25) 2 ∂r RO ∂R ∂r 2πROrH RO ∂r rR O

Multiplying through by (2πR2OH/FL) yields

1 -  QFR ∂y DLr2πH ∂2x DLr2πH 1 ∂x ∂x + ) + + ∂t  C0 ∂t FL FL r ∂r ∂r2 2πHRO ∂DLr ∂x 1 ∂x (26) FL ∂R ∂r r ∂r

(

)

(27)

FL FL ) A|R 2πRH

(28)

Thus, eq 26 becomes

(

)

∂x (1 - )QFR ∂y 1 ∂2x 1 1 ∂x + ) + + -1 ∂t C0 ∂t PeR ∂r2 PeR r ∂r 1 1 ∂DLr ∂x (29) PeR DLr ∂r ∂r Equation 29 is the basis of the theoretical model for ion exchange performance in a radial flow system. It describes the one-dimensional nonlinear process of ion exchange within a fixed resin bed. The ion exchange occurs with radial dispersion and both ion exchange and radial dispersion, develop with time. The solution of this equation predicts the compositions of ions A and B in both the solution and resin phases. The application of the equation is not limited however to exchange between two ions only. Several assumptions are implicit in eq 29: (1) The resin bed is homogeneous with uniform porosity. (2) The resin bed has finite inner and outer radii, RI and RO. (3) The resin bed is fixed and does not move. (4) Both fluid and resin bead compressibilities are negligible, and gravity effects are neglected. (5) Fluid flows at a rate constant with time, under isothermal conditions and is uniformly distributed about the axis. (6) Molecular diffusion is negligible. (7) Dispersion occurs only in the flow direction and is isotropic for the same radius. (8) The dispersion coefficient is independent of the chemical concentration but is a function of the radial distance. Since the resin is initially in the B-form and the solution within the pores of the bed contains B cations only initially, the initial conditions for the partial differential equation (eq 29) are

x ) 0 at t ) 0 for rO g r g rI

(30)

y ) 0 at t ) 0 for rO g r g rI

where r and t are dimensionless distance and time, respectively. Substitution of r and t into eq 22 yields

FL

URR DLr

where the superficial velocity, UR, is a function of radial position and is defined

UR )

QFR ∂y ∂CRA ) ∂τ zA ∂τ

2

PeR )

(17)

where, C0 is the total normality in the solution. C0 does not change when ionic species of differing valencies exchange with one another. So, we have

∂2CA

The Pe´clet number is defined as the ratio of bulk mass transfer to dispersive transfer in the radial direction:

At t ) 0 there is a step change in the composition of the solution injected into the bed. Thereafter the composition and concentration of the injected solution remain constant. The boundary condition at the outlet should be of the form ∂nx/∂rn ) 0, where n is the order of the derivative. We chose the second-order derivative to be zero as higher order derivatives would un-necessarily complicate the model while using the first-order derivative would be physically inappropriate and would significantly distort the model predictions at the outlet. Thus, the boundary conditions for eq 29 then are

x ) 1 at r ) rI for t > 0 2

∂x ) 0 at r ) rO for t > 0 ∂r2

(31)

2362 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

The ion exchange resin bed used in experiments to study the radial ion exchange process is wedge-shaped with an arc of 30°. Thus the total exchange capacity for this resin bed may be defined as

π(R2O - R2I )H 12

QRT ) QFR(1 - )

(32)

Note that QRT can be determined by experiment. Making the above substitutions, eq 29 may be rewritten as

∂x + ∂t

QRT

1 ∂2 x ∂y ) + π(R2O - R2I )H ∂t PeR ∂r2 C0 12 1 1 ∂DLr ∂x 1 1 ∂x + (33) -1 PeR r ∂r PeR DLr ∂r ∂r

[

(

]

)

This equation is used to model the radial ion exchange processes within the wedge-shaped resin bed developed by the authors. The equation may be solved numerically by finite difference techniques once a link is established between the solution and resin phase mole fractions, i.e., between x and y. The numerical solutions predict both the concentration profile and concentration history of the exchanging ions in both the solution and the resin phases. The numerical techniques used to solve the equations are presented by Tsaur (1996). Ion Exchange Equilibrium Model The solution and resin phases are assumed to be in equilibrium with one another. The solid resin phase concentration of the ions, y, is linked to the solution phase concentration, x, by the equilibrium model developed by Mehablia (1994) and co-workers (Mehablia et al., 1994, 1996) and improved by the authors. Nonidealities in both phases are considered by applying the Pitzer (1973, 1991) model and the Wilson (1964) model to calculate the activity coefficients in the solution and resin phases, respectively. The model also takes into account the formation of ion pairs of the electrolytes in the solutions which limit the free ions available for ion exchange (Kester and Pytkowicz, 1969; Johnson and Pytkowicz, 1978). The thermodynamic equilibrium constant is calculated independently using the approach of Argersinger et al. (1950). For ion exchange as defined by eq 1, the thermodynamic equilibrium constant is defined as

KTAB )

( )( )( )( ) ( ) ybA yaB

CaB CbA

γaB γbA

γbRA γaR

≡

γbRA a KAB a γRB

∫01ln KaAB dyIA

Radial Dispersion Model The dispersion coefficient quantifies the extent to which mechanical dispersion arising from back-mixing of the fluid affects the fluid displacement process. Radial dispersion reduces the driving force for mass transfer and eventually diminishes the ion exchange performance of the resin bed. Since the nonlinear radial flow presents a variable velocity field, the dispersion in radial flow varies with position within the bed. The model for pure radial dispersion without any mass transfer (i.e., without ion exchange) can be obtained by simplifying eq 29 taking the packed bed to be chemically-inert:

(

)

∂x 1 ∂2x 1 1 ∂DLr ∂x 1 1 ∂x ) + + -1 2 ∂t PeR ∂r PeR r ∂r PeR DLr ∂r ∂r

(34)

(35)

where yIA is the equivalent ionic fraction of A ions in the resin phase. The integration of the area under a

(36)

The initial condition is

x ) 0 at t ) 0 for r g rI

(37)

and the boundary conditions are

x ) 1 at r ) rI for t > 0

a in which KAB is the experimentally-accessible equilibrium quotient. The solution phase activity coefficients, γi, are calculated by the Pitzer model using free ion concentrations, while the resin phase activity coefficients, γRi, are calculated using the Wilson model. Argersinger et al. (1950) developed a relationship between the thermodynamic equilibrium constant and the equilibrium quotient:

ln KTAB )

a plot of ln KAB against the solid resin phase equivalent ionic fraction yields the thermodynamic equilibrium constant. The equilibrium concentrations of ions A and B in both the solution and resin phases may be determined by experiments. The equilibrium quotient is then determined, using the Pitzer model to calculate the solution phase activity coefficients. When the equilibrium quotient data are integrated over the entire range of resin phase compositions, the thermodynamic equilibrium constant may then be determined using eq 35. Once this is known, the method of Mehablia et al. (1994) is then used to determine the two Wilson binary interaction parameters, ΛAB and ΛBA. The ion exchange equilibrium model is described more fully by Tsaur (1996). As noted above, the model assumes that instantaneous pointwise equilibrium exists between the solution and resin phases at all times throughout the model. This assumption limits the application of the model to flowing solutions with low radial liquid velocities. Around the axis where the flow velocities will be much higher than in the outlying regions of the resin bed, it is possible that equilibrium will not be attained within the two phases.

(38)

∂2x ) 0 at r ) rO for t > 0 ∂r2 where x is the mole fraction of the tracer concentration, t is the dimensionless time, r is the dimensionless radial distance, PeR is the Pe´clet number for radial flow, and DLr is the radial dispersion coefficient. The dependence of dispersion within a porous medium on the flow velocity is very complex. Since the dispersion we model is assumed to arise from mechanical dispersion alone, the dispersion coefficient is mainly affected by the connectivity and the spatial distribution of the flow channels. Consequently, in the tracer experiments the dispersion is very sensitive to flow heterogeneities and spatial velocity variations. Ippolito et al. (1993) recommend that radial dispersion in porous

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2363

media is proportional to the pore velocity:

DLr ) λURP

(39)

where λ is called dispersion length and can be characterized by experiment. In the calculation, the model accounts for the effects of resin swelling or shrinking on the experimentallydetermined porosities and dispersion coefficients. Resin swelling or shrinking which occurs when the form of the resin changes has a considerable influence on dispersion within a porous resin bed. Resin bead size varies with the composition of the ions on the resin phases. This variation in the volume occupied by the resin will affect not only the bed’s apparent porosity but also the extent of dispersion within the bed. Therefore, both the porosity and dispersion coefficient applied in the model calculation vary as functions of the composition of the ions on the bed:

i,n ) Ayi,n + B(1 - yi,n)

(40)

Figure 2. Wedge-shaped resin bed and associated experimental equipment.

A simplified diagram showing the main features of the apparatus is presented in Figure 2. A more detailed description is presented by Tsaur (1996).

and

Di,n ) DAyi,n + DB(1 - yi,n)

(41) Equilibrium Results

where, i,n is the bed’s apparent porosity at point i along its radius and at time t ) n∆t, A and B are the porosities determined by experiments when the bed is completely in the A-form and B-form, respectively, Di,n is the dispersion coefficient at point i along its radius and at time t ) n∆t, DA and DB are the dispersion coefficients determined by experiments when the bed is completely in the A-form and B-form, respectively, and yi,n is the mole fraction of A ions on the resin at point i along its radius and at time t ) n∆t. Experimental Material and Apparatus The resin used in this study is a commerciallyavailable 20-50 mesh (0.3-0.85 mm diameter) cation exchange resin, DOWEX MSC-1. It is a macroporous polystyrene-divinylbenzene resin with active sulphonated groups. The electrolyte solutions are prepared from analytical-grade chemicals and distilled-deionized water. All the solutions contain chloride as the only anion species and are deaerated before use. The radial ion exchange cell consists of a wedgeshaped ion exchange resin bed. The bed is formed from two parallel and horizontal perspex sheets 15.0 mm apart. The two side walls are perpendicular to the two sheets but at an angle of 30° to one another. The radius of the inner curved wall is 30.0 mm while the radius of the outer curved wall is 300.0 mm. The two curved walls are made from sintered glass plates curved to the appropriate radius. These plates act as flow distributors as well as helping to contain the resin in the bed. When packed the resin bed had a porosity varying between 31.0% and 33.0% depending upon the form of the resin. The solution is drawn in through a distributor at the inner face. Eleven identical chambers are distributed uniformly along the length of the outer face of the outer curved glass sheet. Each chamber is connected to flow lines of identical lengths which then pass to a multichannel peristaltic pump. This ensures that the fluid is produced at a rate uniform around the bed’s periphery.

Two binary ion exchange systems were studied experimentally using a batch technique: the Na+-H+ system and the Ca2+-H+ system. In both systems chloride was the only anion present. The cation exchange capacity of the resin was found to be 4.99 ( 0.02 mequiv/g dry weight for the H-form, 4.55 ( 0.02 mequiv/g dry weight for the Na-form, and 3.88 ( 0.02 mequiv/g dry weight for the Ca-form (Tsaur, 1996). Between 0.02 and 50 g of air-dried resin of a known form was carefully weighed out into a volumetric flask. Precisely 100 mL of a prepared electrolyte solution of known composition and concentration was then added to each flask. The capped flasks were shaken without interruption for 72 h at a temperature of 20 ( 1 °C. All equilibrium experiments were repeated at least once. A titration technique was used to determine the concentration of the H+ ions in the solutions after the 72 h. Simple material balances allowed the composition of the ions in the solution and resin phases to be calculated. For the Na+-H+ system the solution concentrations studied were 0.010, 0.10, and 0.50 N, while for the Ca2+-H+ system the concentrations studied were 0.10, 0.50, and 1.0 N. Figures 3 and 4 represent the binary equilibrium data collected for the two systems. As may be seen the data are clearly reproducible. Also presented are the values calculated for the thermodynamic equilibrium constants and the Wilson binary interaction parameters. The values determined for each of the Wilson binary interaction parameters for both the Na+-H+ and the Ca2+H+ systems obey the reciprocity rule: ΛABΛBA ) 1 (Allen et al., 1989). This is in agreement with the thermodynamic aspects of the Wilson (1964) model. Clearly, the ion exchange model is able to model the equilibrium behavior of both systems very well. Dispersion Experiments A series of tracer experiments were conducted to determine the extent of dispersion within the radial ion

2364 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 1. Summary of the Dispersion Experiments Performed in the Wedge-Shaped Resin Bed resin bed form porosity Na

0.330

H

0.310

flow rate (mL/h) 410 1014 408 1010

pore velocity, URP (×10-5 m/s) cell inlet cell outlet 73.2 181 77.6 192

7.32 18.1 7.76 19.2

Pe`clet number, URPR/DLr cell inlet cell outlet 6.00 3.75 3.33 2.31

dispersion coefficient, DLr (m2/s)

60.0 37.5 33.3 23.1

0.005URP 0.008URP 0.009URP 0.013URP

Figure 5. Tracer response curve used to determine dispersion in the wedge-shaped resin bed. Bed in the H-form with the HCl solution injected at a rate of 408 mL/h. Figure 3. Experimentally-determined and model-predicted equilibrium isotherms for the Na-H binary system.

Figure 4. Experimentally-determined and model-predicted equilibrium isotherms for the Ca-H binary system.

exchange bed. Resin in the Na-form was packed uniformly into the wedge-shaped ion exchange bed. The resin was then rinsed with deaerated, distilled, and deionized water. A 0.10 N NaOH solution was injected in place of the water as a tracer at a controlled flow rate. The effluent was collected at 3 min intervals and analyzed for its OH- ion concentration by titration. The experiments were repeated at different flow rates. The resin bed was then put into the H-form, and a 0.10 N HCl solution was injected as the nonabsorbing tracer. The concentration of H+ ions in the effluent was then measured for different injection flow rates. Tracer experiments with the resin in the Ca-form were not performed, as the porosity of the Ca-form resin bed determined is very close to the porosity of the Na-form resin bed. In this study the dispersion in the Ca-form resin bed is taken to be the same as the dispersion in the Na-form resin bed.

The dispersion experiments performed in the wedgeshaped cell for both the Na-form bed and the H-form bed at two different flow rates are summarized in Table 1. Since the cross-sectional area of the resin bed in the wedge-shaped cell is not a constant but varies with the radial distance from the central inlet, the pore velocity is a function of the radial distance. Consequently, the pore velocity-dependent dispersion coefficient is also a function of the radial distance. The values for both the pore velocity and the Pe`clet number at the cell inlet and outlet are listed in the table. The dispersion coefficient expressions presented in the table are the best fit found. Experimental errors are estimated to be less than 2%. Figure 5 compares the experimental data to the fitted model predictions for the dispersion occurring in the H-form resin bed in the wedge-shaped cell at the flow rate of 408 mL/h. The diagram shows good reproducibility between the two experiments performed and excellent curve fitting of the dispersion model. The fitting is performed by history matching using the finite difference form of the simple dispersion model for nonlinear radial flow (eq 36). By varying the Pe`clet number and the mean residence time, the best fit is found. The agreement shown between the experiments and the fitted dispersion model predictions is also very good for the other experiments summarized in Table 1. Ion Exchange Performance Results The experiments conducted to study the ion exchange performance in the wedge-shaped cell resin bed are summarized in Table 2. The operation capacities listed are on a resin bed volume basis and under the specific experimental conditions. The high degree of experimental reproducibility is demonstrated by comparing the repeated experiments for a solution concentration of 0.50 N at the flow rate around 400 mL/h. The operation capacities of both the Na+-H+ and Ca2+-H+ exhaustion runs are affected by the solution concentration and the flow rate. High solution concentration and low flow rate increase the operation capacity of an exhaustion process; low solution concentration and high

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2365 Table 2. Summary of the Wedge-Shaped Resin Bed Ion Exchange Performance Experiments operation cycle

resin bed form

injection solution

solution concentration (N)

flow rate (mL/h)

data points

operation capacity (mequiv/mL)

regeneration exhaustion regeneration exhaustion regeneration exhaustion regeneration exhaustion regeneration exhaustion regeneration exhaustion

Na H Na H Na H Ca H Ca H Ca H

HCl NaCl HCl NaCl HCl NaCl HCl CaCl2 HCl CaCl2 HCl CaCl2

0.530 0.500 0.530 0.500 0.530 0.250 0.530 0.500 0.530 0.500 0.530 0.250

388 393 393 385 947 975 385 376 380 378 989 936

27 19 22 18 28 25 20 13 21 12 20 21

1.95 1.98 1.95 1.99 2.01 1.96 1.59 1.70 1.58 1.68 1.47 1.58

Figure 6. Breakthrough curve for Na-H regeneration runs using a 0.53 N HCl solution. Wedge-shaped resin bed initially in the Na-form.

flow rate decrease the operation capacity of an exhaustion process. All the regeneration runs use 0.530 N HCl as the regenerant. Only the flow rate varies between the runs. When the bed has been treated with 10 L of the regenerant solution, regeneration is nearly completed. The flow rate is then changed to 100 mL/h to continue the regeneration until the concentration of H+ ions in the effluent is almost the same as the concentration of the injection regenerant. For the Na+-H+ regeneration runs, the operation capacities are consistent and very close to the corresponding operation capacities of the Na+-H+ exhaustion runs. Increasing the flow rate has no significant effect on the operation capacity for the Na+-H+ regeneration process; however, the operation capacities of the Ca2+-H+ regeneration runs are lower than the operation capacities of the Ca2+-H+ exhaustion runs for the corresponding solution concentration, and the increase of the flow rate slightly reduces the operation capacity of the Ca2+-H+ regeneration process. The flow rates listed in Table 2 are the average flow rates over the duration of an experiment. Due to resin shrinking and swelling, which often accompanies the ion exchange process, the flow rate has about (5% variation from the average value. Figures 6-9 allow comparisons to be made graphically between the breakthrough curves obtained by experiments and the breakthrough curves predicted using the radial model. The experiments for the exhaustion and regeneration runs for the two binary ion exchange systems are performed cyclically. Exhaustion and regeneration experiments are run for the monovalent Na+-H+ and heterovalent Ca2+-H+ binary ion exchange systems. For each system, the experiments

Figure 7. Breakthrough curve for Na-H exhaustion runs using a 0.50 N HCl solution. Wedge-shaped resin bed initially in the H-form.

Figure 8. Breakthrough curve for the Ca-H system using a 0.53 N HCl solution. Wedge-shaped resin bed initially in the Ca-form.

run at two different solution concentrations and at two different flow rates. The effluent concentrations determined experimentally are the average concentration of the effluent collected over the time interval. All the experiments run at a flow rate of about 400 mL/h are repeated. The experiment results presented in the four diagrams are clearly reproducible. Due to the accuracy of the volumetric measurement technique used, the experimental errors are estimated to be less than 0.5%. The dispersion coefficients used in eq 41 are calculated applying the dispersion coefficient functions, which are independently determined by dispersion experiments in the wedge-shaped resin beds and are listed in Table 1 for the Na-form and the H-form resin beds at different flow rates. By doing so, at any time the dispersion coefficient at any radial position within the resin bed is obtained and then applied to the model

2366 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

Figure 9. Breakthrough curve for the Ca-H system using a 0.53 N CaCl2 solution. Wedge-shaped resin bed initially in the H-form.

calculations to predict ion exchange performance in the wedge-shaped resin bed. The cyclic ion exchange experiments using the wedgeshaped cell begin with the resin bed in the Na-form. In the first experiment the Na-form bed is regenerated into the H-form. Figure 6 presents the breakthrough curves for the regeneration when 0.530 N HCl solution is drawn through the bed. In the first experiment the solution is drawn at a flow rate of 388 mL/h, while in the second experiment the solution is drawn at the slightly higher flow rate of 393 mL/h. The model prediction is based upon an average flow rate of 390 mL/ h. As may be seen by comparing the results from the two separate experiments, the data are reproducible. The agreement between the experimental observations and the model prediction is very good except for the deviations at breakthrough between 1.0 and 1.5 l of effluent volume. The breakthrough curves for the Na+-H+ exhaustion runs are shown in Figure 7 when 0.50 N NaCl solution is drawn through the resin bed initially in the H-form. In the first experiment the solution is drawn at the flow rate 393 mL/h, while in the second experiment the solution is drawn at the slightly lower flow rate 385 mL/ h. The model prediction uses the average flow rate 389 mL/h. Comparing the results from the two separate experiments shows that the data are reproducible and the agreement between the experimental observations and the model prediction is excellent except for the deviations at breakthrough around 1.5-2.0 l of effluent volume. Once the monovalent to monovalent Na+-H+ ion exchange experiments were completed, experiments to study ion exchange between divalent Ca2+ and monovalent H+ ions were conducted. Figure 8 presents the breakthrough curve for the Ca2+-H+ regeneration runs when 0.530 N HCl solution is drawn through the bed initially in the Ca-form. In the first experiment the solution is drawn at the flow rate 385 mL/h, while in the second experiment the solution is drawn at the slightly lower flow rate 380 mL/h. The model prediction is based upon the average flow rate 383 mL/h. Comparing the results from the two separate experiments shows that the data are reproducible and the model prediction and experimental observations are in reasonable agreement. The deviations occur almost over the entire period of breakthrough. The predicted breakthrough curve appears to lag behind the observed results by a constant volume. The breakthrough curve for the Ca2+-H+ exhaustion runs is shown in Figure 9 when 0.50 N CaCl2 solution

is drawn through the H-form resin bed. In the first experiment the solution is drawn at the flow rate 376 mL/h, while in the second experiment the solution is drawn at the slightly higher flow rate 378 mL/h. The model prediction is based upon the average flow rate 377 mL/h. The comparisons of the results from the two separate experiments show that the data are reproducible and the model prediction and the experimental observation are in reasonable agreement. The deviations occur at breakthrough around 1.2 L of effluent volume. Due to space restrictions, diagrams comparing the breakthrough curves obtained by experiments with those predicted using the radial model for the other flow rates are not presented. The agreement between experimental observations and predictions for those other experiments are just as good as those reported in this paper. Tsaur (1996) presents a detailed discussion of all results. It should be noted, however, that the model does not take account of the effect of the resin swelling and shrinking caused by the change of the surrounding solution concentration. The porosities are determined after the resin bed is rinsed with distilled deionized water; the dispersion coefficients are determined by injection of 0.10 N NaOH or HCl solution as the tracer into the water-rinsed resin beds; and ion exchange performance experiments are conducted using solutions of concentration ranging from 0.25 to 1.0 N. As the resin shrinks or swells when the surrounding solution concentration changes, this makes the experimentallydetermined porosities and dispersion coefficients either slightly overestimated or underestimated. Furthermore, the model also does not reflect the variable wall effect due to resin shrinking or swelling. Since an ion exchange process within a resin bed involves the conversion of the resin form and sometimes the change of the concentration of the surrounding solution, the variation of wall effect caused by resin shrinking and swelling is unavoidable. When the resin shrinks, the channels between the cell wall and the resin bed packed in the cell expand, allowing more solution to bypass to the exit without proper contact with the resin. Because of the unchangeable volume of the wedge-shaped cell, the resin bed becomes tighter when resin swells. The channels between the cell wall and the resin bed are reduced, and less solution bypasses to the outlet of the cell. Consequently the contact between the solution and the resin is enhanced. The wall effect varies as an ion exchange front moves, making a notable impact on the ion exchange performance. This may improve an ion exchange process by delaying breakthrough or impair an ion exchange process through an early breakthrough occurrence, both of which lead to the deviations of the model predictions from the experimental observations. Unfortunately, it is very difficult to isolate such wall effects from the ion exchange processes. Concluding Remarks A mathematical model has been developed to simulate ion exchange processes occurring within a fixed radial ion exchange bed. The model takes account of both dispersion and the nonideal behavior of the exchanging ions in both the solution and the exchanger phases. The predictions of the model agree well with the experimental observations for both the exhaustion and regenera-

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tion cycles for the monovalent Na+-H+ binary system and for the heterovalent Ca2+-H+ binary system. The number of potential applications of radial ion exchange is significant. Radial flow units could replace the more traditional column ion exchange beds. The much lower fluid velocities encountered in the outer zones of a radial ion exchange bed will result in improved ion exchange performances when compared to traditional column ion exchange beds. By careful design of the bed geometry, the pressure drop across a radial bed would be significantly lower than that across a comparable column bed. Moreover, the dispersion in a radial bed will grow less quickly than it would in a column, because of the slowing of the fluid away from the bed’s central injection point. Nomenclature a ) valence of ion A AR ) cross-sectional area of the bed b ) valence of ion B C0 ) total solution concentration Ci ) concentration of i ions in the solution phase DLr ) solution phase dispersion coefficient FL ) flow rate H ) thickness of the annular bed JA ) dispersion flux of A ions a ) equilibrium quotient KAB T KAB ) thermodynamic equilibrium constant PeR ) Pe`clet number in radial flow Q ) ion exchange capacity r ) dimensionless radius of the annular bed R ) annular bed radius RI ) inlet radius of the annular bed RO ) outlet radius of the annular bed t ) dimensionless time UR ) superficial velocity URP ) pore velocity VR ) variable volume of the exchanger bed y ) mole fraction of A ions in the resin phase YIA ) equivalent ionic fraction of A ions in the resin phase Zi ) valence of i ions γi ) solution phase activity coefficient of ion i γRi ) resin phase activity coefficient of ion i  ) bed porosity Λij ) Wilson binary interaction parameter FR ) ion exchange resin density τ ) time

Literature Cited Allen, R.; Addison, P. A.; Dechapunya, A. H. The Characterization of Binary and Ternary Ion Exchange Equilibrium. Chem. Eng. Biochem. Eng. J. 1989, 40, 151-158. Argersinger, W. J.; Davidson, A. W.; Bonner, O. D. Thermodynamics and Ion Exchange Phenomena. Trans. Kansas Acad. Sci. 1950, 53, 404-410.

Dorfner, K. Ion Exchangers; Walter de Gruyter Berlin: New York, 1991; Chapter 2, p 677. Hang, S. H.; Lee, W. C.; Tsao, G. T. Mathematical Models for Radial Chromatography. Chem. Eng. Sci. 1988, 38, 179-186. Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962; Chapter 9, p 421. Ippolito, I.; Hinch, E. J.; Daccord, G.; Hulin, J. P. Tracer Dispersion in 2-D Fractures with Flat and Rough Walls in a Radial Flow Geometry. Phys. Fluids A 1993, 5, 1952-1962. Johnson, K. S.; Pytkowicz, R. M. Ion Association of Cl- with H+, Na+, K+, Ca2+ and Mg2+ in Aqueous Solution at 25 °C? Am. J. Sci. 1978, 278, 1428-1447. Kester, D. R.; Pytkowicz, R. M. Sodium, Magnesium and Calcium Sulphate Ion-Pairs in Seawater at 25 °C. Limnol. Oceanogr. 1969, 14, 686-692. Mehablia, M. A. Multicomponent Ion Exchange Equilibria. Ph.D. Dissertation, Department of Chemical Engineering, The University of Melbourne, 1994. Mehablia, M. A.; Shallcross, D. C.; Stevens, G. W. Prediction of Multi-Component Ion Exchange Equilibria. Chem. Eng. Sci. 1994, 49, 2277-2286. Mehablia, M. A.; Shallcross, D. C.; Stevens, G. W. Prediction of Ion Exchange Equilibria for the H+-Na+-K+ and H+-Na+K+-Ca2+ Systems. Solvent Extr. Ion Exch. 1996, 14, 309-322. Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical basis and general equations. J. Phys.Chem. 1973, 77, 268-277. Pitzer, K. S. Activity Coefficients in Electrolyte Solutions; CRC Press: Boca Raton, FL, 1991. Rachinskii, V. V. Basic Principles of Radial Chromatography. J. Chromatogr. 1968, 33, 234-241. Rice, R. G. Approximate Solutions for Batch, Packed Tube and Radial Flow AdsorberssComparison with Experiment. Chem. Eng. Sci. 1982, 37, 83. Rice, R. G.; Heft, B. K. Separations via radial flow chromatography in compacted particle beds. AIChE J. 1991, 37, 629-632. Slater, M. J. Principles of Ion Exchange Technology; ButterworthHeinemann: Oxford, 1991; Chapter 7, p 54. Tan, C. T.; Homsy, G. M. Stability of miscible displacements in porous media. J. Fluids 1987, 30, 1239-1245. Tsaur, Y. Radial Ion Exchange Processes. Ph.D. Dissertation, Department of Chemical Engineering, The University of Melbourne, 1996. Wilson, G. M. Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Energy of Mixing. J. Am. Chem. Soc. 1964, 86, 127-130. Yortsos, Y. C. Dispersion of Driven Instability in Miscible Displacement in Porous Media. Phys. Fluids 1988, 31, 3511-3518. Zimmerman, W. B.; Homsy, G. M. Nonlinear Viscous Fingering in Miscible Displacement with Anisotropic Dispersion. Phys. Fluids 1991, 3, 1858-1872.

Received for review October 4, 1996 Revised manuscript received February 13, 1997 Accepted February 13, 1997X IE960622M

X Abstract published in Advance ACS Abstracts, April 1, 1997.