Multicomponent Space-Charge Transport Model for Ion-Exchange

Jan 15, 2004 - Department of Chemical Engineering, Tulane University, New ... and membrane ion-exchange sites.8,9 The space-charge uptake model was ...
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Ind. Eng. Chem. Res. 2004, 43, 2957-2965

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Multicomponent Space-Charge Transport Model for Ion-Exchange Membranes with Variable Pore Properties Y. Yang†,‡ and P. N. Pintauro*,§ Department of Chemical Engineering, Tulane University, New Orleans, Louisiana 70115, and Department of Chemical Engineering, Case Western Reserve University, Cleveland, Ohio 44106

A multicomponent space-charge transport model for an ion-exchange membrane composed of cylindrical pores with a variable radius and/or a variable wall charge density has been developed and tested for the Donnan dialysis separation of aqueous Cs+/Pb2+ and Na+/Pb2+ mixtures with a Nafion 117 cation-exchange membrane. Model equations for ion and water transport take into account ion/fixed-charge site electrostatic interactions, electric-field-induced water dipole orientation, ion-hydration free-energy changes during ion partitioning, concentration-dependent transport parameters, and ion-pair formation between absorbed Pb2+ and the membrane fixed charges. An excellent match of theory with experimental concentration vs time and water flow vs time data was achieved using pores with a constant radius and a repeating high/low squarewave surface charge distribution (where the low surface charge was a consequence of ion pairing). The model could not reproduce Donnan dialysis transport data when (1) the pore radius and wall charge density were constant and (2) the wall charge density was constant and the pore radius was a linear function of axial position. Pore radius changes had little effect on computed ion fluxes, but did produce large changes in the water flux across a cation-exchange membrane. Introduction Ion-exchange membranes are used in a variety of processes and devices, including dialysis and electrodialysis separations, batteries and proton-exchange membrane fuel cells, electrochemical reactors, and various electrochemical sensors. Mathematical models have been developed to predict ion and solvent fluxes in these membranes, including phenomenological theories based on irreversible thermodynamics1,2 and space-charge models where the membrane microstructure is predefined as an array of cylindrical pores.3-5 Most models consider a single salt electrolyte; the analysis of membrane transport becomes considerably more complicated when the number of mobile ionic species increases, but it is often these multicomponent membrane systems that are of most practical importance. The present authors have developed and tested multicomponent space-charge ion uptake and ion/water transport models for ion-exchange membranes, where the membrane is composed of an array of parallel cylindrical pores with a uniform and continuous distribution of fixed charges on the pore wall (to represent densely packed membrane ion-exchange groups). Both the equilibrium sorption and transport models consider ion/fixed-charge site electrostatic interactions, electricfield-induced water dipole orientation, and ion-hydration free energy changes during ion absorption and transport. With no adjustable parameters, the uptake (partition coefficient) theory predicted accurately the equilibrium absorption of various salt mixtures (e.g., binary mixtures of quaternary ammonium and alkali metal cation salts, as well as and two- and three* To whom correspondence should be addressed. Tel.: 216368-4150. Fax: 216-368-3016. E-mail: [email protected]. † Tulane University. ‡ Present Address: ExxonMobil Upstream Research Company, P. O. Box 2189, Houston, TX 77252. § Case Western Reserve University.

component mixtures of alkali metal salts) in a Nafion 117 cation-exchange membrane.6,7 (Nafion is a registered trademark of E.I. DuPont deNemours and Co., Inc.) When applied to monovalent/divalent cation absorption, the uptake model was modified to account for the formation of contact ion pairs between the +2charged cations and membrane ion-exchange sites.8,9 The space-charge uptake model was combined with ion flux and water flow equations to simulate cation/water transport across a Nafion cation-exchange membrane for single salt solutions and multicomponent electrolyte mixtures of monovalent cation salts.10,11 In the present paper, we extend the space-charge transport model to membranes with nonuniform pore radii and/or a nonuniform distribution of fixed charges on the pore wall. This new model is applied successfully to the Donnan dialysis separation of monovalent/divalent cation salt mixtures with a Nafion 117 cationexchange membrane. Multicomponent Transport Model for Variable Pore Geometry/Variable Wall Charge The space-charge membrane transport model equations for ions and water in the axial (x) and radial (r) pore directions have been presented and discussed in detail elsewhere.11 A listing of the equations follows. 1. A modified Boltzmann equation for ion concentrations (Cj)

Cj(x,r) ) Cbj exp

{

[

]}

Aj -zjF[Φ(x,r) - Φ(x,0)] 1 1 RT RT (x,r) b

(1)

where Φ is the electric potential, zj is the charge number of ion species j, F is Faraday’s constant,  is the solvent dielectric constant, Aj is a solvation (hydration) constant for ion j, and the superscript b refers to the bulk (external) solution. Cation and anion hydration con-

10.1021/ie030558q CCC: $27.50 © 2004 American Chemical Society Published on Web 01/15/2004

2958 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004

∇‚N h j(x) ) 0

Table 1. Ion Radii and Hydration Parameters ion species

ion radius (nm)

hydration parameter, Aj, at 298 K (J/mol)

ion diffusion coefficient,a Doj (cm2/s)

Pb2+ Cs+ Na+ H+ NO3-

0.132b 0.165 0.102 0.158c

1.05 × 106 c 1.92 × 105 d 2.09 × 105 d 2.61 × 105 e 3.48 × 105 c

0.942 × 10-5 2.06 × 10-5 1.33 × 10-5 9.31 × 10-5 1.90 × 10-5

7. The Navier-Stokes equation (with an added electric body force term) and the constant fluid density continuity equation for the water velocity in a membrane pore n

-∇P(x,r) + η∇2v(x,r) - ∇Φ(x,r)F

a

From equivalent ionic conductance data in ref 17. b Unless noted otherwise, ion radii from ref 16. c From ref 9. d From ref 7. e From ref 11.

stants were computed using the theory presented in ref 7, where it was shown that Aj is a function of the ion size and charge and the solvent type. The hard-sphere ion radius and hydration parameters for the cation and anion species examined in the present study are listed in Table 1. 2. Poisson’s equation with a nonuniform dielectric constant

∇‚[(x,r)∇Φ(x,r)] )

F

∑zjCj (x,r)

(2)

where * is the permittivity of vacuum. 3. Booth’s equation,12,13 which describes the decrease in solvent dielectric constant with increasing electric field strength

(x,r) ) ξ2 +

[

β)

Dj(x,r) )

where ζ is the solvent dipole moment and κ is the Boltzmann constant. 4. Nernst-Planck equations for the molar flux (Ni) of all but one mobile ionic species

(9)

Doj ηo

(10)

η(x,r)

where ηo is the viscosity of pure water and

1 + λ(x,r)/2 η(x,r) ) ηo [1 + λ(x,r)]2

(11)

In eq 11, λ(x,r) is the local solute volume fraction (considering all mobile ions in a membrane pore)

]

5ζ 2 (ζ + 2) 2κT

(8)

where P is the pressure and η is the pore fluid viscosity. The concentration dependence of ion diffusivities (Dj) and solution viscosity (η) in a membrane pore was modeled11 using infinite-dilution diffusion coefficient (Doj ) data from the literature, the Nernst-Einstein relationship, and the Einstein viscosity equation

3(b - ξ2) 1 1 (3) β∇Φ(x,r) tanh[β∇Φ(x,r)] β∇Φ(x,r)

In the above equation, ξ is the optical refractive index of the solvent, and

zjCj(x,r) ) 0 ∑ j)1

∇‚v(x,r) ) 0

n

*j)1

(7)

λ(x,r) )

( ) 4

∑j Cj(x,r)N˜ 3πRˆ j3

× 10-3

(12)

where R ˆ is the hard-sphere ion radius and N ˜ is Avogadro’s number. Infinite-dilution diffusion coefficients (in water at 25 °C) for the cation and anion species examined in the present study are listed in Table 1. The boundary conditions for the model equations in the radial pore direction are

At r ) 0,

∂Φ ) 0,  ) b, vr ) 0 ∂r

σ-

Nj(x,r) )

∂Φ

∑j ezjΓj

(13)

zjDjFCj(x,r) -Dj∇Cj(x,r) ∇Φ(x,r) - Cj(x,r) v(x,r) (4) RT

At r ) a,

where D is the diffusion coefficient and v is the velocity. 5. The current density equation

where σ is the surface charge density of ion-exchange sites on the pore wall (based on the membrane’s ionexchange capacity), e is the charge on an electron, and Γj is the number density of multivalent (e.g., divalent) cations with charge zj that bind (ion-pair) to membrane ion-exchange sites. The use of eq 14 presumes that each divalent counterion binds to two ion-exchange sites, thereby forming a charge-neutral moiety. Ion pairing between monovalent counterions and membrane fixed charges was not considered, on the basis of previous modeling and NMR studies. The axial boundary conditions at the pore entrance (x ) 0) and exit (x ) L) are an osmotic pressure relationship and a modified Boltzmann equation for ion concentrations, i.e.

n

I)

zjFN h j (x) ∑ j)1

(5)

where I is the current density (A/m2) and N h j(x) is the average molar flux of ion species j, integrated over a pore cross section with radius a, at a given axial pore position

N h j(x) )

∫0aNj(x,r)2r dr a2

(6)

∂r

)-

, vx ) 0, vr ) 0

*(a)

(14)

n

6. Steady-state conservation of species equations for the ion fluxes

P(x,r) ) Po + RT [

Ci(x,r) - Cbi ] ∑ i)1

(15)

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2959

Figure 1. Membrane pores with variable pore properties. (a) Pore with a monotonically increasing radius. (b) Pore with a squarewave wall charge density. Cylindrical constant radius segments are drawn in part a for the variable-pore-radius case.

and

Cj(x,r) )

{

Cbj exp

[

]}

-zjF[Φ(x,r) - Φb] Aj 1 1 RT RT (x,r) b

(16)

where Po is atmospheric pressure; Φb is the electric potential in the external solution (which is set equal to zero on one side of the membrane in order to define the zero ground potential state); and Φ(x,r) and (x,r) for x)0 and L are found by solving the one-dimensional equations for multicomponent equilibrium ion uptake (i.e., Poisson’s equation, Booth’s equation, and the modified Boltzmann equation for ion concentrations).11,14 In the present study, we sought to apply the above equations to a membrane with a variable pore radius and/or a variable pore-wall charge density (see Figure 1a and b). Our approach was to divide the pore into segments that can be approximated by small sections of straight (constant-radius) cylinders, where the wall charge density for each cylinder is constant (see Figure 1a). Within each of the straight cylindrical segments and at the two membrane/solution interfaces, the transport model equations and boundary conditions presented above are utilized. This technique is similar to the method of multiple scales for modeling fluid flow in tubes of nonuniform radius (see, for example, ref 15). Additional mathematical coupling relations must be established for neighboring cylindrical sections, where discontinuities in pore radius and/or wall charge density might exist. To simplify the notation, a cylindrical pore section is identified as Sk at xk-1 e x e xk for k ) 1, ..., m, as shown in Figure 2. Because thermodynamic equilibrium exists in the radial-pore direction with a constant-radius cylindrical section, discontinuities in the pore radius (a) and/or wall charge density (σ) at x ) xk, k ) 1, 2, 3, ..., m, must result in discontinuities in the electric potential, ion concentration, dielectric constant,

Figure 2. Discontinuities in physical quantities at the right and left boundaries of two neighboring cylindrical segments.

and hydrostatic pressure at the boundary between any two subsections. To relate Cj(x,r), Φ(x,r), and (x,r) at the left and right sides of the intersection of two neighboring cylinders, denoted as Sk and Sk+1 (see Figure 2), thermodynamic equilibrium is assumed at x ) xk. Conceptually, the intersections can be regarded as equilibrium interfaces within the membrane (in addition to the equilibrium uptake conditions that exist at the membrane/bulk solution interfaces). With this assumption, a Donnan relation and an equation for the hydrostatic pressure can be written at the centerline point of the interface between any two neighboring sections, where the superscripts l and r signify the left and the right end points, respectively (see Figure 2).

{

C1j (Sk+1) ) C rj (Sk)exp -

}

zj F 1 [Φ (Sk+1)-Φr(Sk)] RT

(17)

n

P1(Sk+1) - P r(Sk) )

r RT [Cj1(Sk+1) - Cj (Sk)] ∑ j)1

(18)

Coupling between cylindrical sections is completed through mass conservation equations for each ion species and for water

h rj (Sk) N h 1j (Sk+1) ) N

(19)

h rw(Sk) N h 1w(Sk+1) ) N

(20)

where N h j and N h w are the average molar fluxes of ion species j and water, respectively, through a membrane pore (see eq 6). Experimental Section Donnan Dialysis Experiment. Donnan dialysis experiments were carried out using the cell shown in Figure 3. A Nafion 117 membrane (5.1 cm2) separated two compartments, each with a volume of 285 cm3,

2960 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 Table 2. Nafion 117 Membrane Pore Radius, Porosity, and Wall Charge Density for the Final (Equilibrium) Solutions of the Donnan Dialysis Experiments

experiment 1a experiment 2b

pore radius (nm)

porosity

wall charge density (C/m2)

1.95 2.27

0.233 0.294

0.591 0.502

a Solution composition: 0.0625 M CsNO , 0.0625 M Pb(NO ) , 3 3 2 0.125 M HNO3. b Solution composition: 0.0625 M NaNO3, 0.0625 M Pb(NO3)2, 0.125 M HNO3.

(the ion-exchange capacity of Nafion, 0.909 × 10-3 mol/ g; the polymer dry density, 2.1 × 106 g/m3 for Nafion 117; and the membrane pore radius in a water-swollen film, 2.75 nm for Nafion 11719). Membrane porosities and calculated pore radii and pore-wall charge densities are listed in Table 2. Figure 3. Schematic diagram of the Donnan dialysis membrane cell.

containing sampling ports (open to the atmosphere) and Teflon-coated stirrers. One of the cell chambers was filled initially with a dilute (0.25 M) aqueous HNO3 solution, and the other chamber contained an aqueous salt mixture of 0.125 M Pb(NO3)2 with either 0.125 M CsNO3 or 0.125 M NaNO3. The concentrations of protons and cation species in each compartment were determined over the course of an experiment by withdrawing periodically 0.5 mL of solution and analyzing for H+ (by titration with NaOH) and metal cations (by atomic absorption spectrophotometry, AAS). The salt/ acid concentrations were well below Nafion’s ionexchange capacity; hence, the concentration of nitrate anions in the membrane was small and not measured. Experiments were repeated to measure water flow across the membrane, where horizontally positioned, precision-bore glass capillary tubes (3-mm-i.d. and partially filled with fluid) were connected to each compartment’s sampling port. By tracking the fluid position in each capillary tube, liquid volume changes in the reservoirs on either side of the membrane were monitored during a Donnan dialysis experiment. Equilibrium Uptake Experiments. Equilibrium salt uptake experiments were performed with Nafion 117 samples to determine (1) the membrane porosity in a given solution (which is needed to compute the pore radius and pore-wall charge density) and (2) the extent of contact ion-pair formation between absorbed Pb2+ and the SO3- fixed charges. The experiments were carried out using a solution of either 0.0625 M Pb(NO3)2 + 0.0625 M CsNO3 + 0.125 M HNO3 or 0.0625 M Pb(NO3)2 + 0.0625 M NaNO3 + 0.125 M HNO3, i.e., solution compositions that would exist in both the acid and salt compartments of the Donnan dialysis cell at the conclusion of a transport experiment. Standard procedures of membrane pretreatment (boiling in nitric acid and then water), salt-solution soaking, leaching in an aqueous KNO3 solution, solution analysis (by base titration and AAS), and membrane drying were used to determine the concentrations of metal cations and protons in Nafion samples.7,8 As in previous applications of the model,6-11,18 the membrane pore radius (a) and pore-wall charge density (σ) for a given external salt solution were determined using a simple microstructure model for Nafion, experimentally measured membrane porosities when samples were equilibrated in water and the salt solution, and a collection of membrane (Nafion) physical property data

Results and Discussion Extent of Ion-Pair Formation: Equilibrium Uptake Experiments. It has been shown in previous studies that divalent cations form contact ion pairs (coordinate covalent bonds) with the fixed-charge sulfonate sites in a Nafion cation-exchange membrane8,9 (one divalent cation binding to two SO3- groups). The result of such ion-pair formation is a decrease in the pore-wall charge density

σef ) σ -

∑j ezjΓj

(21)

where σef is the “effective” pore-wall charge density, after neutralization of membrane fixed charges. (The right-hand side of eq 21 appears in the numerator of the pore-wall boundary condition, eq 14.) To find the correct value of σef for the Pb2+/Cs+/H+ and Pb2+/Na+/ H+ solutions in the Donnan dialysis experiments, equilibrium uptake experiments were performed where the sorption selectivity of H+ with respect to Pb2+ (denoted as RH+ Pb ) was measured in prepared solutions (as + described above). Here, RH Pb is defined as +

RH pb )

T CbPb CH + b CTPb CH +

(22)

where CTj is the total concentration of H+ or Pb2+ inside the membrane (mobile + ion-paired for the case of Pb2+). A theoretical value of the equilibrium selectivity was obtained by solving the ion uptake model (Poisson’s equation, Booth’s equations, and the modified Boltzmann equation for ion concentrations, as described in ref 7) for a pore with a given radius (from Table 1) and wall charge density σef. By means of a systematic trial-and-error approach, the value of σef that gave the + best match of the theoretical and experimental RH Pb values was found. The results of this match are presented in Table 3. The small value of the absorption selectivity in Table + T T 3 (RH Pb < 1, which means that CPb > CH+) is attributed to ion-pair formation and not to a high concentration of unbound Pb2+ in the pore fluid. The solution-phase concentration of Pb2+ in Nafion is actually less than that of H+, as shown by the computed radial concentration profiles in Figure 4 for Pb2+/Cs+/H+ uptake. The general shape of these profiles can be explained in terms of the modified Boltzmann equation (eq 1). The concentration

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2961

Figure 4. Computed radial-direction concentration profiles in the pores of a Nafion 117 membrane when equilibrated in a solution of 0.0625 M Pb(NO3)2 + 0.0625 M CsNO3 + 0.125 M HNO3. s, Cs+; - - -, H+; s s s, Pb2+; s - s - s NO3 (a ) 1.95 nm, σ ) 0.407 C/m2). Table 3. Effective Pore-Wall Charge Density and Experimental/Theoretical Values of Equilibrium Uptake Selectivities of H+ with Respect to Pb2+ in a Nafion 117 Membrane RH+ Pb RH+ Pb σef

(exp) (theory)

experiment 1a

experiment 2

0.654 0.665 0.407 C/m2

0.49 0.374 0.187 C/m2

Figure 5. Computed radial-direction water dielectric constant profile in a Nafion 117 membrane pore after equilibration in a solution of 0.0625 M Pb(NO3)2 + 0.0625 M CsNO3 + 0.125 M HNO3.

Donnan Dialysis Experiments. To match the theory to experimental concentration vs time data from the batch Donnan dialysis experiments, the transport model equations (which hold only within the membrane) must be combined with mass-balance relationships for the external reservoirs on either side of the membrane

hj d[VCbj ] SθN ) dt τ

a

Solution composition: 0.0625 M CsNO3, 0.0625 M Pb(NO3)2, 0.125 M HNO3. b Solution composition: 0.0625 M NaNO3, 0.0625 M Pb(NO3)2, 0.125 M HNO3.

of mobile cations and anions at any radial-pore location is governed by the combined effects of electrostatic forces (that attract counterions and repel coions) plus a repulsive hydration force that acts on all ions and arises when the dielectric constant of pore water is less than 78. Even when there is substantial neutralization of Nafion’s ion-exchange capacity by ion-paired divalent cations (i.e., σef in Table 3 is less than σ in Table 2), a sufficient number of ion-exchange groups is available to create a strong radial-direction electric field near the pore wall that orients water dipoles, causing the water dielectric constant to fall below its bulk (external) solution value. This point is illustrated in Figure 5, where the computed water dielectric constant is plotted against the radial pore position for the equilibrium uptake of Pb2+, Cs+, and H+. Hydration forces strongly affect ions with a high surface charge density; thus, unbound Pb2+ and, to a lesser extent, H+ are excluded from the pore-wall region (see Figure 4). The lowsurface-charge-density Cs+ ion is essentially unaffected by the low-dielectric-constant water and its concentration monotonically increases as the pore wall is approached. When the membrane was equilibrated in 0.0625 M NaNO3 + 0.0625 M Pb(NO3)2 + 0.125 M HNO3, the predicted concentrations of bound and unbound Pb2+ in the pore fluid were higher than those for the Pb2+/Cs+/H+ case (resulting in smaller values of σef + and RH Pb ). This observation is consistent with prior studies where (1) Cs+ was absorbed preferentially over Na+ in Nafion,7 (2) the concentration of unbound Pb2+ in a Nafion pore decreased as the pore fluid concentration of the coabsorbed monovalent concentration increased,8,9 and (3) the fraction of ion-paired (neutralized) pore-wall sulfonate sites in Nafion correlated with the pore-fluid concentration of unbound divalent cations.9

with

d[V] Sθvj ) dt τ

(23)

where V is the reservoir volume, S is the total membrane area, vj is the pore-averaged axial fluid (water) velocity, N h j is the pore-averaged molar flux of ion species j (see eq 6), and τ is the pore tortuosity. The multicomponent transport model with the following three different pore configurations was matched to experimental Donnan dialysis data: (1) a constantradius pore (given in Table 2) with a constant wall charge density (σef in Table 3), (2) a constant-radius pore (from Table 2) with a variable (repeating) high/low wall charge density that represents localized discrete regions of high ion-pair formation, and (3) a variable-radius pore (monotonically increasing from the salt side to the acid side) with a constant wall charge equal to σef (Table 3). Cases 1 and 2 are treated first in the discussion below. The nonuniform adsorption of divalent ions was modeled by treating the wall charge density, σ(x), as a step function with alternating sections of high (σhigh) and low (σlow) wall charge density (similar to that shown in Figure 1b). A value of σhigh was set equal to the unmodified pore-wall charge density σ (in Table 2), whereas σlow was viewed as an adjustable parameter for fitting the model to data collected in a Donnan dialysis experiment. In such an experiment, Pb2+ was present initially in the salt compartment only, so σlow was used at the membrane/salt solution interface and σhigh was used at the membrane/acid solution interface (for all t g 0). The fractional pore-wall surface area having a high or low wall charge density (denoted as qhigh or qlow, respectively) was calculated so that the overall (average) wall charge density was equal to σef in Table 3

qhigh )

σef - σlow σhigh - σlow

and

qlow )

σhigh - σef (24) σhigh - σlow

2962 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004

Figure 6. Cs+/Pb2+ selectivity results from the Donnan dialysis experiment with a Nafion 117 membrane and initial concentrations of 0.125 M CsNO3 + 0.125 M Pb(NO3)2 and 0.25 M HNO3. Symbols are experimental data, and lines are model predictions with different models for the wall charge density (constant pore radius of 1.95 nm) s, model with alternating sections of high and low wall charge density (see Table 4); - - -, model with constant wall charge density σef (from Table 3).

Figure 7. Effect of the number of high/low wall charge segments on Cs+/Pb2+ selectivity results from the Donnan dialysis experiment with a Nafion 117 membrane and initial concentrations of 0.125 M CsNO3 + 0.125 M Pb(NO3)2 and 0.25 M HNO3. Symbols are experimental data, and lines are model predictions (constant pore radius of 1.95 nm, high/low wall charge densities from Table 4): - - -, total number of segments ) 18; s, total number of segments ) 36; s s s, total number of segments ) 72.

The presence of an immobile ion/fluid layer near the pore wall21 was also incorporated into the transport model, where the layer thickness (h) was set equal to the hydrated radius of an ion-paired divalent cation (0.401 nm for Pb2+, from ref 20). Within this fluid layer (i.e., at a - h < r < a), for cylindrical pore segments with σ ) σlow, the water viscosity was set infinitely large (and the ion diffusion coefficients were zero, as per eq 10). At other radial positions (0 < r < a - h) in σlow pore segments and for the entire pore cross section for those segments where σ ) σhigh, the normal solution viscosity and ion diffusion coefficient relationships were used (see eqs 10-12). The match of experimental data and modeling results for the Donnan dialysis experiment with an initial acid solution of 0.25 M HNO3 and an initial salt mixture of 0.125 M CsNO3 + 0.125 M Pb(NO3)2 is shown in Figure 6. Data are plotted as the concentration ratio of Cs+ to Pb2+ in the acid compartment vs the Pb2+ concentration in the same acid reservoir. It has been shown previously11 that model predictions, when plotted in this way, are independent of the pore tortuosity, which is not known a priori. Thus, our assessment of the model in Figure 6 was not complicated by the need to estimate τ in eq 23. The numbers (m) of σhigh and σlow sections was each set initially to be 18, with the length (∆) of the sections being ∆high ) Lqhigh/m and ∆low ) Lqlow/m (where L is the total pore length, i.e., the wet membrane thickness). As can be seen in Figure 6, there is excellent agreement between the experimental measurements and model predictions with a repeating square-wave wall charge distribution, where σhigh ) 0.591 C/m2 and σlow ) 0.163 C/m2 and where 57% of the pore surface was at the high wall charge density. The computed Cs+/ Pb+ concentration ratio was overestimated when a constant wall charge boundary condition was used in the model (with a constant pore radius). The “best-fit” values of σhigh, σlow, qhigh, and qlow that allowed the model to fit experimental data for the two multicomponent Donnan dialysis experiments are listed in Table 4. The match of theory and experimental data for a bimodal pore-wall surface charge does not preclude the possibility that some other variable surface charge distribution

Table 4. Values of σ and q for the Sections of High and Low Wall Charge Density in a Nafion Pore for the Multicomponent Donnan Dialysis Model with Pb2+ (C/m2)

σhigh σlow (C/m2) qhigh qlow

experiment 1a

experiment 2b

0.591 0.163 57% 43%

0.502 0.125 16% 84%

a Initial solution composition: 0.125 M CsNO , 0.125 M Pb3 (NO3)2, 0.25 M HNO3. b Initial solution composition: 0.125 M NaNO3, 0.125 M Pb(NO3)2, 0.250 M HNO3.

might exist in Nafion pores. No other distribution was examined in the present study. The monovalent/divalent cation separation selectivity in Figure 6 decreased, and the two theoretical curves merged as the batch Donnan dialysis experiment progressed. The results show that ion transport in the pore fluid near the pore wall (where there is discrimination between Cs+ and mobile Pb2+) slowed with time during the batch Donnan dialysis experiment. A similar effect was seen in a previous application of the model to multicomponent salt separation with monovalent cations.11 Although the use of high and low wall charge densities had a strong effect on the modeling results, as compared to the results for a uniform wall charge density, the computed Cs+/Pb2+ transport selectivities were relatively independent of the actual number of cylindrical pore segments used in the calculations. As can be seen in Figure 7, essentially the same computergenerated selectivity vs CPb curve was obtained when the Nafion pore was subdivided into 18, 36, or 72 subsections. The matches of model predictions and experimental measurements for cation and proton concentration changes and water transport as a function of time for the Donnan dialysis experiment using an initial aqueous 0.25 M HNO3 solution and an initial aqueous salt mixture of 0.125 M CsNO3 + 0.125 M Pb(NO3)2 are presented in Figures 8 and 9. For these theoretical calculations, the pore tortuosity was fixed by finding that value of τ that allowed the model to match the first five experimental CPb(t) data points. After selecting τ,

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 2963

Figure 8. Measured and computed external cation and proton concentrations in the salt and acid compartments of the Donnan dialysis cell. Nafion 117 membrane and initial solutions of 0.125 M CsNO3 + 0.125 M Pb(NO3)2 and 0.25 M HNO3. Symbols are experimental data points, and lines are computational results. (a ) 1.95 nm, θ ) 0.223, high/low wall charge densities from Table 4, τ ) 7.5).

Figure 9. Computed and measured water transport rates (from the salt to the acid chamber) during the Donnan dialysis experiment with Nafion 117 and initial solutions of 0.125 M CsNO3 + 0.125 M Pb(NO3)2 and 0.25 M HNO3. Symbols are experimental data points, and line is the model prediction.

the computed concentration vs time and water flow vs time results matched very well with the experimental data. Bulk solution changes in H+ concentration with time and water transport across the membrane were slower for Donnan dialysis with Cs+/Pb2+ than with a salt mixture of Na+ and Cs+ that does not form ion pairs with membrane fixed charges (see Figures 6 and 7 in ref 11). Both the model and experimental data suggest that neutralization of the membrane fixed-charge sites by divalent ions decreases the effectiveness of the ionexchange membrane in transporting ions and water. Computed results and experimentally measured concentrations for Donnan dialysis with an initial aqueous acid solution of 0.25 M HNO3 and an initial salt mixture of 0.125 M NaNO3 + 0.125 M Pb(NO3)2 are presented in Figures 10 and 11. Again, there is very good agreement between the theory and experimental data for both ion and water transport. For the Na+/Pb2+ case, a higher percentage (84%) of the pore wall had a low wall charge density (0.125 C/m2), as compared to the Cs+/Pb2+ simulation. This result is consistent with prior divalent/ monovalent equilibrium uptake studies8 with Nafion

Figure 10. Measured and computed external cation and proton concentrations in the salt and acid compartments of the Donnan dialysis cell. Nafion 117 membrane and initial solutions of 0.125 M NaNO3 + 0.125 M Pb(NO3)2 and 0.25 M HNO3. Symbols are experimental data points, and lines are computational results. (a ) 2.27 nm, θ ) 0.294, high/low wall charge densities from Table 4, τ)3.8).

Figure 11. Computed and measured water transport rates (from the salt to the acid chamber) during the Donnan dialysis experiment with Nafion 117 and initial solutions of 0.125 M NaNO3 + 0.125 M Pb(NO3)2 and 0.25 M HNO3. Symbols are experimental data points, and line is the model prediction.

and with the data in Table 3, which show that there was more lead binding (ion-pairing) when the coabsorbed monovalent cation was Na+ rather than Cs+. As shown in Figure 10, the flux of Pb2+ was initially greater than that of Na+, but at long times, the trend was reversed, resulting in a crossover of the concentration profiles. The migration component of the cation flux (the zjDjFCj/RT∇Φ term in eq 4) has been shown to dominate during the early states of a batch Donnan dialysis experiment.11 Pb2+ has a higher initial migration flux because (1) its charge number (zj) is twice that of Na+ and (2) the majority of the mobile lead ions are located in the center region of the pore (see Figure 4), where the fluid viscosity is low and the Pb2+ diffusion coefficient (which appears in the migration term) is large. As time progresses, the radial-direction concentration profiles for all ionic species at the salt and acid ends of a membrane pore begin to coincide (as more salt enters into the acid compartment and more acid is transported into the salt reservoir).11 Consequently, the electric potential difference across the pore in the axial direction (which is the driving force for ion migration) decreases; transport near the pore wall slows; and diffusion in the

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center region of the pore dominates, in which case NNa > NPb because the diffusion coefficient of Na+ is greater than that of Pb2+ (see Table 1). For completeness, model calculations were carried out for the case of a variable radius pore with a constant pore-wall charge density (σef in Table 3). When the pore radius was allowed to change linearly with axial position from 1.0 to 3.0 nm, for experiment 1 in Table 3 (where the measured pore radius was 1.95 nm), the computed Donnan dialysis ion fluxes were only slightly smaller (