Simple Model for Characterizing a Donnan Dialysis Process

Article pubs.acs.org/IECR

Simple Model for Characterizing a Donnan Dialysis Process David Hasson,*,† Adam Beck,‡ Fiana Fingerman,† Chen Tachman,† Hilla Shemer,† and Raphael Semiat† †

Rabin Desalination Laboratory, Department of Chemical Engineering, TechnionIsrael Institute of Technology, Haifa 32000, Israel Institute for Water Resources and Water Supply, Hamburg University of Technology, D-21073 Hamburg, Germany

‡

ABSTRACT: The objective of this work is to characterize the kinetics of Donnan dialysis process by a simple model. The Nernst−Planck equation is simplified by neglecting the membrane swelling pressure and assuming an average diffusion coefficient for all ions. Experimental data are presented on the permeation of monovalent phosphate ions from a KH2PO4/ NaH2PO4 feed solution to a NaCl receiver solution using two different anion exchange membranes: a low flux membrane (Fumasep FAB) and a high flux membrane (Selemion AMV). Measurements indicate a brief initial ion exchange process in which the membrane is loaded with phosphate ions. Phosphate transport data fit the predicted correlations and provide values of the overall kinetic coefficient. The kinetic coefficients derived from both membranes were found to increase linearly with the membrane phosphate concentration. The phosphate dialysis data of the low flux membrane were found to be controlled by a diffusive process, while the data of the high flux membrane were found to be mass transfer controlled. The increase of the kinetic coefficient with phosphate concentration is shown to be related to the increase in membrane ion diffusivity with its solution concentration.

1. INTRODUCTION Donnan dialysis,1−4 due to its simple operation and low energy requirement, has a high potential for purifying waters from ionic contaminants and for recovery of valuable metals from waste solutions. In this process the solution of the ionic species to be separated, such as an anionic species, is held in a feed compartment separated from a receiver compartment by an anion exchange membrane. The receiver compartment contains a high concentration of a stripping anion. The anion exchange membrane (AEM) prevents cation permeation and enables diffusional transport of the anionic species from the feed to the receiver section with an equivalent opposite migration of counteranions from the receiver to the feed compartment. Donnan dialysis studies with cation exchange membranes explore separation and valuable metal recovery of ions such as cobalt, nickel, silver, copper, zinc, aluminum, and chromium.5−12 Studies with anion exchange membranes explore removal of harmful ions from water such as arsenate, fluoride, nitrate, and bromide.13−20 Further studies examine the possibility of softening hard water by Donnan dialysis of calcium, magnesium, sulfate, and bicarbonate.21−24 As in all membrane separation processes, a key feature of a Donnan dialysis process is prediction of the ionic flux for given feed and receiver compositions and ion exchange membrane properties. Transport of ions through the membrane is a rather complex process which depends on the diffusion coefficients of the ions in the membrane, the ion concentrations at the solution/ membrane interface, and the ion exchange capacity. Evaluation of the diffusion coefficients of ions in the membrane has been investigated by several authors.25−34 The primary means by which an ion exchange membrane regulates anion/cation permeation is by the selective adsorption of counterions versus co-ions at the membrane/solution interface.35 Numerous studies have been carried out over the years to measure and model the equilibrium uptake of anions and cations, mostly in cation exchange membranes. Results of © 2014 American Chemical Society

particular significance for the present work are the effects of the increase in salt solution concentration on the membrane fixed ion concentration, co-ion concentration, counterion concentration, and membrane ion diffusivities. Fixed ion concentration and counterion concentrations show a moderate increase while the increase in co-ion concentration is high.35−37 Salt diffusion coefficient increases with increasing salt concentration from factors which are currently poorly understood.38 The objective of this work is to test the applicability of a simple model for estimating the kinetics of Donnan dialysis. The study is on the separation of phosphate ions. Practical interest in phosphate removal is related to the lack of a reliable antiscalant for controlling calcium phosphate scaling in reverse osmosis desalination of ultrafiltration/microfiltration purified wastewaters39 and for complying with strict phosphorus effluent regulations requiring trace phosphate removal from waters discharged to the environment.40

2. KINETIC MODEL The kinetic system studied is shown in Figure 1. A phosphate solution of initial concentration [KH2PO4]0 is recycled through the feed compartment I of an anion exchange dialysis cell. A sodium chloride solution of initial concentration [NaCl]0 is recycled through the receiver compartment II. The volumes of the two compartments are taken to be equal. 2.1. Flux Rate Equation. The derivation below indicates that the equilibrium concentrations of the system (marked by an asterisk) are given by: Received: Revised: Accepted: Published: 6094

December 18, 2013 March 5, 2014 March 6, 2014 March 6, 2014 dx.doi.org/10.1021/ie404291q | Ind. Eng. Chem. Res. 2014, 53, 6094−6102

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⎧ d[H 2PO4 −]m F JH PO − = −DmH2PO4−⎨ − [H 2PO4 −]m 2 4 ⎩ dx RT dE m ⎫ ⎬ dx ⎭

(4)

⎧ d[Cl−]m F dE m ⎫ ⎬ − [Cl−]m JCl− = −JH PO − = −DmCl−⎨ 2 4 ⎩ dx RT dx ⎭ (5)

where F is the Faraday constant, R is the universal gas constant, T is the absolute temperature, and Dm is the ionic diffusion coefficient. The electrical neutrality inside the membrane is given by

Figure 1. Dialysis system analyzed.

[H 2PO4 −]*2 [Na +]0 [Cl−]*2 = = [K+]0 [H 2PO4 −]1* [Cl−]1*

[H 2PO4 −]m + [Cl−]m = [X m+]

(1)

where [Xm+] is the fixed ion concentration in the membrane. Elimination of the gradient [dE/dx] yields

Thus, if the initial NaCl concentration is higher than the initial KH2PO4 concentration, a driving force is induced acting to transfer the phosphate ions from the feed to the receiver compartment. Derivation of a simplified model describing the kinetics of the process is based on the following assumptions: (i) concentrations are used instead of activities, an assumption justified by the low solution concentrations investigated in this study; (ii) membrane swelling pressure is negligible; (iii) leakage of cations through the anion exchange membrane is assumed to be negligible. Figure 2 depicts concentration profiles in the system and defines the various concentration parameters. The following electroneutrality and mass balance conditions constrain the system:

⎧ ⎪ {[H 2PO4 −]m + [Cl−]m } d[H 2PO4 ]m ⎪ ⎨ =− dx ⎪ [H2PO4−]m + [Cl−]m ⎪ DmCl− DmH2PO4− ⎩ −

JH PO − 2

4

{

−

−

= [H 2PO4 ]1 + [Cl ]1 = [H 2PO4 ]w1 + [Cl ]w1

JH PO − = −Dm

(7)

4

d[H 2PO4 −]m d[Cl−]m = −JCl− = +Dm dx dx (8)

Integration of eq 8 provides expressions describing mass transfer of the phosphate from the feed solution to the membrane interface, diffusive transport through the membrane, and mass transfer of the phosphate from the membrane interface, to the receiver solution:

(2)

[Na +]0 = [Cl−]0 = [Cl−]2 + [Cl−]1 = [Cl−]2 + [H 2PO4 −]2 = [Cl−]w2 + [H 2PO4 −]w2

}

Since except for H+ and OH− ion diffusivities of all species differ only slightly, eq 7 can be greatly simplified by assuming that the phosphate and chloride membrane diffusivities are equal. The basic diffusion equations become

2

−

} {

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

= −JCl−

[K+]0 = [H 2PO4 −]0 = [H 2PO4 −]1 + [H 2PO4 −]2 −

(6)

(3)

JH PO − = k{[H 2PO4 −]1 − [H 2PO4 −]w1 }

where bracketed terms denote molar concentrations. The following Nernst−Planck expressions describe the diffusional fluxes Ji (mol/m2 s) of the ions through the ion exchange membrane under the concentration gradient and the electric potential gradient dEm/dx across the membrane:

2

4

=

Dm {[H 2PO4 −]m1 − [H 2PO4 −]m2 } δ

= k{[H 2PO4 −]w2 − [H 2PO4 −]2 }

(9)

Figure 2. Concentration profiles across the membrane: (a) H2PO4−; (b) Cl−. 6095

dx.doi.org/10.1021/ie404291q | Ind. Eng. Chem. Res. 2014, 53, 6094−6102


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where δ is the membrane thickness and k is the convective mass transfer coefficient. The flux of the chloride ions from the receiver to the feed solution is similarly given by: −

Dvm ⎛ mol ⎞ Pso ⎜ 2 ⎟ = ⎝m h⎠

−

4

=

Dm {[Cl−]m2 − [Cl−]m1 } = k{[Cl−]w1 − [Cl−]1 } δ (10)

Equality of the electrochemical potential of the phosphate ions in solution (μsH2PO4−) with their potential at the membrane− solution interface (μHm2PO4−) relates solution to membrane concentrations through the Donnan potential (EDON). The electrochemical potential equality of the phosphate ions on the feed side yields the following solution−membrane relation: [E DON]1 = Em1 − E1 ⎤ [H 2PO4 −]m1 1⎡ = ⎢RT ln + V[H2PO4−]πs⎥ − F⎣ [H 2PO4 ]w1 ⎦

− (11)

δ (m)

(18)

− d[H 2PO4 −]1 [H 2PO4 −]2 ⎞ S ⎛ [H PO ] = Ps ⎜ 2 + 4 1 − ⎟ dt V ⎝ [K ]0 [Na +]0 ⎠

(19)

Elimination of [H2PO4 ]2 using eq 1 yields the following one variable differential equation: −

⎤ [Cl−]m1 1⎡ −π ⎥ − E1 = ⎢RT ln + V [Cl ] s F⎣ [Cl−]w1 ⎦

d[H 2PO4 −]1 S = Ps {B1[H 2PO4 −]1 − B2 } dt V

(20)

where (12)

Assuming a negligible swelling pressure, the selectivity coefficient (KC) can be taken to be unity: [H 2PO4 −]m1 [Cl−]w1 KC = [H 2PO4 −]w1 [Cl−]m1

B1 =

1 1 + [K+]0 [Na +]0

(21)

B2 =

[K+]0 [Na +]0

(22)

Integration of eq 20 gives a relation describing the decay of the phosphate concentration with time in the feed side section:

= exp[(πs/RT )(VH2PO4− − VCl−)]

− ⎛ S⎞ 1 ⎧ B1[H 2PO4 ]1 − B2 ⎫ ⎬ = −⎜Ps ⎟t ln⎨ − ⎝ V⎠ B1 ⎩ B1[H 2PO4 ]0 − B2 ⎭

(13)

≃1

Equations 2, 6, and 13 provide the following relations between membrane and solution concentrations on the feed side: [H 2PO4 −]m1 [Cl−]m1 [X m+] = = [H 2PO4 −]w1 [Cl−]w1 [K+]0

(14)

[H 2PO4 −]m2 [Cl−]m2 [X m+] = = − − [H 2PO4 ]w2 [Cl ]w2 [Na +]0

⎛ S⎞ 1 ln{1 − B1[H 2P PO4 −]1 } = −⎜Ps ⎟t ⎝ V⎠ B1

(15)

Substituting solution concentrations of eqs 14 and 15 into eq 9, the final flux equation describing diffusion of phosphate ions from the feed to the receiver compartment is given by mol m3

3

3. EXPERIMENTAL SECTION 3.1. Experimental System. The experimental system (Figure 1) consisted of a rectangular dialysis cell having the overall dimensions of 400 × 50 × 20 mm, a 1 L volume vessel holding a recycling feed solution and a 1 L vessel holding the recycling receiver solution. The dialysis cell contained two 8 mm wide compartments separated by an anion exchange membrane having a total area of 0.02 m2. The volume of the solution held in each compartment was 150 mL, and the total solution volume in each recycle sections was 1.15 L in the first series of runs and 0.5

2

S (m ) (16)

where Ps represents an overall kinetic coefficient given by: 1 1 1⎧ 1 1 ⎫ ⎬ = + ⎨ + + Ps Pso k ⎩ [K ]0 [Na +]0 ⎭

(24)

Hence a plot of the receiver experimental [H2PO4−]2 data in the form of (1/B1) ln(1 − B1[H2PO4−]2) versus t should yield a straight line providing an additional estimate of the value of Ps.

( ) V (m )

⎧ [H PO −] [H 2PO4 −]2 ⎫ ⎬ = Ps⎨ 2 + 4 1 − [Na +]0 ⎭ ⎩ [K ]0

(23)

Equation 23 predicts that a plot of the feed experimental [H2PO4−]1 data in the form of (1/B1) ln(B1[H2PO4−]1 − B2) versus t should yield a straight line providing the value of the kinetic coefficient Ps from the slope. An additional determination of the Ps value can be obtained from the receiver [H2PO4−]2 measurements. In this case, by combining eqs 2 and 23, it is found that

An identical derivation for the receiver compartment yields

d[H 2PO4 −]1 ⎛ mol ⎞ JH PO − ⎜ 2 ⎟ = − 2 4 ⎝ dt (s) m h⎠

mol m3

−

where V denotes the ionic molar volume and πs is the swelling pressure. Similarly, the electrochemical potential equality for the chloride ion on the membrane feed side yields the following equation: [E DON]1 = Em1

+

m

Equation 17 shows that the overall kinetic coefficient is determined by the sum of two resistances: a resistance related to the ion diffusion through the membrane and a resistance related to the mass transfer from the solution bulk to the membrane interface. Analysis of eqs 17 and 18 indicates that diffusion control of the dialysis process is promoted by relatively low ion diffusivities, high solution concentrations of the diffusing ions, and high mass transfer coefficients. Mass transfer control is promoted by relatively low solution concentrations, low mass transfer coefficients, and high ion diffusivities. 2.2. Batch Kinetics. The following derivation provides the concentration change with time in batch mode operation of the dialyzer.

JH PO − = JCl− = k{[Cl ]2 − [Cl ]w2 } 2

m2 h

( ) [X ] ( )

(17) 6096



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L in the second series. The recycle flow rate in each section was 1.8 L/min in all runs, providing a Reynolds number in the flow passages of the dialysis cell of 1035. Two anion exchange membranes were used: “Fumasep FAB” supplied by Fumatech Co., Germany, was used in the first series and “Selemion AMV” obtained from Asahi Glass, Japan, was used in the second series. The Fumasep FAB membrane had a thickness of about 100 μm and an ion exchange capacity of about 1.1 mequiv/g. It is classified as a membrane of high proton blocking capability, high selectivity, high mechanical stability, and high stability in acidic and caustic environments. The Selemion AMV membrane thickness was about 120 μm and had an ion exchange capacity of 1.85 mequiv/g. Selemion AMV membrane has traditionally been used for treating metallurgical and metal finishing effluents, seawater desalination, separation of different ions, recovery of boron in aqueous solution, and removal of organic matter from saline wastewater.41−43 3.2. Experimental Conditions. The initial feed solution was prepared by dissolving analytical grade KH2PO4 salt (first series) or NaH2PO4 salt (second series) in deionized (DI) water. The receiver solution was prepared by dissolving of analytical grade NaCl salt in DI water. The initial concentration of the receiver solution was 1000 mg/L (17.1 mmol/L) in the first series and 5850 mg/L (100 mmol/L) in the second series. The initial phosphate concentration in the feed solution was varied in the range 0.58−5.03 mmol/L in the first series and in the range 0.05−2.03 mmol/L in the second experimental series. The present study focused on Donnan dialysis with a monovalent phosphate ion. The pH required for maintaining a high dissociation fraction of H2PO4− can be evaluated from the following expression relating to a dilute solution at a temperature of 25 °C:44 −1 ⎛ [H+] H 2PO4 − 10−7.2 10−19.5 ⎞ = ⎜ −2.1 + 1 + + ⎟ CT [H+] ⎝ 10 [H+]2 ⎠

Table 1. Effect of Initial Feed Concentration on Membrane Phosphate Ion Concentration and on Kinetic Coefficients phosphate concn (mmol/L) final anion exchange membr FAB

AMV

Ps (mmol/m2 h)

init

feed

receiver

feed

receiver

membr phosphate concn (mmol/cm3)

0.58 1.47 2.91 5.03 0.05 0.49 0.98 2.03

0.196 0.598 1.546 3.216 0.001 0.011 0.016 0.075

0.302 0.840 1.237 1.979 0.046 0.456 0.938 1.814

1.15 2.58 3.45 5.05 0.80 7.79 16.91 28.05

0.88 2.55 3.33 5.29 0.79 6.84 14.58 25.88

0.065 0.100 0.185 0.255 0.001 0.008 0.013 0.030

4.1. Phosphate Removal at Various Inlet Concentrations. Figure 3 displays the decay in phosphate content of the

Figure 3. Total phosphate content in the feed section as a function of time at varying initial phosphate concentrations using the FAB membrane.

(25)

Equation 25 shows that, at a pH below 6, over 94% of the phosphate is in the form of H2PO4−. All runs were carried out at a temperature of 22−24 °C with a feed pH in the range 5.7−5.9 ensuring the presence of an almost pure H2PO4− solution. The duration of each run was 24−26 h in the first series and 7 h in the second series. The membrane cell was stored between runs with a NaCl solution of 1000 mg/L and was rinsed with DI water before a fresh experiment was performed. Periodic analyses of the phosphate ion concentrations in the solutions held in the feed and in the receiver section vessels were carried out by the Hach method 8040 using a DR2800 spectrophotometer with PhosVer 3 Ascorbic reagent. The detection limit of this analytical method is 0.02 mg/L PO43−. The measurements of the feed and receiver concentrations as functions of time served to check the predictions of the kinetic model (eqs 23 and 24).

Figure 4. Total phosphate content in the receiver section as a function of time at varying initial phosphate concentrations using the FAB membrane.

4. RESULTS AND DISCUSSION Table 1 summarizes the main results obtained in the two experimental series. The procedures for evaluating the kinetic coefficients from feed and receiver data are described below. The data indicate a potential for phosphate removal by Donnan dialysis with the high flux membrane AMV; a phosphate recovery of about 90% was achieved in all experiments. It is also of interest to note that the percentage phosphate removal from the feed decreased with increasing initial feed concentration. A similar trend was observed in dialysis tests performed by Pintauro and Benlonn.36

feed section, while Figure 4 shows the increase in phosphate content in the receiver section using the FAB membrane. The phosphate mass balance, comparing the initial phosphate content with the sum of the phosphates in the feed and receiver sections, revealed that during the first 3 h the receiver section phosphate content was lower than the phosphate amount depleted from the feed section. Subsequently, the measurements showed full material equality between the phosphate removed from the 6097



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phosphate mass leaving the feed section and entering the receiver solution throughout the experiments amounted to 0.13 mmol. After immersion in the NaCl solution, a virtually identical mass of 0.12 mmol of phosphate was released from the membrane, thus satisfying the mass balance within 2.1%, well within the experimental error. 4.2. Correlation of the Flux Measurements According to the Kinetic Model. All experimental data conformed very well to the linear relationships predicted by the kinetic model of section 2 indicating the validity of the simplifying assumptions for the examined system, notably selectivity coefficients of unity. Figure 7 illustrates correlation of the decay of the phosphates

feed section and the phosphate reaching the receiver section. This process of phosphate intake by the membrane up to a saturation level occurred consistently in all runs and is illustrated in Figure 5 for the FAB membrane.

Figure 5. Typical process of phosphate intake by the FAB membrane (initial phosphate concentration of 1.47 mmol/L in the feed section).

A similar behavior was displayed by the AMV membrane. Lower phosphate content entered the AMV membrane compared to the FAB membrane, and the saturation level of phosphate inside the membrane was reached within 2 h. Figure 6 shows that the phosphate concentration in the membrane increased linearly with the initial phosphate

Figure 7. Linear correlation of feed side data according to eq 21 (initial phosphate concentration of 2.91 mmol/L; FAB membrane).

content in the feed section (eq 21) using the FAB membrane. The data relating to the first 3 h in which phosphate ions entered the membrane are omitted. Figure 8 illustrates correlation of the increase of the phosphate content in the receiver section (eq 22) using the FAB membrane.

Figure 6. Phosphate concentration inside the FAB and AMV membranes at varying initial phosphate concentrations.

concentration. For the FAB membrane the membrane phosphate content increased from 0.065 mmol/cm3 at a feed concentration of 0.58 mmol/L to 0.25 mmol/cm3 at a feed concentration of 5.03 mmol/L. A rough estimate indicates that at the highest feed concentration of 5 mmol/L the membrane phosphate content was about 20% of the ion exchange capacity. For the AMV membrane, the membrane phosphate ranged from 0.001 mmol/ cm3 at a feed concentration of 0.05 mmol/L to 0.030 mmol/cm3 at a feed concentration of 2.03 mmol/L. A linear increase of the counterion membrane concentration with the salt solution concentration has been observed in cation exchange membrane systems.36 Confirmation of the membrane phosphate content data was carried out by immersing the FAB membrane, at the termination of each experiment, in a fresh solution of the initial receiver concentration (17 mmol/L NaCl) and measuring the phosphate content released from the membrane. The released phosphate concentration was found to correspond with the amount of phosphate determined by the mass balance data obtained during the dialysis period. A typical result is presented below, for an initial concentration of 0.58 mmol/L. The difference between the

Figure 8. Linear correlation of receiver section data according to eq 22 (initial phosphate concentration of 2.91 mmol/L; FAB membrane).

Equation 17 indicates that the reciprocal of the overall kinetic coefficient (1/Ps) represents two resistances in series: an ion diffusion resistance within the membrane (1/Pso) and a mass transfer resistance at the solution−membrane interfaces expressed by (1/k){1/[K+]0 + 1/[Na+]0}. The mass transfer coefficient for the solution flow through the rectangular channel of the dialysis cell can be estimated from the Leveque−Sieder−Tate equation for laminar flow at the entrance region of a channel:45 6098


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1/3 ⎛ kd d⎞ = 1.86⎜Re Sc ⎟ ⎝ Ds L⎠

density, measured by weighing a membrane sample, was 1.16 g/ cm3. The membrane diffusivity Dvm can be therefore estimated from the following equation:

(26)

where k is the averaged mass transfer coefficient, Ds is the average solution diffusivity of H2PO4− and Cl− ions (1.45 × 10−5 cm2/s), d is the hydraulic diameter of the rectangular flow passage (1.38 cm), L is the flow passage length (40 cm), Re is the Reynolds number (1035), and Sc is the Schmidt number of the solution (1000). The resulting value of k (4.44 × 10−4 cm/s = 1.60 × 10−2 m/h) was used to evaluate the diffusional kinetic coefficient Pso from eq 18. The data in Table 2 clearly indicate that the phosphate flux in the FAB membrane was controlled by the ion diffusion; the

⎛ cm 2 ⎞ ⎛ mmol ⎞ Psoδ Dvm ⎜ = Pso ⎜ 2 ⎟ (2.17 × 10−10) ⎟= ⎝m h⎠ [X m+] ⎝ s ⎠ (27)

Values of the diffusivity coefficient displayed in Table 3 are seen to increase with feed concentration and, more significantly, with the ion membrane concentration. The increase in diffusion coefficient with both phosphate feed concentration and phosphate membrane concentration is virtually linear (Figures 10 and 11).

Table 2. Values of Intrinsic Diffusional Kinetic Coefficient Pso as a Function of Initial Feed Concentration for Low Flux FAB Membrane Psa and Psob (mmol/m2 h) init concn (mmol/L)

a

feed

receiver

feed KH2PO4

receiver NaCl

Ps

Pso

Ps

Pso

0.58 1.47 2.91 5.03

17.1 17.1 17.1 17.1

1.15 2.58 3.45 5.05

1.32 2.93 3.78 5.50

0.88 2.55 3.33 5.29

0.98 2.89 3.63 5.78

Experimental overall coefficient. bDiffusional coefficient. Figure 10. Membrane diffusion coefficient as a function of phosphate feed concentration and membrane phosphate concentration.

diffusional resistance accounted for over 90% of the total flux resistance. Figure 9 shows that the diffusional kinetic coefficient Pso increases linearly with membrane phosphate content.

Figure 11. Membrane diffusion coefficient as a function of membrane phosphate concentration. Figure 9. Intrinsic kinetic diffusive coefficient as a function of the initial phosphate concentration.

The data in Table 3 show membrane diffusivities lower by a factor of 10−70 relative to solution diffusivity values. This result is similar to data reported by Miyoshi31 using four different cation exchange membranes. The highest decrease factor of 80 was observed with a Selemion CMV membrane and the lowest factor of 20 was seen with a Nafion 417 membrane.

It is of interest to examine values of the FAB membrane ion diffusion coefficient calculated from the intrinsic kinetic coefficient Pso. According to the manufacturer’s data, the ion exchange capacity of the FAB membrane is 1.1 mequiv/g and the membrane thickness is 100 μm. The measured membrane

Table 3. Phosphate Ion Diffusion Coefficient in FAB Membrane as a Function of Membrane Phosphate Content feed init feed concn (mmol/L)

membr phosphate concn (mmol/cm3)

Pso (mmol/m2 h)

0.58 1.47 2.91 5.03

0.065 0.100 0.185 0.255

1.32 2.93 3.78 5.50 6099

receiver

membr Dvm × 10−10 (cm2/s)

Pso (mmol/m2 h)

membr Dvm × 10−10 (cm2/s)

2.86 6.36 8.20 11.94

0.98 2.89 3.63 5.78

2.13 6.27 7.88 12.54



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Increase of the diffusion coefficient with solution concentration was observed by Jan et al.29 in their Donnan dialysis of NaCl and HCl with a Selemion CMV cationic exchange membrane. The Na+ membrane diffusion coefficient increased linearly from 3 × 10−8 cm2/s at a feed concentration of 2.5 mmol/L Na+ to 6 × 10−8 cm2/s at a feed concentration of 10 mmol/L Na+. The increase of the diffusion coefficient with solution concentration is confined to charged membranes. Data reported recently by Geise et al.38 confirm that the diffusion coefficient increases with salt concentration in charged membranes but the coefficient is actually reduced with increasing solution concentration in uncharged membranes. Contrary to the membrane diffusion control mechanism disclosed by the FAB membrane results, the kinetic coefficient data of the AMV membrane indicate mass transfer control. For mass transfer control, the overall kinetic coefficient Ps should closely approach the value predicted by the mass transfer term given by

Figure 12 is therefore simply an expression of the mass transfer control conclusion. Mass transfer control in Donnan dialysis has been previously observed. Ktari et al.46 investigated Donnan dialysis of KCl−HCl using a Neosepta CM2 cation exchange membrane. The feed compartment contained KCl solution at a concentration which was varied in the different runs from 0.001 to 0.010 mol/L. The receiver compartment contained HCl solution at a concentration 100 times higher than the corresponding KCl feed solution (i.e., from 0.1 to 1.0 mol/L). The potassium flux data provided some evidence for a mass transfer controlled process. Conclusive evidence on mass transfer control was obtained by Xue et al.47 in their extensive investigation of the interdiffusion of metal−hydrogen ion couples through a Nafion 117 cation exchange membrane by the rotating diffusion cell technique. Their results clearly showed that at low cation concentration ion transport was controlled by mass transfer and at high cation concentrations ion transport was controlled by membrane diffusion. For instance, using a strip solution of 0.2 M HCl solution and feed solutions containing 2, 10, and 100 mmol/L NaCl, respectively, the measured Na+ fluxes were plotted against √w, the square root of the rotating speed. For the two low feed concentrations, the Na+ flux data increased linearly with √w, as predicted by the Levich mass transfer control equation. For the high feed concentration of 100 mmol/L, increase of the rotation speed had no effect on the flux, indicating that mass transfer resistance was negligible.

1 1⎧ 1 1 ⎫ ⎬ ≅ ⎨ + Ps k ⎩ [NaH 2PO4 ]0 [NaCl]0 ⎭ =

1 ⎧ 1 1 ⎫ ⎨ ⎬ + 0.016 ⎩ [NaH 2PO4 ]0 100 ⎭

(28)

Table 4 shows that the values of the experimental overall kinetic coefficient Ps are virtually identical to values predicted from the above mass transfer control equation.

5. CONCLUSIONS Most of the papers describing the kinetics of Donnan dialysis involve elaborate computational models. The present paper illustrates the possibility of practical estimation of the main kinetic features of a Donnan dialysis process by simplified models. This approach enabled identification of the controlling mechanisms of two different membranes: one of which displayed diffusional control and the other mass transfer control. The results of this study highlight a major mechanism involved in Donnan dialysis: accumulation of the dialyzing ions inside the membrane in proportion to the solution feed concentration and increase of the flux through the increase in ion diffusivity induced by the ion membrane concentration. Another interesting observation of this study is that ion exchange membranes can exhibit very wide differences in diffusional resistance.

Table 4. Comparison of Values of the Experimental Overall Kinetic Coefficients Ps of High Flux AMV Membrane with Predicted Values Based on Mass Transfer Control Ps (mmol/m2 h) exptl overall coeff Ps

init concn (mmol/L) feed NaH2PO4

receiver NaCl

feed

receiver

av

pred mass transfer Ps

0.05 0.49 0.98 2.03

100 100 100 100

0.80 7.79 16.91 28.05

0.79 6.84 14.58 25.88

0.80 7.32 15.75 26.97

0.80 7.80 15.52 31.83

Figure 12 shows that the overall kinetic parameter Ps of the AMV membrane increases linearly with the initial phosphate concentration. For mass transfer control, eq 28 shows that Ps ≅ 0.016[NaH 2PO4 ]0

■

(29)

AUTHOR INFORMATION

Corresponding Author

*Tel.: +972-4-8292936. Fax: +972-4-8295672. E-mail: Hasson@ tx.technion.ac.il. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work forms part of the bachelor research thesis of Fiana Fingerman and Chen Tachman and of the master’s thesis of Adam Beck. Adam Beck’s research at Technion was in the framework of the exchange program between Technion and Hamburg University of Technology (TUHH). The TUHH− Technion program initiated by Professors W. Calmano and M. Ernst of TUHH is funded by the BMBF-MOST Young Scientist Exchange Program.

Figure 12. Kinetic coefficient as a function of initial phosphate solution concentration for the AMV membrane. 6100



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NOMENCLATURE [Cl−]*1 = equilibrium concentrations of chloride in the feed [Cl−]2* = equilibrium concentrations of chloride in the receiver [Cl−]0 = initial concentration of chloride [Cl−]1 = concentrations of chloride in the feed [Cl−]2 = concentrations of chloride in the receiver [Cl−]m = concentrations of chloride inside the membrane [Cl−]m1 = concentrations of chloride inside the membrane on the feed side [Cl−]m2 = concentrations of chloride inside the membrane on the receiver side DmH2PO4− = phosphate ionic diffusion coefficient DmCl− = chloride ionic diffusion coefficient dEm/dx = electric potential gradient across the membrane d[H2PO4−]m/dx = phosphate concentration gradient inside the membrane d[Cl−]m/dx = chloride concentration gradient inside the membrane EDON = Donnan potential E1 = Donnan potential in feed solution Em1 = Donnan potential inside the membrane in the feed side [H2PO4−]*1 = equilibrium concentrations of phosphate in the feed [H2PO4−]2* = equilibrium concentrations of phosphate in the receiver [H2PO4−]0 = initial concentration of phosphate [H2PO4−]1 = concentrations of phosphate in the feed [H2PO4−]2 = concentrations of phosphate in the receiver [H2PO4−]m = concentrations of phosphate inside the membrane [H2PO4−]m1 = concentrations of phosphate inside the membrane on the feed side [H2PO4−]m2 = concentrations of phosphate inside the membrane on the receiver side JH2PO4− = diffusional flux of the phosphate ions through the membrane JCl− = diffusional flux of the chloride ions through the membrane [K+]0 = initial potassium concentration [Na+]0 = initial sodium concentration V[H2PO4−] = phosphate molar volume V[Cl−] = chloride molar volume [Xm+] = fixed ion concentration

Synbols

F = Faraday constant KC = selectivity coefficient Ps = kinetic coefficient, mol/m2 h R = universal gas constant S = membrane area T = absolute temperature V = membrane volume δ = membrane thickness πs = swelling pressure

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REFERENCES

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Simple Model for Characterizing a Donnan Dialysis Process

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