Chapter 30. Electrodialysis and Electrophoresis

Electrodialysis refers to mass transport through stacks of ion-selective membranes under the action of an electric field. It is used to separate an electrolyte feed solution into a dilute and a concentrated solution. A common application is to purify brackish water in connection with desalination, but the technique has found applications in many other fields as well. The use of selective ion-exchange membranes is the separating factor here. In this chapter, we start with the description of the process, application areas, and technological aspects together with some variations. Then a simple model to design or simulate the equipment is shown where we apply mass conservation law, Faraday’s law, and Ohm’s law to determine the voltage needed for the separation and the power requirements. Transport across the membranes is simply lumped into a resistance in this model. A more detailed model uses the transport across charged membranes using the Nernst-Planck equation and a brief discussion and pertinent references are provided. The concept of limiting current is an important design consideration as well and is also discussed briefly.

Electrophoresis refers to movement of charged particles in solution under the action of an electric field. This field is becoming important in what is now called the subject of proteomics, which requires separation of proteins, DNA molecules, and so on. The difference in mobility is used as the separating agent here. Proteins and colloidal materials carry a surface charge and move under the influence of an electric field with the velocity proportional to the mobility and the direction of motion dependent on the type of charge. If now a flow is superimposed, different species will follow different trajectories due to differences in their surface charge or mobility. Hence they separate into “bands” similar to chromatography. This is the basic idea used in the process and there are many variations and designs used in practice. Three such designs are introduced. A simple model is discussed to predict the bandwidth and the extent of separation.

An illustrative diagram of the unit is presented in Figure 30.1. The unit shown has two membrane pairs of anion-selective and cation-selective membranes. The working principle of the separation method can now be discussed using this figure as a reference. Note that in practice a large number of membrane pairs arranged into an alternating manner are used and assembled into a “plate and frame” type of design.

In this arrangement, the feed solution gets diluted in alternative compartments marked 1 and 3 in the figure. The presence of an anion-selective membrane on the left side of compartment 1 and a cation-selective one on the right side removes the anion X^– and the cation M⁺ in compartment 1. The salt concentration in feed solution gets diluted as it flows through compartment 1. A similar process occurs in compartment 3.

In compartment 2 the feed gets concentrated and the reason for this is because the M⁺ coming into compartment 2 from 1 is not able to cross to 3 due to the presence of the anion-exchange membrane as a barrier. Similarly X^– coming from 3 meets a cation exchange barrier and is not able to cross over to compartment 1. Hence a concentrated salt solution results in this compartment.

In addition to the cell pairs, there are two electrodes at the two ends of the of the unit. A electrolyte rinse solution flows in these electrode compartments (marked E in the figure) in order to increase the solution conductivity. (The rinse solution is usually recycled.) Chemical reactions O₂ evolution and H₂ evolution occur at the anode and the cathode, respectively. The anode reaction is

2OH−→12O2+H20+2e−

The following is the cathode reaction:

2H⁺ + 2e^– → H₂

These reactions are needed for the current flow along the external circuit.

Chlorine formation at the anode is a possible side reaction if the feed contains Cl^– ions. This is an anodic reaction represented as

2Cl^– → 2e^– + Cl₂

The extent of these reactions are minimal and only a small fraction of the water gets electrolyzed. The overall material balance is that the brackish water gets separated into relatively deionized water and a concentrated solution. This nature of separation is similar to that in reverse osmosis although the separation principles are different. One is based on pressure-driven transport while the other is based on electricity-driven transport.

• Very dilute salt solution: If the range of concentration of the salt is below 1000 mg/L, then use ion exchange.

• Dilute solution: Electrodialysis fits well and is economical if the range of concentration is within a range of 1000 to 10,000 mg/L.

• Not so dilute solution: If the range of concentration of the salt is above 10,000 mg/L, then reverse osmosis is more economical.

Large capacity systems, where about 220, 000 m³/day of brackish water are treated for potable use, have been designed. Other applications include the food industry (salt removal from cheese whey, soy sauce, etc.), organic chemicals (e.g., organic acids from organic salt wastes), and related applications. Valero et al. (2011) provide more application examples and a review of electrodialysis technology.

The anion-exchange membrane allows negative ions to pass through. The membrane matrix usually has fixed positive charge, for example, quaternary ammonium salts of the generic formula RNH⁺₃, which repels the positive ions. Chloromethylation followed by amination is done on a polymer matix to create these functional groups.

Both membranes should have the following desirable properties:

• Low electrical resistance

• Insoluble in aqueous media

• Mechanical rigidity

• Resistance to change in pH and to osmotic swelling

The membranes can be homogeneous or heterogeneous. In the heterogeneous structure, the ion-selective membranes are embedded in a neutral membrane matrix. Membrane that are selective to univalent ions are also available, for example, NO3− transport is permitted with exclusion or limited transport to SO4−−, and so on. Such selectivity is accomplished by specially treating anion-exchange membranes to provide additional functional groups. Thus a wide range of membranes is available and selection depends on the application areas and relative cost of the equipment. A detailed review on the developments in the preparation, properties, and application of ion-exchange membranes is available in Xu (2005).

The mechanism for water splitting is not clear and could be due to the catalytic effect of the transition layer. Fu et al. (2003) discuss some proposed models for water splitting and they found that PEG (polyethylene glycol) has a strong effect on water splitting. The other solvent studied in BPMs is methanol, which dissociates into CH₃O^– and H⁺ ions.

A common application is splitting the salt into an acid and base. The arrangement is shown in Figure 30.3. The CEM and AEM permit the separation of the cation and anion of the salt (MX) while the bipolar membrane (BPM) causes water to split into H⁺ and OH^– ions. The H⁺ ions permeate through the cation-exchange side of the BPM and form acid HX with the X^– ions provided by permeation through the AEM. Similarly on the other side the OH^– ions (shown as R^– in the figure for generality) from the anion exchange side of the BPM form the base MR (MOH) with M⁺ being supplied via the CEM. The final result is that a salt solution, MX, in the feed is split into an acid (HX) and a base (MOH). Note that this is one type of cell arrangement and a number of alternative designs are available as reviewed in Tongwen and Weihua (2002).

Detailed review of the application of elecrodialysis with a bipolar membrane is presented by Tongwen and Weihua (2002). These membranes are finding applications in wastewater treatment and recovery of value from waste.

Preliminary design is done using mass balance and Faraday’s and Ohm’s laws. Detailed concentration profiles in the membrane and in the boundary layer near the membrane surface are not included. In other words, mass transport across the membrane and in the external film near the membrane are not considered. Finally a correction factor, the current efficiency, is applied to allow for these effects. These design aspects are available in Lee et al. (2002) and Korngold (1982) and the discussion follows the ideas developed in these sources.

where F is the Faraday constant, Q is the flow rate across each cell pair, and ΔC_A is the concentration change from inlet to outlet. The concentration is often expressed in g-equivalent (g-eq per m³ rather than g mole/m³). This takes care of the valency of the cation or anion and n (number of electrons transferred) is then set as one. For a univalent electrolyte such as NaCl, g-eq is the same as g-mol.

The membrane area can then be calculated if an operating current density i is chosen:

where η is a current efficiency factor to account for the various losses in the system.

The voltage drop across each stack pair in accordance with Ohm’s law is

where Σ R is the total resistance for current flow in the cell pair. V is the applied voltage across the system and N is the number of cell pairs in the complete unit.

The total resistance is composed of four individual resistances shown in Figure 30.4; these are added to get the total resistance.

These resistances in series are the following:

1. Resistance of the diluate solution, R₁. This is calculated as h/κ_d where h is the cell width and κ_d is the specific electrical conductivity of the diluate solution, S/m.

2. Resistance of the AEM membrane, R_A, in Ωm².

3. Resistance of the concentrate solution, R₂. This is calculated as h/κ_c where h is the cell width and κ_c is the specific electrical conductivity of the concentrate solution, S/m.

4. Resistance of the CEM membrane, R_C in Ωm².

The total resistance Σ R is equal to R₁ + R_A + R₂ + R_C. Note that these are based on a unit area for transport. Hence the unit is Ωm². Also note that S represents Siemens, an unit for conductance that is the same as Ω^–1 and is denoted as mho in earlier work.

The resistance of the commercial ion-exchange membranes are in the range of 3 to 6 ×10^–4 Δm² and depends on the thickness of the membrane that are in the range of 0.3 to 0.6 mm.

The specific conductivity of the salt solution is concentration dependent and is expressed as

κ = C

where Λ is the equivalent conductance (S m²/mol) and C is the concentration in either the diluate or the concentrate compartment. The concentration changes along the reactor length and an average value between the inlet and outlet is used as an approximation. (Note that a mesoscopic model is required for more detail.)

The equivalent conductance of a salt solution is concentration dependent as well. However, a value of 10.86 S m²/kmol for equivalent conductance appears to be a reasonable average value, as suggested by Lee et al. (2002), and can be used for a preliminary design.

The calculations are demonstrated by Example 30.1.

The operating current density cannot be larger than the limiting current. A value of 0.85 times the limiting current is used as an approximate design criteria.

The equation to calculate the limiting current density is

i_L = nFk_LC_A

where k_L is the mass transfer coefficient near the external film near the membrane. C_A should be the exit concentration since this is where the limiting conditions will happen. It is however difficult to estimate the mass transfer coefficient since it depends on the flow velocity, spacer configuration, and membrane surface roughness. Hence experimentally determined values are used in practical design as suggested by Lee et al. (2002). Typical values are in the range of 30–50 A/m². Additional information is available in Tanaka (2005).

Factors leading to inefficiencies such as diffusion of co-ions through the membranes and water transport across the membranes can also be investigated using the framework of the detailed model. Hence such models provide more phenomenological understanding, which in turn helps in improving the performance of the unit.

It is useful to indicate the direction of the motion of the various types of molecules. This depends on the type of the surface charge, which depends on a number of factors discussed next.

⁺NH₃RCOO^– + H⁺ →⁺ NH₃RCOOH

In alkaline solutions the proteins acquire a negative charge due to the following reaction:

⁺NH₃RCOO^– + OH⁺ → NH₂RCOO^– + H₂O

Hence the nature of the charge of a protein therefore depends on pH. If the pH is increased progressively, the charge changes from positive to zero and then negative. The pH at which the charge is zero is called the iso-electric pH value and the protein will not move in an electric field at that pH. This gives another handle on separation since the relative motion of one protein to another can be controlled by simply changing the pH of the solution. This method is called isoelectric focusing.

The feed solution containing the proteins to be separated is introduced as a point source at a location near the inlet. This is convected by flow in the x-direction and moves to the electrode at y = H by electrophoretic migration. (Here we assume y = H is of the opposite polarity to the charge on the protein.) The computation of the trajectory of a protein and the point at which it will separate is described next by a simple model. The model is for plug flow of the liquid for simplicity but it is easy to extend this to a parabolic profile, which is characteristic of laminar flow in a channel.

A coordinate system in the direction of the flow vector (x-axis) and perpendicular to the flow (y-axis or the direction of the field) is set up as shown in the figure. The species is convected by flow in the x-direction and hence has an x-velocity of v_x. Here we assume no slip and the velocity of the particle is taken to be equal to that of the fluid. The species moves with a y-velocity of μ_pE_y in the y-direction. Hence the velocity of the particle is

v = e_xv_x + e_yμ_pE_y

The angle between the direction of convection and the x-axis can therefore be calculated as

tanθ=μpEyυx

The speed in this direction (the θ-direction), denoted as v_p, is

υp=[υx2+(μpEy)2]

This provides the trajectory of the protein due to the action of the flow and migration and is shown in Figure 30.5. Trajectories are shown in this figure for two proteins with different mobility values.

Different proteins have different mobilities and arrive at different locations as shown in the figure. The fractions are collected at suitable locations along the wall (ports 1 and 2 in the figure).

Superimposed on the trajectory there is a spread due to diffusion. This can be described by a Gaussion distribution:

C=mpυp4πDpxcexp(−υpyc24Dpxc)

where m_p is the source strength equal to the mass or moles of protein being injected at the source point per unit width of the slab. x_c is the distance along the trajectory. y_c is the distance perpendicular to the trajectory; this is the direction where the spread occurs due to diffusion.

The standard deviation for diffusion is

The separability of two protein species by electrophoresis can be assessed by this simple model. In general the standard deviation of 4 will maintain a concentration difference of 0.1%. Otherwise there will be an overlap and the separation will not be sharp.

This discussion provides a basic model for the process and is useful to provide some guidelines in design and selection of suitable equipment. The factors to consider are as follows:

• The length of the device must be reasonable.

• Multiple species must be separable without overlap.

• Joule heating due to the imposed field causes a temperature rise, which can cause protein denaturation. Hence the design should limit the maximum temperature rise in the system and heat transfer models also become important.

• Other effects such as Taylor dispersion and double-layer distortion affecting the mobility of the particle become important in many cases.

The velocity of a protein in the flow direction is now the algebraic sum of the convection and electromigration velocity. The time it takes to arrive at a distance L (the length of the device) is L/v_p; this represents the retention time. Two proteins with different retention times can therefore be separated. The analysis is therefore similar to the chromatography studied in Chapter 29.

An ideal pulse will be observed in the exit stream as a dirac delta function after the elapse of the retention time. In reality band broadening will be observed due to both diffusion and dispersion. The diffusion in the flow direction (y) will cause a spread in the response curve in the y-direction while the axial dispersion will cause a peak broadening in the x-direction as well. The difference in the retention time for separation should be about 4σ apart so that the pulse overlap does not interfere with the separation.

Additional band spreading can occur due to electro-osmosis, and hydrodynamic effects caused by thermal gradients. The thermal gradients are due to Joule heating caused by the applied potential. These can cause protein denturation and control of the temperature peak is important.

Other designs include serpentine channels, studied for example by Wang et al. (2004), and the use of microchannel devices.

30.2 State some applications of electrodialysis.

30.3 What are the two exit streams from an electrodialysis system?

30.4 What is electrodialysis reversal?

30.5 What are some of the applications of electrodialysis with bipolar membranes?

30.6 What is the general chemical composition of a cation-exchange membrane?

30.7 What is the general chemical composition of an anion-exchange membrane?

30.8 Define the limiting current and show its importance in electrodialysis.

30.9 What is the principle behind electrophoresis?

30.10 What is an amphoteric electrolyte?

30.11 What is meant by an isoelectric point?

30.12 What data is needed to find the isoelectric point of an amino acid?

30.13 Differentiate between the field arrangements in the Philpot and Hannig designs of an electrophoresis unit.

30.14 What is the effect of Joule heating in electophoresis on separation efficiency?

30.15 What additional benefit is obtained in the rotating bed over the Hannig design?

30.2 Membrane area needed for separation. An electrodialysis unit has a feed concentration of 58 g eq./m³ and the diluate concentration is reduced by 90%. The recovery factor is 0.75. The production capacity of the plant is 350 m³/day. An operating current density of 40 A/m² is suggested with 200 cell pairs. Find the area of membrane needed. The current efficiency is 0.9.

30.3 Limiting current. If a limiting current of 50 A/m² is measured for a salt concentration of 0.1 M NaCl, estimate the mass transfer coefficient. The data on limiting current at a fixed salt concentration was fitted as i_L = Av^α by Lee et al. (2002) and the exponent α was found to be in the range of 0.41 to 0.52. Determine the exponent if a convection-diffusion model with laminar flow is used.

30.4 Separation distance for proteins. It is required to separate two proteins with mobilities of μ₁ = 8 × 10^–5 m.C/N.s and μ₂ = 6 × 10^–5 m. C/N.s. The diffusion coefficient is 6 × 10^–10 m²/s. The flow velocity is 0.2 mm/s. The applied field is 2000 V/m. The electrodes are placed 1 cm apart. The Philpot design is used with the field perpendicular to the flow. Find the trajectory of the “plumes” of the two proteins and the distance at which the two proteins can be separated. First assume plug flow and use the equations suggested in the text. Second assume laminar flow and calculate the trajectory by solving the following pair of equations. The following equation is applicable for the distance along the elutant flow (same for both proteins):

dxdt=v(y)=G2μLy(d−y)

Here the right-hand side represents the velocity for laminar flow in a channel, G the applied pressure gradient, and μ_L the viscosity of the liquid. For distance across the flow (different for different proteins) the following equation is applicable:

dydt=μpEy

30.5 Retention time and spread in the Hannig design. Two proteins with mobilities of μ₁ = 8 × 10^–5 m.C/N.s and μ₂ = 6 × 10^–5 m.C/N.s are to be separated in a Hannig design arrangement. The diffusion coefficient is 6 × 10^–11 m²/s. The elutant velocity is 1 cm/s. Find the retention time and the spread in the bands if the length of the channel is 100 cm.

30.6 Rotating annular bed. Show that the distance traveled in the angular direction in the rotating annular bed is Rfωt¯A, where t¯A is the retention time for protein A, R_f is the radial position at which the feed is introduced, and ω is the angular velocity. Show that the separation distance between two components A and B is

ΔSAB=Rf(ωt¯A–ωt¯B)

Determine the expression for the retention time for A and B. Find the separation distance for a design with an inner radius = 25 cm, outer radius = 35 cm, length = 50 cm, applied voltage = 60 V, μ_A = 8 × 10^–5 m.C/N.s and μ_B = 4 × 10^–5 m C/N.s, and rotation speed = 15 rpm.