COMSOL 6.4 - Desalination in an Electrodialysis Cell

Electrodialysis is a separation process for electrolytes based on the use of electric fields and ion-selective membranes. Some common applications of the electrodialysis process are:

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Desalination of process streams, effluents, and drinking water

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pH regulation in order to remove acids from, for example, fruit juices and wines

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Metal electrowinning of precious metals

This tutorial demonstrates the basics of electrodialysis in a desalination cell.

Model Definition

Figure 1 shows the basic principle of the desalination stack. The stack has one repetitive cell unit, shown in Figure 2.

Figure 1: Schematic picture of a desalination stack with 3 desalination units (in reality 10 - 20 unit cells are used)

Figure 2: The repetitive unit cell with one desalination unit.

The model geometry for this example is based on the repetitive unit, excluding the inlet and outlet flow regions. The model geometry is shown in Figure 3.

Figure 3: Model geometry.

The cell contains two ion selective membrane domains: the left is mainly permeable to cations, and the right to anions. The middle domain is a free flowing electrolyte domain, where salt is to be removed. This domain is named the diluate domain. The rightmost and leftmost domains are free flowing electrolyte domains, where the ion concentration increases during cell operation. These domains are called the concentrate domains. In the free electrolyte domains, the flow enters at the bottom inlets and exits at the top outlets at an average flow rate is 5 mm·s−1. An analytical expression (Poiseuille flow) for the fluid velocity is used in this model.

The cell is fed with a seawater electrolyte consisting of 0.5 M NaCl and operated at a unit cell voltage of 1.5 V.

Choice of current distribution interface

In this model we make use of the Nernst–Planck equations for ion flux and charge transport by which the following equation describes the molar flux of species i (which is either Cl or Na in this model), Ni, due to diffusion, migration and convection:

The first term is the diffusion flux, Di is the diffusion coefficient (SI unit: m2/s), and ci the species concentration. The migration term consists of the species charge number zi, the species mobility um,i (SI unit: s·mol/kg) and the electrolyte potential (ϕl). In the convection term, u denotes the fluid velocity vector (SI unit: m/s).

The electrolyte current density is calculated using Faraday’s law by summing up the contributions from the molar fluxes, multiplied by the species charges, with the observation that the convective term vanishes due to the electroneutrality condition (see the theory for the Tertiary Current Distribution, Nernst–Planck interface):

(1)

The conservation of current is then used to calculate the electrolyte potential.

This model uses Tertiary Current Distribution, Nernst–Planck interface when solving for the electrolyte potential in the free electrolyte and ion-selective membrane domains.

In the free electrolyte domains, the only ions present are assumed to be Na+ and Cl-. The ion selective membranes also contain additional ions fixed in a matrix. Therefore, the fixed space charge, ρfix, is added while calculating the sum of charges in the electroneutrality condition:

(2)

The fixed space charge is prescribed in terms of the membrane charge concentration.

The checkbox Add Donnan shift to initial values, when enabled, shifts the initial values on the selection of the Ion-Exchange Membrane node in order to fulfill electroneutrality and compliance with the Donnan equilibria.

Note that the ion-selective membrane domains do allow for transport of both Na+ and Cl− but that the sign and magnitude of the fixed charge, in combination with the Donnan boundary conditions (described in the next section) will favor transport of either anions or cations. The sign of the fixed membrane charge is positive in the anion, and negative in the cation-exchange membrane domains, respectively.

Membrane — free Electrolyte Boundary Conditions

The boundary conditions at the boundaries between the membrane and free electrolyte domains are set up in the following way:

The normal electrolyte current density is equal to the current density in the membrane:

(3)

The ion flux for each species is continuous over the membrane-free electrolyte interface:

(4)

Furthermore, we have the following relation between the potentials, ϕl, and the concentrations:

(5)

where ci,m is the species concentration in the membrane, and ci,e the species concentration in the free electrolyte and zi the corresponding charge. The potential shift caused by Equation 5 is called Donnan potential, or dialysis potential.

Equation 3, Equation 4, and Equation 5 above represent a mix of Dirichlet and Neumann conditions for the electrolyte potential and species concentrations. The Ion-Exchange Membrane domain feature in the Tertiary Current Distribution, Nernst–Planck interface is used to define these conditions.

Periodic condition

Use the Periodic Condition feature to set the concentration on the rightmost boundary to equal that of the leftmost boundary and to set the electrolyte potential offset between the rightmost and leftmost boundaries.

Results and Discussion

Figure 4 shows the Na+ ion concentration in the cell for the membrane charge concentration of 1000 mol/m3. The concentration increases in the concentrate domains, and decreases in the diluate domain. A boundary layer with high concentration gradients forms close to the membrane surfaces. In a real desalination cell it is common to add spacers that, apart from adding mechanical stability to the cell, also induce convective mass transport in the direction perpendicular to the main flow, and hence may reduce the thickness of the boundary layer.

Figure 4: Ion concentration.

Figure 5 shows the electrolyte potential along a horizontal line placed at half the cell height. The main part of the potential losses occurs in the membranes. The Donnan potential discontinuity can be seen at the boundaries between the free electrolyte and membrane domains.

Figure 5: Electrolyte potential at half the cell height.

Figure 6 shows the concentration of Na+ and Cl- ions at half the cell height for the membrane charge concentration, 1000 mol/m3. The concentration of both Na+ and Cl-ions is higher in the concentrate domains and lower in the diluate domain, as also shown in Figure 4. The concentration of Na+ is significantly lower than for Cl- in the anion-selective domain (and vice versa), but it is not zero.

Figure 6: Ion concentrations at half the cell height for a membrane charge concentration of 1000 mol/m3.

Figure 7 and Figure 8 show a comparison between the migrative and diffusive fluxes in the free electrolyte for Na+ and Cl-, respectively. The diffusive fluxes get prominent close to the membrane boundaries due to the high concentration gradients. The migrative fluxes govern in the middle of the channels and have different signs for Na+ and Cl- due to the different signs of the ion charges.

Figure 7: Comparison of fluxes, species Na+.

Figure 8: Comparison of fluxes, species Cl-.

Finally, Figure 9 shows the current density along the left free electrolyte–membrane boundary of the anion selective membrane for membrane charge concentration of 1000 mol/m3. Depletion of chloride gets more prominent toward the top of the cell, resulting in a higher Donnan potential shift and a higher negative current density.