Electrodialysis (1D)
Introduction
Electrodialysis, an electrochemical separation technology, has been utilized to desalinate water for decades. Compared with other technologies, such as reverse osmosis (RO), electrodialysis shows advantages in desalinating brackish waters with moderate salinity by its less intense energy consumption, higher water recovery, and robust tolerance for adverse nonionic components (e.g., silica and biological substances) in water. Between the electrodes of an electrodialysis stack (a reactor module), multiple anion and cationexchange membranes are alternately positioned and separated by spacers to form individual cells. When operated, electrodialysis converts electrical current to ion flux and, assisted by the opposite ion selectivity of cation and anionexchange membranes (cem and aem), moves ions from one cell to its adjacent cell in a cellpair treatment unit (Figure 1). The iondeparting cell is called a diluate channel and the ionentering cell a concentrate channel. Recovered (desalinated) water is collected from diluate channels of all cell pairs while the concentrate product can be disposed of as brine or retreated. More overview of the electrodialysis technology can be found in the References.
One cell pair in an electrodialysis stack can thus be treated as a modeling unit that can multiply to largerscale systems. The presented electrodialysis 1D model establishes mathematical descriptions of ion and water transport in a cell pair and expands to simulate a stack with a specified cellpair number. Modeled mass transfer mechanisms include electrical migration and diffusion of ions and osmosis and electroosmosis of water. By its name, “1D” suggests the variation of mass transfer and solution properties over length (l in Figure 1) is mathematically described. The model relies on the following key assumptions:
The concentrate and diluate channels have identical geometry.
For each channel, component fluxes in the bulk solution are uniform in b and s directions in Figure 1 but dependent on the l directions (the 1dimensional assumption).
Steady state: all variables are independent of time.
Cocurrent flow operation.
Electrical current is operated below the limiting current.
Ideality assumptions: activity, osmotic, and van’t Hoff coefficients are set at one.
All ionexchange membrane properties (ion and water transport number, resistance, permeability) are constant.
Detailed concentration gradient effect at membranewater interfaces is neglected.
Constant pressure and temperature through each channel.
Control Volumes
This model has two 1D control volumes for the concentrate and diluate channels.
Diluate
Concentrate
Ports
On the two control volumes, this model provides four ports (Pyomo notation in parenthesis):
inlet_diluate (inlet)
outlet_diluate (outlet)
inlet_concentrate (inlet)
outlet_concentrate (outlet)
Sets
This model can simulate electrodialysis for desalination of a water solution containing multiple species (neutral or ionic). All solution components ( H_{2}O, neutral solutes, and ions) form a Pyomo set in the model. For a clear model demonstration, this document uses an NaCl water solution as an instance hereafter. The user can nevertheless expand the component set as needed to represent other feed water conditions.
This model can mathematically take a multicomponent (i.e., > one salt molecule to be treated) as an input; nevertheless a multicomponent solution creates unknown or difficulttospecify parameters, e.g., the electrical transport numbers through membranes, the multiion diffusivity, etc., and physical relationships, which may result in illposed or illconditioned problems challenging the models’ numerical solutions. While we continuously work on advancing our models to absorb new principles revealed by progressing research, we advise the users be very cautious with simulating multicomponent systems by this programmed model for aspects stated above.
Description 
Symbol 
Indices 

Time 
\(t\) 
[t] ([0])^{1} 
Length_domain 
\(x\) 
\(l \times(0, 1)\) ^{2} 
Phase 
\(p\) 
[‘Liq’] 
Component 
\(j\) 
[‘H2O’, ‘Na_+’, ‘Cl_‘] 
Ion 
\(j\) 
[‘Na_+’, ‘Cl_’] ^{3} 
Membrane 
n/a 
[‘cem’, ‘aem’] 
 Notes
^{1} The time set index is set as [0] in this steadystate model and is reserved for the future extension to a dynamic model.
^{2} By the IDAES convention, the index of length_domain is normalized to a continuous set of (0, 1), which is discretized when differential equations in the model are solved by numerical methods such as “finite difference” discretization. In this documentation, \(x\) refers to the length dimension before normalization and carries a unit of [m].
^{3} “Ion” is a subset of “Component” and uses the same symbol j.
Degrees of Freedom
Applying this model to an NaCl solution yields 33 degrees of freedom (Table 2), among which temperature, pressure, and component molar flow rate are state variables that are fixed as initial conditions. The rest are parameters that should be provided in order to fully solve the model.
Description 
Symbol 
Variable Name 
Index 
Units 
DOF Number ^{1} 

Temperature, inlet_diluate 
\(T^D\) 
temperature 
None 
\(K\) 
1 
Temperature, inlet_concentrate 
\(T^C\) 
temperature 
None 
\(K\) 
1 
Pressure, inlet_diluate 
\(p^D\) 
temperature 
None 
\(Pa\) 
1 
Pressure, inlet_concentrate 
\(p^C\) 
temperature 
None 
\(Pa\) 
1 
Component molar flow rate, inlet_diluate 
\(N_{j, in}^D\) 
flow_mol_phase_comp 
[t], [‘Liq’], [‘H2O’, ‘Na_+’, ‘Cl_‘] 
\(mol s^{1}\) 
3 
Component molar flow rate, inlet_concentrate 
\(N_{j, in}^C\) 
flow_mol_phase_comp 
[t], [‘Liq’], [‘H2O’, ‘Na_+’, ‘Cl_‘] 
\(mol s^{1}\) 
3 
Water transport number 
\(t_w\) 
water_trans_number_membrane 
[‘cem’, ‘aem’] 
dimensionless 
2 
Water permeability 
\(L\) 
water_permeability_membrane 
[‘cem’, ‘aem’] 
\(m^{1}s^{1}Pa^{1}\) 
2 
Voltage or Current ^{2} 
\(U\) or \(I\) 
voltage_applied or current_applied 
[t] 
\(\text{V}\) or \(A\) 
1 
Electrode areal resistance 
\(r_{el}\) 
electrodes_resistance 
[t] 
\(\Omega m^2\) 
1 
Cell pair number 
\(n\) 
cell_pair_num 
None 
dimensionless 
1 
Current utilization coefficient 
\(\xi\) 
current_utilization 
None 
dimensionless 
1 
Channel height 
\(d\) 
channel_height 
none 
\(m\) 
1 
Membrane areal resistance 
\(r\) 
membrane_areal_resistance 
[‘cem’, ‘aem’] 
\(\Omega m^2\) 
2 
Cell width 
\(b\) 
cell_width 
None 
\(\text{m}\) 
1 
Cell length 
\(l\) 
cell_length 
None 
\(\text{m}\) 
1 
Thickness of ion exchange membranes 
\(\delta\) 
membrane_thickness 
[‘cem’, ‘aem’] 
\(m\) 
2 
diffusivity of solute in the membrane phase 
\(D\) 
solute_diffusivity_membrane 
[‘cem’, ‘aem’], [‘Na_+’, ‘Cl_‘] 
\(m^2 s^{1}\) 
4 
transport number of ions in the membrane phase 
\(t_j\) 
ion_trans_number_membrane 
[‘cem’, ‘aem’], [‘Na_+’, ‘Cl_‘] 
dimensionless 
4 
 Note
^{1} DOF number accounts for indices of corresponding variables.
^{2} A user should provide either current or voltage as the electrical input, in correspondence to the “Constant_Current” or “Constant_Voltage” treatment mode (configured in this model). The user also should provide an electrical magnitude that ensures an operational current below the limiting current of the feed solution.
Solution component information
To fully construct solution properties, users need to provide basic component information of the feed solution to use this model, including identity of all solute species (i.e., Na ^{+}, and Cl ^{} for a NaCl solution), molecular weight of all component species (i.e., H_{2}O, Na ^{+}, and Cl ^{}), and charge and electrical mobility of all ionic species (i.e., Na ^{+}, and Cl ^{}). This can be provided as a solution dictionary in the following format (instantiated by a NaCl solution).
ion_dict = {
"solute_list": ["Na_+", "Cl_"],
"mw_data": {"H2O": 18e3, "Na_+": 23e3, "Cl_": 35.5e3},
"electrical_mobility_data": {"Na_+": 5.19e8, "Cl_": 7.92e8},
"charge": {"Na_+": 1, "Cl_": 1},
}
This model, by default, uses H_{2}O as the solvent of the feed solution.
Information regarding the property package this unit model relies on can be found here:
watertap.property_models.ion_DSPMDE_prop_pack
Equations
This model solves mass balances of all solution components (H_{2}O, Na ^{+}, and Cl ^{} for an NaCl solution) on two control volumes (concentrate and diluate channels). Under the 1D treatment, balance equations are expressed as differential algebraic equations (DAE) when concerned variables are functions of length (x). The DAEs are solved in a discretization manner using the “finite difference” or “collocation” method implemented in Pyomo.DAE.
Mass balance equations are summarized in Table 3. Mass transfer mechanisms account for solute electrical migration, diffusion, water osmosis, and electroosmosis. Theoretical principles, e.g., continuity equation, Fick’s law, and Ohm’s law, to simulate these processes are well developed and some good summaries for the electrodialysis scenario can be found in the References.
Description 
Equation 
Index set 

Component mass balance 
\(\left(\frac{\partial N_j (x)}{\partial x}\right)^{C\: or\: D}+J_j(x)^{C\: or\: D} b=0\) 
\(j \in \left['H_2 O', '{Na^{+}} ', '{Cl^{}} '\right]\) 
mass transfer flux, concentrate, solute 
\(J_j^{C} = \left(t_j^{cem}t_j^{aem} \right)\frac{\xi i(x)}{ z_j F}\left(\frac{D_j^{cem}}{\delta ^{cem}} +\frac{D_j^{aem}}{\delta ^{aem}}\right)\left(c_j^C(x)c_j^D(x) \right)\) 
\(j \in \left['{Na^{+}} ', '{Cl^{}} '\right]\) 
mass transfer flux, diluate, solute 
\(J_j^{D} = \left(t_j^{cem}t_j^{aem} \right)\frac{\xi i(x)}{ z_j F}+\left(\frac{D_j^{cem}}{\delta ^{cem}} +\frac{D_j^{aem}}{\delta ^{aem}}\right)\left(c_j^C(x)c_j^D(x) \right)\) 
\(j \in \left['{Na^{+}} ', '{Cl^{}} '\right]\) 
mass transfer flux, concentrate, H_{2}O 
\(J_j^{C} = \left(t_w^{cem}+t_w^{aem} \right)\frac{i(x)}{F}+\left(L^{cem}+L^{aem} \right)\left(p_{osm}^C(x)p_{osm}^D(x) \right)\frac{\rho_w}{M_w}\) 
\(j \in \left['H_2 O'\right]\) 
mass transfer flux, diluate, H_{2}O 
\(J_j^{D} = \left(t_w^{cem}+t_w^{aem} \right)\frac{i(x)}{F}\left(L^{cem}+L^{aem} \right)\left(p_{osm}^C(x)p_{osm}^D(x) \right)\frac{\rho_w}{M_w}\) 
\(j \in \left['H_2 O'\right]\) 
Additionally, several other equations are built to describe the electrochemical principles and electrodialysis performance.
Description 
Equation 

Electrical input condition 
\(i(x) = \frac{I}{bl}\), for ‘Constant_Current’; \(u(x) =U\) for ‘Constant_Voltage’ 
Ohm’s law 
\(u(x) = i(x) r_{tot}(x)\) 
Resistance calculation 
\(r_{tot}(x)=n\left(r^{cem}+r^{aem}+\frac{d}{\kappa^C(x)}+\frac{d}{\kappa^D(x)}\right)+r_{el}\) 
Electrical power consumption 
\(P(x)=b\int _0 ^l u(x)i(x) dx\) 
Waterproductionspecific power consumption 
\(P_Q=\frac{P(x=l)}{3.6\times 10^6 nQ_{out}^D}\) 
Current efficiency for desalination 
\(bi(x)\eta(x)=\sum_{j \in[cation]}{\left[\left(\frac{\partial N_j ^D(x)}{\partial x}\right) z_j F\right]}\) 
All equations are coded as “constraints” (Pyomo). Isothermal and isobaric conditions apply.
Extended simulation
Membrane and interfacial potentials
This model supports extensive simulations of (1) the nonohmic potential across ion exchange membranes and (2) the Nernst diffusion layer. Users can customize these extenions via two configurations: has_nonohmic_potential_membrane that triggers the calculation of nonohmic potentials across ion exchange membranes and has_Nernst_diffusion_layer that triggers the simulation of a concentrationpolarized Nernst diffusion layer including its ohmic and nonohmic potential changes. Based on a electrochemical cell setup in Figure 2 and established theoretical descriptions (References), our model accounts for the crossmembrane diffusion and Donnan potentials (nonohmic), ion concentration polarization in assumed Nernst diffusion layers (NDL), and the ohmic and nonohmic (i.e., diffusion) potentials across NDLs. These extensions make the model closer to the nonideal physical conditions that can be encountered in real desalination practices.
Table 5 presents the equations underlying the two extensions assuming a 1:1 symmetric electrolyte such as NaCl.
Description 
Equation 
Condition 

Limiting current density 
\(i_{lim}(x) = i_{lim,0}\frac{c_b^D(x)}{c_b^D(0)}\) 
has_Nernst_diffusion_layer==True and limiting_current_density_method == LimitingCurrentDensityMethod.InitialValue 
\(i_{lim}(x) = A v^B c_b^D (x)\) 
has_Nernst_diffusion_layer==True and limiting_current_density_method == LimitingCurrentDensityMethod.Empirical 

\(i_{lim}(x) = \frac{Sh F D_b c_b^D(x)}{d_H \left(t_+^{cem}t_+ \right)}\) 
has_Nernst_diffusion_layer==True and limiting_current_density_method == LimitingCurrentDensityMethod.Theoretical 

Nonohmic potential, membrane 
\(\phi_m(x)=\frac{RT}{F} \left( t_+^{iem}  t_^{iem} \right) \ln \left( \frac{c_s^R(x)}{c_s^L(x)} \right)\) 
has_nonohmic_potential_membrane == True 
Ohmic potential, NDL 
\(\phi_d^{ohm}(x)=\frac{FD_b}{\left(t_+^{iem}t_+\right)\lambda}\ln\left(\frac{c_s^L(x)c_b^R(x)}{c_s^R(x)c_b^L(x)}\right)\) 
has_Nernst_diffusion_layer==True 
Nonohmic potential, NDL 
\(\phi_d^{nonohm}(x)=\frac{RT}{F}\left(t_+t_\right) \ln\left(\frac{c_s^L(x)c_b^R(x)}{c_s^R(x)c_b^L(x)}\right)\) 
has_Nernst_diffusion_layer==True 
NDL thickness, cem 
\(\Delta^{L/R}(x) = \frac{F D_b c_b^{L/R}(x)}{\left(t_+^{iem}t_+ \right) i_{lim}(x)}\) 
has_Nernst_diffusion_layer==True 
NDL thickness, aem 
\(\Delta^{L/R}(x) =  \frac{F D_b c_b^{L/R}(x)}{\left(t_+^{iem}t_+\right) i_{lim}(x)}\) 
has_Nernst_diffusion_layer==True 
Concentration polarization ratio, cem 
\(\frac{c_s^L(x)}{c_b^L(x)} = 1+\frac{i(x)}{i_{lim}(x)},\qquad \frac{c_s^R(x)}{c_b^R(x)} = 1\frac{i(x)}{i_{lim}(x)}\) 
has_Nernst_diffusion_layer==True ^{1} 
Concentration polarization ratio, aem 
\(\frac{c_s^L(x)}{c_b^L(x)} = 1\frac{i(x)}{i_{lim}(x)},\qquad \frac{c_s^R(x)}{c_b^R(x)} = 1+\frac{i(x)}{i_{lim}(x)}\) 
has_Nernst_diffusion_layer==True 
Note
^{1} When this configuration is turned off, \(i_{lim}\) is considered as \(\infty\) and the ratio becomes 1.
Some other modifications to previously defined equations are made to accommodate the two extensions. These are shown in Table 6.
Original equation description 
Equation replacement 
Condition 

Ohm’s law 
\(u(x) = i(x) r_{tot}(x) + \phi_m(x) + \phi_d^{ohm}(x) + \phi_d^{nonohm}(x)\) ^{1} 
has_nonohmic_potential_membrane == True and/or has_Nernst_diffusion_layer==True 
Resistance calculation 
\(r_{tot}(x)=n\left(r^{cem}+r^{aem}+\frac{d \Delta_{cem}^L(x)  \Delta_{aem}^R(x)}{\kappa^C(x)}+\frac{d \Delta_{cem}^R(x)  \Delta_{aem}^L(x)}{\kappa^D(x)}\right)+r_{el}\) 
has_Nernst_diffusion_layer==True 
mass transfer flux, concentrate, solute 
\(J_j^{C} = \left(t_j^{cem}t_j^{aem} \right)\frac{\xi i(x)}{ z_j F}\left(\frac{D_j^{cem}}{\delta ^{cem}}\left(c_{s,j}^{L,cem}(x)c_{s,j}^{R,cem}(x) \right) +\frac{D_j^{aem}}{\delta ^{aem}} \left(c_{s,j}^{R,aem}(x)c_{s,j}^{L,aem}(x) \right)\right)\) 
has_nonohmic_potential_membrane == True and/or has_Nernst_diffusion_layer==True 
mass transfer flux, diluate, solute 
\(J_j^{D} = \left(t_j^{cem}t_j^{aem} \right)\frac{\xi i(x)}{ z_j F}+\left(\frac{D_j^{cem}}{\delta ^{cem}}\left(c_{s,j}^{L,cem}(x)c_{s,j}^{R,cem}(x) \right) +\frac{D_j^{aem}}{\delta ^{aem}} \left(c_{s,j}^{R,aem}(x)c_{s,j}^{L,aem}(x) \right)\right)\) 
has_nonohmic_potential_membrane == True and/or has_Nernst_diffusion_layer==True 
mass transfer flux, concentrate, H_{2}O 
\(J_j^{C} = \left(t_w^{cem}+t_w^{aem} \right)\frac{i(x)}{F}+\left(L^{cem} \left(p_{s, osm}^{cem, L}(x)p_{s, osm}^{cem, R}(x) \right)+L^{aem} \left(p_{s, osm}^{aem, R}(x)p_{s, osm}^{aem, L}(x) \right)\right)\frac{\rho_w}{M_w}\) 
has_Nernst_diffusion_layer==True 
mass transfer flux, diluate, H_{2}O 
\(J_j^{D} = \left(t_w^{cem}+t_w^{aem} \right)\frac{i(x)}{F}\left(L^{cem} \left(p_{s, osm}^{cem, L}(x)p_{s, osm}^{cem, R}(x) \right)+L^{aem} \left(p_{s, osm}^{aem, R}(x)p_{s, osm}^{aem, L}(x) \right)\right)\frac{\rho_w}{M_w}\) 
has_Nernst_diffusion_layer==True 
Note
^{1} \(\phi_m(x), \phi_d^{ohm}(x)\) or \(\phi_d^{nonohm}(x)\) takes 0 if its corresponding configuration is turned off (value == False).
Frictional pressure drop
This model can optionally calculate pressured drops along the flow path in the diluate and concentrate channels through config has_pressure_change
and pressure_drop_method
. Under the assumption of identical diluate and concentrate channels and starting flow rates, the flow velocities in the two channels are approximated equal and invariant over the channel length when calculating the frictional pressure drops. This approximation is based on the evaluation that the actual velocity variation over the channel length caused by water mass transfer across the consecutive channels leads to negligible errors as compared to the uncertainties carried by the frictional pressure method itself. Table 7 gives essential equations to simulate the pressure drop. Among extensive literatures using these equations, a good reference paper is by Wright et. al., 2018 (References).
Description 
Equation 
Condition 

Frictional pressure drop, Darcy_Weisbach 
\(p_L=f\frac{\rho v^2}{2d_H}\) ^{1} 
has_pressure_change == True and pressure_drop_method == PressureDropMethod.Darcy_Weisbach 
\(p_L=\) userinput constant 
has_pressure_change == True and pressure_drop_method == PressureDropMethod.Experimental 

Hydraulic diameter 
\(d_H=\frac{2db(1\epsilon)}{d+b}\) 
hydraulic_diameter_method == HydraulicDiameterMethod.conventional 
\(d_H=\frac{4\epsilon}{\frac{2}{h}+(1\epsilon)S_{v,sp}}\) 
hydraulic_diameter_method == HydraulicDiameterMethod.spacer_specific_area_known 

Renold number 
\(Re=\frac{\rho v d_H}{\mu}\) 
has_pressure_change == True or limiting_current_density_method == LimitingCurrentDensityMethod.Theoretical 
Schmidt number 
\(Sc=\frac{\mu}{\rho D_b}\) 
has_pressure_change == True or limiting_current_density_method == LimitingCurrentDensityMethod.Theoretical 
Sherwood number 
\(Sh=0.29Re^{0.5}Sc^{0.33}\) 
has_pressure_change == True or limiting_current_density_method == LimitingCurrentDensityMethod.Theoretical 
Darcy’s frictional factor 
\(f=4\times 50.6\epsilon^{7.06}Re^{1}\) 
friction_factor_method == FrictionFactorMethod.Gurreri 
\(f=4\times 9.6 \epsilon^{1} Re^{0.5}\) 
friction_factor_method == FrictionFactorMethod.Kuroda 

Pressure balance 
\(p_{in}p_L l =p_{out}\) 
has_pressure_change == True 
Note
^{1} As discussed in the last paragraph, in this section we assumed a constant linear velocity (in the cell length direction), \(v\), in both channels and along the flow path. This \(v\) is calculated from the volume flow rate at \(x=0\) by the property package.
Nomenclature
Symbol 
Description 
Unit 

Parameters 

\(\rho_w\) 
Mass density of water 
\(kg\ m^{3}\) 
\(M_w\) 
Molecular weight of water 
\(kg\ mol^{1}\) 
Variables 

\(N\) 
Molar flow rate of a component 
\(mol\ s^{1}\) 
\(J\) 
Molar flux of a component 
\(mol\ m^{2}s^{1}\) 
\(b\) 
Cell/membrane width 
\(m\) 
\(l\) 
Cell/membrane length 
\(m\) 
\(t\) 
Ion transport number 
dimensionless 
\(I\) 
Current input 
\(A\) 
\(i\) 
Current density 
\(A m^{2}\) 
\(U\) 
Voltage input over a stack 
\(V\) 
\(u\) 
xdependent voltage over a stack 
\(V\) 
\(n\) 
Cell pair number 
dimensionless 
\(\xi\) 
Current utilization coefficient (including ion diffusion and water electroosmosis) 
dimensionless 
\(z\) 
Ion charge 
dimensionless 
\(F\) 
Faraday constant 
\(C\ mol^{1}\) 
\(D\) 
Ion Diffusivity 
\(m^2 s^{1}\) 
\(\delta\) 
membrane thickness 
\(m\) 
\(c\) ^{1} 
Solute concentration 
\(mol\ m^{3}\) 
\(t_w\) 
Water electroosmotic transport number 
dimensionless 
\(L\) 
Water permeability (osmosis) 
\(ms^{1}Pa^{1}\) 
\(p_{osm}\) 
Osmotic pressure 
\(Pa\) 
\(r_{tot}\) 
Total areal resistance 
\(\Omega m^2\) 
\(r\) 
Membrane areal resistance 
\(\Omega m^2\) 
\(r_{el}\) 
Electrode areal resistance 
\(\Omega m^2\) 
\(d\) 
Channel height 
\(m\) 
\(\kappa\) 
Solution conductivity 
\(S m^{1}\ or\ \Omega^{1} m^{1}\) 
\(\eta\) 
Current efficiency for desalination 
dimensionless 
\(P\) 
Power consumption 
\(W\) 
\(P_Q\) 
Specific power consumption 
\(kW\ h\ m^{3}\) 
\(Q\) 
Volume flow rate 
\(m^3s^{1}\) 
\(\phi_m\) 
Nonohmic potential across a membrane 
\(V\) 
\(\phi_d^{ohm}\) 
Ohmic potential across a Nernst diffusion layer 
\(V\) 
\(\phi_d^{nonohm}\) 
Nonohmic potential across a Nernst diffusion layer 
\(V\) 
\(\Delta\) 
Nernst diffusion layer thickness 
\(m\) 
\(D_b\) 
General mass diffusivity of the salt molecule in the bulk solution 
\(m^2 s^{1}\) 
\(i_{lim}\) 
Limiting current density 
\(A m^{2}\) 
\(i_{lim,0}\) 
Limiting current density experimentally evaluated from the untreated salt solution 
\(A m^{2}\) 
\(\lambda\) 
Equivalent conductivity of the solution 
\(m^2 \Omega^{1} mol^{1}\) 
\(A\) 
Experimental parameter to calculate \(i_{lim}\) 
\(C mol^{1} m^{1B} s^{B1}\) 
\(B\) 
Experimental parameter to calculate \(i_{lim}\) 
dimensionless 
\(d_H\) 
Hydraulic diameter 
\(m\) 
\(\epsilon\) 
Spacer porosity 
dimensionless 
\(f\) 
Darcy’s fraction factor 
dimensionless 
\(p_L\) 
Frictional pressure drop per unit of length 
\(Pa m^{1}\) 
\(v\) 
Flow linear velocity 
\(m s^{1}\) 
\(\rho\) 
Aqueous phase density 
\(kg m^{3}\) 
\(S_{v,sp}\) 
Volumespecific surface area of the spacer 
\(m^{1}\) 
\(\mu\) 
Dynamic viscosity 
\(Pa s^{1}\) 
\(Re\) 
Renold number 
dimensionless 
\(Sc\) 
Schmidt number 
dimensionless 
\(Sh\) 
Sherwood number 
dimensionless 
Subscripts and superscripts 

\(C\) 
Concentrate channel 

\(D\) 
Diluate channel 

\(j\) 
Component index 

\(in\) 
Inlet 

\(out\) 
Outlet 

\(cem\) 
Cation exchange membrane 

\(aem\) 
Anion exchange membrane 

\(iem\) 
Ion exchange membrane, i.e., cem or aem 

\(L\) 
The left side of a membrane, facing the cathode 

\(R\) 
The right side of a membrane, facing the anode 

\(s\) 
location of the membrane surface 

\(b\) 
location of bulk solution 

\(+\) or \(\) 
monocation or monoanion 
Note
^{1} When no component subscript (j), \(c\) refers to the concentration of the salt molecules (e.g., NaCl)
References
Strathmann, H. (2010). Electrodialysis, a mature technology with a multitude of new applications. Desalination, 264(3), 268288.
Strathmann, H. (2004). Ionexchange membrane separation processes. Elsevier. Ch. 4.
Campione, A., Cipollina, A., Bogle, I. D. L., Gurreri, L., Tamburini, A., Tedesco, M., & Micale, G. (2019). A hierarchical model for novel schemes of electrodialysis desalination. Desalination, 465, 7993.
Campione, A., Gurreri, L., Ciofalo, M., Micale, G., Tamburini, A., & Cipollina, A. (2018). Electrodialysis for water desalination: A critical assessment of recent developments on process fundamentals, models and applications. Desalination, 434, 121160.
Spiegler, K. S. (1971). Polarization at ion exchange membranesolution interfaces. Desalination, 9(4), 367385.
Wright, N. C., Shah, S. R., & Amrose, S. E. (2018). A robust model of brackish water electrodialysis desalination with experimental comparison at different size scales. Desalination, 443, 2743.