OneStack Electrodialysis with a Concentrate Fluid Recirculation
Introduction
Electrodialysis (ED) is a promising technology for desalinating brackish waters and has been deployed at industrial scales [1]. It utilizes electrical potential to drive ion diffusion through anion and cation exchange membranes, resulting in the dilution of the feed stream while producing a concentrated brine. A single ED stack is an assembly of multiple flowby cells separated by alternating cation and anion exchange membranes positioned between a pair of electrodes. When a voltage is applied, ions in the cell pair are driven from one channel (forming the diluate channel) to the other (forming the concentrate channel), thereby desalinating water. A onestack ED desalination system represents a basic ED operation from which more complicated or largerscale systems can be derived. Analyzing a onestack ED system therefore provides valuable information on the technology’s performance and costeffectiveness for a given treatment task.
Equal flow conditions through the diluate and concentrate channels would result in a product water recovery of 50%. Larger water recoveries are commonly achieved by operating the system in a feedandbleed mode, where a portion of the concentrate outlet is recirculated back to its inlet, thus increasing the portion of product water from the diluate outlet. This flowsheet simulates a onestack ED system operated in feedandbleed mode. A simpler ED flowsheet without fluid recirculation is presented as
Implementation
The modeled onestack ED system is illustrated by Figure 1. The feed solution is split into two fluids through a separator unit, entering the diluate and concentrate channels of the ED stack. Two pump units are placed respectively on the two channels entering pipelines to counterbalance the pressure drops across ED stack channels. On the outlet side of the ED stack, all diluate fluids are collected into the total product water, and the concentrate fluids into the total brine stream. A second separator unit takes a portion of the brine back into the concentrate pump inlet, with the rest being sent to brine disposal. Water recovery in this model is volumebased, i.e., the ratio of product volume to total volume of feed solution. The flowsheet relies on the following key assumptions:
supports steadystate only
a property package (i.e., MCAS) is provided for all unit models
The electrodialysis 1D block is set up with the following configuration arguments:
m.fs.EDstack = Electrodialysis1D( property_package=m.fs.properties, operation_mode=ElectricalOperationMode.Constant_Voltage, finite_elements=20, has_pressure_change=True, has_nonohmic_potential_membrane=True, has_Nernst_diffusion_layer=True, limiting_current_density_method=LimitingCurrentDensityMethod.Theoretical, pressure_drop_method=PressureDropMethod.Darcy_Weisbach, hydraulic_diameter_method=HydraulicDiameterMethod.spacer_specific_area_known, friction_factor_method=FrictionFactorMethod.Gurreri, )These configurations enable the electrodialysis unit to use a flowsheetunified property package, set a constant stack voltage, and adopt a favorable number of finite elements for 1dimensional simulation and solving. The overall ED configuration represents the most comprehensive modeling that takes into account the pressure change, diffusion layer phenomenon, and nonohmic potentials in the system.
In the given optimization case, the objective function is to minimize the levelized cost of water, which can be represented by the following equation where \(Q\) represents volumetric flow, \(f_{crf}\) represents capital recovery factor \(C_{cap,tot}\) represents total capital cost, \(C_{op,tot}\) represents total operating cost, and \(f_{util}\) represents the utilization factor:
\[LCOW_{Q} = \frac{f_{crf} C_{cap,tot} + C_{op,tot}}{f_{util} Q}\]The product water salinity is set to 0.1 \(g L^{1}\).
 Documentation for unit models from WaterTAP:
 Documentation for unit models from IDAES:
 Documentation for the property model:
Degrees of Freedom
The number of degrees of freedom (DOF) is associated with the number of fixed variables (parameters) determined by the purpose of the modeling case. There are two implemented modeling cases in the flowsheet: (1) the prediction of desalination outcome (salinity of the product water and saline disposal) and (2) the optimization of key decision variables in system design. In the first case, DOF is set to zero by fixing all initial conditions of the feed solution fluid and definite ED stack parameters. All fixed values are presented in the section to follow. In the second case, the values of those chosen to be the decision variables in the optimization are unfixed. The DOF number is therefore the number of decision variables. In this example, the decision variables are:
stack voltage applied
ED cell pair number
ED cell length
Flowsheet Specifications
Name 
Value 
Unit 
Reference 

Salinity (NaCl) 
\(2\) 
\(g L^{1}\) 
– 
Volume flow rate 
\(5.2 \times 10^{4}\) 
\(m^3 s^{1}\) 

Temperature 
\(298.15\) 
\(K\) 
– 
Pressure 
\(101325\) 
\(Pa\) 
– 
Na^+ diffusivity 
\(1.33 \times 10^{9}\) 
\(m^2 s^{1}\) 

Cl^ diffusivity 
\(2.03 \times 10^{9}\) 
\(m^2 s^{1}\) 

NaCl mass diffusivity 
\(1.60 \times 10^{9}\) 
\(m^2 s^{1}\) 

Cell pair number 
\(56\) 
\(1\) 
– 
Cell length 
\(1.68\) 
\(m\) 

Cell width 
\(0.197\) 
\(m\) 

Channel height 
\(7.1 \times 10^{4}\) 
\(m\) 
– 
Water recovery 
\(70%\) 
\(1\) 
– 
Stack voltage 
\(10\) 
\(V\) 
– 
Thickness, aem and cem 
\(1.3 \times 10^{5}\) 
\(m\) 

Areal resistance, aem 
\(1.77 \times 10^{4}\) 
\(\Omega m^2\) 

Areal resistance, cem 
\(1.89 \times 10^{4}\) 
\(\Omega m^2\) 

Water permeability, aem 
\(1.75 \times 10^{14}\) 
\(m s^{1} Pa^{1}\) 

Water permeability, cem 
\(2.16 \times 10^{14}\) 
\(m s^{1} Pa^{1}\) 

Water transport number, aem 
\(4.3\) 
\(1\) 

Water transport number, cem 
\(5.8\) 
\(1\) 

NaCl mass diffusivity, aem and cem 
\(3.28 \times 10^{11}\) 
\(m^2 s^{1}\) 

Spacer porosity 
\(0.83\) 
\(1\) 

Spacer specific surface area 
\(10400\) 
\(m^{1}\) 

Pump efficiency 
\(0.8\) 
\(1\) 
– 