Electrocoagulation (0D)

The main assumptions of the implemented model are (partially adopted from Dubrawski, et al. 2014):

Steady-state and plug flow
Model dimensionality is limited to a 0D control volume
Single liquid phase and solvent (water) only
The system is insulated and adiabatic
No passivation on electrode surfaces
Negligible internal circuit resistance
Stoichiometric electrochemical reactions occur at cathode and anode
Parallel plate electrodes
Each electrode is the same material and same size

Introduction

Electrocoagulation (EC) is a water treatment process that uses electrical current to destabilize and aggregate suspended particles in water. The process involves the generation of coagulant species via electrochemical reactions at the electrodes. As the cathode is oxidized, metal ions are released in to the water matrix and form hydroxide species, which then interact with and enmesh suspended particles. After formation and agglomeration, the flocculated material is settled (or floated) out of the treated water stream.

EC is an electrochemically complex process. The performance and technoeconomics is influenced by many factors including the composition of the water matrix, the applied current density, the electrode material, and other aspects of the reactor design. Because energy consumption can be a significant component of the overall cost of the process, this model presents three different approaches to estimate the overpotentials associated with the electrochemical reactions.

Model Configurations

The EC model includes different configuration options for the electrode material, reactor material, and the overpotential calculation:

Electrode material: aluminum (default) and iron.
Overpotential calculation: fixed (default), regression approximation, and a detailed calculation.
Reactor material: carbon steel (default), stainless steel, and PVC.

Selecting either electrode material will properly set the WaterTAP model parameters for electrode density (density_electrode_material), molecular weight (mw_electrode_material), charge transfer number (charge_transfer_number), and the stoichiometric coefficient of the electrochemical reaction (stoich_coeff). Any of these parameters can be changed by the user after the model build.

If the user uses the defaut overpotential calculation, the overpotential variable (overpotential) is a degree of freedom and must be fixed.

If the user selects the regression approximation for the overpotential calculation, the model will set default values for the overpotential regression coefficients (overpotential_k1 and overpotential_k2).

If the user selects the detailed calculation for the overpotential calculation, the model will also set default values for the anode cell potential (anode_cell_potential_std), the anode entropy change (anode_entropy_change_std), the anodic exchange current density (anodic_exchange_current_density), and the cathodic exchange current density (cathodic_exchange_current_density) relevant to each electrode material.

Overpotential Calculation

The overpotential is the additional voltage required over the ohmic potential to drive the electrochemical reaction. The electrocoagulation model determines the total cell voltage required to drive the electrochemical reactions according to:

\[E_{cell} = E_{ohmic} + E_{over}\]

Where \(E_{cell}\) is the total cell voltage, \(E_{ohmic}\) is the ohmic potential required, and \(E_{over}\) is the overpotential.

The WaterTAP electrocoagulation model provides three options for calculating the overpotential, outlined as follows.

Fixed

If this overpotential calculation is selected, the user must provide a fixed value for the overpotential variable (overpotential) in volts. This value is used directly in the calculation of the total cell voltage.

Regression Approximation

This overpotential calculation uses a regression adapted from Eq. 18 in Gu et al. (2009) to determine the overpotential:

\[E_{over} = k_1 \text{ln}\left(i \right) + k_2\]

Where \(E_{over}\) is the overpotential, \(i\) is the current density (mA/cm²), \(k_1\) is a regression coefficient (mV), and \(k_2\) is a regression coefficient (mV). The values for the regression coefficients have default values, but users should adjust them based on experimental data for their specific system.

Detailed Calculation

If the detailed calculation is selected, the model will use the Nernst equation to calculate the overpotential based on the standard cell potential, the entropy change, and the exchange current densities for the anodic and cathodic reactions, and will also use the Tafel slope parameter to estimate the activation overpotential.

In general, the overpotential is calculated as follows:

\[E_{over} = |E_c - E_a| + \varphi_a + |\varphi_c| + \psi_a + |\psi_c|\]

Where \(E_c\) is the non-equilibrium electrode potential at the cathode, \(E_a\) is the non-equilibrium electrode potential at the anode, \(\varphi_a\) is the anodic activation overpotential, \(\varphi_c\) is the cathodic activation overpotential, \(\psi_a\) is the anodic concentration overpotential, and \(\psi_c\) is the cathodic concentration overpotential. The electrocoagulation model assumes the concentration overpotential is negligible (i.e., that the electrochemical reactions are not mass transfer limited) and \(\psi_c = \psi_a = 0\).

The non-equilibrium electrode potentials at the cathode and anode are calculated via the Nernst equation:

\[E_a = E_{a}^0 + \frac{\Delta S_a (T - T_0)}{z_a F} - \frac{RT}{z_a F} \text{ln}\left( C_{i}^{-\nu} \right)\]

\[E_c = E_{c}^0 + \frac{\Delta S_c (T - T_0)}{z_c F} + \frac{RT}{z_cF} \text{ln}\left( p_{H_2} \left( C_{OH}\right)^2 \right)\]

Where \(E_{i}^0\) is the standard cell potential, \(R\) is the universal gas constant (8.314 J/(mol K)), \(T\) is the temperature (K), \(z_i\) is the number of electrons transferred in the electrochemical reaction, \(F\) is the Faraday constant (96,485 C/mol), \(\Delta S_i\) is the entropy change for the reaction (J/(mol K)), \(C_{i}\) is the concentration of the reactant species (mol/L), \(C_{OH}\) is the hydroxide concentration (mol/L), and \(p_{H_2}\) is the partial pressure of hydrogen gas (atm).

The anodic and cathodic activation overpotentials are calculated using the Tafel equation:

\[\varphi_a = b_a \text{ln}\left( \frac{i}{i_{a0}} \right)\]

\[\varphi_c = b_c \text{ln}\left( \frac{i}{i_{c0}} \right)\]

Where \(i_{a0}\) and \(i_{c0}\) are the anodic and cathodic exchange current densities (A/m²), \(b_a\) and \(b_c\) are the anodic and cathodic Tafel slope parameters (V), and \(i\) is the current density (A/m²).

Ports

The model provides three ports (Pyomo notation in parenthesis):

Inlet port (inlet)
Outlet port (outlet)
Byproduct port (byproduct)

Sets

The table below outlines example Sets that could be used with the electrocoagulation model. Any component can be included as long as it is properly configured into the property package.

Description	Symbol	Example Indices
Time	\(t\)	`[0]`
Phases	\(p\)	`['Liq']`
Components	\(j\)	`['H2O', 'Cation_+', 'Anion_-', 'Inert']`

Model Components

The electrocoagulation model includes variables and expressions that are common to all configurations. These are provided in the table below.

Description	Symbol	Variable Name	Index	Units
Variables
Inlet temperature	\(T\)	`temperature`	`[t]`	\(\text{K}\)
Inlet pressure	\(p\)	`pressure`	`[t]`	\(\text{Pa}\)
Component mass flow rate	\(M_j\)	`flow_mass_phase_comp`	`[t, p, j]`	\(\text{kg s}^{-1}\)
Phase volumetric flow rate	\(q_j\)	`flow_vol_phase`	`[t, p]`	\(\text{m}^{3} \text{ s}^{-1}\)
Coagulant dose	\(D_c\)	`coagulant_dose`	None	\(\text{g L}^{-1}\)
Electrode thickness	\(d_{electrode}\)	`electrode_thickness`	None	\(\text{m}\)
Electrode mass	\(m_{electrode}\)	`electrode_mass`	None	\(\text{kg}\)
Electrode volume	\(V_{electrode}\)	`electrode_volume`	None	\(\text{m}^3\)
Electrode gap	\(d_{gap}\)	`electrode_gap`	None	\(\text{m}\)
Electrolysis time	\(t_{elec}\)	`electrolysis_time`	None	\(\text{min}\)
Current density	\(i\)	`current_density`	None	\(\text{A m}^{-2}\)
Applied current	\(I\)	`applied_current`	None	\(\text{A}\)
Ohmic resistance	\(R_{ohmic}\)	`ohmic_resistance`	None	\(\Omega \text{ m}^{2}\)
Charge loading rate	\(\text{CLR}\)	`charge_loading_rate`	None	\(\text{C L}^{-1}\)
Current efficiency	\(\eta\)	`current_efficiency`	None	\(\text{dimensionless}\)
Overpotential	\(E_{over}\)	`overpotential`	None	\(\text{V}\)
Cell voltage	\(E_{cell}\)	`cell_voltage`	None	\(\text{V}\)
Anode area	\(A_{anode}\)	`anode_are`	None	\(\text{m}^2\)
Cathode area	\(A_{cathode}\)	`cathode_area`	None	\(\text{m}^2\)
Volume of electrocoagulation reactor	\(V_{r}\)	`cell_volume`	None	\(\text{m}^3\)
Total floc basin volume (flotation + sedimentation)	\(V_{floc}\)	`floc_basin_vol`	None	\(\text{m}^3\)
Floc basin retention time	\(t_{floc}\)	`floc_retention_time`	None	\(\text{min}\)

Expressions
Conductivity	\(\kappa\)	`conductivity`	None	\(\text{S m}^{-1}\)
Theoretical coagulant dose	\(D_{c,t}\)	`theoretical_coagulant_dose`	None	\(\text{kg}\)
Ohmic potential	\(E_{ohmic}\)	`ohmic_potential`	None	\(\text{V}\)
Electrode area total	\(A_{electrode}\)	`electrode_area_total`	None	\(\text{m}^2\)
Total power required	\(P_{tot}\)	`power_required`	None	\(\text{W}\)
Power density Faradaic	\(p_{F}\)	`power_density_faradaic`	None	\(\mu\text{W m}^{-2}\)
Power density total	\(p_{total}\)	`power_density_total`	None	\(\mu\text{W m}^{-2}\)

Common parameters and their initial values are listed below.

Description	Symbol	Variable Name	Index	Units	Default Value
Parameters
Component removal efficiency on mass basis	\(\eta_{j}\)	`removal_frac_mass_comp`	`[j]`	\(\text{dimensionless}\)	0.7
Water recovery on mass basis	\(\eta_{w}\)	`recovery_frac_mass_H2O`	None	\(\text{dimensionless}\)	0.99
Conversion factor for mg/L TDS to S/m	\(x\)	`tds_to_cond_conversion`	None	\(\text{mg m }\text{L}^{-1}\text{ S}^{-1}\)	5e3
Standard temperature	\(T_0\)	`standard_temperature`	None	\(\text{K}\)	298.15
Electrode molecular weight	\(MW\)	`mw_electrode_material`	None	\(\text{kg mol}^{-1}\)	different for electrode material; see table below
Stoichiometric coefficient for electrode material	\(\nu\)	`stoich_coeff`	None	\(\text{dimensionless}\)	different for electrode material; see table below
Charge transfer number	\(z\)	`charge_transfer_number`	None	\(\text{dimensionless}\)	different for electrode material; see table below
Electrode density	\(\rho_{electrode}\)	`density_electrode_material`	None	\(\text{kg m}^{-3}\)	different for electrode material; see table below
Fractional increase in water temperature from inlet to outlet	\(x_T\)	`frac_increase_temperature`	None	\(\text{dimensionless}\)	1.05

If overpotential_calculation is set to regression, the following variables are also created:

Description	Symbol	Variable Name	Index	Units
Variables
Overpotential regression coefficient 1	\(k_1\)	`overpotential_k1`	None	\(\text{mV}\)
Overpotential regression coefficient 2	\(k_2\)	`overpotential_k2`	None	\(\text{mV}\)

If overpotential_calculation is set to detailed, the following variables, parameters, and expressions are also created. Note that many of these parameters are dependent on the electrode material selected.

Description	Symbol	Variable Name	Index	Units	Default Value
Variables
Anodic Tafel slope	\(b_a\)	`tafel_slope_anode`	None	\(\text{V}\)	0.0403
Cathodic Tafel slope	\(b_c\)	`tafel_slope_cathode`	None	\(\text{V}\)	0.0633

Parameters
Cathode surface pH	\(\text{pH}\)	`cathode_surface_pH`	None	\(\text{dimensionless}\)	11
Partial pressure of hydrogen gas	\(P_{H2}\)	`partial_pressure_H2`	None	\(\text{atm}\)	1
Cathodic non-equilibrium cell potential, standard @ 25C	\(E_{c}^0\)	`cathode_cell_potential_std`	None	\(\text{V}\)	-0.83
Cathodic entropy change	\(\frac{\Delta S_c}{z_cF}\)	`cathode_entropy_change_std`	None	\(\text{V K}^{-1}\)	-0.000836
Anodic non-equilibrium cell potential, standard @ 25C	\(E_{a}^0\)	`anode_cell_potential_std`	None	\(\text{V}\)	different for electrode material; see table below
Anodic entropy change	\(\frac{\Delta S_a}{z_aF}\)	`anode_entropy_change_std`	None	\(\text{V K}^{-1}\)	different for electrode material; see table below
Anodic exchange current density	\(i_{a0}\)	`anodic_exchange_current_density`	None	\(\text{A m}^{-2}\)	different for electrode material; see table below
Cathodic exchange current density	\(i_{a0}\)	`cathodic_exchange_current_density`	None	\(\text{A m}^{-2}\)	different for electrode material; see table below

Expressions
Hydroxide concentration at cathode surface	\(C_{OH}\)	`cathode_conc_mol_hydroxide`	None	\(\text{mol L}^{-1}\)
Change in effluent temperature relative to standard	\(\Delta T\)	`temp_diff_std`	None	\(\text{K}\)
Anode equilibrium potential adjusted for outlet temperature	\(E_a^{adj}\)	`anode_cell_potential_temp_adj`	None	\(\text{V}\)
Anode cell potential via Nernst equation	\(E_a\)	`anode_cell_potential`	None	\(\text{V}\)
Cathodic cell potential via Nernst equation	\(E_c\)	`cathode_cell_potential`	None	\(\text{V}\)
Anodic activation overpotential	\(\varphi_a\)	`anode_overpotential`	None	\(\text{V}\)
Cathodic activation overpotential	\(\varphi_c\)	`cathode_overpotential`	None	\(\text{V}\)
Cathode equilibrium potential adjusted for outlet temperature	\(E_c^{adj}\)	`cathode_cell_potential_temp_adj`	None	\(\text{V}\)

For aluminum electrodes, these are the default values used in the model.

Description	Symbol	Variable Name	Units	Default Value
Molecular weight of electrode material	\(MW\)	`mw_electrode_material`	\(\text{kg mol}^{-1}\)	26.98e-3
Charge transfer number of electrode material	\(z\)	`charge_transfer_number`	\(\text{dimensionless}\)	3
Stoichiometric coefficient for electrode material	\(\nu\)	`stoic_coeff`	\(\text{dimensionless}\)	1
Density of electrode material	\(\rho\)	`density_electrode_material`	\(\text{kg m}^{-3}\)	2710
Anodic non-equilibrium cell potential, standard @ 25C	\(E_{a}^0\)	`anode_cell_potential_std`	\(\text{V}\)	-1.66
Anodic entropy change	\(\frac{\Delta S_a}{z_aF}\)	`anode_entropy_change_std`	\(\text{V K}^{-1}\)	5.33e-4
Anodic exchange current density	\(i_{a0}\)	`anodic_exchange_current_density`	\(\text{A m}^{-2}\)	2.602e-5
Cathodic exchange current density	\(i_{a0}\)	`cathodic_exchange_current_density`	\(\text{A m}^{-2}\)	1e-4

For iron electrodes, these are the default values used in the model.

Description	Symbol	Variable Name	Units	Default Value
Molecular weight of electrode material	\(MW\)	`mw_electrode_material`	\(\text{kg mol}^{-1}\)	55.845e-3
Charge transfer number of electrode material	\(z\)	`charge_transfer_number`	\(\text{dimensionless}\)	1
Stoichiometric coefficient for electrode material	\(\nu\)	`stoic_coeff`	\(\text{dimensionless}\)	1
Density of electrode material	\(\rho\)	`density_electrode_material`	\(\text{kg m}^{-3}\)	7860
Anodic non-equilibrium cell potential, standard @ 25C	\(E_{a}^0\)	`anode_cell_potential_std`	\(\text{V}\)	-0.41
Anodic entropy change	\(\frac{\Delta S_a}{z_aF}\)	`anode_entropy_change_std`	\(\text{V K}^{-1}\)	7e-5
Anodic exchange current density	\(i_{a0}\)	`anodic_exchange_current_density`	\(\text{A m}^{-2}\)	2.5e-4
Cathodic exchange current density	\(i_{a0}\)	`cathodic_exchange_current_density`	\(\text{A m}^{-2}\)	1e-3

Degrees of Freedom

Aside from the inlet feed state variables (temperature, pressure, component molar flowrate), the user must specify 8-9 degrees of freedom to fully specify the model, depending on the configuration.

The following degrees of freedom should be specified regardless of the configuration:

electrode_thickness
electrode_gap
electrolysis_time
floc_retention_time

The following degrees of freedom are fixed dependent on the configuration:

overpotential (if overpotential_calculation is set to fixed)
overpotential_k1 and overpotential_k2 (if overpotential_calculation is set to regression)
tafel_slope_anode and tafel_slope_cathode (if overpotential_calculation is set to detailed)

Then, the user can select combinations of three of the following variables to have a fully specified model. The specific combination would be dependent on what the user knows about the system and their modeling objectives.

current_density
applied_current
current_efficiency
cell_voltage
coagulant_dose
charge_loading_rate
anode_area or cathode_area

Solution Component Information

The electrocoagulation model is designed to work with WaterTAP’s multi-component aqueous solution (MCAS) property package. The inlet solute list must contain TDS because the model uses the TDS concentration to calculate the conductivity of the solution. Because the removal efficiency is defined on a mass basis, MCAS must be configured to use mass as the material flow basis.

An example configuration is provided below:

ec_feed = {
    "solute_list": ["TDS", "Ca_2+", "Mg_2+"],
    "mw_data": {
        "TDS": 58.44e-3,
        "Ca_2+": 40.08e-3,
        "Mg_2+": 24.31e-3,
    },
    "material_flow_basis": MaterialFlowBasis.mass,
}

m = ConcreteModel()
m.fs = FlowsheetBlock(dynamic=False)
m.fs.properties = MCASParameterBlock(**ec_feed)
m.fs.unit = Electrocoagulation(
    property_package=m.fs.properties,
    electrode_material="iron",
    overpotential_calculation="detailed",
)

Equations and Relationships

Description	Equation
Common
Conductivity	\(\kappa = C_{TDS} / x\)
Total electrode area	\(A_{electrode} = A_{anode} + A_{cathode}\)
Power required	\(P_{tot} = E_{cell} I\)
Power density Faradaic	\(p_{F} = \frac{E_{over}I}{A_{anode}}\)
Power density total	\(p_{tot} = \frac{P_{tot}}{A_{anode}}\)
Effluent temperature	\(T_{out} = x_T T_{in}\)
Water recovery	\(M_{H_2O, out} = M_{H_2O, in} \eta_w\)
Water mass balance	\(M_{H_2O, out} = M_{H_2O, in} - M_{H_2O, byprod}\)
Component mass balance	\(M_{j, out} = M_{j, in} - M_{j, byprod}\)
Component removal efficiency	\(M_{j, byprod} = \eta_j M_{j, in}\)
Charge loading rate	\(\text{CLR} = \frac{I}{q_{liq}}\)
Floc reactor volume	\(V_{floc} = q_{liq} t_{floc}\)
Faraday’s Law	\(D_c = \frac{I \eta MW}{q_{liq} z F}\)
Theoretical coagulant dose	\(D_{c,t} = \frac{I MW}{q_{liq} z F}\)
Anode area required	\(A_{anode} = \frac{I}{i}\)
Cathode area required	\(A_{cathode} = A_{anode}\)
Ohmic resistance	\(R_{ohmic} = \frac{d_{gap}}{\kappa}\)
Ohmic potential	\(E_{ohmic} = \frac{I R_{ohmic}}{A_{anode}}\)
Cell voltage required	\(E_{cell} = E_{over} + E_{ohmic}\)
Electrode volume	\(V_{electrode} = \left( A_{anode} + A_{cathode} \right) d_{electrode}\)
Electrode mass	\(m_{electrode} = V_{electrode} \rho_{electrode}\)
Reactor volume	\(V_{cell} = q_{liq} t_{elec}\)

Regression
Overpotential regression	\(E_{over} = k_1 \text{ln}(i) + k_2\)

Detailed
Hydroxide concentration at cathode surface	\(C_{OH} = 10^{14 - \text{pH}}\)
Anode equilibrium potential adjusted for outlet temperature	\(E_a^{adj} = E_{a}^0 + \frac{\Delta S_a (T - T_0)}{z_a F}\)
Anodic cell potential	\(E_a = E_a^{adj} - \frac{RT}{z_a F} \text{ln}\left( C_{i}^{-\nu} \right)\)
Cathode equilibrium potential adjusted for outlet temperature	\(E_c^{adj} = E_{c}^0 + \frac{\Delta S_c (T - T_0)}{z_c F}\)
Cathodic cell potential	\(E_c = E_c^{adj} + \frac{RT}{z_cF} \text{ln}(p_{H_2} (C_{OH})^2)\)
Anodic activation overpotential	\(\varphi_a = b_a \text{ln}(i / i_{a0})\)
Cathodic activation overpotential	\(\varphi_c = b_c \text{ln}(i / i_{c0})\)
Overpotential	\(E_{over} = \|E_c - E_a\| + \varphi_a + \|\varphi_c\|\)

References

K. L. Dubrawski, C. Du and M. Mohseni (2014)
General Potential-Current Model and Validation for Electrocoagulation
Electrochimica Acta 2014 Vol. 129 Pages 187-195
DOI: 10.1016/j.electacta.2014.02.089

Z. Gu, Z. Liao, M. Schulz, J. R. Davis, J. C. Baygents and J. Farrell (2009)
Estimating Dosing Rates and Energy Consumption for Electrocoagulation Using Iron and Aluminum Electrodes
Industrial & Engineering Chemistry Research 2009 Vol. 48 Issue 6 Pages 3112-3117
DOI: 10.1021/ie801086c

Bratsch, S. G. (1989).
Standard Electrode Potentials and Temperature Coefficients in Water at 298.15 K.
Journal of Physical and Chemical Reference Data, 18(1), 1-21.
DOI: 10.1063/1.555839

Zhang, F., Yang, C., Zhu, H., Li, Y., & Gui, W. (2020).
An integrated prediction model of heavy metal ion concentration for iron electrocoagulation process.
Chemical Engineering Journal, 391, 123628.
DOI: 10.1016/j.cej.2019.123628

R. Holze (2007)
M.D. Lechner (ed.)
Electrochemical Thermodynamics and Kinetics
Table 5.1. Exchange current densities and rate constants in aqueous systems
Landolt-Börnstein - Group IV Physical Chemistry 9A
DOI: 10.1007/978-3-540-45316-1