Numerical investigation of combined chemical and electrochemical processes in Fe₂O₃ suspension electrolysis

Abstract

Suspension electrolysis is a combined process of chemical and electrochemical reactions. The developed model for a parallel-plate electrochemical reactor is based on mixture model for suspension flow and balance equation for diluted species taking into account the dispersed phase content and ions migration due to the electrolyte current and partial dissolution of suspended particles in the suspension electrolysis. Electrochemical reactions are specified through flux boundary conditions at the electrode/electrolyte interface. The influence of the combined processes is reflected through the distribution of ions concentration profile in liquid phase and current density profile at the electrode surface. Numerical investigation indicates that about 90 % of the iron deposition flux is accommodated by an additional component flux due to the chemical reaction of partial dissolution of α-Fe₂O₃ particles in suspension electrolysis.

Appendices

Appendix 1: Charge balance at the electrode/electrolyte interface

The electric potential fields are governed by the charge conservation equations. The charge balance at interface between electron-conducting and ion conducting media is given by [27]

$$\frac{{\partial Q_{{}} }}{\partial \tau } + \nabla \cdot I_{\text{s}} = \left( {I_{1} - I_{2} } \right)$$

(A.1.1)

where $I_{1}$ is the current in electron-conducting media normal to the boundary; $I_{2}$ is the current in ion –conducting media normal to the boundary; $I_{\text{s}}$ is the superficial current density; and $Q$ is the charge. The interfaces between ionic and electronic media behave like a capacitor in which the charge density is a function of potential difference across the double layer. Charge or discharge rate at the electrode–electrolyte double layer can be defined as

$$\frac{\partial Q}{\partial \tau } = C_{\text{dl}} \frac{\partial \eta }{\partial t}$$

(A.1.2)

where $\eta$ is the potential differences, $\eta = \varphi - \varphi_{m}$. For the potential difference at the electrode/electrolyte interface, the following charge conservation equation is valid

$$C_{\text{dl}} \frac{\partial \eta }{\partial \tau } = - \nabla \cdot I_{\text{s}} + \left( {I_{1} - I_{2} } \right).$$

(A.1.3)

Appendix 2: Auxiliary equations for combined processes

The dissociation reaction of α-Fe₂O₃ in alkaline solution is given by Diakonov et al. [16]

$$0.5{\text{Fe}}_{2} {\text{O}}_{3} + 2.5{\text{H}}_{2} {\text{O}} = {\text{Fe}}({\text{OH}})_{4} + {\text{H}}^{ + }$$

(A.2.1)

The dissociation constant is defined as

$$K_{\text{s}} = a_{{{\text{Fe(OH)}}_{4}^{ - } }} a_{{{\text{H}}^{ + } }}$$

(A.2.2)

where $a_{{{\text{Fe(OH)}}_{4}^{ - } }}$ is the activity of $\rm{Fe(OH)}_{4}^{ - }$ ions and $a_{{{\text{H}}^{ + } }}$ is the activity of H⁺ ions. The equilibrium concentration of $\rm{Fe(OH)}_{4}^{ - }$ ions is a function of H⁺ ions concentration

$$a_{{{\text{Fe(OH)}}_{4}^{ - } }} = {{K_{\text{s}} } \mathord{\left/ {\vphantom {{K_{\text{s}} } {a_{{{\text{H}}^{ + } }} }}} \right. \kern-0pt} {a_{{{\text{H}}^{ + } }} }}.$$

(A.2.3)

Activity of species is associated with the molar concentration $a = 0.001\gamma c$, where $\gamma$ is the activity coefficient. The activity coefficient is calculated using NBS smoothed experimental data [28]. The ionic association in NaOH solution is given by

$$\rm{Na}^{ + } + \rm{OH}^{ - } \leftrightarrow \rm{NaOH}$$

(A.2.4)

The association constant is

$$K_{\text{A}} = \frac{{\left( {1 - \alpha } \right)c^{\text{NaOH}} }}{{\left( {\alpha c^{\text{NaOH}} \gamma^{{{\text{OH}}^{ - } }} } \right)^{2} }}$$

(A.2.5)

where c ^NaOH is the molar concentration of NaOH solution; $\alpha$ is degree of dissociation; and $\gamma^{{{\text{OH}}^{ - } }}$ is activity coefficient of OH⁻ ions. The constant of association for 50 % NaOH solution is taken from [29]. The degree of dissociation is found from solving (A.2.5) under the given electrolyte concentration. The inlet concentration of OH⁻ and Na⁺ ions is calculated using the degree of electrolyte dissociation

$$c_{\text{in}}^{{{\text{OH}}^{ - } }} = c^{\text{NaOH}} \cdot \alpha ,\;c_{\text{in}}^{{{\text{Na}}^{ + } }} = c_{\text{in}}^{{{\text{OH}}^{ - } }}.$$

(A.2.6)

The inlet concentration of $\rm{Fe(OH)}_{4}^{ - }$ ions is set equal to the equilibrium concentration $c_{\text{in}}^{{{\text{Fe(OH)}}_{4}^{ - } }} = 1000 \cdot a^{{{\text{Fe(OH)}}_{4}^{ - } }}$. The next Nernst equation is valid for equilibrium electrochemical reaction (4)

$$E = E^{0} + \frac{\text{RT}}{zF}\ln \left( {\frac{{a^{{{\text{Fe}}^{ + 3} }} }}{{a^{{{\text{Fe}}^{ + 2} }} }}} \right)$$

or

$$a^{{{\text{Fe}}^{ + 2} }} = a^{{{\text{Fe}}^{ + 3} }} \exp \left( { - \frac{zF}{\text{RT}}\left( {E - E^{0} } \right)} \right).$$

(A.2.7)

Fe(III) ions participate in hydrolysis reactions given by [11, 17]

$$\rm{Fe}^{ + 3} + \rm{H_{2} O} \overset {\it{K}_{\rm{1}} } \longleftrightarrow \rm{Fe(OH)}^{ + 2} + \rm{H}^{ + }$$

$$\rm{Fe}^{ + 3} + 2\rm{H_{2} O} \overset {\it{K}_{\rm{2}} } \longleftrightarrow \rm{Fe(OH)}_{2}^{ + } + 2\rm{H}^{ + }$$

$$\rm{Fe}^{ + 3} + 3\rm{H_{2} O} \overset {\it{K}_{\rm{3}} } \longleftrightarrow \rm{Fe(OH)}_{3} + 3\rm{H}^{ + }$$

$$\rm{Fe}^{ + 3} + 4\rm{H_{2} O} \overset {\it{K}_{\rm{4}} } \longleftrightarrow \rm{Fe(OH)}_{4}^{ - } + 4\rm{H}^{ + }$$

$$\rm{Fe(OH)}^{ + 2} + \rm{H_{2} O} \overset {\it{K}_{\rm{5}} } \longleftrightarrow \rm{Fe(OH)}_{2}^{ + } + \rm{H}^{ + }$$

$$2\rm{Fe}^{ + 3} + 2\rm{H_{2} O}\overset {\it{K}_{\rm{6}} } \longleftrightarrow \rm{Fe_{2} (OH)}_{2}^{ + 4} + 2\rm{H}^{ + }.$$

The hydrolysis constants are defined as follows:

$$K_{1} = \frac{{a^{{{\text{Fe}}^{ + 3} }} }}{{a^{{{\text{Fe(OH)}}^{ + 2} }} a^{{{\text{H}}^{ + } }} }},\;K_{2} = \frac{{a^{{{\text{Fe}}^{ + 3} }} }}{{a^{{{\text{Fe(OH)}}_{2}^{ + } }} (a^{{{\text{H}}^{ + } }} )^{2} }},\;K_{5} = \frac{{a^{{{\text{Fe(OH)}}^{ + 2} }} }}{{a^{{{\text{Fe(OH)}}_{2}^{ + } }} a^{{{\text{H}}^{ + } }} }}$$

(A.2.8)

$$K_{3} = \frac{{a^{{{\text{Fe}}^{ + 3} }} }}{{a^{{{\text{Fe(OH)}}_{3} }} \left( {a^{{{\text{H}}^{ + } }} } \right)^{3} }},\;K_{4} = \frac{{a^{{{\text{Fe}}^{ + 3} }} }}{{a^{{{\text{Fe(OH)}}_{4}^{ - } }} \left( {a^{{{\text{H}}^{ + } }} } \right)^{4} }}$$

(A.2.9)

$$K_{6} = \frac{{\left( {a^{{{\text{Fe}}^{ + 3} }} } \right)^{2} }}{{a^{{{\text{Fe}}_{ 2} ( {\text{OH)}}_{2}^{ + 4} }} \left( {a^{{{\text{H}}^{ + } }} } \right)^{2} }} .$$

(A.2.10)

Activity of Fe⁺³ ions can be expressed as follows:

$$a^{{{\text{Fe}}^{ + 3} }} = a^{{{\text{Fe(OH)}}_{4}^{ - } }} K_{4} \left( {a^{{{\text{H}}^{ + } }} } \right)^{4}.$$

(A.2.11)

Taking into account (6), (A.2.11), and (A.2.7), Butler–Volmer equation for the overall cathode electrochemical reaction can be written as

$$I^{\text{Fe}} = I_{0}^{\text{Fe}} \left( {\frac{{c^{{{\text{Fe(OH)}}_{4}^{ - } }} }}{{c_{\text{ref}}^{{{\text{Fe(OH)}}_{4}^{ - } }} }}} \right)^{2} \frac{{c^{{{\text{OH}}^{ - } }} }}{{c_{\text{ref}}^{{{\text{OH}}^{ - } }} }}\left( {\exp \left( {\frac{{\alpha_{\text{A}}^{\text{C}} F}}{\text{RT}}\left( {\eta^{\text{C}} - \eta_{\text{eq}}^{\text{C}} } \right)} \right) - \exp \left( { - \frac{{\alpha_{\text{C}}^{\text{C}} F}}{\text{RT}}\left( {\eta^{\text{C}} - \eta_{\text{eq}}^{\text{C}} } \right)} \right)} \right)$$

(A.2.12)

where $I_{ 0}^{\text{C}}$ is the cathode exchange current density; $c_{\text{ref}}^{{{\text{Fe(OH)}}_{4}^{ - } }}$ is the reference concentration of $\rm{Fe(OH)}_{4}^{ - }$ ions; and $\eta^{\text{C}}$ is the potential difference at the cathode electrode/electrolyte interface.

Hydrogen dissolved in liquid phase is in equilibrium with hydrogen ions at the cathode electrode surface. The equilibrium concentration of hydrogen ions is calculated using Nernst equation defined for equilibrium electrochemical reaction of hydrogen oxidation

$$c_{\text{eq}}^{{{\text{H}}^{ + } }} = {{1000\;a^{{{\text{H}}^{ + } }} } \mathord{\left/ {\vphantom {{1000\;a^{{{\text{H}}^{ + } }} } {\gamma^{{{\text{H}}^{ + } }} }}} \right. \kern-0pt} {\gamma^{{{\text{H}}^{ + } }} }},\;E = \frac{\text{RT}}{F}\ln \left( {\frac{{a^{{{\text{H}}^{ + } }} }}{{a_{{{\text{H}}_{2} }}^{0.5} }}} \right).$$

(A.2.13)

Gas content in equilibrium gas–liquid mixture can be calculated from equilibrium flash equation

$$\sum\limits_{k = 1}^{n} {\frac{{\left( {K^{(k)} - 1} \right)Z^{(k)} }}{{\left( {K^{(k)} - 1} \right)\left( {1 - \gamma_{\text{G}} } \right) + 1}} = 0}$$

(A.2.14)

where $\gamma_{\rm{G}}$ is the local splitting factor; $K^{(k)}$ is the distribution of the each components between the vapor and liquid phases, $K^{(k)} = {{y^{(k)} } \mathord{\left/ {\vphantom {{y^{(k)} } {x^{(k)} }}} \right. \kern-0pt} {x^{(k)} }}$; and $Z^{(k)}$ is the mixture concentration. The linkage between molar splitting factor and gas volume fraction is given by

$$\alpha_{\text{G}} = \frac{{\gamma_{\text{G}} \rho_{\text{L,mol}} }}{{\gamma_{\text{G}} \rho_{\text{L,mol}} + \gamma_{\text{L}} \rho_{\text{G,mol}} }}.$$

(A.2.15)

Interfacial area of solid particles in suspension is

$$a_{\text{v}} = \frac{6}{{d_{\text{p}} }}\varepsilon_{\text{d}}.$$

(A.2.16)

For suspended particles in liquid flow, mass transfer coefficient in liquid phase is calculated from an empirical correlation [30, 31]

$${\text{Sh}} = C \cdot ({\text{Sc}} \cdot {\text{Ar}})^{1/3}$$

(A.2.17)

where C is the constant; Sh is the Sherwood number; Sc is the Schmidt number; and Ar is the Archimedes number. The total component balance is written for parallel-plate reactor as follows

$$M_{\text{deposition}}^{\text{Fe}} = M_{\text{transfer}}^{\text{Fe}} + M_{\text{dissolving}}^{\text{Fe}}.$$

(A.2.18)

Component flow due to the diffusion flow of $\rm{Fe(OH)}_{4}^{ - }$ ions at the cathode electrode is

$$M_{\text{transfer}}^{\text{Fe}} = \int\limits_{S} {\left( { - D_{{{\text{Fe(OH)}}_{4}^{ - } }} \nabla c^{{{\text{Fe(OH)}}_{4}^{ - } }} } \right){\text{d}}S}.$$

(A.2.19)

Component flow due to the iron deposition at the cathode electrode is

$$M_{\text{deposition}}^{\text{Fe}} = \frac{{\nu^{{ ( {\text{Fe)}}}} }}{{n_{e}^{{ ( {\text{Fe)}}}} F}}\left| {I_{\text{avg}}^{\text{C}} } \right| \cdot S_{\text{cell}}.$$

(A.2.20)

The component flow due to dissolving solid particles in suspension electrolysis is

$$M_{\rm{dissolving}}^{\rm{Fe}} = \int\limits_{V} {\beta_{\rm{f,L}} a_{\rm{v}} \left(c_{\rm{eq}}^{{\rm{Fe(OH)}_{4}^{ - } }} - c^{{\rm{Fe(OH)}_{4}^{ - } }} \right)\rm{d}{\it V}}.$$

(A.2.21)

The main conclusion is that the partial dissolution of solid particles compensates the component flow of iron deposition at the cathode electrode

$$S_{\rm{transfer}} + S_{\rm{dissolving}} = 1,$$

(A.2.22)

where S _transfer is the contribution of diffusion process, $S_{\text{transfer}} = {{\left| {M_{\text{transfer}}^{\text{Fe}} } \right|} \mathord{\left/ {\vphantom {{\left| {M_{\text{transfer}}^{\text{Fe}} } \right|} {\left| {M_{\text{deposition}}^{\text{Fe}} } \right|}}} \right. \kern-0pt} {\left| {M_{\text{deposition}}^{\text{Fe}} } \right|}}$ and S _dissolving is the contribution of the dissolution process in combined chemical and electrochemical processes, $S_{\text{dissolving}} = {{\left| {M_{\text{dissolving}}^{\text{Fe}} } \right|} \mathord{\left/ {\vphantom {{\left| {M_{\text{dissolving}}^{\text{Fe}} } \right|} {\left| {M_{\text{deposition}}^{\text{Fe}} } \right|}}} \right. \kern-0pt} {\left| {M_{\text{deposition}}^{\text{Fe}} } \right|}}$.