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Fluid flow effects on diffusion layer and current density for electrochemical systems

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Abstract

The effects of flow field upon the distribution of ionic concentration, electric potential, concentration boundary layer thickness, and electric current density were investigated. A modified numerical scheme is proposed to simulate the corresponding electrochemical system which is governed by nonlinear partial differential equations. Seven types of geometries and various flow fields with Reynolds numbers up to 2100 are considered. The obtained results indicate the current numerical method can successfully simulate the increase of current density on the cathode as the applied potential cell increases, and that rise will continue until the limiting current density is reached. To predict the effect of fluid flow, the proposed scheme is applied for various Peclet numbers. The increase of current density for Peclet numbers between 1 and 104 is quite evident. But for large Peclet numbers between 104 and 107, the current density increases gradually. The results also show that as the anode size is doubled, the maximum current density occurs at the leading and trailing edges. However, if the cathode size is doubled, the maximum current density occurs at the center regions of it. Knowing the regions where current density is extremum helps electochemical system designers to control the parameters of the corresponding process.

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Abbreviations

c m :

concentration of species m [mol/m3]

c ref :

characteristic concentration, cref=1000 [mol/m3]

c*m :

dimensionless concentration of species m \({\rm{c}}_m^* = {{{{\rm{c}}_m}} \over {{{\rm{c}}_{ref}}}}\)

c ∞ m :

bulk concentration of species m [mol/m3]

c *∞ m :

dimensionless bulk concentration of species m, \({\rm{c}}_m^{*\infty } = {{{\rm{c}}_m^\infty } \over {{{\rm{c}}_{ref}}}}\)

\({\rm{c}}_{{1_k}}^*\) :

dimensionless concentration of species 1 at node k

\({\rm{c}}_{{2_k}}^*\) :

dimensionless concentration of species 2 at node k

D m :

diffusion coefficient of species m [m2/s]

D ref :

characteristic diffusion coefficient, Dref=7.2E-10 [m2/s]

D*m :

dimensionless diffusion coefficient of species m, \({\rm{D}}_m^* = {{{{\rm{D}}_m}} \over {{{\rm{D}}_{ref}}}}\)

E ∞ r :

equilibrium open circuit potential of reaction r [volt]

E ∞ r :

dimensionless equilibrium open circuit potential of reaction \({\rm{r,}}\,{\rm{E}}_r^{*\infty } = {{{\rm{E}}_r^\infty } \over {{\Phi _{ref}}}}\)

F:

faraday’s constant, F=96485.34 [C/mol]

h:

height of channel [m]

h i :

step of computational mesh in x-direction at node (i, j)

i :

total electric current density [A/m2]

i* :

dimensionless of total current density, \({{\bf{i}}^*} = {{{\rm{i}}{{\rm{L}}_{ref}}} \over {{\rm{F}}{{\rm{D}}_{ref}}{{\rm{c}}_{ref}}}}\)

\({{\bf{i}}_{{0_r}}}\) :

exchange current density of reaction r [A/m2]

\({{\bf{i}}_{{n_r}}}\) :

normal current density of reaction r [A/m2]

\({\bf{i}}_{{n_r}}^*\) :

dimensionless normal current density of reaction \({\rm{r,}}\,{\rm{i}}_{{n_r}}^* = {{{{\rm{i}}_{{n_r}}}{{\rm{L}}_{ref}}} \over {{\rm{F}}{{\rm{D}}_{ref}}{{\rm{c}}_{ref}}}}\)

k j :

step of computational mesh in y-direction at node (i, j)

Lref :

characteristic length, Lref=h

v*:

dimensionless vertical component of velocity, \({{\rm{v}}^*} = {{\rm{v}} \over {{{\rm{U}}_{ref}}}}\)

V*E :

dimensionless electrode surface potential, \({\rm{V}}_E^* = {{{{\rm{V}}_E}} \over {{\Phi _{ref}}}}\)

w:

width of channel, w=0.1 [m]

x :

horizontal direction [m]

x*:

dimensionless of horizontal direction, \({{\rm{x}}^*} = {{\rm{x}} \over {{{\rm{L}}_{ref}}}}\)

x*electrode :

dimensionless of electrode length

y :

vertical direction [m]

y*:

dimensionless of vertical direction, \({{\rm{y}}^*} = {{\rm{y}} \over {{{\rm{L}}_{ref}}}}\)

z m :

charge number of species m

μ m :

absolute mobility of species m

Φ :

electrolyte potential [volt]

Φ*:

dimensionless of electrolyte potential, \({\Phi ^*} = {\Phi \over {{\Phi _{ref}}}}\)

Φk*:

dimensionless of electrolyte potential at node k

N m :

flux vector of species m [mol/m2·s]

Nm*:

dimensionless flux vector of species m

n*:

dimensionless direction normal to the boundary

nr :

number of electrons transferred in reaction r

Pe:

Peclet number, \({\rm{Pe = }}{{{{\rm{U}}_{ref}}{{\rm{L}}_{ref}}} \over {{{\rm{D}}_{ref}}}}\)

R:

universal gas constant, R=8.314462 [J/k·mol]

Re:

Reynolds number, \({\mathop{\rm Re}\nolimits} = {{4{{\rm{u}}_{ave}}{\rm{w}}{{\rm{L}}_{ref}}} \over {2v\left( {{\rm{w + }}{{\rm{L}}_{ref}}} \right)}}\)

Sc:

Schmidt number, \({\rm{Sc = }}{v \over {{{\rm{D}}_{ref}}}}\)

Sr,m :

stoichiometric coefficient of species m in reaction r

T:

electrolyte temperature [k]

Tconv :

convection term in governing equation

uave :

average of inlet velocity [m/s]

U ref :

characteristic velocity, Uref=6uave

u*:

dimensionless horizontal component of velocity, \(\left( {{{\rm{u}}^*} = {{\rm{u}} \over {{{\rm{U}}_{ref}}}}} \right)\)

uE :

east horizontal velocity, \({{\rm{u}}_E} = {\rm{u}}_{i + {1 \over 2},j}^* = {{{\rm{u}}_{i,j}^* + {\rm{u}}_{i + 1,j}^*} \over 2}\)

uW :

west horizontal velocity, \({{\rm{u}}_W} = {\rm{u}}_{i - {1 \over 2},j}^* = {{{\rm{u}}_{i,j}^* + {\rm{u}}_{i - 1,j}^*} \over 2}\)

V :

vector of velocity [m/s]

V* :

dimensionless velocity, \({{\rm{V}}^*} = {{\rm{V}} \over {{{\rm{U}}_{ref}}}}\)

vN :

north vertical velocity, \({{\rm{v}}_N}{\rm{ = v}}_{i,j + {1 \over 2}}^* = {{{\rm{v}}_{i,j}^* + {\rm{v}}_{i,j + 1}^*} \over 2}\)

vs :

south vertical velocity, \({{\rm{v}}_S}{\rm{ = v}}_{i,j - {1 \over 2}}^* = {{{\rm{v}}_{i,j}^* + {\rm{v}}_{i,j - 1}^*} \over 2}\)

VA :

anode potential [volt]

VC :

cathode potential [volt]

Φ ref :

characteristic potential, \({\Phi _{ref}} = {{{\rm{RT}}} \over {\rm{F}}}\left[ {{\rm{Volt}}} \right]\)

∇:

del operator

∇*:

dimensionless of del operator, ∇* = Lref · ∇

ν :

kinematic viscosity [m2/s]

\({\eta _{{s_r}}}\) :

surface overpotential of reaction r [volt]

ρ 0 :

electrolyte density [kg/m3]

\({\alpha _{{a_r}}}\) :

anodic transfer coefficient of reaction r

\({\alpha _{{c_r}}}\) :

cathodic transfer coefficient of reaction r

βm,r :

constant order of species m in reaction r

μ(m)E :

east artificial viscosity of species m, \(\mu _E^{\left( m \right)} = \max \left\{ {{\rm{D}}_m^*,{\rm{Pe}}{{\rm{h}}_{i - 1}}{{\left| {{{\rm{u}}_E}} \right|} \over 2}} \right\}\)

μ(m)W :

west artificial viscosity of species m, \(\mu _W^{\left( m \right)} = \max \left\{ {{\rm{D}}_m^*,{\rm{Pe}}{{\rm{h}}_i}{{\left| {{{\rm{u}}_W}} \right|} \over 2}} \right\}\)

μ(m)N :

north artificial viscosity of species m, \(\mu _N^{\left( m \right)} = \max \left\{ {{\rm{D}}_m^*,{\rm{Pe}}{{\rm{k}}_{j - 1}}{{\left| {{{\rm{v}}_N}} \right|} \over 2}} \right\}\)

μ(m)S :

south artificial viscosity of species m, \(\mu _S^{\left( m \right)} = \max \left\{ {{\rm{D}}_m^*,{\rm{Pe}}{{\rm{k}}_j}{{\left| {{{\rm{v}}_S}} \right|} \over 2}} \right\}\)

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Acknowledgements

This work was supported by Shahid Chamran University of Ahvaz (Grant no. SCU.EM98.709). The authors would also like to appreciate the Gas Networks Research Center of Shahid Chamran University of Ahvaz for sharing its facilities during the course of this research.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, 61357-83151, Ahvaz, Iran

    Behzad Ebadi, Morteza Behbahani-Nejad & Maziar Changizian

  2. Gas Networks Research Center, Shahid Chamran University of Ahvaz, 61357-83151, Ahvaz, Iran

    Behzad Ebadi, Morteza Behbahani-Nejad & Maziar Changizian

  3. Department of Mathematics, Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084, Cluj-Napoca, Romania

    Ioan Pop

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  1. Behzad Ebadi
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  2. Morteza Behbahani-Nejad
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  3. Maziar Changizian
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  4. Ioan Pop
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Correspondence to Morteza Behbahani-Nejad.

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Ebadi, B., Behbahani-Nejad, M., Changizian, M. et al. Fluid flow effects on diffusion layer and current density for electrochemical systems. Korean J. Chem. Eng. 37, 1453–1465 (2020). https://doi.org/10.1007/s11814-020-0556-8

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  • DOI: https://doi.org/10.1007/s11814-020-0556-8

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