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The effect of convection in the external diffusion layer on the results of a mathematical model of multiple ion transport across an ion-selective membrane

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Abstract

A previously published macrohomogeneous mathematical model of the simultaneous transport of multiple ions across an ion-selective membrane under current load based on the Nernst–Planck equation was extended. A significantly more realistic model is proposed and realised. The change with the most significant impact on the results of the model is consideration of convective mass transfer in the external diffusion layers adjacent to the membrane surfaces. This change results in a reduction of the concentration maximum previously observed in the membrane interior and highlights the importance of the external diffusion layers for ion transport across an ion-selective membrane. Hitherto this has often been underestimated.

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Abbreviations

a :

Activity [1]

A ca :

Clarke’s equation constant [m3 mol−1]

c :

Molar concentration related to the volume of the solution or to the volume of the wet membrane [mol  m−3]

d e :

Equivalent diameter [m]

D :

Diffusivity [m2 s−1]

F :

Faraday number [96,487 C mol−1]

j :

Current density [A m−2]

k :

Membrane permeability [m2]

\(\bar k\) :

Mass transfer coefficient [m s−1]

L :

Membrane length [m]

M :

Molar weight [kg mol−1]

\(\overline M \) :

Mean molar weight of mixture [kg mol−1]

N :

Molar flux [mol m−2 s−1]

p :

Pressure [Pa]

R :

Universal gas constant [8.314 J K−1 mol−1]

T :

Temperature [K]

v:

Fluid flow velocity [m s−1]

V :

Volume [m3]

V ∞ ca :

Clarke’s equation constant [m3 mol−1]

x :

Molar fraction

z :

Charge number [1]

Re :

Reynolds number \(Re=\frac{vd_e \rho }{\eta }\)

Sc :

Schmidt number \(Sc=\frac{\eta }{D\rho }\)

Sh :

Sherwood number \(Sh=\frac{\bar{k}d_e }{D}\)

δ:

Thickness of Nernst diffusion or membrane layer [m]

ϕ:

Source [mol m−3 s−1]

γ:

Activity coefficient [1]

η:

Electrolyte dynamic viscosity [kg m−1 s−1]

φ:

Galvani potential [V]

μ:

Chemical potential [1]

\(\tilde {\mu }\) :

Electrochemical potential [1]

ρ:

Density [kg m−3]

τ:

Time [s]

a :

Anion

c :

Cation

ca :

Apparent component

Don :

Donnan

l i :

Length

m :

Molar

M :

Membrane

x :

Axial coordinate

s :

Solid phase

tot :

All components (including water)

a :

Apparent

l :

Liquid phase

l i :

Length

M :

Membrane

N ions :

Number of ions

N tot :

Number of all components

0:

Standard state

t :

True

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Acknowledgements

Financial support by the Grant Agency of the Czech Republic under project number 203/05/0080 and by the Ministry of Education, Youth and Sports of teh Czech Republic under project number MSM6046137301 is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Inorganic Technology, Institute of Chemical Technology Prague, Technická 5, 16628, Prague 6, Czech Republic

    V. Fíla & K. Bouzek

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  1. V. Fíla
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Fíla, V., Bouzek, K. The effect of convection in the external diffusion layer on the results of a mathematical model of multiple ion transport across an ion-selective membrane. J Appl Electrochem 38, 1241–1252 (2008). https://doi.org/10.1007/s10800-008-9545-z

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