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Poisson–Nernst–Planck model of multiple ion transport across an ion-selective membrane under conditions close to chlor-alkali electrolysis

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Abstract

The assumption of electrochemical equilibrium at membrane–electrolyte interfaces is frequently accepted in a mathematical simulation of multiple ion transport (MIT) across a single-layer perfluorinated sulfonated cation-selective membrane (CM). This assumption is obviously inaccurate at high electric current loads typical of industrial applications, e.g. brine electrolysis. An assessment of this problem is one of the main objectives of this contribution. For this purpose, a one-dimensional stationary Poisson–Nernst–Planck (PNP) model was employed to describe MIT across a CM. The model input parameters used correspond closely to industrial chlor-alkali electrolysis. The model results are compared with those of an equivalent model published earlier that considered the classical Nernst–Planck equation and Donnan equilibrium at the CM–electrolyte interfaces (denoted as the DNP model). Both the DNP and the PNP models provide identical results at low current loads. However, a comparison at high current loads close to ‘industrial scale’ was impossible due to convergence problems of the DNP model. The ion transport numbers and membrane permselectivity were estimated by means of the PNP model. This model predicts conditions at the membrane interfaces close to thermodynamic equilibrium even in a current density range up to 10,000 A m−2. Additionally, the PNP model takes into account the kinetics of water autoprotolysis. It was shown that a high flux of OH− ions across a CM effectively alkalizes the catholyte diffusion layer, ensuring a precipitation of alkaline earth cations outside the CM and thus minimizing internal CM blockage.

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Abbreviations

c :

Molar concentration (mol m−3)

D :

Diffusion coefficient (m2 s−1)

F :

Faraday’s constant (F = 96484.56 C mol−1)

J :

Molar flux density (mol m−2 s−1)

j :

Local current density (A m−2)

k :

Hydraulic membrane permeability (m2)

k b , k f :

Kinetic rate constant of backward and forward reaction (m3 mol−1 s−1)

K w :

Water autoprotolysis equilibrium constant (K w  = 1 × 10−8 mol2 m−6)

N :

Number of ions included in the system

p :

Hydrostatic pressure (Pa)

q :

Space charge density (C m−3)

R :

Universal gas constant (R = 8.314 J K−1 mol−1)

S :

Source term (mol m−3 s−1)

t :

Transport number

T :

Absolute temperature (K)

U :

Voltage (V)

v :

Convective velocity (m s−1)

w :

Width (m)

x :

Coordinate (m)

z :

Valence number

ε r :

Relative permittivity of free water (ε r  = 78.5)

ε 0 :

Permittivity of vacuum (ε 0  = 8.8542 × 10−10 F m−1)

η :

Dynamic viscosity (kg m−1 s−1)

ϕ :

Electric potential (V)

σ :

Conductivity (S m−1)

ADL:

Anodic diffusion layer

CDL:

Cathodic diffusion layer

CM:

Cation-selective membrane

DER:

Donnan exclusion region

fix:

Fixed charge

i:

Ion

p:

Phase

ba:

Bulk anolyte

bc:

Bulk catholyte

p:

Phase

1D:

One-dimensional

ADL:

Anodic diffusion layer

CDL:

Cathodic diffusion layer

CM:

Cation-selective membrane

DER:

Donnan exclusion region

DNP:

Donnan–Nernst–Planck model

DNP1, DNP2:

Donnan equilibrium Nernst–Planck model 1 and 2

ISM:

Ion-selective membrane

MIT:

Multiple ion transport

PDE:

Partial differential equation

PNP:

Poisson–Nernst–Planck model

WSR:

Water splitting/recombination reaction

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Acknowledgments

The authors gratefully acknowledge the financial aid for this research by the Science Foundation (GACR) of the Czech Republic, Project No: 14-17351P.

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Authors and Affiliations

  1. Department of Inorganic Technology, University of Chemistry and Technology Prague, Technická 5, 166 28, Prague 6, Czech Republic

    Roman Kodým, Vlastimil Fíla & Karel Bouzek

  2. Department of Chemical Engineering, University of Chemistry and Technology Prague, Technická 5, 166 28, Prague 6, Czech Republic

    Dalimil Šnita

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  1. Roman Kodým
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  2. Vlastimil Fíla
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  4. Karel Bouzek
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Correspondence to Roman Kodým.

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Kodým, R., Fíla, V., Šnita, D. et al. Poisson–Nernst–Planck model of multiple ion transport across an ion-selective membrane under conditions close to chlor-alkali electrolysis. J Appl Electrochem 46, 679–694 (2016). https://doi.org/10.1007/s10800-016-0945-1

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