(174bd) Modeling of Cation Exchange Membranes Using Maxwell-Stefan Approach for Chlor-Alkali System

The generalized Maxwell-Stefan approach is considered more suitable
than the Nernst-Planck approach to describe multicomponent ion and water
transport inside an ion exchange membrane. This is due to the fact that the
Nernst-Planck approach is limited to dilute ionic systems due to the assumption
of an ideal solution and neglection of ion-ion interactions [1, 2], whereas many electrochemical processes involve highly concentrated
and non-ideal solutions. The Maxwell-Stefan approach takes the interactions of
different components and the non-ideal solutions into account. Moreover, the
Maxwell-Stefan approach includes the water transport via the solvent-ion
interactions whereas the Nernst-Planck model has to introduce a separate
equation (i.e., the Schlögl equation) to account for the water transport [2, 3].

One main challenge in applying the Maxwell-Stefan approach is the lack
of reliable data on diffusivities at high concentrations. Another weakness of a
previously developed Maxwell-Stefan model [4] is the calculation of the
membrane potential gradient by neglecting the concentration gradient and by
using Ohm’s law to derive the potential gradient explicitly. The neutrality
condition is broken by this simplification, which has been numerically proven
during the investigation of the extended Nernst-Planck model [5].

In this presentation, we will show that semi-empirical correlations are
suitable to define the Maxwell-Stefan diffusivities inside the membrane. The newly developed model [6] has been used to
predict transport numbers of ions and water for the chlor-alkali process and
has been validated using the available experimental data (Figures 1a, 1b, and 2). With the diffusivities presented in this work, the
model shows a better fit to the experimental data than with previously reported
fitted diffusivities [4, 7].

Figure 1. Modeled and experimental data
(Yeager et al. [11, 12] and T. Berzins [12]) of the sodium transport number as a
function of catholyte concentration. (a) current density = 2 kA.m^-2,
 EW=1150, membrane thickness = 0.25 mm, 25 wt% NaCl and temperature = 80 ^oC.
(b) current density = 3 kA.m^-2, EW=1100, membrane thickness = 0.1
mm, 25 wt% NaCl and temperature = 80 ^oC.

Figure 2. Modeled and
experimental data (Yeager et al. [11]) of the relative water transport number as a function of catholyte
concentration using different values of Maxwell-Stefan diffusivites at 2 kA.m^-2.
EW=1150, membrane thickness = 0.25 mm, temperature = 80 ^oC and 25
wt% NaCl.

Also, we applied the Augmented matrix method [8] to solve both the
concentration and the potential gradients simultaneously using the built-in
partial differential equation parabolic elliptic (pdepe) solver in Matlab®. By adopting this method, no further assumption about the potential
gradient is needed. The values of diffusivities also affect the membrane
potential drop as shown in Figure 3. Our model shows a reasonable match with
the experimental values of the membrane potential at 80 ^oC around
0.291 V at 3 kA.m^-2[9] and 0.51 V at 6 kA.m^-2[10]. This confirms the strong influence of the values of the
Maxwell-Stefan diffusivities on the Maxwell-Stefan model. 

Figure 3. Membrane potential drop as a function of current density using
different values of Maxwell-Stefan diffusivites. The experimental data of
Bergner et al.[13]:
NaCl = 18 wt%, NaOH = 33 wt%, temperature = 90 ^oC,  EW=1100,
membrane thickness = 0.29 mm (properties of Nafion N954 [14])
. The typical value of membrane potential at 80 ^oC  for current
densities of 3.5 kA.m^-2 and 6 kA.m^-2  are  0.291 V [9]
and  0.51 V [10]
respectively.

References

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[2]      F. Helfferich, Ion Exchange. McGraw-Hill
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[3]      R. Schlögl, “The significance of convection
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[9]      T. F. O’Brien, T. V. Bommaraju, and F. Hine, Handbook
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[10]     Asahi Kasei Chemicals Corporation, “Recent
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consumption,” 2009.

[11]     H. L. Yeager, “Sodium Ion Diffusion in
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[12]     H. L. Yeager, “Transport Properties of
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[13]     D. Bergner, M. Hartmann, and H. Kirsch,
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[14]     W.Grot, Fluorinated Ionomers, Second edi.
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[15]     A. Nidola, “Zero Gap Membrane Cell and Spe
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