Abstract
Energy storage in porous capacitor materials, capacitive deionization (CDI) for water desalination, capacitive energy generation, geophysical applications, and removal of heavy ions from wastewater streams are some examples of processes where understanding of ionic transport processes in charged porous media is very important. In this work, one- and two-equation models are derived to simulate ionic transport processes in heterogeneous porous media comprising two different pore sizes. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without Faradaic reactions or specific adsorption of ions. A two-step volume averaging technique is used to derive the averaged transport equations for multi-ionic systems without any further assumptions, such as thin electrical double layers or Donnan equilibrium. A comparison between both models is presented. The effective transport parameters for isotropic porous media are calculated by solving the corresponding closure problems. An approximate analytical procedure is proposed to solve the closure problems. Numerical and theoretical calculations show that the approximate analytical procedure yields adequate solutions. A theoretical analysis shows that the value of interphase pseudo-transport coefficients determines which model to use.
Highlights
In systems comprising different phases, conservation equations for the properties under study are normally derived for each separate phase
A one-equation model can be extracted from a multi-equation model based on the principle of local equilibrium (Whitaker, [2,3]; Kaviani, [4]; Quintard and Whitaker, [5,6]; del Rio and Whitaker, [7,8]; among others)
We show typical variation in the effective diffusivities with the porous medium void fraction. In all these Figures, we plot the value of the isotropic effective diffusivity versus the ratio of the porous solid and liquid phases effective diffusivities (δ = εαDe f f,XX/Dγ) using the liquid phase void fraction as a fixed parameter
Summary
In systems comprising different phases, conservation equations for the properties under study are normally derived for each separate phase. This approach leads to the formulation of a multi-equation model. In order to simplify the math, we will use the assumptions of no convective flow inside the macropores (νs = 0) and that all ionic bulk diffusivities are equal (Di = constant) These assumptions allow us to drop the i-subscripts in the effective diffusivity tensors. In order to do so, we will solve the closure problems in isotropic spatially periodic unit cells The use of these cells will reduce the problem to calculation of only one component of the effective diffusivity tensors (Dij,xx = iDiji)
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