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Volume Averaging Study of the Capacitive Deionization Process in Homogeneous Porous Media

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Ion storage in porous electrodes is important in applications such as energy storage by supercapacitors, water purification by capacitive deionization, extraction of energy from a salinity difference and heavy ion purification. A model is presented to simulate the charge process in homogeneous porous media comprising big pores. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without faradaic reactions or specific adsorption of ions. A volume averaging technique is used to derive the averaged transport equations in the limit of thin electrical double layers. Transport between the electrolyte solution and the charged wall is described using the Gouy–Chapman–Stern model. The effective transport parameters for isotropic porous media are calculated solving the corresponding closure problems. The source terms that appear in the average equations are calculated using numerical computations. An alternative way to deal with the source terms is proposed.

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\(A_{\upalpha \upbeta }\) :

Interphase area \(\upalpha \)\(\upbeta \) (m\(^{2}\))

\(a_{v}\) :

Effective area (m\(^{2}\,\)m\(^{-3})\)

\(c_{i}\) :

Ion concentration (mol m\(^{-3})\)

\(C_{\infty }\) :

Salt bulk concentration (mol m\(^{-3})\)

\({\tilde{c}}_\upalpha \) :

Deviation salt concentration in \(\upalpha \)-phase (mol m\(^{-3})\)

\(\langle c_\upalpha \rangle ^{\upalpha }\) :

Intrinsic phase average concentration (mol m\(^{-3})\)

\({\underline{\underline{D}}}_\mathrm{eff}\) :

Effective diffusivity tensor (m\(^{2}\,\mathrm{s}^{-1})\)

\(D_{i}\) :

Diffusion coefficient (m\(^{2}\,\hbox {s}^{-1})\)

\(\underline{f}_{1}\) :

Closure vector field (m)

\(f_{2}\) :

Closure scalar field (s m\(^{-1})\)

\(F\) :

Faraday constant (C mol\(^{-1})\)

\(\underline{g}_{1}\) :

Closure vector field (m)

\(g_{2}\) :

Closure scalar field \((\hbox {V m}^{2}\,\hbox {s mol}^{-1})\)

\({\underline{\underline{I}}}\) :

Unit tensor (dim.)

\(\underline{I}_{e}\) :

Ionic current per unit area \((\hbox {A m}^{-2})\)

\(J_\mathrm{charge}\) :

Charge-transfer flux \((\hbox {C m}^{-2}\,\hbox {s}^{-1})\)

\(J_\mathrm{salt}\) :

Salt molar flux \((\hbox {mol m}^{-2}\,\hbox {s}^{-1})\)

\(l_{\upalpha }\) :

Microscopic length scale (m)

\(\underline{l}_{i}\) :

Lattice vectors (m)

\(L\) :

Macroscopic length scale (m)

\(L_{c}\) :

Macroscopic length scale for the gradient (m)

\(\underline{n}_{\upalpha \upbeta }\) :

Unit normal vector from the \(\upalpha \) into the \(\upbeta \)-phase (dim.)

\(N_{i}\) :

Ion flux \((\hbox {mol m}^{-2}\,\hbox {s}^{-1})\)

\(q\) :

Excess charge density \((\hbox {C m}^{-2})\)

\(\langle q\rangle _{\upalpha \upbeta }\) :

Excess charge area averaged \((\hbox {C m}^{-2})\)

\(r_{o}\) :

Radius of the representative elementary volume (m)

\(t\) :

Time (s)

\(t_{C}^{\bullet }\) :

Characteristic time for the supercapacitor regime (s)

\(t_{D}^{\bullet }\) :

Characteristic time for the desalination regime (s)

\(u_{i}\) :

Isotropic mobility \((\hbox {m}^{2}\,\hbox {V}^{-1}\,\hbox {s}^{-1})\)

\({\underline{\underline{U}}}_\mathrm{eff}\) :

Mobility tensor \((\hbox {m}^{2}\,\hbox {V}^{-1}\,\hbox {s}^{-1})\)

\(V_{\upalpha }\) :

\(\upalpha \)-Phase in REV \((\hbox {m}^{3})\)

\(V_\mathrm{T}\) :

Thermal voltage (V)

\(w\) :

Excess salt density \((\hbox {mol m}^{-2})\)

\(\langle w\rangle _{\upalpha \upbeta }\) :

Excess salt adsorption area average \((\hbox {mol m}^{-2})\)

\(x\) :

Spatial position (m)

\(z_{i}\) :

Ionic charge number (dim.)

\(\Delta \phi _\mathrm{D}\) :

Diffuse-layer potential difference (V)

\(\Delta \phi _\mathrm{Stern}\) :

Stern-layer potential difference (V)

\(\varepsilon _{\upalpha }\) :

\(\upalpha \)-Phase volume fraction (dim.)

\(\phi \) :

Electrostatic potential (V)

\(\tilde{\phi }_\upalpha \) :

Potential deviation in \(\upalpha \)-phase (V)

\(\langle \phi _\upalpha \rangle ^{\upalpha }\) :

Intrinsic phase average potential (V)

\(\kappa \) :

Effective conductivity \((\hbox {A V}^{-1}\,\hbox {m}^{-1})\)

\(\lambda _\mathrm{B}\) :

Bjerrum length (m)

\(\lambda _\mathrm{D}\) :

Debye length (m)

\(\psi _{\upalpha }\) :

Generic variable in \(\upalpha \)-phase

\(\underline{\nabla }\) :

Nabla operator \((\hbox {m}^{-1})\)

\(i\) :

Component \(i\)

\(\upalpha \) :

\(\upalpha \)-Phase

\(\upbeta \) :

\(\upbeta \)-Phase

\(\upalpha \upbeta \) :

\(\upalpha \)\(\upbeta \) interphase

\(\upalpha \) :

\(\upalpha \)-Phase

\(\upbeta \) :

\(\upbeta \)-Phase


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This research was partially conducted at the Oak Ridge National Laboratory (ORNL) and supported by the Laboratory Director’s Research and Development Seed Program of ORNL. ORNL is managed by UT-Battelle, LLC, under Contract DE-AC05-0096OR22725 with the US Department of Energy.

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Correspondence to Jorge Gabitto.



Order of Magnitude Analysis in Eq. (18)

$$\begin{aligned} \varepsilon _{\upalpha } \partial \langle {c}_\upalpha \rangle ^{\upalpha }/\partial t= & {} \underline{\nabla } \bullet ({\underline{\underline{D}}}_{\mathrm{eff}} \bullet \underline{\nabla }\langle {c}_\upalpha \rangle ^{\upalpha }) + \underline{\nabla } \bullet (D \varepsilon _{\upalpha } \underline{u} \partial \langle {w}\rangle _{\upalpha \upbeta } /\partial t) - a_{v} \partial \langle {w}\rangle _{\upalpha \upbeta } /\partial t, \\&O\left( \frac{{D}{\left\langle {{c}_{\upalpha }} \right\rangle }^{\upalpha }}{{L L}_{c}}\right) , \quad O\left( \frac{{1}}{{L}}\frac{\partial {\left\langle {w} \right\rangle }_{{\upalpha \upbeta }}}{\partial {t}}\right) , \quad O\left( \frac{{1}}{{l}_\upalpha }\frac{\partial {\left\langle {w} \right\rangle }_{{\upalpha \upbeta }}}{\partial {t}}\right) .\nonumber \end{aligned}$$

Using, \(- \underline{n}_{\upalpha \upbeta }\bullet \underline{\nabla }{f}_2 = 1/D\), we get \({f}_2 \approx O({l}_\upalpha /D)\) because \(f_{2}\) varies with \(l_{\upalpha }\). From \(\underline{{u}}=\frac{{1}}{{V}_{\upalpha }}\int \limits _{{A}_{{\upalpha \upbeta }}} {{\underline{{n}}}_{{\upalpha \upbeta }} {f}_{2} { \hbox { d}A}} \), using mean value theorem and \({f}_2 \approx O({l}_\upalpha /D)\) we get, \(\underline{{u}}\approx O{(1 / D)}\). Here, we used \(A/V_{\upalpha } = a_{V}\) and \({a}_{V} \approx O({l}_\upalpha ^{{-1}})\).

Order of Magnitude Analysis in Eq. (32)

$$\begin{aligned}&\underline{\nabla } \bullet \{{\underline{\underline{U}}}_{\mathrm{eff}} \bullet \langle {c}_\upalpha \rangle ^{\upalpha } \underline{\nabla }\langle \phi _\upalpha \rangle ^{{\upalpha }}\} + \underline{\nabla }{\bullet }\left\{ \frac{{\varepsilon }_{\upalpha } { U}{\left\langle {{c}_{\upalpha }} \right\rangle }^{\upalpha }}{{V}_{\upalpha }}\left( \int \limits _{{A}_{{\upalpha \upbeta }}} {\underline{{n}}}_{{\upalpha \upbeta }} { g}_{2} \hbox { d}A\right) \frac{\partial {\left\langle {q} \right\rangle }_{{\upalpha \upbeta }}}{\partial {t}}\right\} \nonumber \\&\quad O\left( \frac{{U \varepsilon }_\upalpha {\left\langle {{c}_{\upalpha }} \right\rangle }^{\upalpha } {\left\langle \phi \right\rangle }^{\upalpha }}{{L L}_{c}}\right) , \quad O\left( \frac{{1}}{{L}}\frac{\partial {\left\langle {q} \right\rangle }_{{\upalpha \upbeta }}}{\partial {t}}\right) ,\nonumber \\&\qquad +\, \underline{\nabla } \bullet \left\{ \frac{{\varepsilon }_{\upalpha } { U}}{{V}_{\upalpha }}\int \limits _{{V}_\upalpha } {{{\tilde{c}}}_\upalpha \underline{\nabla }{\tilde{\phi }}_\upalpha { \hbox { d}V}} \right\} - a_{v} \quad \frac{\partial {\left\langle {q} \right\rangle }_{{\upalpha \upbeta }}}{\partial {t}} = 0,\\&\qquad O\left( \frac{{U \varepsilon }_\upalpha {\left\langle {{c}_{\upalpha }} \right\rangle }^{\upalpha } {\left\langle \phi \right\rangle }^{\upalpha }}{{L L}_{c}}\frac{{l}_\upalpha }{{L}}\right) , \quad O\left( \frac{{1}}{{l}_\upalpha }\frac{\partial {\left\langle {q} \right\rangle }_{{\upalpha \upbeta }}}{\partial {t}}\right) .\nonumber \end{aligned}$$

The order of magnitude for the deviation variables \({\tilde{c}}_{\upalpha } \) and \(\tilde{\phi }_{\upalpha } \) were obtained using Eqs. (23) and (36). We use \({g}_2 \approx O\left( {l}_\upalpha / \left( U{\left\langle {{c}_{\upalpha }} \right\rangle }^\upalpha \right) \right) \) estimated from Eq. (41).

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Gabitto, J., Tsouris, C. Volume Averaging Study of the Capacitive Deionization Process in Homogeneous Porous Media. Transp Porous Med 109, 61–80 (2015).

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  • CDI
  • Volume average
  • Porous media