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Jump Condition for Diffusive and Convective Mass Transfer Between a Porous Medium and a Fluid Involving Adsorption and Chemical Reaction

Abstract

In this paper, mass transfer at the fluid–porous medium boundaries is studied. The problem considers both diffusive and convective transport, along with adsorption and reaction effects in the porous medium. The result is a mass flux jump condition that is expressed in terms of effective transport coefficients. Such coefficients (a total dispersion tensor and effective reaction and adsorption coefficients) may be computed from the solution of the corresponding closure problem here stated and solved as a function of the Péclet number (Pe), the porosity and a local Thiele modulus. For the case of negligible convective transport (i.e., \({Pe\ll 1}\)), the closure problem reduces to the one recently solved by the authors for diffusion and reaction between a fluid and a microporous medium.

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Abbreviations

A :

Surface of the large-scale averaging region V

\({{\fancyscript{A}}_\infty}\) :

Superficial area of the large-scale averaging region V , m2

\({\fancyscript{A}_j}\) :

Area of the j-region contained in A , j = ω, η, m2

\({\fancyscript{A}_{\eta \omega}}\) :

Area of the inter-region, m2

a v (x):

Position-dependent interfacial area per unit volume, m−1

a v ω :

Interfacial area per unit volume in the homogeneous ω-region, m−1

b β :

Vector that maps \({\nabla \left\langle {c_{A\beta}}\right\rangle^{\beta}}\) into \({\tilde {c}_{A\beta}}\), m

c A β :

Molar concentration of species A in the β-phase, mol/m3

\({\left\langle {c_{A\beta}}\right\rangle_s^\beta}\) :

Surface average concentration, \({\int_{A_{\omega \eta}^\ast} {\left\langle {c_{A\beta}}\right\rangle _s^\beta {d}A} =\int_{V_\omega} {\left[{\left\langle {c_{A\beta}}\right\rangle ^{\beta}-\left\langle {c_{A\beta}}\right\rangle _\omega ^\beta}\right]{d}V} + \int_{V_\eta} {\left[{\left\langle {c_{A\beta}}\right\rangle ^{\beta}-\left\langle {c_{A\beta}}\right\rangle _\eta ^\beta}\right]{\rm d}V} ,{\rm mol/m}^{2}}\)

\({\left\langle {c_{A\beta}}\right\rangle _j^\beta}\) :

Intrinsic average of the concentration of species A in the homogeneous j-region, j = η, ω, mol/m3

\({\left\langle {c_{A\beta}}\right\rangle^\beta}\) :

Intrinsic average of the concentration of species A, mol/m3

\({\tilde {c}_{A\beta}}\) :

Spatial deviations of the concentration of species A the β-phase, mol/m3

D(x):

Position-dependent diffusion tensor, m2/s

D(x ω ):

Diffusion tensor in the homogeneous ω-region, m2/s

D β (x):

Position-dependent total dispersion tensor, m2/s

\({{{\bf D}}_\beta^\ast \left({\bf x}\right)}\) :

Position-dependent dispersion tensor, m2/s

\({{\fancyscript{D}}_\beta}\) :

Molecular diffusion, m2/s

d β :

Closure variable that maps \({\partial \left\langle {c_{A\beta}}\right\rangle ^{\beta}/\partial t}\) into \({\tilde {c}_{A\beta}}\), s

h :

Half of the vertical length of the unit cell for the inter-region, m

K β :

Position-dependent permeability tensor, m2

K β ω :

Norm of the Darcy’s law permeability tensor in the ω-region, m2

K :

Pseudo-heterogeneous adsorption coefficient, m

\({K_{eff_{\vphantom{\int}}}^{rx}}\) :

Reaction rate effective coefficient for the inter-region, m/s

\({K_{eff}^{ads}}\) :

Adsorption rate effective coefficient for the inter-region, m

\({\tilde {K}}\) :

Spatial deviations of K, m

k r :

Pseudo-heterogeneous reaction rate coefficient, m/s

\({\tilde{k}_r}\) :

Spatial deviations of k r , m/s

\({\left\langle k \right\rangle_{\eta\omega}}\) :

Area average of k = k r , K

L C :

Characteristic length of \({\nabla \left\langle {c_{A\beta}}\right\rangle ^{\beta}}\), m

L C1 :

Characteristic length of \({\nabla \nabla \left\langle {c_{A\beta}}\right\rangle ^{\beta}}\), m

l i :

Unit cell vectors (i = 1,2), m

l β :

Characteristic length of the β-phase, m

l :

Length of the base of the unit cell used to solve the closure problem, m

\({{\bf n}_{\beta \sigma}}\) :

Unit normal vector directed from the β-phase, toward the \({\sigma}\) -phase

n η ω :

Unit normal vector directed from the η-region toward the ω-region

P e :

Particle Péclet number

r 0 :

Radius of the averaging volume, m

s β :

Closure variable that maps \({\left\langle {c_{A\beta}}\right\rangle _\beta}\) into \({\tilde{c}_{A\beta}}\)

t :

Time, s

t * :

Process characteristic time, s

\({\fancyscript{V}}\) :

Volume of the averaging region, m3

\({\fancyscript{V}_\infty}\) :

Volume of the large-scale averaging region, m3

\({\fancyscript{V}_j}\) :

Volume of the j-region contained within V , j = ω, η, m3

V j :

Space of the j-phase contained within the averaging region, \({j=\beta ,\sigma}\)

v β :

Velocity vector in the β-phase, m/s

\({{\tilde {\bf v}}_\beta}\) :

Spatial deviations of v β , m/s

x :

Position vector that locates the centroid of the averaging volume, m

y β :

Position vector relative to the centroid of the averaging volume, m

β :

Adjustable coefficient in the jump condition by Ochoa-Tapia and Whitaker (1995a)

\({\delta _\beta}\) :

Characteristic length associated with the spatial variations of the concentration, m

\({\varepsilon_\beta \left({\bf x}\right)}\) :

Position-dependent volume fraction of the β-phase

\({\varepsilon_{\beta,\omega}}\) :

Volume fraction of the β-phase in the ω-region

\({\phi}\) :

Local Thiele modulus

ψ β :

Arbitrary function associated with the β-phase

A :

Relative to species A

C :

Relative to the concentration

s :

Surface property

\({\varepsilon}\) :

Relative to the porosity

η :

Relative to the homogeneous η-region

ω :

Relative to the homogeneous ω-region

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Correspondence to J. Alberto Ochoa-Tapia.

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Valdés-Parada, F.J., Alvarez-Ramirez, J., Goyeau, B. et al. Jump Condition for Diffusive and Convective Mass Transfer Between a Porous Medium and a Fluid Involving Adsorption and Chemical Reaction. Transp Porous Med 78, 459–476 (2009). https://doi.org/10.1007/s11242-009-9343-z

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Keywords

  • Jump condition
  • Adsorption
  • Reaction
  • Closure problem
  • Volume averaging