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Interfacial Mass Transport in Porous Media Augmented with Bulk Reactions: Analytical and Numerical Solutions

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Abstract

The focus of this work is on the rate of interfacial mass transfer at the interface between two immiscible fluids in porous media, subjected to variations in velocity and first-order consumption in the bulk medium. Numerical and analytical solutions are presented. We quantify the entrance length scale, which is the typical length in the flow direction over which the local and equilibrium Sherwood numbers (\(Sh\)) become identical, for Darcy–Brinkman flows in the presence of first-order bulk reactions (presented by Damköhler (\(Da\)) number). The study considers the effect of Schmidt number (\(Sc\)), viscosity ratios, permeability and bulk reaction coefficients. Results suggest a closed form solution for the entrance length scale. It is also observed that the assumption of equilibrium conditions prior to approaching this length may be violated for lower bulk reaction rates. Numerical and analytical results are in good agreement and suggest limited dependency of equilibrium \(Sh\) and \(Da\), as \(Da\) diminishes.

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Abbreviations

\(\alpha \) :

Parameter defined in [38]

\(\beta \) :

Parameter defined in [38]

\(\delta _\mathrm{{m}}\) :

Mass transfer length scale

\(\delta _\mathrm{{v}}\) :

Velocity length scale

\(\epsilon \) :

Porosity

\(\eta \) :

Parameter defined in [1]

\(\gamma _{1,2}\) :

Parameters defined in [55]

\(\hat{c}\) :

Changed concentration in [31]

\(\iota _{1,2}\) :

Parameters defined in [59, 60]

\(\lambda \) :

Parameter defined in [36]

\(\left\langle \ell \right\rangle \) :

Crosswise averaging operator for \(\ell \)

\(\mu _\alpha \) :

Viscosity

\(\nu _\alpha \) :

Kinematic viscosity

\(\overline{\ell }\) :

Equivalent value for \(\ell \) in normalized coordinate

\(\phi \) :

Parameter defined in [13]

\(\rho \) :

Density

\(\mathtt e \) :

Relative error

\(\theta \) :

Inclination angle

\(\tilde{\ell }\) :

Approximating operator for \(\ell \)

\(c\) :

Dissolved concentration

\(C_{1-\alpha }\) :

Homogeneous solution coefficient

\(C_{2-\alpha }\) :

Homogeneous solution coefficient

\(c_\mathrm{{{eq}}}\) :

Dissolved equilibrium concentration

\(D\) :

Effective diffusion coefficient

\(Da\) :

Damköhler number

\(f_\alpha \) :

Body force

\(g\) :

Gravitational acceleration

\(h_\alpha \) :

Thickness of fluid

\(J\) :

Mass flux

\(k\) :

Permeability

\(k_\mathrm{{b}}\) :

Bulk demand coefficient

\(k_\mathrm{{f}}\) :

Mass transfer coefficient

\(L\) :

Entrance length

\(P\) :

Index of Bessel function

\(P_\alpha \) :

Particular solution

\(Pe\) :

\(Pe={ P \acute{ e } clet }\) number

\(Sc\) :

Schmidt number

\(Sh\) :

Sherwood number

\(Sh_\mathrm{{{eq}}}\) :

Equilibrium Sherwood number

\(u_\alpha \) :

Velocity

\(X\) :

New streamwise coordinate

\(x\) :

Streamwise coordinate

\(Y\) :

New crosswise coordinate

\(y\) :

Crosswise coordinate

\(y_\mathrm{{v}}\) :

Velocity normal unit

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Acknowledgments

This study was partially funded by CRC CARE.

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Authors and Affiliations

  1. Centre for Environmental Risk Assessment and Remediation (CERAR), University of South Australia, Mawson Lakes Campus, Mawson Lakes, SA, 5095, Australia

    Kaveh Sookhak Lari

  2. CSIRO Land and Water, Private Bag No. 5, Wembley, WA, 6913, Australia

    Kaveh Sookhak Lari, Colin D. Johnston & Greg B. Davis

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  1. Kaveh Sookhak Lari
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  2. Colin D. Johnston
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Correspondence to Kaveh Sookhak Lari.

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Sookhak Lari, K., Johnston, C.D. & Davis, G.B. Interfacial Mass Transport in Porous Media Augmented with Bulk Reactions: Analytical and Numerical Solutions. Transp Porous Med 106, 405–423 (2015). https://doi.org/10.1007/s11242-014-0407-3

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