Abstract
We propose a novel macroscopic model for conjugate heat and mass transfer between a \emph{mobile region}, where advective transport is significant, and a set of \emph{immobile regions} where diffusive transport is dominant. Applying a spatial averaging operator to the microscopic equations, we obtain a \emph{multi-continuum} model, where an equation for the average concentration in the mobile region is coupled with a set of equations for the average concentrations in the immobile regions. Subsequently, by mean of a spectral decomposition, we derive a set of equations that can be viewed as a generalisation of the multi-rate mass transfer (MRMT) model, originally introduced by Haggerty & Gorelick. This new formulation does not require any assumption on local equilibrium or geometry. We then show that the MRMT can be obtained as the leading order approximation, when the mobile concentration is in local equilibrium. The new Generalised Multi-Rate Mode (GMRM) has the advantage of providing a direct method for calculating the model coefficients for immobile regions of arbitrary shapes, through the solution of appropriate micro-scale cell problems. An important finding is that a simple re-scaling or re-parametrisation of the transfer rate coefficient (and thus, the memory function) is not sufficient to account for the flow field in the mobile region and the resulting non-uniformity of the concentration at the interfaces between mobile and immobile regions.
Highlights
Conjugate transfer in heterogeneous media is of pivotal importance for a wide range of applications ranging from dispersion of contaminants in aquifers [1,2,3,4,5] and stagnation/recirculation zones [6,7,8,9] to heat transfer in granular media and suspension flows [10], or colloid interface reactions [11,12]
It is important to notice that in our model described by Eqs. (3) and (4), the diffusive modes are mostly excited by the conjugate transfer with the immobile regions and not by source terms due, for example, to bulk reactions
We propose a novel approach to derive the multirate mass transfer model that is different from that of the memory function or that of Haggerty and Gorelick
Summary
Conjugate transfer in heterogeneous media is of pivotal importance for a wide range of applications ranging from dispersion of contaminants in aquifers [1,2,3,4,5] and stagnation/recirculation zones [6,7,8,9] to heat transfer in granular media and suspension flows [10], or colloid interface reactions [11,12]. We propose a novel general derivation of the multirate mass transfer model that address the following modeling issues: (i) providing a unique way of calculating the model parameters, like a set of equations that can be solved once for a whole class of problems; (ii) including the effect of advective transport on the conjugate transfer in a way that is mathematically formal and physically sound; and (iii) Derivation from first principles containing a limited and clear set of assumptions This with the aim of facilitating any extension in future works. Additional details on the homogenisation procedures can be found in Appendix
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