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We propose a novel macroscopic model for conjugate heat and mass transferbetween a \emph{mobile region}, where advective transport is significant, and aset of \emph{immobile regions} where diffusive transport is dominant. Applyinga spatial averaging operator to the microscopic equations, we obtain a\emph{multi-continuum} model, where an equation for the average concentrationin the mobile region is coupled with a set of equations for the averageconcentrations in the immobile regions. Subsequently, by mean of a spectraldecomposition, we derive a set of equations that can be viewed as ageneralisation of the multi-rate mass transfer (MRMT) model, originallyintroduced by Haggerty & Gorelick. This new formulation does not require anyassumption on local equilibrium or geometry. We then show that the MRMT can beobtained as the leading order approximation, when the mobile concentration isin local equilibrium. The new Generalised Multi-Rate Mode (GMRM) has theadvantage of providing a direct method for calculating the model coefficientsfor immobile regions of arbitrary shapes, through the solution of appropriatemicro-scale cell problems. An important finding is that a simple re-scaling orre-parametrisation of the transfer rate coefficient (and thus, the memoryfunction) is not sufficient to account for the flow field in the mobile regionand the resulting non-uniformity of the concentration at the interfaces betweenmobile and immobile regions.

Generalised Multi-Rate Models for conjugate transfer in heterogeneous materials