Generalized multirate models for conjugate transfer in heterogeneous materials

Abstract

We propose a novel macroscopic model for conjugate heat and mass transfer between a mobile region, where advective transport is significant, and a set of immobile regions where diffusive transport is dominant. Applying a spatial averaging operator to the microscopic equations, we obtain a 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 [1]. 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 nonuniformity of the concentration at the interfaces between mobile and immobile regions.

Description of Chemical Transport in Laboratory Rock Cores Using the Continuous Random Walk Formalism

“…In this context, some authors have assumed that the single‐rate mass transfer coefficient (∝ t 2 ) varies with time (Fernàndez‐Garcia & Sanchez‐Vila, 2015). Others have adopted a formulation where the concentration “seen” by the immobile regions depends on the fluid velocity and differs from the bulk or average concentration in the mobile region (Municchi & Icardi, 2020). We note that Equation could in principle be used to explicitly describe diffusive transport within the immobile zone, in a formulation that mimics a multirate mass transport process, albeit with an increase of the number of fitting parameters.…”

Macroscopic models for filtration and heterogeneous reactions in porous media

“…This confirms the power and accuracy of this homogenisation-based approach to extend the applicability of macroscopic transport theory beyond the 'standard' problems that rely on solenoidal velocity field and slow reactions. It is important to notice that our approach does not include 'conjugate' transfer (i.e., transport inside the solid grain) but the present approach can be conveniently coupled with the recently proposed generalised multi-rate transfer model [16].…”

“…We are interested in the fluid flow and scalar transport inŶ f , neglecting transport within the solid regionŶ s , that is here represented only through its interfaceΓ. We do not discuss the upscaling of systems with full conjugate transfer, which has been extensively studied in a previous work [16]. If such flow is well described by the Stokes equations (i.e., incompressible, low Reynolds number), it has been shown that the homogenisation procedure leads to the Darcy equation [11].…”

Heterogeneous Multi-Rate mass transfer models in OpenFOAM®

“…In this work we propose a novel numerical implementation of the MRMT model, based on the generalisation proposed in an earlier work of some of the authors [44]. This numerical implementation is written as a new library within the C++ opensource finite volume library OPENFOAM ® [50].…”

“…Usually, the region occupied by the fluid phase (mono or multi-component) is called mobile region whereas the remaining region, occupied by the matrix, is the immobile region. It is often assumed that the dominant transport process in the immobile regions is diffusion [44], while the mobile region can exchange mass and energy with the immobile region. Figure 1 depicts a typical domain composed of a mobile region and several immobile regions, similar to those often found in subsurface flow applications.…”