Generalized Taylor-Aris dispersion in spatially periodic microfluidic networks. Chemical reactions

Macrotransport theory governing solute transport in spatially periodic networks is extended so as to account for first-order, irreversible chemical reactions occurring within the network. The otherwise locally continuous interstices of the spatially periodic medium are modeled as a discrete graphical network by the expedient of dividing the repetitive unit cell into a finite number of subvolume elements i (i = 1, 2, ..., n) representing the nodes of the graph. The solute is assumed to be depleted at the uniform rate k(i) when present in node i, i.e., each node i is modeled as a continuous stirred-tank flow reactor. The edges of the graph embody the solute transport processes occurring between nodes, either via "piggy-back" entrainment in a flowing fluid or external force-driven animation, or both, as well as by molecular diffusion. A Taylor-Aris-like "method-of-moments" scheme is applied to homogenize the resulting master equation governing solute transport within the network, thereby explicitly furnishing (i) a pair of adjoint matrix eigenvalue problems for computing the node-based macrotransport fields P₀^∞ (i) and A(i) (ultimately required to calculate the mean solute velocity Ū*), as well as the network-scale, effective first-order irreversible reaction rate constant K̄*; (ii) a matrix equation for computing the third node-based macrotransport field B(i) (ultimately used to determine the Taylor-Aris solute dispersivity D̄*); and (iii) edge-based summations of the three preceding nodal fields, used to calculate the network-scale solute velocity vector Ū* and dispersivity dyadic D̄*. The computational simplicity of this graphical network scheme, in contrast with the original interstitially continuous Taylor-Aris macrotransport paradigm, is demonstrated in the context of an elementary geometric model of a porous medium.