Moment analysis for predicting effective transport properties in hierarchical retentive porous media

Interesting observations on the importance of the detailed architecture of HPLC particles show that a deep understanding of the band broadening processes in liquid chromatography is necessary to further progress HPLC column technology and maximize the speed with which liquid chromatography can be carried out. [1], [2], [3], [4], [5].

Over the past 20 years, research in the field of liquid chromatography (LC) has also been directed toward the development of new columns with the goal of significantly increase the packing uniformity [6], [7], [8], [9], [10], [11], [12], [13] far beyond that of the currently used packed-bed and monolithic silica columns. It is precisely by moving in this direction that new LC columns have been designed and realized by in situ etching a 2-D array of micromachined pillars, using a lithographic mask to precisely define the position and the dimension of the pillars [9], [14], [15], [16], [17], [18], [19]. If pillars are non-porous, like in COMOSS columns, the surface available for adsorption is too small to be competitive with packed-bed columns, so that porous shell pillars have been introduced, as obtained by electrochemical anodization of the solid silicon pillars obtained using deep reactive ion etching[20], [21]. Porous pillars have 250 to 500 times the retention surface area of non-porous pillars, and this factor can further increase by increasing the shell thickness.

These newer LC columns are typical examples of porous media with a hierarchical pore-structure characterized by large inter-pillar macropores and small intra-pillar micropores. They have the dual advantage of an ordered macro-porous structure and large surface area for adsorption/desorption processes occurring at the microscale. The regular and ordered structure of the macro-pore network permits the identification of a unit periodic cell as the basic element representative of the entire column and various homogenization techniques can be adopted to estimate the effective transport parameters, focusing exclusively on the unit cell instead of the entire domain.

Multiple-scale expansion [22], [23], [24], volume averaging [25], [26], [27], [28], [29], [30], [31] and moment analysis [32], [33], [34], [35], [36], [37], [38], [39], [40], [41] are all equivalent strategies, alternative to stochastic Lagrangian simulations [42], [43], [44], [45], [46], [47], [48], CFD/DNS and Lattice-Boltzman simulations [7], [8], [11], [49], [50], [51], [52], [53] to investigate long-range/large-distance asymptotic dispersion properties. However, the study of the transient behaviour of dispersion properties is gaining more and more attention [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68] due to the miniaturization of analytical/separation devices. Indeed, when the time scales of experimental observations reduce together with the device dimensions, the accurate description of transient phenomena becomes more and more important for a correct interpretation of experimental results [69], [70], e.g. for the estimate of the time/length scales to achieve asymptotic conditions.

The homogenization method proposed in this paper allows us to investigate both the time evolution and the asymptotic values of the effective velocity vector and dispersion tensor characterizing solute transport in hierarchical porous media in which diffusion and adsorption/desorption occur in the micro-porous stationary phase.

The method, grounded on exact local and integral moments [71], [72], [73], [74], represents a generalization of the earlier work of Venditti et al. [75] in which a non-porous adsorbing stationary phase was considered.

The article is organized as follows. Section 2 introduces the general transport problem under investigation. Sections 3 introduces the definitions of exact local and integral moments accounting for the presence of two phases: a fluid (mobile) macro-porous phase and a micro-porous adsorbing stationary phase. Section 3 also presents the transport equations and boundary conditions for the local moments and discusses how the temporal evolution of the effective velocity V1(t) and dispersion coefficient D1,1(t) along the main flow direction can be evaluated on the basis of the time-dependent lower-order moments. Section 4 derives the asymptotic expression for the dispersion properties. A very compact expression for the dispersion coefficient D∞1,1 along the main fluid-flow direction is presented, clearly showing that D∞1,1 is controlled by the square norm of the gradients of the stationary b-fields, that are the basic quantities to be estimated for the evaluation of the dispersion tensor in asymptotic conditions. Section 6 focuses on the comparison between results obtained via DNS and with the exact-moment approach, in transient and asymptotic conditions. Specifically, three case studies are considered: 1) solute dispersion in a 2-d slit with micro-porous walls; 2) solute dispersion in a 2-d triangular array of porous pillars with circular/elliptical shape; 3) solute dispersion in a 3-d face-centered cubic array of spheres. In all the three cases, the macro-porosity ε is set to 0.4 as the typical packed-bed porosity. Lastly, Section 7 focuses on the 3-d version of the 2-d chromatographic column consisting of an array of porous pillars. The influence on the separation performance of the no-slip impermeable top/bottom walls (needed in reality to close-off the pillar array) is investigated in detail. It was found that there is no appreciable difference between the 2-d and the 3-d h curves. A physical explanation of this finding is proposed, supported by direct analysis of the b-fields.

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