The Maxwell equation relates the permeability P of a dispersion of particles A (modeled as hard congruent non-interacting spheres) in a continuous matrix B, to the ratio of the permeabilities of the components α=PA/PB and the corresponding volume fractions vA, vB (=1−vA). Originally devised for use at vA→0, the validity of the said equation for practical purposes was shown, by analytical methods, to extend to congruent weakly interacting spheres packed in regular cubic lattices, in the low to medium vA range; while the equation itself extends further up to the correct upper limit P=PA at vA=1. Replacing a simple cubic (s.c.) lattice of spheres with an identical lattice of cubes, overcomes this limitation (because cubes can pack up to vA=1) but only at the expense of losing analytical mathematical tractability. However, an analytical limiting form of the s.c. lattice of cubes could be derived near the limit vA→1 and shown to agree with the Maxwell equation. Even so, a large gap of unexplored territory was obviously still left in the remaining medium to high vA range. The gap in missing data was successfully filled in Part I by the use of a numerical computation approach, which showed that the result obtained at vA→1 is valid for practical purposes throughout the medium to high vA range; thus confirming the applicability of the Maxwell equation to sc lattices of cubes (and correspondingly of the Wiener equation in the case of anisometric particles) in this vA region. The gap in analytical treatment is filled to a large extent, in the present companion paper, by the development of an approach combining two different analytical limiting forms of the s.c. lattice-of-cubes model (as well as comparable limiting forms applicable to anisometric particles treated by the Wiener equation). We show here that this approach (referred to in the main text as the BP II model) provides (to a high degree of approximation in the practically important region of 0≤α≤10) a satisfactory analytical theoretical basis for meaningful application of the Maxwell equation in the higher vA range, comparable with that afforded by the existing lattice-of-spheres analytical treatments in the lower vA region.
Papadokostaki, K., Minelli, M., Doghieri, F., Petropoulos, J. (2015). A fundamental study of the extent of meaningful application of Maxwell's and Wiener's equations to the permeability of binary composite materials. Part II: A useful explicit analytical approach. CHEMICAL ENGINEERING SCIENCE, 131, 353-359 [10.1016/j.ces.2015.03.031].
A fundamental study of the extent of meaningful application of Maxwell's and Wiener's equations to the permeability of binary composite materials. Part II: A useful explicit analytical approach
MINELLI, MATTEO;DOGHIERI, FERRUCCIO;
2015
Abstract
The Maxwell equation relates the permeability P of a dispersion of particles A (modeled as hard congruent non-interacting spheres) in a continuous matrix B, to the ratio of the permeabilities of the components α=PA/PB and the corresponding volume fractions vA, vB (=1−vA). Originally devised for use at vA→0, the validity of the said equation for practical purposes was shown, by analytical methods, to extend to congruent weakly interacting spheres packed in regular cubic lattices, in the low to medium vA range; while the equation itself extends further up to the correct upper limit P=PA at vA=1. Replacing a simple cubic (s.c.) lattice of spheres with an identical lattice of cubes, overcomes this limitation (because cubes can pack up to vA=1) but only at the expense of losing analytical mathematical tractability. However, an analytical limiting form of the s.c. lattice of cubes could be derived near the limit vA→1 and shown to agree with the Maxwell equation. Even so, a large gap of unexplored territory was obviously still left in the remaining medium to high vA range. The gap in missing data was successfully filled in Part I by the use of a numerical computation approach, which showed that the result obtained at vA→1 is valid for practical purposes throughout the medium to high vA range; thus confirming the applicability of the Maxwell equation to sc lattices of cubes (and correspondingly of the Wiener equation in the case of anisometric particles) in this vA region. The gap in analytical treatment is filled to a large extent, in the present companion paper, by the development of an approach combining two different analytical limiting forms of the s.c. lattice-of-cubes model (as well as comparable limiting forms applicable to anisometric particles treated by the Wiener equation). We show here that this approach (referred to in the main text as the BP II model) provides (to a high degree of approximation in the practically important region of 0≤α≤10) a satisfactory analytical theoretical basis for meaningful application of the Maxwell equation in the higher vA range, comparable with that afforded by the existing lattice-of-spheres analytical treatments in the lower vA region.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
