Collisional motion of a granular material composed of rough inelastic spheres is analysed on the basis of the kinetic Boltzmann–Enskog equation. The Chapman–Enskog method for gas kinetic theory is modified to derive the Euler-like hydrodynamic equations for a system of moving spheres, possessing constant roughness and inelasticity. The solution is obtained by employing a general isotropic expression for the singlet distribution function, dependent upon the spatial gradients of averaged hydrodynamic properties. This solution form is shown to be appropriate for description of rapid shearless motions of granular materials, in particular vibrofluidized regimes induced by external vibrations.The existence of the hydrodynamic state of evolution of a granular medium, where the Euler-like equations are valid, is delineated in terms of the particle roughness, β, and restitution, e, coefficients. For perfectly elastic spheres this state is shown to exist for all values of particle roughness, i.e. − 1≤β≤1. However, for inelastically colliding granules the hydrodynamic state exists only when the particle restitution coefficient exceeds a certain value em(β)< 1.In contrast with the previous results obtained by approximate moment methods, the partition of the random-motion kinetic energy of inelastic rough particles between rotational and translational modes is shown to be strongly affected by the particle restitution coefficient. The effect of increasing inelasticity of particle collisions is to redistribute the kinetic energy of their random motion in favour of the rotational mode. This is shown to significantly affect the energy partition law, with respect to the one prevailing in a gas composed of perfectly elastic spheres of arbitrary roughness. In particular, the translational specific heat of a gas composed of inelastically colliding (e = 0.6) granules differs from its value for elastic particles by as much as 55 %.It is shown that the hydrodynamic Euler-like equation, describing the transport and evolution of the kinetic energy of particle random motion, contains energy sink terms of two types (both, however, stemming from the non-conservative nature of particle collisions) : (i) the term describing energy losses in incompressibly flowing gas; (ii) the terms accounting for kinetic energy loss (or gain) associated with the work of pressure forces, leading to gas compression (or expansion). The approximate moment methods are shown to yield the Euler-like energy equation with an incorrect energy sink term of type (ii), associated with the ‘dense gas effect’. Another sink term of the same type, but associated with the energy relaxation process occurring within compressed granular gases, was overlooked in all previous studies.The speed of sound waves propagating in a granular gas is analysed in the limits of low and high granular gas densities. It is shown that the particle collisional properties strongly affect the speed of sound in dense granular media. This dependence is manifested via the kinetic energy sink terms arising from gas compression. Omission of the latter terms in the evaluation of the speed of sound results in an error, which in the dense granular gas limit is shown to amount to a several-fold factor.
Flow fields within spatially periodic arrays of cylinders arranged in square and hexagonal lattices are calculated, with microscale Reynolds number ranging between zero and 200, employing a finite element numerical scheme. The terminology of an ‘‘apparent permeability’’ is introduced to establish a relationship existing between mean velocity and macroscopic pressure gradient characterized by a finite Reynolds number flow. In contrast with the low Reynolds number ‘‘true ’’ permeability, the apparent permeability is shown here to generally depend upon the direction of the applied pressure gradient, owing to nonlinearities existing within the local fluid motion. The orientation-dependent permeabilities of both square and hexagonal monodisperse arrays are observed to diminish with increasing Reynolds number. Similar behavior is also observed for a bidisperse square array, though the apparent permeability of the latter is shown less sensitive to Darcy velocity orientation at large Reynolds numbers in comparison to the corresponding monodisperse square array, for all cylinder concentrations examined.
Effective thermophysical properties of ceramic materials (mainly insulating materials) with porosity (II) >30% are reviewed. Nonmonotonic pressure and temperature dependences of the effective thermal conductivity (X) are analyzed, based on the ceramic microstructure (pores, cracks, and grain boundaries present in many industrial refractories) and several heat‐transfer mechanisms in composite multiphase materials. These mechanisms include heat conduction in solid and gas phases, thermal radiation, gas convection, and the mechanism originating from intrapore chemical conversion processes accompanied by gas emission. For high temperatures, λ of porous insulations is governed by thermal radiation. Contact‐heat‐barrier resistances play a less‐important role in highly porous ceramics than in their dense counterparts. This underlies a weaker pressure dependence at low temperatures (<500°C) of λ of the majority of industrial insulating materials than in dense materials possessing microcracks and small pores in the grain‐boundary region. For high gas pressure, λ of porous insulating materials is governed by free convective‐gas motion. For low gas pressures (normally <1 kPa), where heat transfer in pores occurs in the free‐molecular regime, X is controlled by the pressure‐dependent mean free path of gas molecules in pores. A classification of the porous material structure and thermophysical properties is proposed, based on the geometric model described in Part 1 of this series.
Taylor dispersion of a passive solute within a fluid flowing through a porous medium is characterized by an effective or Darcy scale, transversely isotropic dispersitivity D*, which depends upon the geometrical microstructure, mean fluid velocity, and physicochemical properties of the system. The longitudinal, D~'t and lateral, D* dispersivity components for two-dimensional, spatially periodic arrays of circular cylinders are here calculated by finite element techniques. The effects of bed voidage, packing arrangement, and microscale Prclet and Reynolds numbers upon these dispersivities are systematically investigated.The longitudinal dispersivity component is found to increase with the microscale Prclet number at a rate less than Pe 2. This accords with previous calculations by Eidsath et al. (1983), although the latter calculations were found to yield significantly lower longitudinal dispersivities than those obtained with the present numerical scheme. With increasing Prclet number, a Pe 2 dependence is, however, approached asymptotically, particularly for square cylindrical arrays -owing to the creation of a linear streamline zone between cylinders.Increasing tortuosity of the intercellular flow pattern reduces the longitudinal dispersivity component and enhances the lateral component. Longitudinal dispersivities for square and hexagonal arrays are found to be quite similar at high porosities; yet they diverge dramatically from one another with decreasing porosity. The longitudinal dispersivity is found to increase markedly with increasing Reynolds number. Comparison of this longitudinal dispersivity with available experimental results shows that/)~'1 experimentally measured for three-dimensional arrays of spheres may be correlated by the present two-dimensional model by an appropriate choice of the array's packing arrangement. In general, the calculated dispersivities were found to be sensitive to the bed packing arrangement and apparently no rationale exists for choosing any one particular geometric microstructure over another for a comparison with existing experimental data. It is thus concluded that existing experimental data pertaining to three-dimensional beds of spherical particles cannot rationally provide a basis for verification of two-dimensional, circular cylindrical dispersion models.The finite-element scheme employed in this work was tested in the purely diffusive, nonflow limit by calculating the composite diffusivities of square cylindrical arrays for different volume fractions and various dispersed solid-continuous phase diffusivity ratios, subsequently comparing these with existing analytical results. An additional test was provided by comparing calculated with analytical axial dispersivities for transport of a dissolved solute in a Poiseuille flow between two parallel plates. O. Notationa B(r), B(r) D Dd, Dc f(r) h il, i2 Ii, 12 !1,12 hi, n2 Pe = 2a V/D 2a12 e Pep-D 1-e lr R Rn Re = 2ap V/# Sp t v -v(r)
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