On the validity of the Carman-Kozeny equation in random fibrous media

The transverse permeability for creeping flow through unidirectional random arrays of fibers/cylinders has been studied numerically using the finite element method (FEM). A modified Carman-Kozeny (CK) relation is presented which takes into account the tortuosity (flow path) and the lubrication effect of the narrow channels. The proposed relation is valid in a wide range of porosities compared to the classical CK equation. The proposed general relationship for the permeability can be utilized for composite manufacturing and also for validation of advanced coarse models for particle-fluid interactions

Micro-Macro relations for flow through random arrays of cylinders

The transverse permeability for creeping flow through unidirectional random arrays of fibers with various structures is revisited theoretically and numerically using the finite element method (FEM). The microstructure at various porosities has a strong effect on the transport properties, like permeability, of fibrous materials. We compare different microstructures (due to four random generator algorithms) as well as the effect of boundary conditions, finite size, homogeneity and isotropy of the structure on the macroscopic permeability of the fibrous medium. Permeability data for different minimal distances collapse when their minimal value is subtracted, which yields an empirical macroscopic permeability master function of porosity. Furthermore, as main result, a microstructural model is developed based on the lubrication effect in the narrow channels between neighboring fibers. The numerical experiments suggest a unique, scaling power law relationship between the permeability obtained from fluid flow simulations and the mean value of the shortest Delaunay triangulation edges (constructed using the centers of the fibers), which is identical to the averaged second nearest neighbor fiber distances. This universal lubrication relation, as valid in a wide range of porosities, accounts for the microstructure, e.g. hexagonally ordered or disordered fibrous media. It is complemented by a closure relation that relates the effective microscopic length to the packing fraction

Carrying capacity of edge-cracked columns under concentric vertical loads

This paper analyses the carrying capacity of edge-cracked columns with different boundary conditions and cross sections subjected to concentric vertical loads. The transfer matrix method, combined with fundamental solutions of the intact columns (e.g. columns with no cracks) is used to obtain the capacity of slender prismatic columns. The stiffness of the cracked section is modeled by a massless rotational spring whose flexibility depends on the local flexibility induced by the crack. Eigenvalue equations are obtained explicitly for columns with various end conditions, from second-order determinants. Numerical examples show that the effects of a crack on the buckling load of a column depend strongly on the depth and the location of the crack. In other words, the capacity of the column strongly depends on the flexibility due to the crack. As expected, the buckling load decreases conspicuously as the flexibility of the column increases. However, the flexibility is a very important factor for controlling the buckling load capacity of a cracked column. In this study an attempt was made to calculate the column flexibility based on two different approaches, finite element and J-Integral approaches. It was found that there was very good agreement between the flexibility results obtained by these two different methods (maximum discrepancy less than 2%). It was found that for constant column flexibility a crack located in the section of the maximum bending moment causes the largest decrease in the buckling load. On the other hand, if the crack is located just in the inflexion point at the corresponding intact column, it has no effect on the buckling load capacity. This study showed that the transfer matrix method could be a simple and efficient method to analyze cracked columns components

FEM-DEM simulation of two-way fluid-solid interaction in fibrous porous media

Fluid flow through particulate media is pivotal in many industrial processes, e.g. in fluidized beds, granular storage, industrial filtration and medical aerosols. Flow in these types of media is inherently complex and challenging to simulate, especially when the particulate phase is mobile. The goals of this paper are twofold: (i) the derivation of accurate correlations for the drag force, taking into account the effect of microstructure, to improve the higher scale macro-models and (ii) incorporating such closures into a “compatible” monolithic multi-phase/scale model that uses a (particle-based) Delaunay triangulation (DT) of space as basis – in future, possibly, involving also multiple fields

Towards unified drag laws for inertial flow through fibrous materials

We give a comprehensive survey of published experimental, numerical and theoretical work on the drag law correlations for fluidized beds and flow through porous media, together with an attempt of systematization. Ranges of validity as well as limitations of commonly used relations (i.e. the Ergun and Forchheimer relations for laminar and inertial flows) are studied for a wide range of porosities. The pressure gradient is linear in superficial velocity, U for low Reynolds numbers, Re, referred to as Darcy’s law. Here, we focus on the non-linear contribution of inertia to the transport of momentum at the pore scale, and explain why there are different non-linear corrections on the market. From our fully resolved finite element (FE) results, for both ordered and random fibre arrays, (i) the weak inertia correction to the linear Darcy relation is third power in U, up to small Re ∼ 1–5. When attempting to fit our data with a particularly simple relation, (ii) a non-integer power law performs astonishingly well up to the moderate Re ∼ 30. However, for randomly distributed arrays, (iii) a quadratic correction performs quite well as used in the Forchheimer (or Ergun) equation, from small to moderate Re. Finally, as main result, the macroscopic properties of random, fibrous porous media are related to their microstructure (arrangement) and porosity. All results (Re < 30) up to astonishingly large porosity, ε ∼ 0.9, scale with Reg, i.e., the gap Reynolds number that is based on the average second nearest neighbour (surface to surface) distances. This universal result is given as analytical closure relation, which can readily be incorporated into existing (non)commercial multi-phase flow codes. In the transition regime, the universal curve actually can be fitted with a non-integer power law (better than ∼1% deviation), but also allows to define a critical Regc ∼ 1, below which the third power correction holds and above which a correction with second power fits the data considerably better

Fibrous random materials: From microstructure to macroscopic properties

Fibrous porous materials are involved in a wide range of applications including composite materials, fuel cells, heat exchangers and (biological)filters. Fluid flow through these materials plays an important role in many engineering applications and processes, such as textiles and paper manufacturing or transport of (under)ground water and pollutants. While most porous materials have complex geometry, some can be seen as two-dimensional particulate/fibrous systems, in which we introduce several microscopic quantities, based on Voronoi and Delaunay tessellations, to characterize their microstructure. In particular, by analyzing the topological properties of Voronoi polygons, we observe a smooth transition from disorder to order, for increasing packing fraction. Using fully resolved finite element (FE) simulations of Newtonian, incompressible fluid flow perpendicular to the fibres, the macroscopic permeability is calculated in creeping flow regimes. The effect of fibre arrangement and local crystalline regions on the macroscopic permeability is discussed and the macroscopic property is linked to the microscopic structural quantities

Upscaling and microstructural analysis of the flow-structure relation perpendicular to random, parallel fiber arrays

Owing largely to multiscale heterogeneity in the underlying fibrous structure, the physics of fluid flow in and through fibrous media is incredibly complex. Using fully resolved finite element (FE) simulations of Newtonian, incompressible fluid flow perpendicular to the fibers, the macroscopic permeability is calculated in the creeping flow regime for arrays of random, ideal, perfectly parallel fibers. On the micro-scale, several order parameters, based on Voronoi and Delaunay tessellations, are introduced to characterize the structure of the randomly distributed, parallel, non-overlapping fiber arrays. In particular, by analyzing the mean and the distribution of the topological and metrical properties of Voronoi polygons, we observe a smooth transition from disorder to (partial) order with decreasing porosity, i.e., increasing packing fraction. On the macro-scale, the effect of fiber arrangement and local crystalline regions on the macroscopic permeability is discussed. For both permeability and local bond orientation order parameter, the deviation from a fully random configuration can be well represented by an exponential term as function of the mean gap width, which links the macro- and the micro-scales. Finally, we verify the validity of the, originally, macroscopic Darcy's law at various smaller length scales, using local Voronoi/Delaunay cells as well as uniform square cells, for a wide range of porosities. At various cell sizes, the average value and probability distributions of macroscopic quantities, such as superficial fluid velocity, pressure gradient or permeability, are obtained. These values are compared with the macroscopic permeability in Darcy's law, as the basis for a hierarchical upscaling methodology