2 [ ] dS 1 2 Theory 2 The theory for the flow - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS MODELLING PARTICULATE FLOW DURING IMPREGNATION OF DUAL-SCALE FABRICS V. Frishfelds and T.S. Lundström* Division of Fluid Mechanics, Luleå University of Technology, SE-97187 Luleå, Sweden *Corresponding author( HTU staffan.lundstrom@ltu.se UTH ) Keywords : Manufacturing, particles, impregnation, dual-scale fabrics, modelling P Summary 2.1 Flow field Filtration of particles during impregnation of dual- Flow perpendicular to two-dimensional systems of scale fabrics is studied numerically for a number of fibres clustered in bundles is considered. Since the configurations with a previously derived model. The fibres themselves are impermeable to the fluid flow, initial position and size of the particles are varied. the stream function at the surface of each fibre is The main result is that structural composites can be constant according to ψ = ψ R i R , where i = 1… n is the tailor-made as to additional properties by such an index of the fibres. The difference in stream function approach. between any two fibres is determined by the flow rate in the gap between the two fibres in question. 1 Introduction To derive the distribution of flow, the system is Fabrics used in modern composite materials have divided into n parts with a modified version of often dual-scale porosity, < 10 μ m inside the fibre Voronoi diagram so that each part contains one bundles and > 100 μ m between the bundles. The fibre, see [8-10]. The fibres are assumed to be detailed geometry of such fabrics is, for instance, of stationary and non-slip boundary conditions are importance for applications when the resin is doped applied. At the crossing between the centre to centre with particles to create multifunctional composites, lines of fibres i and j with the Voronoi lines the [1-3]. The added particles can, for example, enhance value of the stream function and the vorticity the fire resistance, toughen the material, introduce      are denoted  R ij R and  R ij ω ( ψ ) R , RR 0 RR 0 electrical conductivity and shielding properties to respectively, see Fig. 1. Using this definition the the material [4-7]. In order to achieve satisfactory quadratic average of vorticity in an area S R ij R at fibre i properties of these functional materials, it is vital to adjacent to fibre j may be written as: have a known spatial distribution of the functionality    (1) throughout the material and it is of great interest to   ij 0 i C develop methods to control the distribution of ij ij 2 d particles during manufacturing. With a controlled ij 0 particle distribution the functionality sought for can where A R ij R originate from the average vorticity in a be optimized without sacrificing other properties. small area S R ij R ; d R ij 0 R is the distance between fibre i and The model here described can be used to increase the Voronoi line that separates fibres i and j ;  the understanding of particulate flow during denotes the case of equal sized fibres i and j . The manufacturing of structural composites as shown in total dissipation rate of energy approaches a [8]. We will here continue to demonstrate the minimum, [11], so the following sum over the total capability of the model by studying the filtration of area should be minimised: particles at fully saturated conditions for different size and position distributions of the particles. , (2)       2 [ ] dS 1 2 Theory 2 The theory for the flow field is based on [9] and will where  is the viscosity of the percolating fluid. The shortly be repeated while the background to the total sum is minimized with respect to all discrete motion of the particles was outlined in [10].

values of the stream function and the vorticity. The Computational Fluid Dynamics (CFD). The total force on each fibre is derived from the vorticity simulations are carried out with boundary near the fibres and by accounting for the viscous and conditions representing a well structured normal forces according to : repeatable material with varying geometry as described in [4].    ( 3 )    ij 0 ij   f r τ B , i i ij ij ij 2 d 2.2 Motion of particles j ij 0 Following [8] the fibres are modelled to be stiff and stationary. The particles, in their turn, move with the fluid and redistribute with the flow field according to a quasi-stationary approach. It implies that the where B R ij R : fluid flow and the corresponding Stokes drag force is calculated assuming that the particles at that   arc 2 d        ij 0   moment are stationary implying that inertial effects arc B r   (4)   ij    ij i   n  are negligible but that the may have a major   ij 0 ij ij influence on the flow field. In next step the particles take new positions depending on the Stokesian drag is a dimensionless variable characterising the ratio force on the individual particles. The total force between the average vorticity along the arc at the results from drag and friction according to: border of the fibres and difference of the stream function at the positions indicated, see Fig. 1. The sum of all drag forces equals the driving pressure 2 r difference if wall effects are neglected. Hence, the  fr f k i v , (6) permeability K follows from Darcy law i i 2 r i   . ( 5 )     v K f S where k is coefficient characterising viscous i ij i ij friction.Particles are not allowed to move into each or into fibres. This, however, creates a contact force to another particle or to a fibre. Using the Monte Carlo Metropolis algorithm for relaxation of stresses, this force relaxes to equilibrium. Friction between bodies (particle-particle, particle-fibre) is  R ij0 R ,  R ij0 not included because the bodies are always assumed to be separated by a tiny layer of fluid   R c q R 2 q R 1  R j  R i d R ji0 d R ij0 Notice that we model a fully two-dimensional case S R ij S R ji and there is very little flow between two closely packed particles or between particles being located Fig. 1 –Modified Voronoi diagrams (solid), close to fibres. This differs from a three dimensional Delaunay triangles (dashed), and small triangles set of spherical particles where the flow often can, (dotted) between three fibres. more easily pass. A consequence is that the fluid flow is more dependent on the fibre distribution in a The dimensionless variables A , B in Eqn. 2-5 two-dimensional case. A previous study on two and are obtained from simulations with three dimensional systems shows that the dimension