Ph.D. Crina Gudelia Costea

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1 Politecnico di Torino - Dipartimento di Scienza dei Materiali ed Ingegneria Chimica Ph.D. Crina Gudelia Costea Adisor: prof. Marco Vanni

2 Introduction Morphology of aggregates Porosity and permeability of aggregates Method of reflections Results Conclusions

3 Introduction Hydrodynamics of aggregates Modeling processes sedimentation flotation agglomeration motion of blood cells spray drying

4 Introduction The aim of work to ealuate in detail the hydrodynamics inside the aggregates in order to calculate the drag force the force exerted on each particle to use these information for analysing the break-up phenomena

5 Morphology of aggregates Aggregates are recognized as fractal objects (Meakin, 1988) What are fractal aggregates? Fractal aggregates can be defined as a disordered systems with a nonintegral dimension The structure of a fractal aggregate is characterised by its fractal dimension, D f The fractal dimension aries from 1 to 3, with a alue of 3 corresponding to a homogeneous structure Fractal aggregates hae two important properties: self-similarity and a power law behaior

6 Morphology of aggregates 1. The essence of self-similar aggregates is that there is a continuum of leel from large-scale structures down to indiidual primary particles. 2. Properties like mass and therefore, density, obeys a power law relation, M α R D f ρ E R D f 3

7 Morphology of aggregates The magnitude of fractal dimension is determined by the mechanism of growth. There are two mainly type of mechanism: diffusion-limited aggregation (DLA) when all collisions lead to a permanent bond reaction-limited aggregation (RLA) when only a fraction of the contacts results in irreersible adhesion between the colliding aggregates There are two mainly type of collision: particle-cluster mechanism cluster-cluster mechanism

8 Morphology of aggregates Fractal aggregates can be characterised by the following parameters: radius of aggregate, R radius of gyration, r radius of primary particle, a porosity, permeability, An aggregate structure

9 Porosity and permeability of aggregates The porosity can be defined as the fractional oid space with respect to the bulk olume constituted by interconnecting pores. The porosity of the aggregate can be calculated in terms of the number of primary particles in the aggregate, N, olume of a primary particle, V p, and the olume of the aggregate, V a, as follows: 1 ε N V V a p

10 Porosity and permeability of aggregates Correlations for calculating the permeability: Kozeny-Carman model Dilute limit model Brinkman s model Happel s model Howells, Hinch, and Kim and Russel s model Neale and Nader s model

11 Method of reflections was inaugurated by Smoluchowski (1911) and continued by Happel and Brenner (1962) proide a systematic scheme of successie iterations, whereby the boundary-alue problem may be soled to any degree of approximation by considering boundary conditions associated with one particle at a time gies detailed information on the flow field inside aggregate and on the forces applied to each particle

12 Method of reflections For a system of n spherical particles we hae to sole the following system of equations: r n r U b r U a r U p n b a µ boundary conditions Stokes equation continuity equation

13 Method of reflections According to the method of reflections the system can be soled as follows: the local elocity and pressure fields may be decomposed into a sum of fields because the equations of motion and boundary conditions are linear, p (1) p (1) + + (2) p (2) + + (3) p + (3) + (4) p + (4) + where, ( (j), p (j) ) - separately satisfies the equations of motion and anishes at infinity further each of these pairs are subdiided into a finite sum of terms, ( k (j),p k (j) ), also satisfying the goerning differential equations and anishing at infinity.

14 Method of reflections Let say that we hae an aggregate formed by a,b,,n particles, if we take the particle a, and define ( (1), p (1) ) by the boundary condition, (1) U a on a The reflection of this field from particle b is then defined by the boundary condition, (2) b U b In general, the reflection of (1) from any of the n-1 particles is defined by (1) on b (2) k U k (1) on k ( k b, c,, n) Thus, the reflection of (1) from all the remaining n-1 particles is gien approximately, (2) k b Following this algorithm we can obtain the force exerted by fluid on an aggregate. n (2) k

15 Results Using FORTRAN software we hae implemented a program based on the algorithm proide by method of reflections. input parameters radius of primary particle, a radius of aggregate, R number of primary particles, N undisturbed elocity field of the fluid, U, V, respectiely, W coordinates centers of particles, x o, y o, z o output parameters forces exerted by fluid on each particle, f i force exerted by fluid on the aggregate, F, (F Σ f i ) Inestigations hae been carried out to ealuate the drag on: A. well-ordered aggregate structures B. random aggregate structures C. fractal aggregate structures

16 Results A. The structure of the well-ordered aggregates inestigated R A 2D section through a simple cubic structure (a) (b) R A 2D section through a face centered cubic structure (c) (d)

particles) b) sphere SC-structure (306 particles) fluid

17 Results The fluid ector elocities through the well-ordered aggregate structures undisturbed a) SC-structure (729 particles) b) sphere SC-structure (306 particles) fluid elocity c) FCC-structure (2457 particles) d) sphere FCC-structure (1062 particles)

18 Results The force exerted by fluid on the central particles of the aggregate a) SC-structure (729 particles) b) FCC-structure (2457 particles)

19 Results B. Random aggregate structures a) The structure of a random aggregate (987 particles) b) The fluid ector elocities through the random aggregate

$Fractal aggregate structure a) The structure of a fractal aggregate$ $(1292 particles) b) A section through a fractal aggregate structure D$

20 Results C. Fractal aggregate structure a) The structure of a fractal aggregate (1292 particles) b) A section through a fractal aggregate structure D f 2.79 c) The number of primary particles s. the aggregate radius d) The fluid ectors elocities through a fractal aggregate structure

21 Results Calculation of drag force F D 6πµRUΩ where, from literature, assuming homogeneous porous structure + Ω β β β β β β tanh tanh κ β R 3 5 / / 3 1/ 2 ) 2(1 (3 ) 3(1 2 ) 9(1 2 ) 9(1 3 ) 18(1 ε ε ε ε ε κ + + p Happel d

22 Results N cst R cst N cst R cst a) b) The drag force obtained from literature s. the drag force obtained from program: a) well-ordered aggregate structures b) random aggregate structures

23 Conclusions Using the method of reflections, the drag force was obtained on different aggregates structures. The results are quite good for well-ordered aggregates structures but for random aggregate structures there is a difference. This could be caused by the way in which was calculate the porosity and the permeability. In the future we intend to inestigate more different structure of aggregates and use another way to calculate the porosity and the permeability of the aggregates, in order to be able to find a relation which can predict where and how the aggregates are broken up.

Porosity-Permeability Relations in Granular, Fibrous and Tubular Geometries

Porosity-Permeability Relations in Granular, Fibrous and Tubular Geometries November 21 st Feng Xiao, Xiaolong Yin Colorado School of Mines American Physical Society, 64 th Annual DFD Meeting, Baltimore,