Politecnico di Torino - Dipartimento di Scienza dei Materiali ed Ingegneria Chimica

Ph.D. Crina Gudelia Costea Advisor: prof. Marco Vanni

Introduction

Morphology of aggregates

Porosity and permeability of aggregates

Method of reflections

Results

Conclusions

Introduction

Hydrodynamics of aggregates

Modeling processes

sedimentation

flotation spray drying

motion of blood cellsagglomeration

to evaluate 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

The aim of work

Introduction

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 anonintegral dimension

The structure of a fractal aggregate is characterised by its fractal dimension, Df

The fractal dimension varies from 1 to 3, with a value of 3 corresponding to a homogeneous structure

Fractal aggregates have two important properties: self-similarity and a power law behavior

1. The essence of self-similar aggregates is that there is a continuum of ‘level’ from large-scale structures down to individual primary particles.

2. Properties like mass and therefore, density, obeys a power law relation,

fDRM α3−= fD

E Rρ

Morphology of aggregates

Morphology of aggregates

The magnitude of fractal dimension is determined by the mechanismof growth. There are two mainly type of mechanism:

diffusion-limited aggregation (DLA) when all collisions lead to apermanent bond

reaction-limited aggregation (RLA) when only a fraction of thecontacts results in irreversible adhesion between the colliding aggregates

There are two mainly type of collision:

particle-cluster mechanism

cluster-cluster mechanism

Morphology of aggregates

radius of aggregate, R

radius of gyration, r

radius of primary particle, a

porosity,

permeability,

An aggregate structure

Fractal aggregates can be characterised by the following parameters:

Porosity and permeability of aggregates

The porosity can be defined as the fractional void space withrespect to the bulk volume constituted by interconnecting pores.

The porosity of the aggregate can be calculated in terms of the number of primary particles in the aggregate, N, volume of a primary particle, Vp, and the volume of the aggregate, Va, as follows:

a

p

V

VN ⋅=−ε1

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

Method of reflections

was inaugurated by Smoluchowski (1911) and continued by Happel and Brenner (1962)

provide a systematic scheme of successive iterations, wherebythe boundary-value problem may be solved to any degree ofapproximation by considering boundary conditions associated withone particle at a time

gives detailed information on the flow field inside aggregate and on the forces applied to each particle

Method of reflections

For a system of n spherical particles we have to solve the following system of equations:

∞→→==

====

=⋅∇

∇=∇

rv

nrUv

brUv

arUv

v

pv

n

b

a

0

..............................

0

12

µ

boundary conditions

Stokes’ equation

continuity equation

Method of reflections

According to the method of reflections the system can be solved as follows:

++++=++++=

)4()3()2()1(

)4()3()2()1(

ppppp

vvvvv

where, (v(j), p(j)) - separately satisfies the equations of motion and vanishesat infinity

the local velocity and pressure fields may be decomposed into a sum offields because the equations of motion and boundary conditions are linear,

further each of these pairs are subdivided into a finite sum of terms, (vk

(j),pk(j)), also satisfying the governing differential equations and

vanishing at infinity.

Method of reflections

Following this algorithm we can obtain the force exerted by fluid on an aggregate.

aonUv a=)1(

Let say that we have an aggregate formed by a,b,…,n particles, if we take the particle a, and define (v(1), p(1)) by the boundary condition,

The “reflection” of this field from particle b is then defined by the boundary condition,

bonvUv bb)1()2( −=

In general, the reflection of v(1) from any of the n-1 particles is defined by

),,,()1()2( ncbkkonvUv kk =−=

Thus, the reflection of v(1) from all the remaining n-1 particles is given approximately,

=

=n

bkkvv )2()2(

Results

Using FORTRAN software we have implemented a program based on the

algorithm provide by method of reflections. input parameters

• radius of primary particle, a

• radius of aggregate, R

• number of primary particles, N

• undisturbed velocity field of the fluid, U, V, respectively, W

• coordinates centers of particles, xo, yo, zo

output parameters

• forces exerted by fluid on each particle, fi

• force exerted by fluid on the aggregate, F, (F = Σ fi )

A. well-ordered aggregate structures

B. random aggregate structures

C. fractal aggregate structures

Investigations have been carried out to evaluate the drag on:

Results

A 2D sectionthrough asimple cubic structure

A 2D sectionthrough a facecentered cubic structure

A. The structure of the well-ordered aggregates investigated

R

(a)

R

(c)

(b)

(d)

Results

The fluid vector velocities through the well-ordered aggregate structures

a) SC-structure (729 particles) b) sphere SC-structure (306 particles)

c) FCC-structure (2457 particles) d) sphere FCC-structure (1062 particles)

undisturbed fluid velocity

Results

a) SC-structure (729 particles) b) FCC-structure (2457 particles)

The force exerted by fluid on the central particles of the aggregate

Results

B. Random aggregate structures

b) The fluid vector velocities through therandom aggregate

a) The structure of a random aggregate (987 particles)

Results

b) A section through a fractal aggregate structurea) The structure of a fractal aggregate (1292 particles)

d) The fluid vectors velocities through a fractalaggregate structure

c) The number of primary particles vs. the aggregateradius

Df = 2.79

C. Fractal aggregate structure

Results

Calculation of drag force

FD=6πµRUΩ

where,

from literature, assuming homogeneous porous structure

−+

−=Ω

βββ

βββ

tanh132

tanh12

2

2

κβ R=

3/5

23/53/12

)1(23(

)1(32

)1(9

2

)1(93

)1(18

ε

εεεε

κ−+

−−−+−−

−=

p

Happel

d

The drag force obtained from literature vs. the drag force obtained fromprogram:

a) well-ordered aggregate structures

b) random aggregate structures

Results

a) b)

N = cst

R ≠ cst

N ≠ cst

R = cst

Conclusions

Using the method of reflections, the drag force was obtained ondifferent 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 thepermeability.

In the future we intend to investigate more different structure ofaggregates and use another way to calculate the porosity and thepermeability of the aggregates, in order to be able to find a relationwhich can predict where and how the aggregates are broken up.