Volume 33, Issue 7, May 1999, Pages 1579–1592

Review paper

Flocculation modelling: a review

  • a School of Water Sciences, Cranfield University, Bedford, UK
  • b Research and Process Development, Yorkshire Water, Bradford, UK

Abstract

The modelling of the flocculation process is reviewed. Recent developments in this area are discussed with reference to the classical analytical expression of Smoluchowski defining collision frequency and originally published in 1917. The constraints imposed by six principal assumptions made by Smoluchowski are considered individually, with the key models that have been developed to address specific limitations discussed in detail. These assumptions comprise: (1) all particle collisions lead to attachment, (2) fluid motion is limited to laminar shear, (3) particles are monodispersed (i.e. all of them are the same size), (4) no breakage of flocs occurs, (5) all particles are spherical in shape and remain so after collision and (6) collisions take place only between two particles. The discussion incorporates an examination of particle dynamics (i.e. rectilinearity vs curvilinearity), particle surface chemistry (van der Waals attraction and electrostatic repulsion), mixing parameters (mixing intensity and the Camp number) and the key floc growth parameter of fractal dimension D. In doing so limitations of modernised theories are identified. It is concluded that constraints imposed on the interpretation of models based on microscopic aspects of the system, pertaining mainly to those phenomena presiding at the particle:solution interface, severely restrict their application in real systems. The more recent microscopic approach based on characterisation of the system through determination of the fractal dimension as a function of time offers the opportunity of a simpler yet more representative modelling, but none-the-less, currently relies on empirical measurement using fairly sophisticated experimental techniques.

Keywords

  • flocculation;
  • coagulation;
  • modelling;
  • fractal;
  • particles

1. Nomenclature

a=radius of primary particle, L
D=fractal dimension
di=diameter of particle i, L
k=Boltzmann constant (M L2 T−2 K−1)
G=local root-mean-square velocity gradient (T−1)
G*=global root-mean-square velocity gradient (T−1)
ni=concentrations of particles of size i (L−3)
Nt=total concentration of particles at time t (L−3)
nv(t)=concentration of particles of volume v at time t (L−3)
T=absolute temperature (K)
v=particle volume (L3)
α=collision efficiency
β(i, j)=rate of collision between particles of size i and j (L3 T−1)
ε=local rate of energy dissipation (L2 T−3)
ε*=global rate of energy dissipation (L2 T−3)
φ=solid fraction of particles
j=total volume of aggregates (L3)
y=self-similar size distribution function
κ=aggregate permeability (L2)
λ=Kolmogorov microscale (L)
μ=viscosity of water (M L−1 T−1)
Full-size table

2. Introduction

The mathematical representation of flocculation, i.e. the process whereby destabilised suspended particles are aggregated, has conventionally been based on considering the process as two discrete steps: transport and attachment. The transport step, leading to the collision of two particles, is achieved by virtue of local variations in fluid/particle velocities arising through (a) the random thermal “Brownian” motion of the particles (perikinetic flocculation), (b) imposed velocity gradients from mixing (orthokinetic flocculation) and (c) differences in the settling velocities of individual particles (differential sedimentation). Attachment is then contingent upon a number of short range forces largely pertaining to the nature of the surfaces themselves.

The two precepts are most succinctly expressed mathematically as a rate of successful collision between particles of size i and j:

equation1
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where α is the collision efficiency, β(i, j) is the collision frequency between particles of size i and j, and ni, nj are the particle concentrations for particles of size i and j, respectively.

The collision frequency β is a function of the mode of flocculation, i.e. perikinetic, orthokinetic or differential sedimentation. The collision efficiency, α (taking values from 0 to 1), is a function of the degree of particle destabilisation: the greater the degree of destabilisation, the greater the value of α. Thus, in effect, β is a measure of the transport efficiency leading to collisions, whilst α represents the percentage of those collisions leading to attachment.

Nearly all flocculation models are based upon this one fundamental equation. The values of the parameters α and β are dependent upon a large number of factors ranging from the nature of the particles to the method of destabilisation and the prevailing flow regime during flocculation. Much of the research in flocculation modelling has been directed at establishing equations and specific values for these two parameters. It is important, however, not to forget the importance of the terms ni and nj in the equation, as the overall rate always increases with particle concentration.

The interpretation of α and β given above implies that the two parameters are independent of one another. However, there is a second interpretation of α and β which makes the distinction between them less clear cut. One could consider α, besides allowing for the degree of particle destabilisation, to be an experimental correction factor compensating for weaknesses in the theoretical representation of β, such that values for α are no longer confined to be between 0 and 1.

3. Classical expressions

The first major attempt at modelling the flocculation process was made by Smoluchowski (1917). Since the equations in Smoluchowski's model have formed the core of almost all subsequent research into flocculation modelling, subsequent developments can be considered with specific reference to each of the assumptions made by Smoluchowski.

The basic equation developed by Smoluchowski is given by

equation2
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Subscripts i, j and k represent discrete particle sizes. The first term on the right hand side defines the increase in particles of size k by flocculation of two particles whose total volume is equal to the volume of a particle of size k. The second term on the right hand side describes the loss of particles of size k by virtue of their aggregation with other particle sizes. The factor of one half in front of the first term on the right hand side ensures that over the summation the same collision is not counted twice. The overall equation thus defines the rate of change in the number concentration of particles of size k.