### 1. Introduction

### 2. Structure Formation Processes in Dispersed Systems

### 2.1. Colloidal Stability and Interfacial Forces

*Born repulsion*) [31] repulsive force due to the electrical double layer and Van der Waals attractive force of the dispersion.

*V*

*or the so-called*

_{T}*Gibbs interaction energy*as shown in the energy profile in Fig. 1 [32, 28, 33–35, 25] which, if overcome by the kinetic energy of the moving particles, will result in aggregation. This is expressed mathematically in Eqs. (1–2), where

*V*

*,*

_{A}*V*

*, ζ refers to the energy barrier due to Van der Waals attraction and electrostatic repulsion and zeta potential respectively. In the case of submicron particles, the kinetic energy is derived primarily from Brownian diffusion and less from hydrodynamic (fluid shear) or gravitational forces (differential settling). However, for larger or coarser particles, hydrodynamic interactions such as turbulence or fluid shear and gravitational forces imparts much of the kinetic energy needed to overcome the flocculation barrier [36, 33, 37]. A critical shear rate is required to overcome the energy barrier in order to allow flocculation and floc formation, and this is dependent on the surface charge and the size of particles. In general, the higher the charge on the particles, and the smaller the particle size, the higher the shear rates that are required to bring about flocculation [21, 35, 38].*

_{R}### 2.2. Kinetics of Fine Particle Aggregation

*k*

*can be represented by the expression in Eq. (3) [31, 45].*

^{f}*k*as they aggregate with respect to time in terms of the collision frequency or rate function

*β*originally for Brownian diffusion and laminar or uniform shear flow, and later for differential settling assuming binary collisions between the particles [40, 45, 47–49]. Although the Smoluchowski equation has been modified in line with recent findings to include several other parameters, the general form of the expression is presented in Eq. (4). The quantity

*β*

*is the collision frequency or rate function for collisions between ith and jth sized particles while*

_{ij}*α*

*is the dimensionless collision efficiency factor. The collision frequency depends on the transport mechanisms of Brownian motion, fluid shear and differential settling, whereas the collision efficiency is a function of the degree of particle destabilization and it gives the probability of collision leading to attachment with values ranging from 0 to 1 [47, 50–53].*

_{ij}##### (4)

$$\frac{d{n}_{k}}{dt}=\frac{1}{2}\sum _{i+j=k}\beta (i,j){n}_{i}\hspace{0.17em}{n}_{j}-\sum _{i=1}^{\infty}\beta (i,k)\hspace{0.17em}{n}_{i}\hspace{0.17em}{n}_{k}$$*k*by flocculation of two particles whose total volume is equal to the volume of a particle of size

*k*, while the second term on the right hand side describes the loss of particles of size

*k*due to aggregation with particles of other sizes. The general form of the equation expresses the rate of change in the number concentration of particles of size

*k*. In arriving at Eq. (4), Smoluchowski made a number of key assumptions listed below:

### 2.3. Flocculation Transport Processes

*β*for these three mechanisms can be expressed mathematically in Eqs. (5–7) where

*β*

*,*

_{BM}*β*

*,*

_{SH}*β*

*refers to the collision frequency function for Brownian motion, fluid shear, and differential settling respectively. The total collision rate or frequency*

_{DS}*β*

*is the sum of contributions from each of the three transport mechanisms (Eq. (8)) [3, 5, 10, 46, 48, 51, 54–56].*

_{T}##### (7)

$${\beta}_{DS}=\frac{\pi g}{72{v}_{w}}{({d}_{i}+{d}_{j})}^{2}(\mathrm{\Delta}{\rho}_{i}{d}_{i}^{2}-\mathrm{\Delta}{\rho}_{j}{d}_{j}^{2})$$#### 2.3.1. Perikinetic particle aggregation

#### 2.3.2. Classical orthokinetic aggregation

*spatial and temporal*) [59, 60]. Camp and Stein in their critique of the Smoluchowski approach to shear flocculation modelling [61], introduced the concept of root-mean-square (R.M.S.) or absolute velocity gradient

*Ḡ*to account for the variations in the shear rate. [26, 46, 58, 62, 63]. Considering the angular distortion of an elemental volume of fluid arising from the application of tangential surface forces,

*Ḡ*is defined according to Eqs. (9–10) as the R.M.S. velocity gradient in the mixing vessel, where

*u*,

*v*,

*w*are the components of the fluctuating velocity while

*x*,

*y*,

*z*refers to the 3-D Cartesian coordinate system,

*P*is the power dissipated,

*V*is the volume of the mixing vessel,

*ε*is the energy dissipation rate per unit mass,

*μ*s the dynamic viscosity and

*v*is the kinematic viscosity [26, 64].

##### (9)

$$G={\left\{{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^{2}{\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial z}\right)}^{2}{\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)}^{2}\right\}}^{{\scriptstyle \frac{1}{2}}}$$*Ḡ*due to the fluctuations in the energy dissipation within the mixing vessel, an average velocity gradient

*G*

*has often been used in place of the absolute value*

_{ave}*Ḡ*[59, 60] (Eq. (11)). Korpijarvi et. al. [65], in their study of mixing and flocculation in a jar using computational fluid dynamics (CFD) reported a large variation in the local velocity gradient

*G*

*within the mixing vessel. In similar studies conducted by Kramer and Clark [26, 66] as well as Mühle [67], a lower estimate of the absolute velocity gradient was proposed (Eqs. (12–13)) to account for the variation of the velocity gradient throughout the fluid where quantities*

_{L}*G*

*,*

_{L}*P*

*,*

_{ave}*ε*,

*V*,

*μ*,

*v*represent the local velocity gradient at different points within the mixing vessel, average power consumption, volume of the mixing vessel, kinetic energy dissipation rate, dynamic and kinematic viscosity respectively.

### 2.3.1.1. Particles in laminar shear

### 2.3.1.2. Particles in turbulent shear

*λ*

_{0}of turbulence, while for particle larger than this length scale, the flocculation mechanism is similar to that of Brownian diffusion [21, 26].

#### 2.3.2. Extended orthokinetic aggregation

*G*where

*m*,

*R*,

*G*and

*A*represent the mass of the floc, radius of the flow trajectory, velocity gradient and the area of the floc projection onto a plane oriented normally to the vector of the resultant force respectively (Eq. (16)). In practice, pelleting flocculation is realized through the selection of an appropriate process engineering conditions and stirrer-vessel system (geometry and configuration) in order to obtain the necessary conditions suitable for rolling and collision-mediated floc pelletization. [83, 91].

#### 2.3.3. Polymer-mediated interactions

*Ø*

*at the critical particle concentration*

_{pt}*N*

*above which pellet flocs are formed as a function of the initial particle concentration*

_{c}*N*

_{0}, polymer concentration

*C*

*, charge density*

_{p}*p*, and the molecular weight

*M*, where

*N*

*is the Avogadro constant (Eq. (17)).*

_{A}### 3. Hydrodynamics of Fine Particle Aggregation

*orthokinetic*aggregation of particles [33, 36, 116]. The floc growth and stability in any flocculated system has been suggested to be a function fluid-particle interactions and the intrinsic physicochemical properties of the floc [75, 77, 117]. The dynamics of these interactions affects all facets of the flocculation process and the degree of aggregation in any sheared system [118]. In the case of hydrodynamic interactions, induced velocity gradient promote the aggregation process but might also be responsible for floc breakage as a result of increased viscous shear stress [119]. Consequently, in the case of shear-induced collisions, the hydrodynamic effect can be very significant [57].

### 3.1. Fluid Mixing and Particle Dispersion

^{e}> 10

^{4}) [120], can be viewed as a hierarchy of irregular, rotational and dissipative motion containing vorticity (curl or rotation of the velocity vector) on different scales or eddy sizes [25, 121]. In turbulent flows such as those encountered in pipe or tube flow, channel flow and stirred tanks or mixing chambers, energy transfer occurs on different eddy scales [64, 67]. Eddies are spatial recognizable flow patterns that exist in a turbulent flow for at least a short time in which there is a correlation between the velocities at two different points [26]. Fluid deformation causes vortices to stretch and vorticity and kinetic energy to be transported from larger to smaller eddies [25, 121]. The turbulent vortex in such fluid motion is generally propagated in the tangential and axial direction than in the radial direction with the vorticity increasing with decreasing eddy size [64, 121, 122].

_{0}(Kolmogorov microscale) expressed in Eq. (18) and at the final scales of molecular diffusivity (Batchelor scale) [120, 123–126]. Micromixing describes the process homogenization of liquid balls with their surroundings on a molecular level. In typical mixing conditions, the dividing line between micro and macro scale is between 100 and 1000 μm respectively [127].

*λ*

_{0}at the beginning of viscous dissipation range [128], there exist many eddies of other scales smaller than the integral scale Λ that transfer kinetic energy continually through the other length scales. The Batchelor

*λ*

*and Taylor*

_{B}*λ*

*scales expressed in Eqs. (19–20) are the examples of other important length scales where*

_{T}*D*is the diameter of the flocs while

*ε*is the turbulent kinetic energy dissipation rate.

*λ*

*is an intermediate length scale in the viscous subrange that is representative of the energy transfer from large to small scales, but it is not a dissipation scale and does not represent any distinct group of eddies [129]. Batchelor micro-scale on the other hand is a limiting length scale where the rate of molecular diffusion is equal to the rate of dissipation of turbulent kinetic energy and represent the size of the region within which a molecule moves due to diffusional forces [123, 125, 128, 130].*

_{T}### 3.2. Energy Dissipation in Turbulent Flow

*d*

*) draw energy from the fluid motion, while the smaller eddies transfer that energy gradually and continually to the smallest eddy, where the energy is ultimately dissipated into heat by friction [27]. The turbulence parameters of interest in any agitated micro-system (with respect to the mixer capacity and performance) are the Kolmogorov micro scale λ*

_{A}_{0}, turbulence kinetic energy dissipation rate ɛ, and agitator tip velocity

*V*

*[122, 124, 131].*

_{tip}*λ*

_{0}is an indicator of the rate of micro mixing and mode of agglomerate deformation due to turbulent shear while the energy dissipation rate ɛ is the rate of the dissipation of kinetic energy. The tip velocity

*V*

*is a measure of the tangential velocity imparted by the flow inducer, an indicator of the strength of the vortex generated by the flow. Peripheral velocity*

_{tip}*V*

*on the other hand gives an indication of the velocity of the flow vortex at Kolmogorov micro-scale [76, 122, 124, 131]. In typical laboratory mixing experiments (D < 1m), the micro-scale of turbulence predominates. Under such conditions, particle collision is promoted by eddy size similar to those of the colliding particles [47].*

_{pkol}### 4. Floc Stability in Sheared Systems

*F*

*, and hydrodynamic breaking force*

_{B}*F*

*[64, 81, 116, 132–135]. While the binding force is determined by the flocs’ structure and physicochemical attributes, flow turbulence is the principal factor in the case of hydrodynamic force [59, 75, 117]. Of these two governing factors of floc stability, turbulence is the least understood owing to its complex nature [45, 136]. Therefore, a detailed analysis of floc stability under turbulent conditions often encountered in natural and agitated systems is not only difficult but often time consuming [45].*

_{H}### 4.1. Aggregate Strength and Hydrodynamic Stress

*F*

*and the global hydrodynamic stress σ acting on a spherical aggregate in the inertia and viscous domain of turbulence (Table. 1).*

_{H}*F*

*using fractal dimension approach while Lu et al. [35] in a similar study presented a theoretical model for the aggregate binding force*

_{B}*F*

*of spherical mono-disperse particles both of which are presented in Eq. (21–22). Yuan & Farnood [143], as well as Jarvis et. al. [144], in their review of aggregate strength and breakage gave an estimate of the floc strength τ and global hydrodynamic stress σ obtained from empirical studies of various types of flocs.*

_{B}##### (21)

$${F}_{B}={48}^{-\frac{2}{3}}\hspace{0.17em}{\pi}^{\frac{5}{3}}\hspace{0.17em}{d}^{\frac{2}{3}}\hspace{0.17em}\sigma {\left({\rho}_{o}-{\rho}_{w}\right)}^{\frac{2}{3}}{d}^{\left(1+\frac{{D}_{f}}{3}\right)}$$*r*

*[67, 149].*

_{w}### 4.2. Fluid-Particle Interactions and Floc Stability

*d*

*, while the largest size*

_{p}*d*

*is determined by the balance of floc growth and rupture within the fluid [64, 139, 150, 151]. The floc growth and breakage is known to occur simultaneously—growing flocs are subjected to breakage while fragments of broken flocs undergo re-agglomeration until a levelling off of the floc sizes at steady state when the maximum stable size*

_{Fmax}*d*

*is attained [54, 139, 152, 153].*

_{Fmax}*N*

*,*

_{A}*N*

*,*

_{B}*σ*

*,*

_{S}*τ*

*, represent change in the particle number concentration per unit volume for aggregation and breakage, the aggregate shear strength and the shearing stress respectively.*

_{S}##### (25)

$$\frac{dN}{dt}=-\frac{2}{3}\hspace{0.17em}0{\left(\frac{\varepsilon}{v}\right)}^{\frac{1}{2}}{d}_{A}^{3}\hspace{0.17em}{N}_{A}^{2}+{\beta}_{1}\frac{{\tau}_{s}}{{\sigma}_{s}}{\left(\frac{\varepsilon}{v}\right)}^{\frac{1}{2}}\hspace{0.17em}{d}_{B}^{2}\hspace{0.17em}{N}_{B}$$*F*

*is greater than or equal to the floc binding or cohesive force*

_{H}*F*

*(B ≤ 1).*

_{B}*d*

*. Few of such expressions are presented for the inertia and viscous subrange of turbulence in Table 2 [143, 144].*

_{Fmax}##### (27)

$${\beta}_{floc}=({\alpha}_{ijBM}{\beta}_{BM}+{\alpha}_{ijSH}{\beta}_{SH}+{\alpha}_{ijDS}{\beta}_{DS})-{\beta}_{br}$$##### (28)

$$\begin{array}{l}B=\frac{Floc\hspace{0.17em}binding\hspace{0.17em}force}{Hydrodynamic\hspace{0.17em}force}=\frac{{F}_{B}}{{F}_{h}}\\ =\frac{{48}^{-\frac{2}{3}}\hspace{0.17em}{\pi}^{\frac{5}{3}}\hspace{0.17em}{d}^{\frac{2}{3}}\hspace{0.17em}\sigma {({\rho}_{o}-{\rho}_{w})}^{\frac{2}{3}}{d}^{\left(1+\frac{{D}_{f}}{3}\right)}}{\frac{\pi {\rho}_{w}^{2}{\mu}^{\frac{2}{3}}\hspace{0.17em}{G}^{\frac{4}{3}}\hspace{0.17em}{d}^{\frac{8}{3}}}{4}}\end{array}$$##### (29)

$$\begin{array}{l}B=\frac{Floc\hspace{0.17em}binding\hspace{0.17em}force}{Hydrodynamic\hspace{0.17em}force}=\frac{{F}_{B}}{{F}_{h}}\\ =\frac{{48}^{-\frac{2}{3}}\hspace{0.17em}{\pi}^{\frac{5}{3}}\hspace{0.17em}{d}^{\frac{2}{3}}\hspace{0.17em}\sigma {\left({\rho}_{o}-{\rho}_{w}\right)}^{\frac{2}{3}}{d}^{\left(1+\frac{{D}_{f}}{3}\right)}}{\frac{\pi {\rho}_{w}{d}^{4}\hspace{0.17em}{G}^{2}}{60}}\end{array}$$*σ*

*on the aggregate in the viscous subrange reaching a maximum value when the floc size is roughly equal to the eddy scale (Kolmogorov micro scale) [118]. Similarly, fluctuating fluid velocities normal to the floc surface or dynamic pressure fluctuations acting on opposite sides of an aggregate will results in normal or bulk pressure stress*

_{s}*σ*

*(tensile or compressive) [53]. In addition, turbulent drag forces*

_{t}*F*

*acting on the surface of an aggregate which originates from the local motion of the fluid relative to the motion of the aggregates will results in instantaneous surface shear forces and shear stress respectively [77]. It has been shown that for flocs in the inertial subrange of turbulence (*

_{D}*d*

*>*

_{F}*λ*), tensile stress will predominate causing wholesale fracture or fragmentation [35], while for those in the viscous subrange (

*d*

*<*

_{F}*λ*), shear stress will cause erosion of the particles, floc shell or floc surface as shown schematically in Fig. 5 [47, 53].

### 4.3. Mechanisms of Aggregate Disruption

*σ*

*≥*

_{t}*τ*

*) in the inertia domain of turbulence (*

_{t}*d*

*>*

_{F}*λ*), subjected to the shearing action of fluid motion, the turbulent hydrodynamic tensile stress

*σ*

*causing bulk rupture or fragmentation by floc splitting and the aggregate tensile strength*

_{t}*τ*

*resisting the fluctuating pressure on the floc may be expressed mathematically in Eqs. (30–31).*

_{t}*σ*

*≥*

_{s}*τ*

*) in the viscous subrange (*

_{s}*d*

*<*

_{F}*λ*); the turbulent hydrodynamic shear stress

*σ*

*eroding the surface of the aggregates and the corresponding aggregate shear strength*

_{s}*τ*

*resisting the viscous shear forces can be expressed mathematically in Eqs. (32–33) [35].*

_{s}*τ*

*resisting floc shell erosion due to pseudo-surface tension force*

_{y}*γ*

*eroding the particle chain of the outer floc surface in the viscous domain of turbulence (*

_{F}*σ*

*≥*

_{s}*τ*

*) (Eq. (34)). The yield stress approach was also presented by Liu et. al. [147] for calculating the maximum floc tensile yield stress*

_{y}*σ*

*at which breakage is likely to occur in the inertia subrange (*

_{y}*σ*

*≥*

_{t}*σ*

*) (Eq. (35)).*

_{y}##### (35)

$${\sigma}_{y}=0.5\hspace{0.17em}\alpha \hspace{0.17em}\rho \hspace{0.17em}{\varepsilon}^{\frac{2}{3}}{\left[\frac{2\pi}{{d}_{f}}\right]}^{-\frac{2}{3}}$$*V*

*above which there will be floc breakage by estimating floc yield stress*

_{c}*σ*

*(tensile or compressive) resulting from dynamic pressure acting on the floc (Eqs. (36–37)). The kinetic equations for fragmentation and surface erosion mechanisms are presented in Eqs. 38–39 based on the empirical results from flocculation in stirred reactor. The quantities*

_{y}*N*

*,*

_{A}*N*

*,*

_{B}*σ*

*,*

_{s}*τ*

*, represent change in the particle number concentration per unit volume for aggregation and breakage, the aggregate shear strength and the shearing stress respectively [35, 49].*

_{s}