# A Semi-Empirical Model to Estimate Maximum Floc Size in a Turbulent Flow

## Abstract

**:**

_{max}= C.G

^{−}

^{γ}) is only applicable for low velocity gradients (<500 s

^{−1}) and is often used for shear rates that are not representative of the global phenomenon. This paper presents a semi-empirical model that is able to predict mean floc size in a very broad shear range spanning from aggregation to floc fragmentation. Theoretical details and modifications relating to the orthokinetic flocculation output are also provided. Modelling changes in turbidity in relation to the velocity gradient with this model offer a mechanistic approach and provide kinetic agglomeration and breakage index k

_{a}and k

_{b}. The floc breakage mode is described by the relationship between the floc size and the Kolmogorov microscale. Shear-related floc restructuring is analysed by monitoring the fractal dimension. These models, as well as those used to determine floc porosity, density and volume fraction, are validated by the experimental results obtained from several flocculation operations conducted on ultrafine kaolin in a 4-litre reactor tank compliant with laws of geometric similarity. The velocity gradient range explored was from 60 to 6000 s

^{−1}.

## 1. Introduction

_{0}is the initial number concentration of primary particles per unit volume; n is the number concentration of primary particles per unit volume at time t; G is the velocity gradient; k

_{a}is is an index of particle aggregation rate; k

_{b}is an index of floc breakage and erosion rate; d

_{max}is the maximum floc diameter; C is the floc strength coefficient that depends on both properties of the floc and fluid and the exponent γ depends on both the breakup mode and size regime of eddies that cause disruption [7]. γ and p are similar. (k

_{a}nG) is the aggregation function; (k

_{b}n

_{0}G

^{P}) is the break-up and erosion function. k

_{a}and k

_{b}depend on the physico-chemical properties of the suspension and the nature of the flocculant used [1,8].

_{0}/n

_{nf}, assuming that the concentration of the number of residual non-flocculated primary particles (n

_{nf}) is proportional to the turbidity of the supernatant, without taking into account the proportionality constants that can be used to calculate the turbidity from the number of particles. The effect of this approximation may be detrimental to the determination of the kinetic agglomeration and breakage index k

_{a}and k

_{b}. The second model (Equation (2)) for agglomerate breakage under the effect of hydrodynamic stress is only applicable for low velocity gradients (<500 s

^{−1}) and in the shear ranges produced by jar tests. In order to conduct comprehensive monitoring of the different flocculation phases in terms of turbidity and size in a sufficiently broad velocity gradient range, more suitable models must be developed. This research is entirely in line with this objective.

^{−1}.

## 2. Theory

_{F}and d

_{F}are the floc mass and diameter, m

_{P}and d

_{P}are the mass and diameter of the primary particle which contributes to the formation of the floc, D

_{f}is the mass fractal dimension of the floc and k

_{α}is the prefactor. The latter depends on parameters including the floc compactness, in other words on its fractal dimension and the size and concentration of primary particles. This prefactor is equal to the base prefactor k

_{0}when the diameter of gyration is considered to be the diameter of the floc, otherwise, k

_{α}= k

_{0}.α

^{Df}. Equation (3) provides the expressions of floc porosity, density and relative volume fraction [4,9]:

_{F}is the average floc porosity; ρ is the average density (F: floc, L: liquid, S: solid); Φ

_{F}is the volume fraction of the flocs; Φ

_{P}is the volume fraction of the initial primary particles; Φ

_{FP}is the relative volume fraction of flocs (Φ

_{FP}= Φ

_{F}/Φ

_{P}).

_{g}is the radius of gyration of floc, r

_{P}is the primary particle radius and D

_{f}is the mass fractal dimension of floc. q is the magnitude of the scattering wave vector. q is a function of the incident beam wavelength (λ = 632.8 nm), the refractive index of the scattering medium and the scattering angle. The textural information is obtained from the analysis of the modulus variation of the scattering wave vector in the q-range of 1.81 × 10

^{−3}to 4.53 × 10

^{−1}µm

^{−1}of the Mastersizer S granulometer long bench version (1000 mm lens − 32 detectors).

_{g}<< q << π/r

_{P}from the negative slope of the linear region of the logarithmic plot of I(q) versus q. Experimental observation showed that this dimension might have values ranging between 1.4 and 2.8 [18,19]. The radius of gyration can be determined in the Guinier region of the scattering curve from the slope of the linear part of the logarithm plotted for scattered intensity I as a function of q

^{2}where qR

_{g}<< 1 (very low q values) using the Guinier approximation [14]:

_{P}as a function of the Reynolds number R

_{e}. It is produced experimentally by measuring the torque applied to the impeller’s drive shaft at different velocities, and possibly different viscosities and densities of the stirred fluid. By modelling this curve, we were able to establish the following empirical function:

_{e}≤ 100000:

_{1}= 0.93002471, P

_{2}= −0.04623452, P

_{3}= −2.738211 × 10

^{−6}, P

_{4}= 1.5310496,

_{5}= 0.0023565875 and P

_{6}= 0.23500056

_{P}is the power number; P is the power dissipated by the agitator into the fluid; D is the impeller diameter; N is the rotational speed of the agitator; ρ

_{p}is the density of the suspension; R

_{e}is the Reynolds number.

_{e}and power P dissipated by the impeller are determined for a given rotation speed by solving the combination of this set of equations using Mathcad. The value for the dynamic viscosity of the flocculated suspension is considered to be independent of the velocity gradient; it is estimated using the Thomas equation [21]:

_{F}is the volume concentration of flocs in the pulp, and η

_{0}is the dynamic viscosity of water at 20 °C. The ratio between the volume of the flocs and that of the suspension represents the volume concentration ∅

_{F}occupied by the flocs in the flocculated suspension with the initial solids concentration C

_{S}. This volume concentration is determined by the expression [2]:

_{S}is the solid density and ε

_{F}is the average floc porosity.

_{p}is the density of the suspension; D

_{i}is the internal diameter of the flocculation reactor; P is the power dissipated by the agitator into the fluid.

_{0}. By doing so, we obtain:

_{0}is the number of initial primary particles before flocculation and n

_{nf}is the number of unflocculated primary particles.

_{a}, k

_{b}and p. Following flocculation, by measuring the residual turbidity we are able to determine the number of non-flocculated particles. Assuming that the residual suspension is monodispersed with n

_{nf}particles per unit volume, i.e., all particles have the same diameter, light attenuation at wavelength λ

_{0}on the turbidity meter by n

_{nf}particles is determined by [9,29]:

_{0}is the intensity of the incident light; I

_{T}is the intensity of the transmitted light; L is the optical path length; C

_{ext}is the effective extinction cross-section of particles. C

_{1}, C

_{2}, C

_{3}and p are three constants that can be accessed using data processing software (Origin, TableCurve, etc.).

_{F}occupied by the flocs in the suspension, normalised by the volume fraction Φ

_{P}initially occupied by the non-flocculated primary particles, can be modelled using the Quemada relationship [21,30]:

_{FP}is the relative volume fraction of flocs (Φ

_{FP}= Φ

_{F}/Φ

_{P}); Φ

_{FP,}

_{0}is the value of Φ

_{FP}at G = 0 s

^{−1}(no shear); Φ

_{FP,}

_{∞}is the value of Φ

_{FP}when G tends to ∞ (ultimate relative volume fraction); s is a parameter that accounts for the shear strength of the flocs.

_{a/b}) which separates the dominant aggregation range from the erosion and/or splitting range. The value of this threshold depends on the concentration of solids.

_{a}characterises the collision efficiency. These two parameters are related by the following relationship [1,5,31]:

_{F}is the volume fraction of the flocs. β

_{T}is the collision efficiency reflecting the hydrodynamic interaction.

_{0}is the initial number concentration of primary particles per unit volume; n

_{nf}is the number of unflocculated primary particles; n

_{p}is the number of primary particles in the floc; n

_{j}is the number of flocs (all have the same diameter).

_{F}is the average volume diameter of the flocs; d

_{p}is the diameter of the primary particle; k

_{a}is is an index of particle aggregation rate; a is a constant that depends on the initial number of primary particles, the number of flocs, the prefactors k

_{α}and the fractal dimensions; b is a constant that depends on the critical dimension separating the two ranges, the initial diameter (d

_{0}), the prefactor k

_{α}and the fractal dimension of the second range; c is a constant that depends on the initial diameter (d

_{p}), the ultimate diameter given when G→∞, the prefactor k

_{α}and the fractal dimension of the second range; s is an exponent whose value is a function of splitting and/or erosion; β

_{1}= 1/D

_{f}

^{a}et; β

_{2}= 1/(3 − D

_{f}

^{b}). D

_{f}

^{a}is the average fractal dimension of the flocs formed during the aggregation phase. D

_{f}

^{b}is the average fractal dimension of the flocs formed during the breaking and/or erosion phase.

## 3. Results and Discussion

^{2+}and Mg

^{2+}in tap water decreases the zeta potential. The flocculant used, SNF Floerger AN 934 MPM, is an anionic acrylamide-sodium acrylate copolymer of 35% anionicity and medium molecular weight. This flocculant is commonly used in industrial water treatment processes. Flocculation by this organic polymer occurs due to interparticle bridging following hydrogen bond adsorption of individual polymer chains onto several particles simultaneously, and following neutralisation of the positive sites generated by the adsorption of Ca

^{2+}and Mg

^{2+}on the kaolinite at pH 8 in tap water, by the carboxyl groups (COO

^{−}) of the flocculant. These divalent cations form bridges between the anionic surfaces of the kaolin particles and the anionic molecules of the flocculant [33,34].

^{−1}increased from 10 to 0.7 s for a volume of 4.2 L for all three concentrations. The imposed flocculant conditioning time was 15 s. The ratio of conditioning time over the estimated theoretical mixing time was between 1.5 and 21. Based on the values of this ratio, it is possible to achieve a homogeneous flocculant distribution in the pulp and efficiency in terms of floc growth, taking into account the stirred volume.

^{2}and the normalised relative deviation between the experimental and predicted values ΔY:

_{exp}and Y

_{cal}are the experimental and model calculated values, respectively.

#### 3.1. Turbidity and Flocculation Mechanisms

^{2}= 0.99, Δτ = 14.2%). The deviations between the experimental and predicted values are acceptable in as far as the turbidity meter used is not very accurate in measuring deviations between very low turbidities. As indicated, the flocculation parameters are affected by the initial concentration of solids C

_{Si}and the shear stress. The linearity between C

_{Si}and minimum residual turbidity indicates that there is a number of particles that escape flocculation and that this number increases with the concentration.

_{si}and the velocity gradient corresponding to the minimum residual turbidity implies that floc cohesion and the interactions between the primary particles inside the flocs increase with the concentration of solids:

^{−1}, the residual turbidity changed very little (near-stationary) while the floc size continued to evolve remarkably, showing a decreasing trend according to the velocity gradient applied (later discussed in Section 3.2.6). This phenomenon indicates that floc splitting is dominant at this concentration. The decrease in p as the initial concentration of solids C

_{si}increased can be attributed to the flocs generated at a concentration of 25 g/L being larger than those obtained at 6.25 and 12.5 g/L. The larger floc size at a high C

_{si}value could promote floc fragmentation by splitting over floc erosion. These behaviours imply that p increases as the splitting/erosion ratio increases. A p value between 1 and 2.3 may indicate the existence of two parallel floc breakage modes at play. When p = 1, splitting is dominant; however, erosion increases as p rises. The floc splitting and erosion rate index k

_{b}increases rapidly as C

_{si}rises and p decreases. At this level, k

_{b}mainly characterises floc splitting. Conversely, the particle aggregation rate index k

_{a}changes very little with C

_{si}. This also shows that k

_{b}is strongly dependent on the hydrodynamic characteristics of the reactor tank. Floc erosion is higher at 6.25 than at 25 g/L. The dominance of one breakage mode over the other depends on the relative relationship that can exist between floc size and eddy size. In the case of flocs smaller than the turbulence microscale (α), erosion will be the main breakage mode. Conversely, when the floc size is between the microscale and the macroscale, fragmentation by splitting will be the main breakage mode. Before exploring this aspect, we note that this analysis will be performed in relation to the volume mean diameter d

_{F}and the volume diameter d10 for which the cumulative volume function is 10%. These diameters are indicated without any additional processing by the particle size analyser. The volume mean diameter will therefore be favoured over the diameter of gyration. The relationship between these two diameters is shown in Figure 4. The diameters of gyration are determined from the light scattering curve obtained from the Mastersizer S particle size analyzer using the Guinier approximation (Equation (8)). They are 1.4% larger than the mean diameters.

_{a}and k

_{b}are the kinetic agglomeration and breakage index; p is the floc breakup index that depends on both the breakup mode and size regime of eddies that cause disruption; τ

_{min}is the minimum residual turbidity; G

_{τ}

_{min}(s

^{−1}) is the velocity gradient corresponding to the minimum residual turbidity.

^{2}= 0.99 for the two relationships, and the normalised relative deviations in relation to the predicted values are 4.8, 5.5 and 5.5% for d10/α as a function of G and 2.4, 2.7 and 4.6% for d

_{F}/α as a function of G.

_{F}/α with slopes that are more or less identical for the three concentrations of solids, involves the formation of micro-flocs by fragmentation of the large flocs. The third phase, whose start point depends on the initial concentration of solids, consists of a phase of strong hydrodynamic stress exerted on the surface of the flocs and micro-flocs, resulting in erosion that is inversely proportional to C

_{si}.

#### 3.2. Floc Structure

^{−1}with a constant flocculant dosage (400 g/t) and constant agitation time (15 s). Each velocity gradient corresponds to mean porosity, density, volume fraction, size and fractal dimension values.

#### 3.2.1. Floc Porosity

_{F}/d

_{P}by Equation (4). The coefficient of determination R

^{2}= 0.99 for the three concentrations of solids (6.25, 12.5 and 25 g/L), and the normalised relative deviations in relation to the predicted values are 0.2, 0.1 and 0.3%. A perfect prediction is obtained with this model. As indicated, porosity increases with floc size due to a primary particle arrangement that traps water (Figure 6A). The higher the floc porosity, the higher their water content will be. This specificity means that flocs will occupy an increasing volume fraction as their size increases (Figure 6C). This phenomenon leads to a gradual reduction in the mean distance between flocs, thus increasing the hydrodynamic surface stress and promoting contact between flocs, which in turn leads to hydrodynamic erosion that increases with the velocity gradient. From 6 to 13 g/L, the porosity is between 0.920 and 0.987. At a concentration of 25 g/L, this parameter drops significantly and lies between 0.855 and 0.980 for 25 g/L. At a constant shear rate, the porosity decreases from a critical volume fraction of primary particles. Flocculation at 25 g/L generates flocs that are significantly more compact than at 6.25 and 12.5 g/L. While the average fractal dimensions obtained using Equation (4) are statistically comparable for the three initial concentrations of solids, the slightly smaller fractal dimension at C

_{si}= 25 g/L is consistent with the decrease in this dimension in the case of density variation as a function of d

_{F}/d

_{P}at C

_{si}= 25 g/L (Figure 6A,B). This can be explained by the formation of primary flocs, which form the final flocs. The prefactors k

_{α}for the three concentrations are 1.0, 0.9 and 2.5 for porosity and 1.0, 0.7 and 2.6 for density respectively. The difference between the values of the k

_{α}parameter of the fractal relationship could reflect the more compact arrangement of the initial particles in the primary flocs at 25 g/L compared to 6.25 and 12.5 g/L.

#### 3.2.2. Floc Density

^{2}= 0.99 for the three concentrations of solids, and the normalised relative deviations in relation to the predicted values are 0.3, 0.2 and 0.4%; the model is perfectly suited. At all concentrations of solids, this parameter decreases as the mean floc size increases (Figure 6B). This continuous, monotonic decrease with size reflects an increase in the mean porosity. As with porosity, this decrease depends on the volume fraction of the primary particles. At 25 g/L, flocculation produces denser flocs with a significantly lower mean fractal dimension. Low density combined with high porosity is likely to affect the floc settling rate and reduce the flocs’ resistance to hydrodynamic stress.

#### 3.2.3. Volume Fraction vs. Average Diameter

_{F}/d

_{P}is obtained using Equation (6). The coefficient of determination R

^{2}= 0.99 for the three concentrations of solids (6.25, 12.5 and 25 g/L), and the normalised relative deviations in relation to the predicted values are 2.3, 2.6 and 4.4%. An excellent prediction is obtained with this model (Figure 6C). The mean fractal dimensions obtained by this model are statistically comparable and follow the same trends observed for porosity and density. Floc growth results in an increase in the flocs’ porosity, a decrease in their density and an increase in their volume concentration. At a constant velocity gradient, the drop in the volume fraction as the concentration of solids rises is due to the increase in the number of primary particles in each floc, resulting in a decrease in porosity and an increase in density.

#### 3.2.4. Fractal Dimension

_{F}as a function of the velocity gradient is presented in Figure 6D. With each flocculation operation, the fractal dimension is deduced from the floc diffusion diagram (Equation (7)). This dimension lies between 2.3 and 2.5 for concentrations of 6.25 and 12.5 g/L, and between 2.2 and 2.7 for a concentration of 25 g/L. The latter concentration generates flocs that are significantly more compact. Floc compactness increases with D

_{F}. Above a value of 2.4, flocs can be considered to be dense and compact [4,9,35]. Variation of D

_{F}implies that the flocs formed are sensitive to the shear stress imposed. As expressed in Figure 6D, the fractal dimension contains information on structural changes in the flocs in relation to the velocity gradient and the initial concentration of solids. Changes in state are indicated by the presence of one or more peaks. For the first peak, its rising slope characterises a phase dominated by the aggregation of primary particles and micro-flocs. At its tip, the system is in equilibrium between cohesion and breakage forces. The following drop in the fractal dimension marks the second phase. For concentrations of solids of 6.25 and 12.5 g/L, this phase leads to the formation of micro-flocs through the fragmentation of large flocs, followed by the restructuring of the flocs into a more compact form clearly shown by the presence of a second peak. At the tip of this peak, D

_{F}equals 2.5–2.52. The third phase for these two concentrations of solids is dominated by the splitting of the restructured flocs. At 25 g/L, there is only one peak. The rising slope, as previously mentioned, indicates the dominance of aggregation and the slope after this peak decreases continuously and monotonically with floc breakage being dominant at the beginning and erosion increasing with the velocity gradient. At 25 g/L, erosion becomes dominant from around 2000 s

^{−1}. The predominant mechanisms depend strongly on the binding forces of the aggregate in relation to the hydrodynamic forces encountered.

#### 3.2.5. Volume Fraction vs. Velocity Gradient

^{2}= 0.99 for the three concentrations of solids (6.25, 12.5 and 25 g/L), and the normalised relative deviations in relation to the predicted values are 2.0, 4.4 and 3.4%. Based on these results, the model is rigorous and is able to determine the critical gradients and ultimate volume fractions (Figure 7). In the given order of the three initial concentrations of solids, the critical gradients are 1025, 959 and 974 s

^{−1}respectively. At 6.25 and 12.5 g/L, these gradients correspond to the start point for the restructuring of flocs that have been fragmented and perfectly match the beginning of the second fractal dimension peaks discussed above (Figure 6D). At 25 g/L, the critical gradient at 974 s

^{−1}is attributed to the start of erosion. The ultimate diameters, deduced by combining the values of the ultimate relative volume fractions and Equation (4), are, in increasing order of the concentrations of solids, 44.5, 30.3 and 9.3 µm, bearing in mind that the average primary particle size is 1.25 µm. According to these results, the flocculation operation generates very strong micro-flocs.

#### 3.2.6. Modelling of the Evolution of the Mean Floc Diameter vs. Velocity Gradient

_{max}= C.G

^{−}

^{γ}[6], is not suitable for modelling experimental results in the range for which it was developed (Figure 8A). This model for agglomerate breakage under the effect of hydrodynamic stress is only applicable for low velocity gradients (<500 s

^{−1}), in particular, in the shear ranges produced by the jar tests. Conversely, the model developed through this research and formulated by Equation (29) applies perfectly and covers the two main flocculation ranges for a very wide range of velocity gradients (Figure 8B). This expression was able to be executed using TableCurve (2D Version 5.01 SYSTAT Software Inc., 2002). With a view to obtaining a global prediction, it seemed appropriate to introduce the mean fractal dimension of the aggregation phase and the fragmentation and erosion phase into the equation developed (Equation (29), β1 and β2). These means are calculated for each concentration of solids from the results obtained from the processing of the light scattering curves measured by the particle size analyser. In this case, the number of parameters in this equation is reduced from eight to six. The main results obtained are presented in Table 2. The coefficient of determination R

^{2}= 0.99 for the three concentrations of solids (6.25, 12.5 and 25 g/L), and the normalised relative deviations in relation to the predicted values are 7.4, 4.2 and 4.5%. An excellent prediction is obtained with this model.

_{si}during both the aggregation period and the splitting/erosion period. The variation of this mean size as a function of G reaches a maximum whose abscissa G

_{a/b}decreases as C

_{si}increases. The dimensions of the flocs that increase with G when G < G

_{a/b}correspond to the dominant aggregation range. Above G

_{a/b}, floc dimensions decrease as G increases, indicating the dominant presence of splitting and/or erosion in this shear range. The value of this threshold G

_{a/b}depends on the initial concentration of solids. Concomitantly, above this threshold, the increase in the velocity gradient leads to a reduction in floc size, volume fraction and porosity, and an increase in density (constant flocculant dosage and constant conditioning time). The decrease in the parameter s as C

_{si}increases is accompanied by an increase in the flocs’ mechanical strength, which accounts for an increase in the density of the bonds between the polymer and particle surface.

_{a}is assumed to be constant during the conditioning period. This hypothesis is an approximation because k

_{a}depends on the floc dimensions and the arrangement of the particles within the flocs, in relation to an aggregate density and a collision radius that are dependent on the fractal dimension. This hypothesis steers the research towards the determination of an adjustment value k

_{a}which may be far from the mechanistic value close to the actual value provided in Table 1. Interpretation becomes difficult but the order of magnitude is given. The model is able to very accurately predict the maximum diameter d

_{max}, the velocity gradient for the transition between the dominant phases G

_{a/b}and the parameter indicating floc shear strength s.

## 4. Materials and Methods

^{−1}. A short 15-s conditioning period with a commercial anionic flocculant AN 934 MPM at a concentration of 400 g/t was applied to each flocculation operation. The size and texture of the flocs (porosity, density, fractal dimension and volume fraction) were determined by laser diffraction with the same Malvern MasterSizer S particle size analyser but using its long bench, a 10 mm optical path measurement unit and a gravity-fed stream via a reactor tank also with a capacity of 4.2-litres, fitted with four orthogonal riffles, placed on top of the particle size analyser (Figure 9B). At the end of each flocculation operation performed at a given velocity gradient, a sample of the flocculated suspension was sampled with a calibrated ladle (known volume and concentration). This floc sample was carefully placed in the reactor tank of the particle size analyser. The new floc suspension, diluted to a measurement concentration for which the attenuation effects of the central laser beam do not exceed 28% obscuration [9], was gently stirred (G ≤ 20 s

^{−1}) by an impeller identical to the flocculation impeller. With this configuration, it was possible to determine the particle size distributions between 4.2 and 3473 µm. It eliminates recirculation of the suspension by pump during measurements, as per the conventional configuration of the particle size analyser, which significantly affects floc size and texture. After a 1min settling period in the flocculation reactor tank, the residual turbidity of the supernatant was measured using the Turb 555 IR turbidity meter (λ

_{T}= 860 nm). Each sample was taken at a depth of 3 cm.

## 5. Conclusions

_{a}and k

_{b}. With the model for floc aggregation and breakage under the effect of hydrodynamic stress developed through this research, it possible to predict the maximum diameter d

_{max}, the velocity gradient G

_{a/b}that separates the aggregation phase from the floc breakage and/or erosion phase, and the parameter indicating floc shear strength “s”. This model shows that the flocs’ mechanical strength increases with increasing concentration. Floc growth results in an increase in the flocs’ porosity, a decrease in their density and an increase in their volume concentration.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

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**Figure 1.**Laboratory conditioning system. (

**A**) Geometrical configuration of the reactor; (

**B**) Power diagram of the single-blade sheet-type agitator.

**Figure 2.**Illustration of the decrease of the number of free primary particles before (

**A**) and during flocculation (

**B**).

**Figure 5.**Relative ratio of floc quartile diameter d10. (

**A**) and mean floc diameter d

_{F}; (

**B**) to Kolmogorov microscale as a function of the velocity gradient.

**Figure 6.**Textural parameters of flocs determined through laser diffractometry. (

**A**) Porosity vs. d

_{F}/d

_{P}; (

**B**) Density vs. d

_{F}/d

_{P}; (

**C**) Relative volume fraction of flocs vs. d

_{F}/d

_{P}; (

**D**): Fractal dimension vs. velocity gradient.

**Figure 8.**Variation of mean floc diameter with increasing velocity gradient. (

**A**) Application of Parker’s model (d

_{max}= C.G

^{−}

^{γ}); (

**B**) Application of the model proposed in this study.

**Figure 9.**(

**A**) photo of the flocculation reactor; (

**B**) photo of the setup implemented for floc texture analysis.

**Table 1.**Results of applying the proposed model to the analysis of residual turbidity variation with velocity gradient.

C_{si} (g/L) | k_{a} | k_{b} (s) | p | τ_{min} (NTU) | G_{τmin} (s^{−1}) | R^{2} | Δτ (%) |
---|---|---|---|---|---|---|---|

6.25 | 4.99 × 10^{−4} | 1.47 × 10^{−9} | 2.32 | 6.7 | 726 | 0.997 | 14.0 |

12.5 | 5.35 × 10^{−4} | 1.91 × 10^{−7} | 1.62 | 7.6 | 1105 | 0.997 | 14.0 |

25 | 7.65 × 10^{−4} | 1.89 × 10^{−5} | 1.00 | 9.3 | 1621 | 0.998 | 14.2 |

**Table 2.**Parameters from the proposed model (Equation (29)) for the relationship between floc size and velocity gradient.

Csi (g/L) | D_{F}^{a} | D_{F}^{b} | G_{a/b} (s^{−1}) | k_{a} | s |
---|---|---|---|---|---|

6.25 | 2.5 | 2.46 | 300 | 0.60 × 10^{−4} | 2.28 |

12.5 | 2.44 | 2.39 | 259 | 0.20 × 10^{−4} | 1.48 |

25 | 2.41 | 2.47 | 174 | 0.25 × 10^{−4} | 1.13 |

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**MDPI and ACS Style**

Bizi, M.
A Semi-Empirical Model to Estimate Maximum Floc Size in a Turbulent Flow. *Molecules* **2022**, *27*, 5550.
https://doi.org/10.3390/molecules27175550

**AMA Style**

Bizi M.
A Semi-Empirical Model to Estimate Maximum Floc Size in a Turbulent Flow. *Molecules*. 2022; 27(17):5550.
https://doi.org/10.3390/molecules27175550

**Chicago/Turabian Style**

Bizi, Mohamed.
2022. "A Semi-Empirical Model to Estimate Maximum Floc Size in a Turbulent Flow" *Molecules* 27, no. 17: 5550.
https://doi.org/10.3390/molecules27175550