A combined experimental and numerical approach to the assessment of floc settling velocity using fractal geometry

2020 ◽  
Vol 81 (5) ◽  
pp. 915-924 ◽  
Author(s):  
R. B. Moruzzi ◽  
J. Bridgeman ◽  
P. A. G. Silva
Keyword(s):  
Particle Size ◽  
Fractal Dimension ◽  
Primary Particle ◽  
Settling Velocity ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Numerical Approach ◽  
Primary Particle Size ◽  
Fractal Aggregates ◽  
Water And Wastewater Treatment

Abstract Sedimentation processes are fundamental to solids/liquid separation in water and wastewater treatment, and therefore a robust understanding of the settlement characteristics of mass fractal aggregates (flocs) formed in the flocculation stage is fundamental to optimized settlement tank design and operation. However, the use of settling as a technique to determine aggregates' traits is limited by current understanding of permeability. In this paper, we combine experimental and numerical approaches to assess settling velocities of fractal aggregates. Using a non-intrusive in situ digital image-based method, three- and two-dimensional fractal dimensions were calculated for kaolin-based flocs. By considering shape and fractal dimension, the porosity, density and settling velocities of the flocs were calculated individually, and settling velocities compared with those of spheres of the same density using Stokes' law. Shape analysis shows that the settling velocities for fractal aggregates may be greater or less than those for perfect spheres. For example, fractal aggregates with floc fractal dimension, Df = 2.61, floc size, df > 320 μm and dp = 7.5 μm settle with lower velocities than those predicted by Stokes' law; whilst, for Df = 2.33, all aggregates of df > 70 μm and dp = 7.5 μm settled below the velocity calculated by Stokes' law for spheres. Conversely, fractal settling velocities were higher than spheres for all the range of sizes, when Df of 2.83 was simulated. The ratio of fractal aggregate to sphere settling velocity (the former being obtained from fractal porosity and density considerations), varied from 0.16 to 4.11 for aggregates in the range of 10 and 1,000 μm, primary particle size of 7.5 μm and a three-dimensional fractal dimension between 2.33 and 2.83. However, the ratio decreases to the range of 0.04–2.92 when primary particle size changes to 1.0 μm for the same fractal dimensions. Using the floc analysis technique developed here, the results demonstrate the difference in settlement behaviour between the approach developed here and the traditional Stokes' law approach using solid spheres. The technique and results demonstrate the improvements in understanding, and hence value to be derived, from an analysis based on fractal, rather than Euclidean, geometry when considering flocculation and subsequent clarification performance.

Hydrodynamic Drag and Electrophoresis of Suspensions of Fractal Aggregates

Fractals ◽  
1997 ◽  
Vol 05 (03) ◽  
pp. 507-522 ◽  
Author(s):  
D. Coelho ◽  
J.-F. Thovert ◽  
R. Thouy ◽  
P. M. Adler
Keyword(s):  
Fractal Dimension ◽  
Stokes Equations ◽  
Volume Fraction ◽  
Mean Field ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Hydrodynamic Drag ◽  
Hydrodynamic Properties ◽  
Fractal Aggregates ◽  
Dilute Limit

The hydrodynamic properties of fractal aggregates are investigated numerically by solving the three-dimensional Stokes equations. Aggregates are built with a variety of sizes and fractal dimensions. Their hydraulic radii conductivities and mobilities are evaluated as functions of their volume fraction in the suspension. An influence of the fractal dimension is visible on the hydrodynamic and electrokinetic characteristics of the aggregates. In the dilute limit, our results compare well with the predictions of various mean-field arguments.

scholarly journals Rat pulmonary responses to inhaled nano-TiO2: effect of primary particle size and agglomeration state

2013 ◽  
Vol 10 (1) ◽  
pp. 48 ◽  
Author(s):  
Alexandra Noël ◽  
Michel Charbonneau ◽  
Yves Cloutier ◽  
Robert Tardif ◽  
Ginette Truchon
Keyword(s):  
Particle Size ◽  
Primary Particle ◽  
Primary Particle Size ◽  
Nano Tio2 ◽  
Agglomeration State

scholarly journals Association of Cryptosporidium with bovine faecal particles and implications for risk reduction by settling within water supply reservoirs

2006 ◽  
Vol 4 (1) ◽  
pp. 87-98 ◽  
Author(s):  
Justin D. Brookes ◽  
Cheryl M. Davies ◽  
Matthew R. Hipsey ◽  
Jason P. Antenucci
Keyword(s):  
Particle Size ◽  
Water Supply ◽  
Risk Reduction ◽  
Settling Velocity ◽  
Three Dimensional ◽  
Three Dimensional Model ◽  
Particle Size Fractions ◽  
Size Classes ◽  
Large Particles ◽  
Water Supply Reservoirs

Artificial cow pats were seeded with Cryptosporidium oocysts and subjected to a simulated rainfall event. The runoff from the faecal pat was collected and different particle size fractions were collected within settling columns by exploiting the size-dependent settling velocities. Particle size and Cryptosporidium concentration distribution at 10 cm below the surface was measured at regular intervals over 24 h. Initially a large proportion of the total volume of particles belonged to the larger size classes (>17 μm). However, throughout the course of the experiment, there was a sequential loss of the larger size classes from the sampling depth and a predominance of smaller particles (<17 μm). The Cryptosporidium concentration at 10 cm depth did not change throughout the experiment. In the second experiment samples were taken from different depths within the settling column. Initially 26% of particles were in the size range 124–492 μm. However, as these large particles settled there was an enrichment at 30 cm after one hour (36.5–49.3%). There was a concomitant enrichment of smaller particles near the surface after 1 h and 24 h. For Pat 1 there was no difference in Cryptosporidium concentration with depth after 1 h and 24 h. In Pat 2 there was a difference in concentration between the surface and 30 cm after 24 h. However, this could be explained by the settling velocity of a single oocyst. The results suggested that oocysts are not associated with large particles, but exist in faecal runoff as single oocysts and hence have a low (0.1 m d−1) settling velocity. The implications of this low settling velocity on Cryptosporidium risk reduction within water supply reservoirs was investigated through the application of a three-dimensional model of oocyst fate and transport to a moderately sized reservoir (26 GL). The model indicated that the role of settling on oocyst concentration reduction within the water column is between one and three orders of magnitude less than that caused by advection and dilution, depending on the strength of hydrodynamic forcing.

scholarly journals The relation between the turbulent Mach number and observed fractal dimensions of turbulent clouds

2019 ◽  
Vol 488 (2) ◽  
pp. 2493-2502 ◽  
Author(s):  
James R Beattie ◽  
Christoph Federrath ◽  
Ralf S Klessen ◽  
Nicola Schneider
Keyword(s):  
Fractal Dimension ◽  
Star Formation ◽  
Mach Number ◽  
Column Density ◽  
Velocity Dispersion ◽  
Error Function ◽  
Empirical Relation ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Cloud Kinematics

Abstract Supersonic turbulence is a key player in controlling the structure and star formation potential of molecular clouds (MCs). The three-dimensional (3D) turbulent Mach number, $\operatorname{\mathcal {M}}$, allows us to predict the rate of star formation. However, determining Mach numbers in observations is challenging because it requires accurate measurements of the velocity dispersion. Moreover, observations are limited to two-dimensional (2D) projections of the MCs and velocity information can usually only be obtained for the line-of-sight component. Here we present a new method that allows us to estimate $\operatorname{\mathcal {M}}$ from the 2D column density, Σ, by analysing the fractal dimension, $\mathcal {D}$. We do this by computing $\mathcal {D}$ for six simulations, ranging between 1 and 100 in $\operatorname{\mathcal {M}}$. From this data we are able to construct an empirical relation, $\log \operatorname{\mathcal {M}}(\mathcal {D}) = \xi _1(\operatorname{erfc}^{-1} [(\mathcal {D}-\operatorname{\mathcal {D}_\text{min}})/\Omega ] + \xi _2),$ where $\operatorname{erfc}^{-1}$ is the inverse complimentary error function, $\operatorname{\mathcal {D}_\text{min}}= 1.55 \pm 0.13$ is the minimum fractal dimension of Σ, Ω = 0.22 ± 0.07, ξ1 = 0.9 ± 0.1, and ξ2 = 0.2 ± 0.2. We test the accuracy of this new relation on column density maps from Herschel observations of two quiescent subregions in the Polaris Flare MC, ‘saxophone’ and ‘quiet’. We measure $\operatorname{\mathcal {M}}\sim 10$ and $\operatorname{\mathcal {M}}\sim 2$ for the subregions, respectively, which are similar to previous estimates based on measuring the velocity dispersion from molecular line data. These results show that this new empirical relation can provide useful estimates of the cloud kinematics, solely based upon the geometry from the column density of the cloud.

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