
MBW:Bacterial Community AggregationFrom MathBioArticle review by Erin Byrne. Han, Mooyoung & Lawler, Desmond F. "The (Relative) Insignificance of G in Flocculation."^{[1]}
ContentsBackgroundFor over 50 years, it was theorized that the rate of aggregation of particles in suspension was primarily determined by the mean velocity gradient (G) of the surrounding fluid. In 1943, Camp & Stein ^{[2]} derived both the equations for G and for the rate of flocculation by fluid shear and stated that "the speed of flocculation is directly proportional to the velocity gradient". Consequently, the design of water treatment facilities (as well as the associated theory) had been based on this relationship. (For more information on bacterial aggregates, please see APPM4390:Bacterial Aggregates in the Bloodstream). In this paper, Han & Lawler took a closer look at two particle collisions. They used the mathematical and theoretical understanding of these collisions in flocculation, considering specifically changes in the fluid motion and shortrange forces as the particles get close to one another. They found that the effects of these changes diminished significantly the role of G in the rate of flocculation. TheoryThere are three primary ways an individual particle can become part of an aggregate: Brownian motion, differential sedimentation, and fluid shear. Each method is dominant in different regions of relative particle to particle size and absolute particle size. The rate of is described using a collision frequency function, β(i,j), that describes collisions between two particles of size i and j, commonly taken as a sum of the collision frequency functions for the three collision mechanisms. The traditional form of β(i,j) is found in Equations (2)(5) of Han and Lawler ^{[1]}. There are several assumptions that are important to point out for this traditional model: \
Han & Lawler analyze the rate of flocculation by addressing the first two assumptions, as well as maintaining the heterodisperse suspension assumption. Most water treatment plants have assumed a monodisperse suspension where all particles are considered to have roughly the same diameter and to be of reasonable size (> 1 micron). In this realm, fluid shear is the dominant mechanism for aggregation and the overall frequency of collisions between particles was taken to be proportional to G. By considering heterodisperse suspensions, Brownian motion and differential sedimentation are incorporated. For more indepth model development and explanation in the formation of aggregation and the effects of flocculation, see MBW:Flocculation Dynamics  A PDE Model Rectilinear v. CurvilinearAs previously described, the rectilinear model does not take into account shortrange forces or the changes in fluid dynamics as the two particles approach each other. The curvilinear model considers both in the following three ways:
In the rectilinear model, the critical separation distance between two interacting particles centers of mass is given by the sum of their radii since any smaller distance will result in a collision. In the curvilinear model, the critical separation distance is much smaller, as shown in Figure 1. The value of the critical separation distance was determined by repeated simulation, exploring various distances and which would result in collision and which would not. Collision EfficiencySince the rectilinear model incorporated the shortrange effects in terms of a collision efficiency factor, Han & Lawler chose to do the same. They developed new factors α_{Br}(i,j), α_{Sh}(i,j), and α_{DS}(i,j) for each collision mechanism based on the three curvilinear assumptions listed above. For fluid shear and differential sedimentation, the collision efficiency factor is taken to be proportional the ratio of critical separation distances for the rectilinear and curvilinear models: For Brownian motion, the results are slightly different. The collision efficiency factor is again a multiplicative correction to the collision frequency function, but the form is taken to be a third order polynomial whose coefficients depend on particle diameter. The full form is given in Han & Lawler ^{[1]} in Table 2. It is notable that the greatest correction occurs when the two particles in consideration are the same size, which for small particles also corresponds to the realm in which Brownian motion is the dominant collision mechanism. Collision FrequencyThe corrected curvilinear collision frequency functions are obtained by applying the collision efficiency factor for each collision mechanism to its corresponding rectilinear collision frequency function. The overall collision frequency function is then the sum of these individual functions. The curvilinear collision frequency function for Brownian motion most closely resembles the rectilinear version. The other two mechanisms show dramatic reductions in magnitude. Their forms are shown in Figures 2, based on one particle having a fixed diameter of 2 microns and allowing the second diameter to vary. You can click on each image to see a larger, more detailed version. The dominant region for each collision mechanism has now shifted, with Brownian motion having expanded and fluid shear contracted. A broader look at these regions (where now both particles are allowed to vary in size) is shown in Figure 3. In Figure 3, the dark lines denote the rectilinear division of dominance. Region A is one is which fluid shear was dominant in the rectilinear model and Brownian motion is dominant in the curvilinear model. Similary, Region B denotes the change in dominance from fluid shear to differential sedimentation, and Region C from differential sedimentation to Brownian motion. The unshaded area is the only portion that remained dominated by fluid shear, and its region of dominance is restricted to particles of similar size. As discussed earlier, water treatment facilities (the audience of this paper) operate with a wide range of particle sizes. Since fluid shear now dominates a relatively small region of particle interactions, the velocity gradient G has become "(relatively) insignificant"^{[1]}. Summary & ConclusionsHan & Lawler examined the effects of hydrodynamic forces and van der Waals attraction in the analysis of two particle collisions and their role in flocculation. The following conclusions were reached:
Project AdditionsMathematics Used Han and Lawler calculated collision frequency by applying curvilinear correction factors to already established collision equations modeling Brownian motion, fluid shear, and differential sedimentation:
Type of Model This is a particle model, where particle size of the colliding particles is the dominant variable , the subscript indicating size class. Biological System Studied Flocculation is generally performed using a bacterial coagulant, but this work is specific to that 'biological system' only in as much as the chemical properties of the bacteria could affect the Van der Waals forces between particles/molecules, and hence change the curvilinear correction factors (). Further information can be found at APPM4390:Bacterial Aggregates in the Bloodstream Discussion of Recent Paper Citing Han & LawlerZhang, Jianjun and Xiaoyan Li. (2003). 'Simulation and verification of particle coagulation dynamics for a pulsed input'. Journal of Water and Environment Technology, Vol 1, No 1 Asian Waterquality '01, 8590. In this more recent paper, Zhang and Li compare the response of the flocculation process to changes in mixing time, floc shape, and shear rate using a rectilinear collision model, a curvilinear model, and experimental data. Curvilinear collision was modeled using Han and Lawler's equation and correction factors, but with one significant difference. Han and Lawler assumed flocs to be conserved spheres, with no porosity, whereas Zhang and Li modeled flocs as having fractal geometry, based on work by Jiang and Logan (1991). The resulting simulations demonstrated that the rectilinear and curvilinear models produce similar floc size distributions, but on vastly different time scales (curvilinear model takes ~10x as long to produce the same result). The curvilinear model also proved more sensitive than the rectilinear model to the magnitude of floc fractal dimension (the degree of fractal behavior) , which changes the mass to size ratio of the floc, as well as porosity. As Han and Lawler predicted, the curvilinear was less sensitive to changes in fluid shear. The curvilinearfractal model showed good agreement with experimental results, indicating the model could be useful for operators of water and waste water treatments plants, would could simulate the flocculation response to changes in coagulation (mixing) rate. Wastewater treatment plants are the primary application for floc research, and can be read about here References
Jiang Q. and Logan B.E. (1991). 'Fractal dimensions of aggregates determined from steadystate size distributions'. Environ. Sci. & Technology., 31, 12291236. Zhang, Jianjun and Xiaoyan Li. (2003). 'Simulation and verification of particle coagulation dynamics for a pulsed input'. Journal of Water and Environment Technology, Vol 1, No 1 Asian Waterquality '01, 8590 