Mathematical Model

The mathematical model considers the motion of sediment particles in the rotating circular flume in two stages: a transport or a settling stage and a flocculation stage. The settling stage is modeled using an unsteady advection-diffusion equation. For flow conditions that exist in the rotating flume, the equation can be simplified to a one dimensional form as follows:

where Ck is the volumetric concentration of sediment of the kth size fraction and wk is the fall velocity of the same fraction. D is the turbulent diffusion coefficient in the vertical direction; t is time and z is the vertical distance from the water surface. This equation was solved using a finite difference scheme proposed by Stone and Brian,18 which minimizes the numerical dispersion. The boundary conditions specified for solving the equation are, (a) no net flux at the water surface and (b) the net upward flux at the sediment water interface is calculated as the difference between the erosion flux and the deposition flux. A uniform concentration of sediment over the water column was used as the initial condition for the model.

The flocculation stage was modeled using a coagulation equation shown in the following equation:

-d—^- = -PN (i, t)J2 K (i, j)N ( j, t) + 2 K (i - j, j)N (i - j, t)N ( j, t) (8.2) j=i j=i

This equation expresses the number-concentration balance of particles undergoing flocculation as a result of collisions among particles. The terms N(i, t) and N( j, t) are number concentrations of particles in size classes i and j, respectively at time t; K (i, j) is the collision frequency function, which is a measure of the probability that a particle of size i collides with a particle of size j in unit time, and i is the collision efficiency, which defines the probability that a pair of collided particles coalesce and form a new particle. The collision efficiency parameter i accounts for the coagulation properties of the sediment-water mixture. This includes the bacterial bond and the bacterial "glue" referred to earlier.

The first term on the right-hand side of Equation (8.2) gives the reduction in the number of particles of size class i by the flocculation of particles in class i and all other size class particles. The second term gives the generation of new particles in size class i by the flocculation of particles in smaller size classes. In this process, it is assumed that the mass of the sediment particles is conserved.

Equation (8.2) was solved after simplifying it into a discrete form by considering the particle size space in discrete size ranges. Each range was considered as a bin containing particles of certain size range. The size ranges in various bins were selected in such a way that the mean volume of particles in bin i is twice that of the preceding bin. When the particles of bin i flocculate with particles of bin j (j < i), the newly formed particles will fit into bins i and i + 1. The proportion of particles going to bins i and i + 1 is calculated by considering the mass of the particles before and after flocculation.

The collision frequency function, K(i, j) assumes different functional forms depending on the type of the collision mechanism considered. The collision mechanisms that were considered in the model were: (a) Brownian motion (Kb); (b) turbulent fluid shear (Ksh); (c) inertia of particles in turbulent flows (KI); and (d) differential settling of particles (Kds). An effective collision frequency function Kef was calculated in terms of the individual collision functions as follows:

The geometric addition in Equation (8.3) above is necessary because of the geometric addition of velocity vectors involved in the last three collision frequency functions (Huebsh).19

The collision frequency functions for the different collision mechanisms considered assume the following functional forms (Valioulis and List20):

In the above equations, k is the Boltzmann constant, T is the absolute temperature in Kelvin, \x is the absolute viscosity of the fluid, v is the kinematic viscosity of the fluid, s is the turbulent energy dissipation rate per unit mass, p and pf are densities of fluid and sediment flocs, respectively, and g is the acceleration due to gravity.

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