# Sludge Settling Model

In this article, Senior Developer, Dr. Gengsheng Wei, discusses the new Sludge Settling Model, available in the upcoming release of FLOW-3D v12.0.

Settling is the first stage for sewage sludge treatment, followed by digestion and disposal. In FLOW-3D v12.0, a sludge settling model has been developed to simulate sludge settlement in septic tanks, clarifiers and other sewage treatment equipment. The sludge motion is governed by both sewage flow and gravity-driven settling. The transport equation for sludge is

$\displaystyle \frac{{\partial c}}{{\partial t}}+\nabla \cdot \left( {C{\mathbf{u}_{{sldg}}}} \right)={{\nabla }^{2}}(DC)$

where C denotes sludge concentration that is defined as sludge mass per unit volume of sludge-water mixture; D is the sludge diffusion coefficient; $\displaystyle {{\mathbf{u}_{{sldg}}}}$ is the total sludge velocity and can be written as

$\displaystyle {{u}_{{sldg}}}=\mathbf{\overline{u}}+{\mathbf{u}_{{settle}}}$

where $\displaystyle \mathbf{\overline{u}}$ is the velocity of sludge-water mixture and $\displaystyle {\mathbf{u}_{{settle}}}$ is the sludge settling velocity that is always in the direction of gravity.

In general, the $\displaystyle {\mathbf{u}_{{settle}}}$ value decreases with increasing C. It drops to zero when C reaches a critical value, Cmax, which is the maximum sludge concentration in settling tanks. In the fully settled regions with C= Cmax, sludge stops settling but its advection with fluid flow continues. Cmax is a user-specified input parameter in the model.

Users can define $\displaystyle {{u}_{{settle}}}$ as a function of C using either tabulated data or the Vesilind equation (1968)1,

${{u}_{{settle}}}={{U}_{0}}{{e}^{{-nC}}}$

where ${{U}_{0}}$ is the maximum sludge velocity, and n is Vesilind parameter with the dimension of volume/mass. When C= Cmax, $\displaystyle {{u}_{{settle}}}$ is set to zero by the model.

The animation above is the simulation result of sludge settling in a septic tank. Sewage flows into the tank from the inlet for 8 sec at the sludge concentration of 5 kg/m3.  Cmax is specified as 10 kg/m3 and can be found in the fully settled regions at the bottom of the tank. U0=1.144×10-3 m/s and n=0.4317 m3/kg are used in the simulation. It is seen that the newly added sludge largely stays in the main chamber of the tank, with a small amount of it entering the secondary settlement chamber through the hole on the separation wall. The initially-settled sludge is disturbed by the incoming sludge but then resettles with it in about 5 min.

The new sludge settling model will help wastewater engineers to better analyze flow and settling process of sludge and improve their designs for wastewater treatment equipment.

### Reference

1Vesilind P. A., 1968, Design of prototype thickeners from batch settling tests. Water Sewage Works 115(7), 302-307.