Theory of settlement; Stoke's law

Theory of settlement of discrete particles

A discrete particle is a particle that keeps its size, shape and weight while settling (e.g. sand, silt). In a quiescent liquid, it first accelerates under gravity, then very quickly reaches a terminal (settling) velocity when the net downward force is balanced by drag.

Stokes' law

For a spherical particle falling freely at constant (terminal) velocity $v_s$, the weight minus buoyancy equals the fluid drag.

$$\begin{aligned} \text{Net weight} &= \frac{\pi d^3}{6}(\rho_s-\rho)\,g \\ \text{Drag (viscous, } Re<1) &= 3\pi\mu d\,v_s \end{aligned}$$

Equating the two:

$$v_s = \frac{g\,(\rho_s-\rho)\,d^2}{18\,\mu} = \frac{g\,(S_s-1)\,d^2}{18\,\nu}$$

where $d$ = particle diameter, $S_s$ = specific gravity of the particle, $\nu=\mu/\rho$ = kinematic viscosity.

Conditions of application in design

• The particle must be spherical, discrete (non-flocculent) and settle freely without being hindered by neighbours.
• The flow around it must be laminar, with Reynolds number $Re=\dfrac{v_s d}{\nu}<1$ (roughly particle diameter below about 0.1 mm for silt of $S_s=2.65$).
• The liquid is at rest or flows at very low velocity (no turbulence, no short-circuiting).
• For larger particles ($1<Re<1000$) the drag coefficient is taken as $C_d=\dfrac{24}{Re}+\dfrac{3}{\sqrt{Re}}+0.34$ and $v_s=\sqrt{\dfrac{4g(S_s-1)d}{3C_d}}$ is found by trial. For $Re>1000$, $v_s=1.8\sqrt{g d (S_s-1)}$ (Newton).

In an ideal rectangular tank the removal depends only on the overflow rate $v_0=Q/A=H/t$: a particle with $v_s\ge v_0$ is entirely removed.

Effect of temperature

Temperature changes the viscosity, and so the settling velocity ($v_s\propto 1/\nu$). The kinematic viscosity at temperature $T$ (°C) can be calculated from

$$\nu = \frac{0.01775}{1+0.0337\,T+0.000221\,T^2}\ \text{cm}^2/\text{s}$$

and for the same particle:

$$\frac{v_{s,T_2}}{v_{s,T_1}} = \frac{\nu_{T_1}}{\nu_{T_2}}$$

Typical values:

Hence the settling velocity at 20 °C is about 1.3 times that at 10 °C. For design, the lowest expected temperature is used, so the tank is also effective in warm weather.