Solids suspended in a fluid will settle at a velocity dependent upon their size and density and the density and viscosity of the fluid. The process occurs naturally in the deposition of suspended matter when a river enters a lake, because as the horizontal velocity is reduced, there is more opportunity for the solids to reach the bottom. Sedimentary rocks are another natural example of settlement processes, although here the time-scale for deposition is very long, so that small particles are deposited. Many waters and wastewaters contain suspended solids which can be removed by sedimentation, although it must be appreciated that as the settling velocity of particles becomes smaller with reducing size and density, the process reaches its practical limit because long retention times are uneconomic.
Suspensions may be composed of discrete particles, which have a fixed rigid surface and
do not readily agglomerate, or of flocculent particles, which have non-rigid surfaces and agglomerate when brought into contact with each other. As demonstrated in Figure 1, a suspension of discrete particles has a constant settling velocity with depth, whereas a flocculent suspension shows an increasing settling velocity with depth due to the growth in size of individual particles by collisions with less rapidly settling particles.
Simple settling theory is based on the behaviour of discrete particles, although in many waters and wastewaters the bulk of solids may well be flocculent in nature so that in practice the theory requires some modification. When a discrete particle is placed in a fluid of lower density it will accelerate until a terminal velocity is reached when the gravitational force is equal and opposite to the frictional drag force:
gravitational force = (ρs – ρw) g V
ρs = density of particle
ρw = density of fluid
g = acceleration due to gravity
V = volume of particle
Frictional force is a function of settling velocity, size of particle, and density and viscosity of fluid, and by dimensional analysis it can be shown that
frictional force = CD Ac ρw νs2 / 2
CD = Newton’s drag coefficient
Ac = cross-sectional area of particle
νs = terminal settling velocity
For spherical particles with Reynolds Number (RN) ≤ 1
CD = 24 / RN
For spherical particles with RN > 1
CD = 24/RN + 3/RN0.5 + 0.34
By equating gravitational and frictional forces an expression for the terminal settling velocity can be obtained.
νs = [2g V (ρs – ρw) / CD Ac ρw]0.5
For spheres Ac = πd2/4, V = πd3/6
νs = [4g d (ρs – ρw) / 3CD ρw]0.5
In laminar flow conditions (RN ≤ 1)
CD = 24v / νs d
where v = kinematic viscosity
Hence Stoke’s Law is
νs = (gd2/18v) [(ρs – ρw)/ρw
A basic understanding of the process of sedimentation can be obtained from a consideration of the concept of an ideal settling basin as illustrated in Figure 2. The assumptions made for such an idealized situation include quiescent conditions in the settling zone, uniform flow across the
settling zone, uniform solids concentration both vertically and horizontally at the entrance to the settling zone, and no resuspension from the sludge zone. Given these idealized conditions, a discrete particle which enters at the top of the settling zone and just reaches the bottom at the outlet end of the settling zone will have a settling velocity ν0
ν0 = distance settled / time = h0/t0
t0 = tank volume / flow rate = Ah0/Q
ν0 = Q/A
The expression Q/A is termed the surface overflow rate, and it can be seen that for discrete particles, removal by sedimentation is controlled by the surface area of the tank and is independent of depth. For particles with a settling velocity νs less than ν0 some removal will occur depending on the depth at which they enter the tank. Removal will occur if particles enter the tank at a distance from the bottom of the settling zone of h0 or less where h= νs t0.
Thus the proportional removal of particles with νs < ν0 will be in the ratio of νs/ν0.