Preamble:

Sedimentation is the second unit operation in the water treatment process. It means that sedimentation comes after the screening. Sedimentation involves the removal of suspended and colloidal materials present in water and wastewater. It is used to separate suspended matter that is heavier than water and cannot float. That is to clarify raw water, effluent, and raw wastewater.

Sedimentation or gravitational settling is usually done in a sedimentation tank and used to remove sand, total suspended solids (TSS), and flocs from the supernatant. The principle behind the operation of a sedimentation tank is to reduce the velocity of the flow of water in order to allow the settling of the major amount of suspended solids at a velocity, known as settling/limiting/terminal velocity.

Types of Sedimentation

There are four main types of sedimentation/gravitational settling:

Discrete settling: this refers to solids in suspension whose concentration is low. Here constant acceleration, size, and weight are assumed and there is usually no agglomeration. The colloids go out as they come in. They are most common in the removal of sand particles. Discrete settling is an ideal case and may or may not happen.

Flocculent settling: this occurs in a dilute suspension of particles that coagulate and flocculate increasing their size and weight and acceleration. They are most common in TSS removal in wastewater. It can occur in coagulation basins and plain sedimentation basins.

Hindered or zone settling: this occurs when suspensions are inside the liquid in which interparticle forces present or hinder them from settling.

Compression settling: this refers to particles that form a structure. Further settling comes by compression of the structure. That compression weight usually comes from the supernatant (clarified liquid).

Theory of Sedimentation

The theory of sedimentation states all particles moving at velocities higher than the terminal velocity or equal to it will settle faster while those moving at a velocity less than the terminal velocity settle at the ratio of their velocity to the terminal velocity.

Laws of Sedimentation

There are two laws that govern sedimentation:

1. Newton’s law: In a sedimentation tank, the gravitational force is trying to settle the particles while the limiting drag force is opposing it. The sum of the gravitational force and limiting drag force is Newton’s law.

2. Stoke’s law: According to Stoke’s law, for settling to take place, F_G = F_D.
When F_G = F_D, the terminal velocity, v_{p (t)} = √ (4/3) (g/C_D) ((ρ_s – ρ_w)/ ρ_w) d_p ≈ √ (4/3) (g/C_D) (S_gp – 1) d_p

Where,
S_gp = specific gravity of particle in question and
d_p = diameter of particles.

In the determination of the terminal or settling velocity, the effect of flow regimes and particle shape are considered.

Flow Regimes

Three flow regimes are usually encountered in water flow problems, laminar flow, transitional flow, and turbulent flow. These flow regimes have an effect on the sedimentation of water and they are designated based on Reynold’s number, N_R

N_R = V_pd_pρ_w/µ = V_pd_p/ν

Where,
Vp = volume of particles (m3)
ν = µ/ρ_w = kinematic viscosity (m²/s)
µ = dynamic viscosity (Ns/m²)
d_p = diameter of particles

For laminar flow, Reynold’s number, N_R ˂ 1

For transitional flow, N_R = 1 to 2000

For turbulent flow, N_R ˃ 2000

Effect of Particle Shape on Drag Coefficient, C_D

Particles have an effect on the drag coefficient, although particles are usually assumed to be spherical.

For spherical particles and laminar flow conditions (N_R < 1),
rag coefficient, C_D = 24/N_R

For spherical particles and transitional flow conditions (N_R > 2000),
Drag coefficient, C_D = 24/N_R + 3/√N_R + 0.34

For spherical particles and turbulent flow conditions (N_R = 1 to 2000),
Drag coefficient, C_D = 0.4

For non-spherical particles, the shape factor is included in the determination of terminal velocity
For spheres, shape factor, ϕ = 1
For sand, ϕ = 2
For other particles, ϕ varies even up to 20 or more

If the flow is laminar and particles are less than 0.1 mm, then terminal or settling velocity, v_p(t) = (g (ρs – ρw)) d_p²/ 18µ ≈ (g (S_gp – 1) d_p²)/ 18ν. Stokes found out that F_D = 3πµv_pd_p (for laminar flow conditions)

Design of Sedimentation Tanks: Ideal case of Discrete Particle Settling

To design the sedimentation tank, first, set a terminal velocity (note: terminal velocity is not dependent on the depth of the tank but the area of the tank and flow rate).

V_c = Q/A = surface overflow rate (m³/m²/day)

When discrete particles are in the sedimentation tank, all the particles with velocity ≥ terminal velocity will settle. Only a fraction with V ˂ terminal velocity will give the total amount of solid to be removed from the system which is the efficiency of the system.

X_r = V_s/V_c

Fraction removed =

The efficiency of the system = fraction removed x 100

Alternatively, the fraction removed for discrete particles can be determined by:

Where,
V_ni = average velocity of particles in the i^th velocity range
n_i = no. of particles in the i^th velocity range
V_c = terminal velocity

Examples

Example 1. Determine the removal efficiency for a sedimentation basin with a critical velocity V_o of 2 m/hr in treating wastewater containing particles whose settling velocities are distributed as given below, plot the particle histogram for the influent and effluent wastewater.

Solution
1. Create a table of six (6) columns as shown below.

2. Calculate the average particle settling velocity for each velocity range by taking the average of the range limits (of Column 1) to create Column 2.
3. Calculate the fraction removed for each velocity by dividing the average settling velocity by the critical overflow velocity, V_o = 2 m/hr. Where any result is greater than 1, write ‘1’ because it means that all the particles have been removed.
4. Determine the number of particles removed by multiplying the number of influent particles by the percent removal (Columns 3 x 4) that is fraction removed.
5. Total Column 5 and Column 3 and write under them.
6. Column 6 = Column 3 – Column 5
7. Percentage removed = (effluent/influent) x 100 = (Column 5/Column 3) x 100 = (372.5/460) x 100 = 80.99 % ≈ 81%.
8. Plot the histogram of influent vs effluent.

Example 2. A particle size distribution has been obtained from a sieve analysis of sand particles for each weight fraction. An average settling velocity has been calculated. What is the overall removal efficiency for an overflow rate of 4000m³/m²/day?

Solution Guide
1. Plot the wet fraction remaining against the settling velocity to get a curve (NB: overflow rate gives overflow velocity in m/min).
2. Get X_o and V_s/V_c from the curve and use the equation below to determine the fraction removed.