1. Filtration Processes Introduction
Filtration processes are used principally for the removal of particulate material in water including clays and silts, micro-organisms and precipitates of organics and metal ions.
The particles that are removed are typically much smaller (0.1–50 m) than the size of the filter media (500–2000 m)

2. Filtration through granular media, particularly sand, is one of the oldest and most widely used water treatment processes.
Fig. Schematic of a deep bed filter

3. Filtration process is a batch operation that typically lasts between 12 and 36 hr.
The filtration process can be divided into seven distinct zones:
1. Lag phase, where clean backwash water passes out of the filter from the under-drains.
2. Pre-ripening phase, where the effluent quality becomes poorer, caused by dirty backwash water remnants within the bed.
3. Second pre-ripening phase, where the effluent quality further deteriorates because of dirty backwash water remnants above the media.

4. Fig. A typical filter cycle.
4. Ripening phase, where the filtrate quality starts to improve to its steady-state value.

5. 5. Effective filtration, where the filter is operating at its optimum level and represents the main period of operation within the cycle.
6. Breakthrough phase, where the filter quality begins to deteriorate as the filter reaches its capacity.
7. Spent phase, where the filter effluent concentration has reached a maximum and wormhole flow may be occurring.

6. 2. Process science 2.1 Removal Collision
The capture of particles within the filter bed involves a two-step process of collision and attachment.
Collision
three mechanisms in collision
(1) Diffusion
All particles will be moved in a random pattern away from their streamline due to the bombardment of surrounding molecules, known as Brownian motion.

7. particle diffusion is only applicable
for particles of less than 1 m in diameter:

8. 2. Interception
If the particle radius is greater than the distance between the flow streamline and the grain, the particle will contact the grain as the fluid flows past
whilst remaining on the streamline:

9. (3) Sedimentation
When the fluid flow is directed downwards sedimentation effects will cause the particles to settle vertically onto the grains. In this way the individual
filter media are acting like very small sedimentation tanks.
The probability of capture is characterised by the dimensionless ratio of the Stokes settling velocity and the approach velocity of the fluid:

10. Where dp is the particle diameter (m), d is the diameter of the filter media (m), u is the filtration velocity (m3 m−2 hr−1), m is the viscosity (kg m−1s−1), K is Boltzman’s constant and T is the absolute temperature (K).

11. Fig. Removal efficiency as a function of particle size.

12. Attachment Analysis of these expressions suggests that the
probability of capture will be reduced by high filtration rates (u−1, u−2/3), large media diameters (d−2, d−2/3), colder temperatures (−1, −2/3) and smaller light particles (dp2, (p– )) for post 1 m particles.
Attachment
When a particle approaches a grain to closer than 100 nm, short-range forces begin to become important.

13. Filtration descriptions
Incorporation of the mechanistic expressions into an analysis of the trajectories of the streamline enables the performance of the filter to be predicted
where  is the attachment efficiency, ε is the bed porosity, 0 is the collision efficiency and L is the bed depth (m).

14. Fig. Filter performance according to fundamental (microscopic) models
Fig. Filter performance according to fundamental (microscopic) models. U = 0.14 cm s−1,L = 60 cm, ε = 0.4, dg = 0.5 mm, T = 293 K, p = 1.05 g.cm−3 (n.b. numbers typical of water treatment).

15. The alternative approach to describing filtration performance is described as phenomenological modelling as it makes no attempt to consider the mechanisms of particle removal.
This approach describes the performance by conducting a mass balance over the filter in conjunction with an empirical
rate expression.

16. where  is the specific deposit (volume of deposit per filter volume),  is the filter coefficient and represents the key descriptor of any individual filter’s ability to remove particles, y is the filter depth and t is the run time.
Comparing the two approaches:

17. As the bed becomes laden with captured particles it becomes less efficient at removal as the streamlines straighten out and the local velocity within the pores increases generating more shear forces.
2.2 Hydraulics
The head loss (i.e. pressure drop) that occurs as water passes through a clean filter bed can be calculated from the well-known Kozeny–Carman equation:

18. where K is the Kozeny constant which is 5 for a fixed bed or slowly moving bed and 3.4 for a rapidly moving bed, dV/dt is the volumetric throughput (flow rate if constant) and Sv is the specific surface area of the media.
Within the typical ranges used in water treatment a simplified expression can be used as the pressure drop increases in an approximately linear fashion with velocity:

19. where kh is a head loss coefficient specific for a particular media (size and type)