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From Christopher R. Schulz and Daniel A. Okun, we learned the following useful information with regards to filtration:
• Filtration is the separation of suspended impurities from water by passage through porous media.
• Slow sand filtration consists of slowly filtering water through a layer of ungraded fine sand. Periodically, the top layer is clogged by impurities and is skimmed off the top.
• Rapid sand filtration rapidly conducts filtration in depth as compared to the slow sand filter which uses only the top layer to capture suspended particles. A lighter anthracite coal layer with larger pore spaces than sand is used on top of a sand layer to capture larger particles while allowing the smaller particles passage to be captured by the lower sand layer.
• Backwashing is the act of removing the captured impurities in the filter bed by introducing enough water, usually from the effluent end, to fluidize and expand the bed and wash away the now released impurities. Backwashing is an art. There is neither a set time nor a set amount of backwash water required. If a filter is heavily clogged, significant length of backwash and greater backwash water for greater bed expansion are required. If the influent water is relatively low in NTU, less clogging may occur and a shorter backwash and less water may be required. Consequently, rapid filtration and required backwash operations necessitate a well trained crew of operators. This finding leads us to define our objective as not only providing a clear well design that works but a set of instructions in operating that design.
• Headloss via expanded media can be calculated as below:

{latex}
\large
$$
h = D(1 - f)(p - 1)
$$
{latex}

where:
h = headloss across the fluidized bed (m)
D = unexpanded bed depth (m)
f = porosity of unexpanded bed (dimensionless)
p = specific gravity of the filter medium (dimensionless)
Once fluidized, headloss through an expanded bed is constant.

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• Minimal Fluidization Velocity Equation:

{latex}
\large
$$
V_f  = {{0.00381(d_{60} ){}^{1.82}\{ \omega _s (\omega _m  - \omega _s )\} ^{0.94} } \over {\mu ^{0.88} }}
$$
{latex}

where

{latex}\large$$V_f $${latex}

= fluidization velocity, gpm/square feet

{latex}\large$$\omega _s $${latex}

= specific weight of water, lb/cubic feet

{latex}\large$$\omega _s $${latex}

= specific weight of water, lb/cubic feet

{latex}\large$$d_{60}$${latex}

= diameter of which 60% of the media is equal to or smaller, mm

{latex}\large$$\mu $${latex}

= viscosity of water, centipose

• Bed Expansion Equation:

{latex}
\large
$$
\overline \varepsilon   = 1 - {D \over {D_e }}(1 - \varepsilon )
$$
{latex}

where

{latex}\large$$\overline \varepsilon$${latex}

= porosity of expanded bed

{latex}\large$$\varepsilon $${latex}

= porosity of unexpanded bed

D = depth of unexpanded bed

{latex}\large$$D_e$${latex}

= depth of expanded bed

• Fluidization Velocity Equation:

{latex}
\large
$$
V = K_e (\overline \varepsilon  )^{n_e }
$$
{latex}

where

V = fluidization velocity

{latex}\large$$K_e,n_e$${latex}

= Constants derived experimentally or empirically as shown below

{latex}\large$$
n_e  = 4.45{\mathop{\rm Re}\nolimits} _0^{ - 0.1}
$$
{latex}

where

{latex}\large$$
{\mathop{\rm Re}\nolimits} _0  = {{\rho _l  \cdot 8.45 \cdot V_f  \cdot d_{60} } \over \mu }
$$
{latex}

where:
μ = the dynamic viscosity of the fluid (Pa- s or N- s/m² or kg/m- s)
p= density of the fluid (kg/m^3)
d60= the particle diameter at which 60% of the particles are smaller or equal to.
Vf= minimum fluidization velocity of the media

and

{latex}\large$$
K_e  = {{V_f } \over {\varepsilon ^{n_e } }}
$$
{latex}