Water tools

Filter backwash: minimum fluidisation, rate for a target expansion, expansion at a rate

A bed fluidises when the head loss through it equals its buoyant weight. From that balance come the rate the coarsest grains need, the rate a target expansion needs, and what a given rate does.

Medium

Grain diametermmUse the d90 for the minimum fluidisation velocity, the effective size for expansion.

Media densitykg/m³Sand 2,650; anthracite 1,400 to 1,800.

Fixed-bed porosity

Fixed-bed depthm

Backwash water temperature°C

Target expansion%About 25% for anthracite, 37% for sand.

Backwash rate to evaluatem/h30 to 60 m/h water only; 15 to 45 with air scour.

Target expansion

Rate for the target expansion

56.6m/h

Expanded porosity at target

0.631

Minimum fluidisation

Minimum fluidisation velocity

33.2m/h

The backwash rate must exceed this for the coarsest grains, usually the d90.

At the evaluated rate

Expansion at the evaluated rate

7.9%

Expanded porosity

0.555

Expanded depth

2.16m

How it works

Fluidised, the head loss equals the buoyant weight of the media, so setting the Ergun loss equal to it gives a quadratic in the Reynolds number for any porosity:

κ_{I} R e^{2} + κ_{V} (1 - ε) R e - β = 0 β = \frac{g ρ_{w} (ρ_{p} - ρ_{w}) d^{3} ε^{3}}{μ^{2}} v = \frac{μ R e}{ρ_{w} d}

(1)

Expansion and porosity are linked by conservation of media, and for a given rate the porosity is the real root of a cubic:

\frac{L_{E}}{L_{F}} = \frac{1 - ε_{F}}{1 - ε_{E}} ε_{E} = \sqrt[3]{X + \sqrt{X^{2} + Y^{3}}} + \sqrt[3]{X - \sqrt{X^{2} + Y^{3}}}

(2)

X = \frac{μ v}{2 g (ρ_{p} - ρ_{w}) d^{2}} (κ_{V} + \frac{κ_{I} ρ_{w} v d}{μ}) Y = \frac{κ_{V} μ v}{3 g (ρ_{p} - ρ_{w}) d^{2}}

(3)

where

$κ_{V}, κ_{I}$: viscous and inertial coefficients of the medium
$R e$: grain Reynolds number
$ε$: bed porosity
$β$: dimensionless group of the fluidised bed
$g$: gravity, 9.81 m/s²
$ρ_{w}, ρ_{p}$: water and media density, kg/m³
$d$: grain diameter, m
$μ$: water viscosity, kg/m·s
$v$: backwash velocity, m/s
$L_{F}, L_{E}$: fixed and expanded bed depth, m
$ε_{F}, ε_{E}$: fixed and expanded porosity
$X, Y$: intermediate groups of the porosity cubic

The same expansion needs a higher rate in warm water. Keep the expanded bed below the wash trough lip and stop air scour before the water reaches it. Dual media: compute each layer and let the larger requirement govern.

Filter backwash: minimum fluidisation, rate for a target expansion, expansion at a rate

How it works

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