Environmental Engineering FOURTH EDITION

Water Treatment 145

to the number of particles times the volume of the particles. The total volume is
thus

NVP

1 -e’

where N = number of particles and Vp = volume occupied by each particle. The total
channel volume is
NVP
e-
1 -e’
and since the total wetted surface is NAP, where Ap is the surface area of each particle,
the hydraulic radius

volume e- 1 NVP -e - ( e ) (%)

R= area - - -

wetted- perimeter area NAP 1-e

For spherical particles, VplAp = d/6, but for particles that are not true spheres,
VplAp = 4(d6), where 4 is a shape factor. For example, 4 = 0.95 for Ottawa sand,
a common filter sand. We thus have

R = (L) 1-e (4%).

Substituting Eqs. (7.4a), (7.4b), and (7.5) into Eq. (7.3b),

hL = f (b) (9) (5)

The friction factor may be approximated as

1-e

f = 150 (7) + 1.75,

(7.5)

(7.7)

where R is the Reynolds number and

The preceding discussion is applicable to a filter bed made of only one size particle.
For a bed made of nonuniform sand grains,

(7.8)