Environmental Engineering FOURTH EDITION

146 ENVIRONMENTAL ENGINEEWG

where Vavg = average volume of all particles and A, = average area of all particles.
Substituting Eq. (7.8) into Eq. (7.6) yields

and

(7.10)

where x = weight fraction of particles retained between any two sieves and d' =
geometric mean diameter between those sieves

1-e

hL = f (:) ( T) (5) ;.

(7.11)

This equation holds for nonstratifid beds, such as are found in a slow sand filter, since
the fraction factor does not vary with depth. When particles are stratified, as in a rapid
sand filter, we can write Eq. (7.11) as

-- dhL - [(f> (9) ($)If$.

dL

The total head loss is thus

(7.12)

hL = 1" dhL = [ (i) (9) ($)] 1 L1 f ;i' dL.

Since d L = L dx, where a!x is the proportion of particles of size d

and if the particles between adjacent sieve sizes are assumed uniform,

hL= [($) (G) ($)]E$.

(7.13)

(7.14)