Environmental Engineering FOURTH EDITION

178 ENVIRONMENTAL ENGINEERING

as shown in Fig. 9-8. In other words, the particle is just barely removed. Had the particle
entered the settling zone at any other height, its trajectory would always have carried it
into the sludge zone. Particles having this velocity are termed criticaZparticZes because
particles with lower settling velocities are not all removed. For example, the particle
having velocity us, entering the settling zone at the surface, will end up in the effluent
zone and escape. If this same particle had entered at height h, it would have just been
removed. Any of these particles that happen to enter the settling zone at height h or
lower would be removed, and those entering above h would not. Since the particles
entering the settling zone are equally distributed, the proportion of those particles with
a velocity u that are removed is

V, = h/H, (9.6)

where His the height of the settling zone. With reference to Fig. 9-8, similar triangles

yield

The time that the critical particle spends in the settling zone is

r=----- -LhH --.

Vh us vo

Hence

H

t

vo = :.

(9.7)

(9.8a)

(9.8b)

The time r is also equal to the hydraulic retention time or VIQ, where Q is the flow rate

and Vis the volume of the settling zone. Moreover

V =AH, (9.9a)

where A is the surface area of the settling zone. Thus

(9.9b)

Equation (9.9b) gives the ovegow rate, an important design parameter for settling

tanks. The units of the overflow rate are

m Q m31s

vo = - = - = -

s A m2'

Although overflow rate is usually expressed as gallondday-foot2, it implies a velocity

and is equal to the velocity of the critical particle. Thus, when the design of a clarifier

is specified by the overflow rate, the critical particle is thereby defined.

Where any two of the parameters - overflow rate, retention time, and depth -

are specified, the remaining parameter is also fixed.