Page 439 - Water and wastewater engineering

11-12 WATER AND WASTEWATER ENGINEERING

The mass balance for a differential element may be expressed as

C
v a (11-5)

t z
where specific deposit
mass of accumulated particles per filter bed volume, mg/L
2
3
v a filtration rate, m /s · m of filter surface area, also m/s
t time, s
By combining Equations 11-4 and 11-5 , the basic phenomenological model is

v C (11-6)
a
t
A simplified steady state model allows the computation of the time to breakthrough ( t B ) and
the time to the limiting head ( t HL ):

t B B D (11-7)
v C C E )
( 0
a
( H T hD
)
L
t HL (11-8)
C
( k HL )( )( 0 C E )
v a
where B specific deposit at breakthrough, mg/L
D depth of filter bed, m

C 0 influent concentration of particles, mg/L
C E effluent concentration of particles, mg/L
H T limiting head, m

h L initial clean bed headloss, m
k HL headloss rate constant, L · m/mg
A regression analysis of data collected from actual or pilot filters is used to estimate B and k HL .
With these “constants,” the length of the filter run and the time to reach terminal headloss can be
estimated. In addition, with pilot data, the optimum filter depth and run time can be estimated for
a given solids loading.

11-5 THEORY OF GRANULAR FILTER HYDRAULICS
The hydraulic issues to be considered in the design of a filter system include: headloss through a
clean filter bed, headloss resulting from the accumulation of particles in the bed, the fluidization
depth of the bed during backwashing, and headloss in expanding the filter bed.

Clean Filter Headloss
Although the equations describing headloss are limited to clean filter beds, they provide an oppor-
tunity to examine the initial stages of filtration and the effects of design variables on headloss.
A number of equations have been developed to describe the headloss of clean water flow-
ing through a clean porous medium, such as a granular filter. Several of these are summarized in
Table 11-2 . These are derived from the Darcy-Weisbach equation for flow in a closed conduit and