Page 872 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix E: Porous Media Hydraulics 827

0.35–0.45; silt, 0.35–0.50; etc. Regarding filtration practice, The chloride ion is an excellent tracer since it is largely
Chang et al. (1999) noted that the rate at which backwash is nonreactive. Also, the larger the column, the less is the error
terminated has a major effect on the porosity of the filter bed. of measurement of V(voids).
A sudden stop of backwash, they noted resulted in higher bed
porosity than a gradual termination.
E.5 APPLICATION OF DARCY’S LAW
In flow of water through a bed of porous media the rate of
E.4.3 POROSITY MEASUREMENT headloss with respect to bed depth is constant if the media is
uniform and is ‘‘clean.’’ Figure E.6 illustrates such condition
Two methods to determine porosity are (1) determining the
for t ¼ 0. In rapid filtration, however, floc particles enter the
volume of the media, and (2) determining the volume of the
bed and attach to the grains with the highest attachment
pores. The media volume method requires measuring the bulk
density at the top, declining exponentially with depth. Such
volume of the media in place, V(bulk), and the oven-dry mass
clogging causes k to decline with depth in the same fashion.
of the media, M(media), placed in the column and knowledge
Since Q must be constant if the flow does not change, then v
of its specific gravity, SG(media). The relation is basically
must increase and so dh=dz increases, which is depicted in the
that the bulk volume equals the volume of solids plus the
hydraulic profiles of Figure E.6.
volume of pores,
Figure E.6 shows a filter bed oriented vertically, as in
M(media) practice, with the velocity vector down. Piezometer taps are
(E:6)
P ¼ 1 located at A, B, C, D. A series of hydraulic grade lines (HGL)
SG(media)g V(bulk)
w
are shown, for t ¼ 0, 2, 4, 6 h. Note also that the bed is
oriented vertically and so the hydraulic gradient is not seen as
in which
clearly is in Figure E.1, where the bed is on its side. At t ¼ 0h,
P is the porosity of media
the clean bed headloss is 4 units with the length of the column
M(media) is the mass of media (kg)
12 units; therefore, Dh=DL ¼ 4=12 ¼ 0.33 units of head=unit
SG(media) is the specific gravity of media, e.g., about 2.65
of length. A valve is located at E and the excess headloss
for sand and 1.4 for anthracite
3 between the headwater and tailwater is taken up by the valve
r w is the specific mass of water (kg=m )
3
V(bulk) is the bulk volume of media (m ) (as shown on the right side of the HGL). The HGL at times
2, 4, 6 h shows the advance of the clogging front. Finally at
6 h, the valve at E is 100% open and the entire headloss is
A second method is to determine the pore volume by first filling
taken up by the media. As seen, most of the headloss is taken
the media from the bottom with a concentrated salt solution.
up by the clogged part of the media and the slope, Dh=DL,
Briefly, the procedure is to fill the column from the bottom with
is highest at the top of the bed. At the bottom of the bed
a solution of NaCl at known concentration, C(voids), say 2000
the slope, Dh=DL, remains as it was at t ¼ 0 h. A valve at
mg=L, after first purging the media of residual water. The
point E in Figure E.6 is common in filter design; the valve
solution is brought exactly to the surface of the column. The
opens as the bed clogs, based upon maintaining constant
volume of the column is calculated from its dimensions and
flow. A simple design would be to omit the valve and let
the void space due to the media support and tubing is deter-
the water level rise as the filter bed clogs. The bed would have
mined by water displacement. The salt solution is then
to be designed, however, with a higher weir crest so that the
purged with distilled water by displacing several volumes of
bed would have sufficient water depth above the media to
void space and the volume is collected and measured to give
avoid hydraulic scour.
V(purge). The chloride concentration of the dilute solution is
measured by titration to give C(purge). Since the mass of salt
is constant, the void volume can be calculated, E.6 MODELS OF PERMEABILITY
A quest of porous media modeling has been to calculate k
V(voids) C(voids) þ V(tubes) C(voids)
from first principles. Such a quest is like seeking the Grail and
¼ V(purge) C(purge) (E:7) in the case of porous media, like other modeling efforts, there
is always one coefficient remaining that must be determined
in which empirically. Then to determine that coefficient, one may as
3
V(voids) is the volume of voids in column of media (m ) well have conducted the basic laboratory testing to get k in the
C(voids) is the concentration of Cl in voids measured first place.
3
(kg=m ) Like many mathematical models, however, a more basic
V(purge) is the volume of solution collected after purging understanding is the fulfillment. Also, we may examine trends
3
column with distilled water (m ) with mathematical models and do sensitivity analysis even if
C(purge) is the concentration of Cl in V(purge) measured that one last coefficient is not determined (we can assume a

3
(kg=m ) number such as ‘‘1’’ for the purpose of exploration of trends).
V(tubes) is the volume of tubes and other support space The starting point for most models of flow through porous
3
under the media (m ) media is the Hagen–Poiseuille equation in which the pores of