Page 864 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix E: Porous Media Hydraulics

Porous media for this discussion includes any granular mater- d 50 is the mean diameter of granular media (m)
ial that comprises a bed for treatment (but may include various m is the dynamic viscosity of fluid at a given temperature
2
kinds of materials with a pore structure that permits a flow (Ns=m )
under a pressure gradient). The hydraulics of porous media r is the dynamic viscosity of water at a given temperature
3
flow is reviewed (Box E.1); the idea is to be able to estimate (kg=m )
headloss for a given hydraulic loading rate.
In the groundwater field, the characteristic length is d 50 , but
d 10 is used here since it is used in the water treatment field to
E.1 APPLICATIONS characterize media, along with the uniformity coefficient
(UC). Conversion from d 10 to d 50 can be estimated by the
Treatment processes that utilize granular media include
relation, d 50 d 10 [1 þ 0.8(UC 1)]. If a sieve analysis is
adsorption by activated carbon, ion-exchange, and deep-bed
done, the d 50 can be obtained directly.
filtration. Also diatomaceous earth, smaller in size and only
5–50 mm in depth of cake, is a porous medium. The charac-
teristic sizes of media are about 0.5 mm for granular activated E.2.2 FOUR FLOW REGIMES
carbon (GAC), 0.5 mm for sand in deep-bed filtration, with Table E.1 shows the four flow regimes—laminar, inertial,
about 0.9 mm for anthracite. In deep beds, such as say 3 m transition, and turbulent, based upon R(porous media) criteria
depth, a coarse medium such as 2 mm anthracite may be as given by Trussell and Chang (1999). The characteristics of
likely. In groundwater flow, the media may range from clays each regime are described in the following enumeration:
to sand and gravel, with a size range from microns to centi-
meters. In filter beds, the granular media is characterized by 1. Laminar: The ‘‘laminar’’ flow regime, i.e., R(porous
d 10 and d 50 ; in other applications, the mean diameter is used media) 1, is characterized by viscous effects gov-
commonly. Figure C.4 provides a graph that may be used for erning flow and is described mathematically by
plotting the results of a sieve analysis to obtain these param- Darcy’s law (Box E.3), i.e.,
eters. In addition, membrane filtration (Chapter 17) occurs
under a hydraulic gradient in which Darcy’s law is applicable. dh
v ¼ K (E:2)
dz
in which
E.2 HYDRAULIC FLOW REGIMES
v is the superficial velocity, i.e., hydraulic load
Hydraulic flow regimes for porous media range from laminar ing rate, Q=A (m=s)
to turbulent and are characterized by the Reynolds number. K is the hydraulic conductivity of porous medium
The starting point is to define the Reynolds number. Then, (m=s)
Darcy’s law, the cornerstone for hydraulic theory (Freeze, h is the hydraulic head, i.e., (p=g þ z) (m)
1994), can be defined. Box E.2 is a slight diversion from the z is the coordinate is direction of velocity
main theme, which is intended to clarify terms, some of which vector, generally vertical in filtration (m)
have emerged since the 1990’s.
In the laminar regime, a dye injected at some point
can be observed to follow a streamline path.
E.2.1 REYNOLDS NUMBER 2. Inertial:As R increases slightly, the laminar condi-
tion will prevail, i.e., a dye will continue to follow
The Reynolds number for flow through porous media,
the streamline path but the v versus dh=dz starts to
(Box E.2) is defined as
deviate from linearity and is described mathematic-
ally by the Forchheimer equation,
rvd
(E:1)
R(porous media) dh L
m ¼ a F v þ b v 2 (E:3)
F
dz
in which in which
R(porous media) is the Reynolds number for porous media a F is the coefficient related to linear head loss
flow (s=m)
v is the superficial velocity defined as Q=A(cross section) b F is the coefficient related to nonlinear head
2
2
(m=s) loss (m =s )
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