Page 866 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix E: Porous Media Hydraulics 821

3. Transition: The transition regime is characterized by
BOX E.2 (continued) VELOCITIES
the transition from inertial flow to full turbulence. At
Convention uses superficial velocity, ‘‘v’’: The porous the lower end of the regime, turbulence is just begin-
media Reynolds number, R(porous media), uses ‘‘v,’’ ning to appear in some of the cells; at the upper end,
the superficial velocity, as is the velocity, ‘‘v,’’ in Darcy’s turbulence is present in most cells. The upper limit of
law, i.e., v ¼ K [Dh=DZ]. The use of superficial vel- this regime is not well defined but is above R 300
ocity, ‘‘v,’’ as opposed to ‘‘v(pore),’’ is by convention. and is likely in the range 600 < R < 800. The For-
chheimer equation form remains, but the constants a
and b change to a T and b T .
4. Turbulent: Full turbulence is present with random
fluctuating micro-velocities about the mean through-
TABLE E.1
out the media. The Forchheimer type equation
Flow Regimes for Porous Media Flow with Governing applies.
Equations
R(porous Equation Designs in water filtration are in the Reynolds number range,
Flow Regime media) Equation Statement 0.5 < R < 50, meaning that they are in either the ‘‘laminar’’ or
‘‘inertial’’ ranges (Trussell and Chang, 1999). The Darcy
Laminar 1 Darcy v ¼ K(dh=dZ)
Inertial 1–100 Forchheimer dh=dZ ¼ a F v þ b F v 2 equation applies in the laminar regime while the Forchheimer
Transition 100–800 Forchheimer dh=dZ ¼ a F v þ b F v 2 equation applies throughout both the laminar and the inertial
Turbulent >800 Forchheimer dh=dZ ¼ a F v þ b F v 2 regimes. As examples of R values that occur at the extremes
of rapid rate filtration practice

2
1. Let v ¼ 6.1 m=h (2.5 gpm=ft ) with d 10 ¼ 0.5 mm and
T ¼ 208C, then R 0.9.
BOX E.3 DARCY’S LAW 2
2. Let v ¼ 37 m=h (15 gpm=ft ) with d 10 ¼ 2.0 mm and
Equation E.2 is the cornerstone of porous media T ¼ 208C, then R 20.
hydraulics. It’s an understatement merely to state the
equation without some discussion. For the upper limit for most conventional designs, e.g., say
2
As seen in Equation E.2, Darcy’s law is a simple v ¼ 24 m=h (10 gpm=ft ) with d 10 ¼ 1.0 mm and T ¼ 208C, then
statement. It’s an empirical relationship, discovered by R 7 and Darcy’s law can be applied with little deviation from
3
Henry Darcy in 1856, that relates flow (m =s) per unit of measured data. The Forchheimer equation, on the other hand,
2 2
gross cross-section area (m ) through a column of por- extends to the laminar range of R, since the v term has little
ous medium to the hydraulic gradient across the col- effect at small R. The latter two points are discussed subse-
umn. Mathematically, its Q=A ¼ K Dh=DZ. Darcy’s quently. First, however, consider the equations applicable to
law applies to any ‘‘boundary conditions,’’ such as laminar flow, i.e., the forms of the Darcy equation.
found in geologic formations, as well as to the simple
ones of a column of filter sand or anthracite, or an ion- E.2.3 EXPERIMENTAL DEMONSTRATION OF DARCY’S LAW
exchanger, or an activated carbon column, or any other
FOR FILTER MEDIA
‘‘packed-bed’’ reactor. As noted by Trussell and Chang
(1999), applications have taken two parallel but inde- Data that illustrate the range of the applicability of Darcy’s
pendent tracks by such groups as hydro-geologists and law for filter media were obtained by Chang et al. (1999)
civil engineers, respectively. The first group has focused and are shown in Figure E.1a and b for 6 of 30 tests (the
on situations found in nature while the second has been 30 tests involved 3 sizes of sand, 3 sizes of anthracite and glass
concerned with engineered systems. beads, with tests for each media conducted for 3 or more poros-
ity values). Figure E.1a shows that the h L =Dz versus v for 0.47
mm sand is linear throughout the range of data, i.e., v 0.01
2
m=s, or 36 m=h (15 gpm=ft ). Also of interest, R 4.7 at
At low R the dh=dz is linearly related to v with small v ¼ 0.01 m=s; thus Darcy’s law is applicable as a means to
v 2 dependence but as R increases, v 2 becomes predict headloss, for R < 5(thehighest R for the data available).
dominant in the relationship. The first appearance Figure E.1b shows the same kind of h L =Dz versus v plot
of true turbulence, i.e., inertial effects, occurs at for 1.47 mm anthracite. The three curves start to deviate from
2
40 < R < 140, based upon visual studies. The inertial linearity at about v 0.005 m=sor 18m=h (7.4 gpm=ft )at
effects are due to the changes in the velocity vector, which R 7. Therefore, at R > 7, the Forchheimer equation
i.e., (1) changes in magnitude due to expansions and would be increasingly important as a means to describe the
contractions as the flow enters and exits from various h L =Dz versus v relationship. But at R 7, Darcy’s equation
cells, and (2) changes in direction due to curvilinear may be applied, which greatly simplifies the calculations of
flow around media particles. headloss.