Page 876 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix E: Porous Media Hydraulics 831

Required ing clarifications, and provided key references on hydraulic
Clean-bed headloss, h L conductivity, on Darcy, and on the formulation of his well-
Solution known law on flow through porous media. The author is
1. Constants Let a ¼ 215 and b ¼ 3.5 and P ¼ 0.47 responsible for the interpretation of her advise.
2. At 208C, m ¼ 0.001002 N s=m 2 and r w ¼ 998.2
kg=m 3
3. Compute Dh L from Equation E.6, GLOSSARY

Absolute temperature: Defined: T(K) ¼ 273.15þ 8C; T(R) ¼
2 2
Dh L 0:001 (1 0:47) 1 459.6 þ 8F.
¼215 0:010185
2:54 998:2 9:81 0:47 3 0:00155 Darcy: Refers to Darcy’s law stating that flow through sand is
proportional to the hydraulic gradient. The results of

1 (1 0:47) 1
þ3:5 0:010185 2 Henry Darcy’s (1803–1858) experiments, using a
9:81 0:47 3 0:00155
2.50 m column 0.35 m diameter fitted with two
(ExE:2:1)
manometer near the top and bottom, respectively,
¼ 0:372 m=m (ExE:2:2) were published in 1856 in Paris, buried in a report
of 647 pages that he had prepared that dealt with the
Dh L ¼ 0:946 m (ExE:2:3)
development of a water supply for the City of Dijon.
His work in pipe flow developed conclusive evi-
Discussion dence that resistance to flow depends on the type
Table CDE.4 is a spreadsheet that provides a means to and condition of the pipe and is usually linked with
calculate Dh L for any conditions. For the conditions stated, Weisbach. He was a part of the Corps des Pont et
the distribution of headloss is 0.251 m=m laminar and Chauseés, an elite fraternity of engineers and a gov-
2
0.121 m=m turbulent. At HLR ¼ 24.4 m=h (10 gpm=ft ) ernment agency that gave engineers considerable
the distribution is 0.167 m=m laminar and 0.054 m=m status as intellectuals and professionals. Darcy was
turbulent.
many things as a professional: the designer of the
water supply for Dijon which was started in 1830
with water delivery in 1840, the administrator a large
E.8 HYDRODYNAMICS
regional engineering office, a leader of the commu-
Steady flow through homogeneous isotropic porous media nity, and a researcher. (The foregoing from Freeze,
can be described mathematically (see Muskat, p. 129) by the 1994; see also Brown, 2002)
hydrodynamic relation (the Laplace equation), Darcy: A unit of intrinsic permeability used sometimes by
persons in the ground water field. The equivalent is:
2
1 Darcy ¼ 0.987 10 12 m . In other words, multiply
2
r F ¼ 0 (E:17) a value in Darcys by the factor 0.987 10 12 m to
2
obtain, k, the intrinsic permeability. For example
Table 14.1 gives the permeability of Filter Cel as
in which F is the hydraulic potential (m). 12 2
0.07 Darcys; then k ¼ 0.07 Darcy 0.987 10 m =
The velocity at any point is proportional to the negative 12 2
Darcy ¼ 0.07 10 m .
potential gradient. The ‘‘solution’’ to Equation E.6 can be seen
d 10 : In a sieve analysis this is the particle size in which 10%
graphically as a ‘‘flow net’’ which is characterized by all
of the particles are smaller; the d 10 size is called also
potential lines and all streamlines crossing normal to one
the ‘‘effective size.’’ The numbers d 10 , d 60 , and UC
another with the ‘‘stream tubes’’ conveying the same incre-
are used to characterize media size distribution in
ment of flow and the DF for adjacent potential lines being
granular media filters used in water treatment.
equal. In a column, such as rapid rate filter or a pilot plant
d 60 : In a sieve analysis this is the particle size in which 60%
filter, the flow net, looking at a side view in two
of the particles are smaller.
dimensions, is simply a rectangular or square grid. Equation
Dispersion: Super-position of random motion at the
E.6 applies for the laminar flow regime, and as noted, prob-
micro-level on the general advective transport of a
ably could be extended into the inertial regime as long as the
fluid. The random motion is due to fluid turbulence
linear relationship between v and dh=dz is a reasonable
in pipE.flow or open-channel flow, or atmospheric
approximation.
advection of air masses. Although molecular
motion is also random and has the same effect, its
effect is small except in laminar flow. The random
ACKNOWLEDGMENTS
motion results in a ‘‘normal’’ (i.e., Gaussian) distri-
Dr. Deanna Durnford, professor of civil and environmental bution about the mean flow. The standard deviation
engineering (Emeritus), Colorado State University, helped to of the normal distribution increases with the number
set straight some of the nomenclature in Darcy’s law, suggest- of ‘‘steps,’’ of which elapsed time is a surrogate