Page 865 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

820 Appendix E: Porous Media Hydraulics

BOX E.1 NOMENCLATURE ON PRESSURE
depth, DZ, and is the difference in elevation between
The idea of pressure is simple, but its not uncommon to the headwater and the tail-water in the column, or
see different terms used to mean the same thing, which Dh ¼ h(tailwater) h(headwater). The gradient, Dh=DZ,
can make one wonder what is meant. The explanation is the ‘‘hydraulic-gradient,’’ which may also be called the
here is done to excess if one is familiar with the teach- ‘‘hydraulic-head-gradient,’’ or the ‘‘potential-gradient.’’
ings of a first course in fluid mechanics. If not, the As noted, in a rapid filter, the hydraulic-gradient starts
discussion may serve as a refresher and also may pro- out uniform with depth in the filter bed but as clogging
vide a common understanding for the terms used. occurs, the gradient becomes steepest at the top and
Pressure is a common measurement, done tradition- declines to a clean-bed gradient. After some hours of
ally by a ‘‘piezometer,’’ a ‘‘manometer,’’ or a pressure operation, there are no net deposits of floc in the upper
gage, or more recently, a pressure transducer. A piez- few centimeters of the filter bed and the hydraulic
ometer is merely a tube inserted at a point ‘‘A’’ where gradient is constant (in the upper few centimeters).
the measurement is to be taken, such as in a pipeline or
a bed of porous medium. The liquid rises to a level,
‘‘h A ,’’ above point ‘‘A,’’ where the pressure, p A ,at ‘‘A’’
equals h A g w ,or p A ¼ h A g w , where g w is the spe-
cific weight of water. A calibrated pressure gage meas- BOX E.2 VELOCITIES
ures p A at directly as does a pressure sensor. The
Two velocities in porous media flow are: (1) superficial
measurement, h A , is the ‘‘hydraulic head at ‘‘A’’
velocity, v, and (2) interstitial velocity, v(pores). In
(p A =g w ). A ‘‘manometer’’ measures the difference in
addition, the term, ‘‘specific-discharge,’’ q, emerged
pressure between two points (such as in a pipeline).
during the 1990s, which has the same mathematical
The commonly used term, ‘‘hydraulic-head’’ refers
definition as ‘‘superficial velocity,’’ v, but it has a dif-
the sum of the elevation head, ‘‘z A ,’’ at the point of
ferent connotation, important in the field of ground-
measurement, ‘‘A,’’ in a given ‘‘flow-field,’’ plus the
water hydraulics.
pressure head, ‘‘p A =g w ,’’ where p A is the pressure in
2
(N=m or Pa) at point ‘‘A’’ and g w is the specific weight Superficial velocity, ‘‘v’’: Consider a vertically oriented
3
of water in N=m . In other words, h A ¼ p A =g w þ z A ; empty column of cross-sectional area, A. Suppose the
velocity head is negligible. For reference, keep in top half of the column is empty and the bottom half is
3
packed with sand. Let a flow, Q, enter the top of the
mind that g w (208C) ¼ r w (208C) g ¼ (998.2 kg=m )
3
2
(9.807 m=s ) 9789 N=m . The elevation head, ‘‘z A ,’’ column and exit at the bottom. The calculated average
is the vertical distance (m) with respect to any arbitrary velocity of water in the top half of the column is,
datum (a convenient reference elevation). v ¼ Q=A. In the lower part of the column, this same
In any flow, pipeline or ground water, the hydraulic calculation is used to characterize the velocity within
head gradient is usually called, simply, the ‘‘hydraulic the sand-bed, i.e., v(superficial) ¼ Q=A(column). To
gradient.’’ The ‘‘velocity-head’’ ‘‘v A =2g’’ is also emphasize the point, the ‘‘superficial-velocity is a
included the definition for h A , which is the kinetic ‘‘pseudo-velocity.’’ To further emphasize the point, the
energy of the water at ‘‘A’’; usually its negligible in ‘‘superficial-velocity,’’ v, is also called the ‘‘approach-
flow through porous media. velocity,’’ or the ‘‘face-velocity.’’
In ground water flow, h A is also termed hydraulic
Interstitial velocity, ‘‘v(pores)’’: The real average vel-
potential with the symbol f.In a ‘‘flow-net,’’ the locus
ocity in the sand bed described is the ‘‘interstitial’’
of points of constant potential is called a ‘‘potential’’
velocity and is termed here, v(pores), calculated as,
line. In ground water flow, the ‘‘flow-field,’’ as depicted
v(pores) ¼ Q=(A P), where P is the average porosity
by a ‘‘flow-net’’ may have curvature, depending on the
of the sand bed. Porosity is the volume of voids divided
‘‘boundary-conditions.’’ In a ‘‘clean-bed’’ (homoge-
by the total volume of voids plus solids. Of course the
neous) rapid filter, the ‘‘flow-field’’ is characterized by
pores of any packed-bed are not uniform and so, at the
an hydraulic-head gradient that is linear with depth and
streamlines that are vertical. The associated ‘‘flow-net’’ microscopic level, v(pores) varies within the pore space,
which is random.
is a rectangular grid. As the media ‘‘clogs’’ with floc,
the hydraulic head gradient becomes steeper at the top Specific discharge, ‘‘q’’: The term, ‘‘specific-
where the floc deposits have a higher concentration. In a discharge,’’ q, is used commonly in the ground water
slow sand filter most of the headloss occurs at the field i.e., q ¼ Q=A. Mathematically q is the same as
surface, where a ‘‘schmutzdecke’’ usually forms. the superficial velocity, v. But its conceptually different
The terms used in hydraulic gradient for a column in that it’s unequivocally ‘‘true.’’ Also, explication of q
are: Dh, which is the headloss across the sand bed of is not necessary.