Turbulent rapid mixing in direct filtration
by Scott Lee Trusler
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Civil
Engineering
Montana State University
© Copyright by Scott Lee Trusler (1983)
Abstract:
The fundamental physical processes which occur in the initial mixing step of coagulation are not fully
understood at this time. This work focuses on initial mixing within a direct filtration treatment scheme,
in order to gain some insight into the nature of the mixing process. A direct filtration pilot plant
incorporating a variable speed, baffled tank mixer was operated with various mixing intensities during
the investigation. Filtrate quality and filter head-loss data from the pilot plant were used to establish the
effectiveness of a given mixing condition. Also, mixing within a hydraulic jump was similarly
investigated in order to compare hydraulic mixing to mechanical (stirred baffled tank or backmix
reactor) mixing.
An idealized mixing model, which was based on the interaction of the coagulant carrying turbulent
microscales with colloid particles, was proposed. The results of the pilot plant studies indicate that a
colloid-microscale size correlation similar to the correlation proposed by the mixing model may exist.
Also, a method_was proposed for calculating the mean velocity gradient (G-value) of a hydraulic jump
occurring on a sloping channel. The usefulness of this calculation method was confirmed in this study.
Finally, the pilot plant data collected during this investigation indicated that the hydraulic jump was
comparable to the baffled tank for initial rapid mixing (based on similar velocity gradients). However,
the inflexibility of the hydraulic jump, in regard to variable degrees of mixing, was also evident. TURBULENT RAPID MIXING
IN DIRECT FILTRATION
by
Scott Lee Trusler
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Master of Science
-
in
Civil Engineering
MONTANA STATE UNIVERSITY
Bo ze ma n, Montana
May 1983
MAIN LIB.
N 3*7%
T fPtI (0
6op. Si
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of a thesis submitted by
Scott Lee Trusler
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iii
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Any copying or use of the material in this thesis
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iv
ACKNOWLEDGEMENTS
I wish to express my appreciation to Dr. A. Amirtharajah
for his
guidance,
assistance
and encouragement.,' throughout
this investigation.
Also,
I wish
to thank Mary Mullen,
for her assistance
during the collection of data, Mrs. Bonnie Recktenwald, who
typed this
thesis
and
my
family,
for their
encouragement.
F i n a l l y , I am grateful for the support provided by the
Engineering Experiment
and the National
Station
Science
Project, grant No.
at Montana State University
Foundation/State of Montana MONTS
ISP-8011449.
I
V
TABLE OF CONTENTS
Page
LIST OF FIGURES
. ...................
LIST OF T A B L E S ......... ..
NOMENCLATURE
ABSTRACT
vii
. ............ ..
. ...
ix
.............................
X
...................... . . . . ' ...........
xii.
1.
I N T R O D U C T I O N ...................... ................
I
2.
OBJECTIVES
. . . . .
3
3.
PREVIOUS RELATED STUDIES
.........................
4
..............................
Turbulent Initial Mixing
.......................
Mixing in baffled tanks
4
................
Mixing in hydraulic jumps
.........
5
. . . .
8
Mixing in Water Treatment
................
10
Initial mixing studies
.............
10
Turbulent flocculation
Filtration Analysis
. . . . .
..........
13
.........
14
............................
16
. . . . . . . .
4.
IDEALIZED MIXING MODEL
5.
EXPERIMENTAL W O R K ......................... . . . . .
Plan
..................
22
Direct Filtration Pilot Plant
Baffled tank reactor
22
................
22
. . . . . . . . . . .
^
"
................
27
-
Details of flume
Dual media f i l t e r .................. ..
.\
.
29
31
V
.
vi
TABLE OF CONTENTS— Continued
Page
Filter Performance Data
.......................
34
Dye Trace S t u d i e s ......................
6.
RESULTS AND DISCUSSION
............................
Calculated Mixing Parameters
Velocity gradients
38
. . .....................
38
..................
40
...........
44
Filter response to baffled tank mixing
. .
Filtrability numbers for baffled
tank mixing
..................
Hydraulic Mixer Comparisons
44
52
Electrophoretic Mobility D a t a ........... ..
7.
38
..................
Hydraulic residence times
Filtration Results
37
.
...................
C O N C L U S I O N S ...........
56
5.7
63
■
R E F E R E N C E S ..................................
APPENDIX - Sample Calculations
.........
65
. . . .
68
vi i
LIST OF FIGURES
Page
1.
Power Dissipation Zones in a Stirred Tank
* .........
(Cutter [5] ) ................ ..
7
Design and Operation Diagram for Alum
T r e a t m e n t .......................................
12
Idealized Colloid-Microscale Interaction
Visualization ...............
17
Schematic Diagram-of Direct Filtration
Pilot Plant .....................................
23
5.
Size Distributions for Turbidity Particles
26
6.
Details of Baffled Tank Reactor and DiscTurbine I m p e l l e r ..............................
28
7.
Details of Hydraulic Jump F l u m e ......... ..
30
8.
Details of Dual Media F i l t e r ....................
32
9.
Residence Time Distributions for the
Baffled Tank R e a c t o r .........................
41
Residence Time Distributions for Two
Hydraulic J u m p s ............. ...................
43
Typical Filter Data for Baffled Tank
Mixing with Alum Treatment
■ ’
(n = 700 R P M ) ................... . ............ ..
45
Typical Filter Data for Hydraulic Jump
Mixing with Ferric Chloride Treatment
(Slope = 2.0 in/ft) ................ ..
46
2.
3.
4.
10.
11.
12.
13.
14.
. . .
...
. .
. .
Rate of Headless and Average Effluent
Turbidity for Baffled Tank Mixing
with Alum Treatment ..............
48
Rate of Headloss and Effluent Turbidity.
for Baffled Tank Mixing with Ferric
Chloride Treatment
............ . . . . . . .
49
viii
LIST OF FIGURES— Continued
Page
15.
.16.
17.
18.
19.
20.
21.
22.
Rates of Headless and Average Effluent
Turbidity for Baffled Tank Mixing
with Ferric Chloride Treatment. ..............
51
Filtrability Numbers for Baffled Tank
Mixing with Alum T r e a t m e n t .................. .
53
Filtrability Numbers for Baffled Tank
Mixing with Ferric Chloride Treatment
;. . . .
54
Filtrability Numbers for Baffled Tank
Mixing with Ferric Chloride Treatment
. . . .
55
Rates of Headless for Hydraulic Jump
and Open Channel Mixing with Ferric
Chloride Treatment
.........................
*
59
Average Effluent Turbidities for Hydraulic
Jump and Open Channel Mixing with
Ferric Chloride Treatment ....................
61
Filtrabiiity Numbers for Hydraulic Jumji
and Open Channel Mixing with Ferric
Chloride Treatment
.............. ...........
62
Hydraulic Jump Length, L, for Jumps in
Sloping C h a n n e l s ..............^...............
70
ix
LIST OF TABLES
<
1.
Page
Water Analysis of Feed Water for Direct
Filtration Pilot Plant
.......................
...........
2.
Size Characteristics of Filter Media
3.
List of Pilot Plant Runs Including
Coagulant Dosages and pH Conditions
4.
5.
33
.........
Calculated Velocity Gradients for Baffled
Tank Mixer
................................ ..
24
.36
.
39
Calculated Velocity Gradients for Hydraulic
Mixers
. * ....................................
40
6,. Mean Hydraulic Residence Times for Baffled
.....................................
Tank Mixer
42
7.
Electrophoretic Mobility Data for Treatment
of M30 and SB325 Colloids . . . . . . . .
58
X
NOMENCLATURE
C
Average effluent turbidity - TU
Co
Filter influent turbidity - TU
Eiji
Channel flow energy - ft
F
Filtrability number
Fl
Upstream Froude number for hydraulic jump
G
Average velocity gradient - s~l
Hip .
Total filter headless at time T - in
No
Number; of particle collisions
N
Number of colloid-microscale
■P
interactions
Average power dissipated - ft*Ib/s
Po
Power number
0
Flow rate - cfs
Re
Impeller Reynolds number
T
Filter run length - hr
V
Power dissipation volume - ft^
Vr
Filtration rate - in/hr
aI
Colloid diameter
d
Impeller diameter - ft; Water depth in flume-ft
g
Acceleration of gravity - ft/s^
n
Impeller speed - rev./s
n I , n2
Colloid number counts
ri r r2
Colloid radii
-
xi
NOMENCLATURE— Continued
t
Theoretical mixing time - V/0 - s
u, v, w
Respective velocity components
X, y, z
Respective coordinate directions
Y2
Water depth downstream of jump - ft
z
Channel elevation - ft
n
Microscale
E
Power dissipation per unit mass. - ft^/s^
Y
Specific weight of water - Ib/ft^
y
Fluid viscosity - l b •s/ft^
V
Fluid kinematic viscosity - ft^/s
P
Fluid density - slugs/ft3
B
Flume slope angle
S
xi i
ABSTRACT
The fundamental physical processes which occur in the
initial mixing step of coagulation are not fully understood
at this time.
This, work focuses on initial mixing within a
direct filtration treatment scheme, in order to gain some
insight into the nature of the mixing p r o c e s s . A direct
filtration pilot plant incorporating a variable speed, baf­
fled tank mixer was operated with various mixing intensities
during the investigation.
Filtrate quality and filter headloss data from the pilot plant were used to establish the
effectiveness of a given mixing condition. . Also, mixing
within a hydraulic jump was similarly investigated in order
to compare hydraulic mixing to mechanical (stirred baffled
tank or backmix reactor) mixing.
An idealized mixing model, which was based on the in­
teraction of the coagulant carrying turbulent microscales
with colloid particles, was proposed.
The results of the
pilot plant
studies . indicate
that
a colloid-microscale
size correlation similar to the correlation proposed by
the mixing model may e x i s t .
Also, a method_was proposed for calculating the mean
velocity gradient (G-value) of a hydraulic jump occurring
on a sloping channel.
The usefulness of this calculation
method was confirmed in this study.
Finally, the pilot plant data, collected during this
investigation indicated that the hydraulic jump was compar­
able to the baffled tank for initial rapid mixing (based on
similar velocity gradients).
However, the inflexibility of
the hydraulic jump, in regard to variable degrees of mix­
ing, was also evident.
I
CHAPTER I
INTRODUCTION
In this day and a g e , the treatment of water to provide
a potable
supply
for consumers has developed
faceted science incorporating
and fluid dynamics.
certain processes
Within
and
ideas from chemistry, physics
the overall
treatment p i c tu re ,
treatment methods have received con­
siderable attention over the past d e c a d e .
which is
filtration
into a multi­
not preceded
by
Direct filtration,
sedimentation, is one
treatment scheme where attention has been focused.
Another
aspect which has received considerable attention is that of
chemical reactions within agitated vessels.
the role of the mixing
intensity
In particular,
itself within the overall
chemical process is of considerable interest.
In addition
a b o v e , turbulent
at te nt io n.
ing in
studied.
jump is
The
water
A
to
the
hydraulic
usefulness
treatment
rational
important
hydraulic jump
leads to high
agitated
mixing
mixing
of
has,
design
for
reliability.
has
mixing
also
mentioned
received
some
the hydraulic
jump
for mix­
however,
not
been
recently
procedure
for
the
several
is
vessel
reasons.
operationally
Also,
hydraulic
First
of
simplistic
hydraulic mixers
all,
which
usually
require less total power input than mechanical mixers which
2
further enhance their operational attractiveness.
son for the
lower power
requirements
The rea­
are two fo l d .
First,
the power dissipation volumes for hydraulic mixers are usu­
ally much
smaller than the
Secondly, hydraulic mixers
volumes
lack
for mechanical
the
frictional
cies common to motor driven mixers.
hydraulic mixing
has
its, share
primary disadvantage with
mixers.
inefficien­
There is no doubt that
of drawbacks
the hydraulic
as
well.
jump being the
The
in­
ability to change, the mixing intensity to accomodate varia­
tions in the influent %ater quality.
The present
study
focused
on
initial mixing
direct filtration treatment scheme.
red baffled
tank)
and
were investigated.
on the
need
mixing step
will,
no
for
of
doubt,
The
a
hydraulic
importance
better
lead
Both mechanical
(hydraulic
to more
jump)
(stir­
mixing
of this study was based
understanding
coagulation.
within a
This
of
better
efficient
use
the
initial
understanding
of
employed in water treatment mixing operations.
the
energy
3
CHAPTER 2
OBJECTIVES
The overall objective of this study was to investigate
the initial mixing step of coagulation and to gain some in­
sight into
the
fundamental
nature
of
the
mixing
process.
The following were the specific objectives:
1. To
study
the
initial mixing
stirred baffled
the role
of
tank
mixing
step
reactor
intensity
charge neutralization using
in
in
of coagulation
order
to
in a
determine
destabilization
a direct
by
filtration pilot
plant.
2. To attempt to relate the results of the mixing study to
the microscale of turbulence.
3. To propose
a methodology
for design of hydraulic
jump
mixers.
4. To
compare
(backmix)
hydraulic
jump
reactor mixing.
mixing
with
baffled
tank
4
CHAPTER 3
PREVIOUS RELATED STUDIES
The theoretical studies which are related to this work
can be divided
into three
categories.
First ,, there
is the
material which deals with turbulence and mixing in general.
Second, are the studies which have focused on mixing in the
water treatment field.
studies are of
Both flocculation and initial mixing
importance here.
Finally,
some aspects re­
garding filtration analysis will be presented.
TURBULENT INITIAL MIXING
As defined by Hinze [1] , turbulence is a flow condition
characterized by irregular motion in which various parameters
vary in time and
space.
the random variations,
obtainable.
The
Hinze
also points out that within
statistically distinct averages are
concepts
underlying
turbulence,
such
as
fluctuating velocities and turbulent intensities, are fully
z-'
.
developed in the literature [I, 2] and will not be discussed
here.
Of more concern to the present study, are the funda­
mentals of
energy
dissipation,
within
a
turbulent
flow.
In any given turbulent flow it can be shown that regions
of high
velocity
approximation,
correlation
these
regions
exist.
where
In
all
a
the
somewhat
fluid
rough
exhibits
5
a similar velocity can he interpreted as eddies or vo rt i c e s .
Kolmogorof £ [3,4]
from an eddy
or
tegral length
was
the
length
scale
first
scale
and
to
rationalize
viewpoint
a microscale.
turbulence
by defining
The
an
integral
in­
scale
is related to large eddies which carry energy and the micro­
scale is the eddy size at which energy dissipation by viscous
friction b e g i n s .
Kolmogoroff noted that these small eddies
(microscale) would
have
to
be
in
a
state
of
equilibrium
and their
size would be a function of the energy
the fluid
viscosity.
By
dimensional
analysis
input and
Kolmogoroff
quantified the microscale as
(I)
n =
where H is the microscale, V
the fluid and
Since mixing
is the kinematic viscosity of
e is the power dissipated per unit mass.
is usually
power dissipation,
it
interpreted as being a function of
is
the
turbulent
microscale
which
is of particular importance here.
Mixing
in Baffled Tanks
Several ideas in regard to mixing in baffled tanks are
important to this investigation.
of Cutter
[5]
which
dissipation within
distinct regions
a
of
First'there are the results
illustrate
the
baffled
tank.
power
nonuniformity
dissipation
Cutter
to
of power
found
exist
three
within
a
6
baffled tank.
He
impeller stream
trates the
labeled
zone
three
and
zones
tion within each zone,
the
regions
the
bulk
and
the
as the tip
zone.
Figure
zone,
I
the
illus­
approximate power dissipa­
in relation to the mean dissipation.
More recently, O k a m o t o , Nishikawa and Hashimoto [6] have con­
firmed the existence of the
impeller stream and bulk
zones
of a baffled tank as described by Cutter.
Long before Cutter's work. Camp and Stein [7] formulated
the velocity g r ad i en t, or G —v a l u e , as a design parameter for
determining the
Stein utilized
power
requirements
laminar
fluid
shear
of
a mixer.
concepts
Camp
and
to derive
the
velocity gradient as
G =
0.5
(2 )
where G is the average velocity gradient, P is the power dis­
sipated within volume V and U
is the fluid viscosity.
Camp
and Stein also generalized the velocity gradient expression
for turbulent
flows by defining the root mean
square
(RMS)
velocity gradient as
G
3v. 2 +
3x'
(3)
where u, v and w are the velocities in the x, y and z direc­
tions respectively.
Equation (2) thus represents an average
value within a turbulent field.
ZONE OF
MAXIMUM
TURBULENCE
IMPELLER ZONE
TOTAL VOLUME = V= V, ♦
V3
AVERAGE POWER DISSIPATION = P
FIGURE I.
Pm3 ^
° - 25P
V3 ^
° - 9V
Power Dissipation Zones in a Stirred Tank (Cutter [5]).
8
Another parameter which is important in tank mixing is
the dimensionless power number, P 0 .
explain the
power
number
concept
Leentvaar and Ywema [8]
in
detail
and
show
the
quantified expression as
— — I —3 “5
P = Pp n d
o
(4)
where P is again the average power dissipated, p is the den­
sity of
the
fluid,
impeller diameter.
tion,
the
power
n
is
the
impeller
speed
and
d
is
the
For a given reactor^impeller configura­
number
remains
constant
beyond
a
certain
Reynolds number Re (^ssIO4 for cylindrical baffled tanks) where
(5)
R
e-
and all
symbols
are as previously defined.
power number of a given
reactor
T h u s , once the
is known, power inputs and
corresponding G-values can be computed for various impellers
and rotation speeds.
Mixing in Hydraulic Jumps
At p r e s e n t , mixing in hydraulic jumps is somewhat less
well defined than mixing in baffled tanks.
all the
tionale.
tools
necessary
to formulate
Chow [9] presents
a possible design ra­
The hydraulic jump can be defined as the mechanism
by which an open channel flow transforms from supercritical
9
to subcritical
of the
jump
the power
flow..
Looking
from, a
G-value
dissipated
at
the
mixing
capabilities
s t a n dp oi nt , necessitates
within
a
specific
volume
be
that
related
to the jump chara ct eri st ics .
Power dissipation in a hydraulic jump is actually well
understood and
the
mechanism
is
undoubtedly
Theory pertaining to energy losses
in a jump
detail by
steep
Chow
for
both
mild
and
turbulence.
is derived
channels.
in
Actual
energy losses are easily computed from water depth measure­
ments in the vicinity of the j u m p , since
2
E = z
T
where E? is the total
+ dCosG + —
(6)
2g
flow energy,
z is the channel eleva­
tion, d is the water d e p t h , .6 is the slope angle,
v is the
flow velocity and g is the acceleration due to g r a v i t y . Thus,
by computing E t upstream and downstream of the jump, the loss
of energy in the jump can be estimated.
Then utilizing the
expression for power
''-P = YQ(AEt )
where
(7)
y is the specific weight of water, and 0 is the flow
rate and
Et
is
power dissipated
The volume
most likely
the
in
in
related
energy
the
which
to
lost
in. the
jump, P
the
the
power
is
ju m p , the
determined
dissipation
characteristic
r
average
as
well.
occurs
jump
is
length.
10
Chow illustrates
that
the
the available flow energy
nel slope)
jump ■ length
L
is
a
function'of
(upstream Froude number and chan­
and presents empirical data which can be used to
estimate the length of a jump which occurs in a sloping chan­
nel.
The
actual power dissipation
occur in the
fluid
volume
can then
be
in the
jump
contained
assumed
to
length L .
Since the power and dissipation volume are k n o w n , a G-value
for the jump can be com put ed .
MIXING IN WATER TREATMENT
Mixing studies
numerous, however,
within
only
a
the
few
water
treatment
studies
are
field
relevant
are
here.
These relevant investigations may be split into- those which
have dealt with initial rapid mixing directly and those which
have dealt with flocculation.
Initial Mixing Studies
Wilson
[10] attempted to define optimum rapid mixing by
studying the
flocculation
different mixing
methods.
efficiencies
The
resulting
actual
from
optimization
two
was
based on a floe strength model where optimum mixing was de­
fined as
the
within the
condition
which produced
flocculator.
reactor and.a tubular
that uniform,
Wilson
reactor
the
employed
in his
strongest
a
studies
baffled
and
floes
tank
concluded
instantaneous, plug-flow mixing was the opti­
mum condition based on floe strength.
J
11
Sometime la te r , Stenquist
a series
pared a
mixer
of continuous
flash
(pipe
iency was
Their
backmixing
for
used
alum
as
the results
of alum
of
the
coagulation
[11]
a. multiple
results
was
completed
to
when
■Even
though
to
plugeffic­
Stenquist
th os e• of
been more
of
grid
conclusive
flocculation
studies may have
than
no
inferior
contradict
chemistry
orifice
produced
inherently
ga u g e .
seem
both
to
treatment-,
Kaufman's conclusions
Kaufman
flow experiments in which they com­
(backmix)
flow).
evidence that
flow mixing
mixer
and
mixing
and
Wilson,
a function
itself.
The
actual effects of alum dosage in relation to mixing are dis­
cussed below.
Amirtharajah and Mills [12] developed the design diagram
for alum
treatment
as
shown
in
Figure
2.
The
diagram
is
based on thermodynamic principles and delineates the region
of solid phase aluminum hydroxide as a function of pH.
Also
shown on the diagram are regions where specific coagulation
mechanisms predominate.
the diagram
to
various degrees
rapid
Amirtharajah and Mills then applied
mixing
by
performing
jar
(G-values) of rapid m i x i n g .
tests
with
They concluded
that intense, short duration mixing was. essential if adsorp­
tion-destabilization was
ation.
the
primary
mechanism
of
coagul­
They also found that when sweep coagulation was the
predominant mechanism,
mixing
important to n o t e , that Wilson
effects were m i n i m a l .
and Stenquist
It is
both operated
Co Kju afi>n
300
IOO
IO
(boundori
with col
3
I
Adsorption
0.3
pH
FIGURE 2
OF MIXED SOLUTION
Design and Operation Diagram for Alum Treatment
ALJUM os Aj(SO4)s.l4.3H20-mg/l
30
Zone
13
their mixers
in the sweep zone during their investigations.
Amirtharajah
[13]
also
relationship between
suggested
the
that
a
fundamental
turbulent microscale
and
size
the par­
ticles which were to be destablized may e x i s t .
Only one
tiveness as
Ellms
[14]
an
investigation
of
initial mixer
conducted
full
the hydraulic
has
been
scale
effec­
reported. . Levy and
alum
treatment
utilizing a hydraulic jump for rapid m i x i n g .
that the
jump's
studies
They concluded
hydraulic
jump
was
a
very
effective
means
for
mixing co ag ul an t s.
For
the
energy
expended.
Levy
and
Ellms concluded that the jump produced mixing of great rapid­
ity and thoroughness.
Turbulent Flocculation
Of primary importance to this investigation are particle
collision theories which were first, introduced by von Smoluchowski
[15] .
He derived the collision theories for laminar
shear but the original formulations have now been generalized
for turbulent flow conditions.
Saffman and Turner [16] uti­
lized Smoluc how ski 's ideas to predict.the collision of fain
droplets
in a turbulent
cl o u d .
They quantified the
number
of coll isi ons , N0 , as
'
.3 Sire 0.5 .
Np = nln2 (rl + r 2 (l5^'
(8)
14
where
and ng are respective drop number counts, rj and r 2
are respective drop radii, and all other symbols are as pre­
viously defined.
Spielman
[17]
has
also pointed
out
that
Saffman and Turner's Equation (8) is applicable to turbulent
flocculation.
Most
recently,
Smolu cho wsk i's ideas
theory which
into,
incorporates
Adler
a
[18]
general
has
incorporated
particle
hydrodynamic
effects
collision
as
well
electric field, double layer and Van der W a a l s ' forces.
as
He
subdivides coagulation into homocoagulation and heterocoagu­
lation, where
homo^
particles and
hetero-
sized particles.
almost always
refers
the above
better,
at
to
least
his
von
turbulent
coagulation
that
equal-sized
within
the
develdment.
seem
to
of
unequal­
homocoagulation
Smol uc ho ws ki 's
flow,
of
coagulation
concluded
of
studies,
extension to
refers
Adler
physical constraints
to
chemical
and
Considering,
all
ideas,
be
is
far
and
their
reaching
and
may thus be applicable to the rapid mixing process.
FILTRATION ANALYSIS
Since this study is not an investigation of filtration
fundamentals per se, the only literature of interest is that
which deals with
response of
a
analysis
filter
to
of
the
filtration data.
depostion
media is a function of both headless
Several researchers
have
attempted
to
The overall
of mass
within
the
and effluent quality.
define
the
response
of a filter to mass loading by combining headless and efflu­
15
ent quality data
21, 22, 23].
into a single
filtrability index
[19, 20,
Recently, Janssens, Adam and Buekens
[24] did
a statistical analysis on various filtrability indexes uti­
lizing direct
filtration pilot plant data as a base.
were interested
best tool
that the
for
in
determining
filter
response
Filtrability
the most useful.
which
Number,
index
analysis.
proposed
by
would
They
be
the
They
concluded
Iyes
[21] , was
The expression for the Filtrability Number
F , is given by
F
V
(9)
V r
where T is the filter run length defined by effluent quality
degradation, H t
is the
total headloss
at time T , C
is the
average filter effluent turbidity through time T , C0 is the
filter influent
turbidity
Consistent units
inspection of
minimized,
V
is
the
F a dimensionless
the arrangement
in the' expression
F is
make
and
of
the
filtration
parameter.
rate.
Close
individual parameters
for F , leads to the. conclusion that when
an
optimum
filtration
condition
exists.
16
CHAPTER 4
IDEALIZED MIXING MODEL
The first
step
model involved
in
developing
an
a visualizatiori process
idealized
whereby
mixing
the micro­
scopic aspects underlying mixing could be postulated.,
Since
mixing is most often associated with power dissipation,
teractions between
colloid particles
in­
in suspension and the
turbulent microscale can be rationalized as being of primary
importance. . The visualization was confined to a homogeneous
turbulent field
consisting
represented by
of
the microscale
eddies
whose
formulation
size
(I).
c o u l d .be
Also,
the
development was restricted to the adsorption-destabilization
process where
a
uniform
particle
treated with a,chemical co ag ul an t.
scale eddies
for the
in
the
transport
field
of
the
were
suspension
was
Furthermore,
assumed
coagulant
to
be
to
be
the micro­
responsible
hydrolysis
species,
which are most likely in incipient form.
Turning to the visualization.
simplistic particle-microscale
Case I represents
Figure 3 illustrates two
interaction
possibilities.
a condition where the microscale
larger than the colloids and thus,
ded within the eddy.
Particles
is much
the particles are imbed­
in the eddy will experience
a laminar like shear field and are also exposed to the coagu-
COLLOID DIA. a
MICROSCALE \
IDEALIZED
MICROSCALE-COLLOID
INTERACTION
©
COLLOID
COAGULANT
CASE 2
t) I r 2
^
I -
O
POWER DISSIPATION
FIGURE 3.
Idealized Colloid-Microscale Interaction Visualization
18
Iant.
Under the conditions of Case I, the efficiency of the
rapid mix
(based on destabilization)
would
seem to be con­
trolled by the coating, mechanisms which cause the coagulant
to be adsorbed
onto
the
colloid.
The exact
nature
of the
coating mechanisms is presently not fully understood.
As more power is made available to the system,
sition is made to smaller and smaller eddies.
a tran­
Finally,
the
condition exists where the microscale and colloids are equal
in size and this
The Case
condition is shown as Case 2 in Figure 3.
2 condition
is such that the particles
and eddies
can be considered as separate entities and in order for the
colloids to
sorts must
be
exposed
occur.
to
Thus,
the
the
coagulant,
efficiency
of
a
collision
mixing
in
of
the
Case 2 situation may be controlled by a collision or inter­
action phenomenon.
In looking more closely at the transition zone between
Case I and Case 2, Amirtharajah
lowing colloid-microscale
of the
idea was
the colloid
that
[25] has proposed the fol­
interaction
theory.
The
basis
for colloid destabilization to occur,
particles
must
be
coated
with
the
positively
charged aluminum hydroxide solid phase species which are incipiently formed
in
transported by the
the
fluid.
fluid eddies
with the colloid particles.
These
species
have
so that they will
to
be
interact
19
Starting with Saffman and Turner's Equation
is applicable
for the condition where
scale is larger than the particles
(8), which
the turbulent micro­
and with some minor re­
arrangement.
N = Icn1Ii2 Ca1 + n)3 (^)
(10)
I
where N
is
the
number
of
colloid-microscale
i n te ra ct io ns ,
k is a constant, ai is the colloid diameter, n is the micro­
scale,
v is the fluid kinematic viscosity,
dissipation per
and microscale
mixer,
stant.
unit
mass
and
number counts
ni
e is the power
t\2 are
and
respectively.
the
colloid
Within a given
the number count of colloids can be considered con­
Furthermore,
in
this
first
approximation
the number count of microscale eddies
even though
microscale
there
is little doubt
is reduced,
crease within
the
that as the
constant,
size
of the
the number count of eddies will in­
mixing 'volume.
tions, Equation (10)
is assumed
model,
With
the
above
assump­
is reduced to the form
N = k 1 (a 1 + n)3 (^)°*5
(ID
20
Rearranging the microscale exp re ss io n, Equation (I),
in the
form
(12)
e =
and substituting into Equation
N = R1Vta1
(11) yields
(13)
n)3'(i)2
The fluid viscosity is assumed to be independent of q and is,
therefore,
incorporated into the constant yielding
3,1.2
N = R2 (a1 + n) (^)
(14)
as the proposed mixing theory expression.
In order
theory,
to
Equation
identify the extreme points of the mixing
(14) was expanded
and differentiated with
respect to n giving
dN
dn
4 -> 2
(15)
21
Setting Equation
(15) equal to zero leads to
ai T
aI 2
2 (-^) 3 + 3
" I = 0
and a real root of
(a]/n) = 0.5.
(16)
Taking the second deriva­
tive of Equation 14,
3-4
6k2 (ajn
and substituting
Equation
14.
=
2-3
+ a^n
0 . 5 n gives
(17)
)
a
positive
value
for
(17) for all n indicating a minimum for Equation
Thus,
when the microscale
minimum number
of
is twice the colloid
colloid—microscale
interactions
size a
should
occur.
Although the above derivation
orous, as a first approximation,
is not complete
or rig­
it provides some interest­
ing insights into the initial mixing process.
22
CHAPTER 5
EXPERIMENTAL WORK
PLAN
The experimental portion of this study consisted pri­
marily of measuring the response of a dual media
various methods of rapid mixing the coagulant.
filter to
Both mechan­
ical and hydraulic mixing were utilized during the investi­
gation.
A
baffled
tank
reactor
served
as
the mechanical
mixer and hydraulic mixing was provided by a flume apparatus.
Mixing within
the
flume
was
produced
by
either
turbulent
open-channel flow or a hydraulic j u m p .
The sensitivity of a dual media filter to small changes
in the
known
characteristics
[26] .
of
its
influent
suspension
is well
T h u s , the filter makes an ideal gauge for measur­
ing the effectiveness of the rapid mix.
DIRECT FILTRATION PILOT PLANT
A direct filtration pilot plant was constructed in order
to study the
initial mixing
step of coagulation.
illustrates the pilot plant flow scheme.
2.0 gpm
(8 g pm/ft 2 filtration
all pilot plant runs.
to insure constant
rate)
Figure 4
A constant flow of
was maintained through
Both hot and cold, tap water were used
temperature
conditions.
A Powers
Foto-
h c
A.
B.
C.
D.
E.
F.
G.
H.
THERMAL BLENDING VALVE
20 MICRON FILTER
RAW WATER PREPARATION
BUFFER FEED
TURBIDITY FEED
COAGULANT FEED
RAPID MIXER
DUAL MEDIA FILTER
0 0
PUMP
FLOW METER
BACKWASH
WASTE
FIGURE 4
Schematic Diagram of Direct Filtration Pilot Plant
24
guard Model
hot and
440-1500
cold
thermostatic blending
streams
feed water.
The
for all plant
to provide
temperature
runs.
of
valve mixed the
the
constant
the
feed
temperature
water
A chemical analysis of the
was performed during
the
investigation and the
was
80°F
feed water
results are
listed in Table I.
Table I.
Water Analysis of Feed Water for Direct Filtration
Pilot Plant.
PARAMETER
QUANTITY
Total Hardness
Total Alkalinity
Total Dissolved Solids
Turbidity
PH
105 mg/1 as CaCOg
80 mg/1 as CaCOg
150 mg/1
0j5 to 1.0 TU
7,4 - 7.7
-2
8.0 m g /1
SO 4
—3
PO4
< 0.2 mg /1
None of
the
analysis
except for the
results
are
particularly
significant
sulfate concentration of 8.0 mg/1.
Sulfate
concentrations of 8.0 mg /1 and up have been shown to have a
considerable impact on the effectiveness of alum when it is ,
used as a coagulant.
It is believed that the sulfate anion
neutralizes the charge on the aluminum hydroxide solid phase .
which greatly
reduces
its
destabilization
potential
[27] .
The remainder of the raw water preparation was carried
out in a 20 gallon continuously stirred tank.
The raw.water
tank was fed continuously with the constant temperature.feed
water as well as a turbidity slurry and pH buffer solution.
25
Hydrochloric acid
(Baker R e a g e n t , 37% H C l ), at a concentra­
tion of
30 milliliters
for the
buffer
water tank
acid
solution.
provided
a
per
liter
Feeding
near
the
constant
of
water,
buffer
raw
was
used
into the raw
water
pH
through
all plant r u n s .
A suspension of Min-U-Sil
bidity slurry.
Two
(Si 0 2 ) was used for the tur­
different
particle
size
distributions
were used in the experimental work and Figure 5 illustrates
the distributions. . It is
tions by
weight
bers are
shown
obtained by
as
well
important
as distributions
in Figure, 5.
standard
Omnimet Image Analyzer
to note
by particle
num­
The weight distributions were
hydrometer
(IA)
that distribu­
tests.
(ASTM
D422) .
An
by Buehler (Bausch & L o m b ) was
employed to provide the number count distributions.
The IA
consists of a microscope which is coupled to a visual monitor.
The lA's electronics scan for shading differentiation (par­
ticles against background) and then count the particles which
are greater than a specified size.
Samples for"the IA were diluted samples of the turbidity
slurry.
The IA samples were analyzed in suspension form be^
cause drying
the
samples
caused
particles
to
agglomerate
which led to erroneous results.
The Min-U-Sil material
itself is a product of the Pen­
nsylvania Glass Sand Corporation (PGS).
the coarsest
grind
manufactured
by
Min-U-Sil 30 (M 3 0 ),
P G S , served
as
one
of
26
A SB 325
A
•
o
M 30
A
o % BY WEIGHT
% BY NUMBER
PERCENTAGE
FINER
Ae
PARTICLE
FICUPE 5.
Size Distributions
SIZE (;jm )
for Turbidity Pa rt i c l e s .
27
the particle distributions.
was obtained
from Berkeley 325 which is t h e , feed
the M30 grinding mills..
below, was
The other distribution,
employed
SB325,
stock for
A sedimentation process,
outlined
to produce the SB325 from the Berkeley
325 material.
Suspensions of 50 grams per liter of Berkeley 325 were
prepared in 5 gallon buckets and allowed to sit for one h o u r .
The particles
which
settled
in this
time period
were
lected and dried resulting in the SB325 distribution.
sedimentation process was developed
col­
This
based on the ASTM D422
hydrometer test and was designed to remove a major fraction
of the particles smaller than 5 micrometers.
F r o m .the
raw
water
t a n k , the
rapid mix unit where either alum
chloride
(FeCl3
water
proceeded
to
the
(AlgfSO^j^'lGHgO) or ferric
was introduced to destabilize the sus­
pension, by charge
neutralization.
After
rapid m i x i n g , the
destabilized suspension passed directly into the dual media
filter.
The details of each of the- rapid mix u n i t s , as well
as the details of the filter, are described below.
Baffled Tank Reactor
Based on
the designs
scale up considerations,
structed as
the details
for mixing
of
in
the
the
research
[5, 6 , 8 ] and
the baffled tank reactor was
shown in Figure
of
other
6..
Also
discturbine
baffled
tank.
con­
shown in Figure 6 are
impeller
Both
which
the
tank
was
and
used
the
28
IMPELLER PLANE
BUSHING
D= 8 .0
H= 12.0
R= 4.0
(a)
POZ O
FIGURE 6 .
Details of Baffled Tank Reactor and Disc-Turbine
Impeller.
/
29
impeller were constructed of p l e x i g l a s .
A constant water depth of 8.0 inches was maintained in
the reactor
during
all pilot plant
runs.
The
influent
to
the reactor was fed 6.0
inches above the reactor bottom and
the effluent
off
was
This reactor
drawn
configuration
inches
2.0
forces
all
above
the
the
flow
bottom.
to
pass
through the impeller plane center line which was 4.0 inches
above the
bottom.
system on
the
The
coagulant
impeller
plane
inches from the blade tips.
was
center
fed
into
line
at
the reactor
a point
0.25
A Bodine NHS-54 variable speed
motor was used to drive the impeller.
Details of Flume
As a comparison for the baffled t a n k , a hydraulic mix­
ing device which incorporated a narrow flume and a hydraulic
jump was
constructed.
figuration and
slopes could
sliding block
2.5
all
be
on
Figure 7 illustrates the
important
attained
the
dimensions.
by
adjusting
aluminum
base.
flume
Various
the
channel
position
Slopes
of
con­
1.5,
of
a
2.0,
and 3.5 inches per foot were utilized in the experiments.
These slopes produced highly
conditions within
the c ha nn el , which
the
turbulent,
channel.
essentially
By
inserting
acted
hydraulic jump could be produced.
supercritical
as
a
a
flow
barrier
sluice
in
gate, a
FIGUPF 7.
Details of Hydraulic Jump Flume.
31
The flume-jump apparatus was placed in exactly the same
location as the.baffled tank within the overall plant s c h e m e . .
Raw water entered
the
channel
via a stilling
with stones to disperse the flow.
basin
filled
Once the water passed in-
v*
to the channel
section
of the coagulant.
it was destabilized by the addition
When a hydraulic jump was being used, the
coagulant feed point was approximately
of the
jump toe.
During plant
alone provided the mixing,
used
inches upstream
runs where the channel
flow
the coagulant feed point was mid­
way along the channel length.
flume was
1.5
It should be noted, that the
to treat the M30
suspension o n l y .
Finally,
the destabilized suspension passed directly into the filter
rise tube as shown in Figure 7.
-
. .
Dual Media Filter
Since the response of the filter was to be used as the
principal method
of
determining
mixing
filter itself was carefully designed.
of the
filter
is shown
rapid mixer enters the
tube.
in the
in Figure 8 .
An overall schematic
The effluent
from the
filter via a 2.0 inch diameter rise
As the filter media clogs during a. run, the water level
rise
tube
increases, thus
filtration with influent control.
of 16
ef fe ct iv en es s, the
inches
of anthracite
coal
sand.
L
providing
constant
rate
The filter media consisted
over 8.0
inches of silica
32
INFLUENT
RISE TUBE
BACKWASH
FILTER BOX
LEVELS
COAL -
•
SAND
I I II I
MANOMETER
BOARD
DRIP SAMPLES
ORIFICE PLATE
UNDERDRAIN
FIGURE 8
- BACKWASH
EFFLUENT
Details of Dual Media Filter
33
Table 2 .
Size Characteristics of Filter Media.
SIZE
CHARACTERISTIC
(Effective Size)
SAND
COAL
0.46
0.8 6
D 10
(mm)
d6 0
(mm)
0.62
1.25
d90
(mm) ■
0.70
1.52
1.35
1.46
Uniformity Coefficient
Table 2 lists the size characteristics of the sand and coal
media.
The
literature
media
sizes
were
carefully
chosen,
based
on
[28] guidelines, to insure a good zone of inter­
mixing and compatible expansion characteristics.
A 6x6x48
box.
This
feet of
inch
plexiglas
configuration
filter
(6 x 6
surface
which
flow
2.0
based on a plant
column
of
inch)
served
provided
yielded
gpm,
as
a
the
filter
0.25
square
filtration
of 8 gpm/ft^.
rate,
The
box
was divided into five levels with a continuous-drip turbid­
ity sampler
and manometer port placed
le v e l s .
Level
(I inch
above
were spaced
one
the
was
25
media)
at 5 inch
inches
and
the
intervals.
at each
above
the
successive
Also,
of the
five
orifice
plate
lower
levels
a sixth
level
was
established below the orifice plate to collect effluent tur­
bidity and total headless d a t a .
The drip sampler consisted
of an aluminum tube extending into the center of the filter
box.
The
manometer
ports
were, similarly
constructed
but
.
34
only projected 0.5 inches into the media.
with twenty
five 0.25
orifice plate.
A plexiglas plate
inch diameter holes was used
for the
The function of the orifice plate was to sup-
V
I
port the media
(in addition to a nylon mesh) and to provide
even filtration.
Also, the orifice plate provided even dis­
persion of the backwash water during cleaning.
Cleaning of the filter was systematically performed by
using both air scour and water b a c k w a s h .
Air was blown intp
the submerged filter media through each of the drip samplers.
The necessity for air scour was established after water wash­
ing alone proved to be ineffective in breaking up the chunks
of material which were sheared off the m e d i a . x After the air
flow was st o p p e d , the filter was washed with water for an ex­
tended period of time..
The washing
flow rate employed was
sufficient to provide a fifty percent expansion of the bed.
FILTER PERFORMANCE DATA
To insure
the
stability
of
the
pilot
plant
and
to
determine the filter res po n s e , various physical and chemical
parameters were
each run.
recorded
at half hour
throughout
Measurements of pH were made on the raw water and
filter effluent
(level
6)
in
order
chemistry within the flow system.
from Beckman
ments.
intervals
Also,
Instruments
the
raw
was
water
used
to
maintain
constant
An Altex digital pH meter
for all the pH measure­
turbidity
was
monitored
to
insure constant mass loading of the rapid mixer and filter.
35
Filter response to the various rapid mix conditions was
established by
each of
the
recording
drip
d a t a , headless
the
turbidity
samplers.
data
was
In
of
addition
obtained
by
the
to
water
the
monitoring
from
turbidity
the
total
available head at each level in the filter.
The electrophoretic mobility of the particles after destabilization
parameter.
(level I)
was
also
monitored
as a secondary
A G.K. Turner Zeta Meter was employed to measure
the mobility of the particles.
The mobility measurement pro­
cedure consisted
t h e . average
of
recording
mobility
of ten
particles found in each of the samples collected at half hour
i nt erv als .
The
reflect the
ten
range
particles
of
were
mobilities
and
randomly
selected
particle
sizes
to
that
were seen in the sample.
A list of all the pilot plant r u n s , including coagulant
dosages and
pH
conditions
total number ■ of
plant
runs
can
be
found
in
where . coagulant
Table
3.
The
was
used
was
thirty-eight.
An additional run without coagulant was also
made in
to
order
efficiency.
establish
the
filter's
baseline
removal
All the pilot plant runs were between three and
five hours in length .
36
Table 3.
List of Pilot Plant Runs Including Coagulant
Dosages and pH Conditions.
COAGULANT
DOSAGE/
COLLOID
MIXER
VELOCITY
GRADIENT-G
(s- 1 )
75
Alum8 mg/1 @ pH 6.9
Baffled
Tank
M30
Ferric Chloride
8 mg/1 @ pH 6.2
Baffled
Tank
M30
Ferric Chloride
8 mg/1 @ pH 6.3
Baffled
Tank
SB325
Ferric Chloride
@ pH 6.2
Hydraulic
8 mg/1
Jump
M30
Ferric Chloride
8 m g / 1 @ .pH 6.2
Open
Channel
M30
2
2
210
I
38 5
810
1640
3000
3800
75
' 210
810
1640
2290
3000
3800
NUMBER
OF
RUNS
2
2
I
2
2
I
I
I
2
I
,
210
810
1640
3000
3800
2
2
2
2
I
I
705
690
680
870
I
I
I
I
1080
1360
1700
2300
I
I
I
I
Alum Dosage = 8 mg/1 as A l 2 (SO4) 3 -14.3H20
Ferric Chloride Dosage = 8 mg/1 as Fe C l 3 *6H20
,
37
DYE TRACE STUDIES
In order to determine the macroscopic mixing character­
istic of both the baffled tank and the hydraulic jump, stan­
dard dye trace studies were pe r f o r m e d .
By utilizing fIuor-
ometric techniques and common analysis procedures explained
by Weber [29], a fairly good representation of the residence
time distribution
(RTD)
for the
given
mixer
was
obtained.
Weber also explains a numerical integration procedure which
yields the centroid of the R T D .
By definition,
the centroid
of the rector RTD is the mean hydraulic residence time (MHRT)
for the reactor.
Furthermore, Weber illustrates a theoret­
ical RTD calculation for a complete mix reactor.
Rhodamine W T , a nonreactive
in the
tracer studies.
was introduced
During
into the mixers
lant feed conduits.
The
dye,
was used
the tracer studies,
as a pulse,
Samples were
effluent after the dye was
Turner Fl uorometer.
fluorescent
collected
via
the
the dye
coagu­
from the mixer
injected and analyzed
on a G.K.
fluorometer detected R h o d a m i n e 'WT
at concentrations as low as 0.0001 mg/1.
The data from the
f luorometer was later plotted and statistically analyzed as
a described
by Weber
[29],
to establish the mean hydraulic
residence time for the given mixer.
38
CHAPTER 6
RESULTS AND DISCUSSION
The results of this investigation fall into three major
categories.
F i r s t > there is the calculated mixing parameters
for the various mixers.
filtration results
are
Secondly,
the pilot plant data and
considered.
Finally,
the hydraulic
and mechanical mixer comparisons are made.
CALCULATED MIXING PARAMETERS
The parameters which were chosen to describe the mixing
in the baffled
tank
and hydraulic
jump were mean hydraulic
residence time (MHRT) and the average mixer, velocity gradient
(G) .
For comparison purposes, Gt values have also been cal­
culated.
regard to
the value of t used was V/Q for the
the
MHRT for the
MHRTs and
hydraulic
given
G-values
Studies section
and
jump,
jump.
were
the
t was
The methods
outlined
in
assumed
for
reactor.
equal
In
to
the
calculating
the
the. Previous
following paragraphs
focus
Related
on the
results of the above mentioned calculations.
Velocity Gradients
Power dissipation within the baffled tank can be calcu­
lated by using the power number Equation
tor power number is kn o w n .
Leentvaar
(4) once the reac­
[8 ] reported that the
I
39
power number
impeller, was
for
a
cylindrical
5.0.
It
is
baffled
important
t a n k , with
to
note
a disc
that
power
losses computed using the power number expression represent
average losses within the entire tank volume.
Thus, by using
the power number losses in the expression for G, Equation (2) ,
an average
found.
velocity
gradient
for
the
baffled
tank
can
be
Table 4 lists the results of the power loss, G-value
and Gt-value calculations for the baffled tank.
Table 4.
Calculated
Velocity
Gradients
50
0 .0.2
, 75
100
0.18
0.62
2.74
210
150
250
400
500
600
700
Baffled
VELOCITY
GRADIENT-G
(s- 1 ) '
POWER
DISSIPATION
(ft'lbs/s)
IMPELLER
RPM
for
Gt
3,900
10,900
385
810
1640
•2290
3000
3800
11. 2 1
21.90
37.84
60.09
Tank.
20, 0 0 0
42,100
85,300
119,100
156,000
197,600
G-values and Gt values for the hydraulic jump and open
channel,
A sample
for the
for various
channel
calculation
jump
can
be
of
found
the
slopes , are shown
power
in the
loss
and
Appendix.
in Table 5.
the
It
G-value
should
be
noted here that the open channel G-value for the 3.5 inches
per foot slope,
is related
to
may
the
be
in error.
extreme
The reason
curvature
in the flume at this large s l o p e .
of
the
for the error
water
surface
Since the flume was only
40
Table 5.
Calculated Velocity Gradients for Hydraulic M i x e r s .
CHANNEL
SLOPE
(in/ft)
FROUDE
NUMBER
1.5
3.41
4.49
6 .29
9.68
3.41 .
4.‘4 9
6.29
9.68
■
F1
2.0
HYDRAULIC
JUMP
2.5
3.5
1.5
2.0
OPEN
CHANNEL
1.0
.2.5
3.5
-
POWER
DISSIPATION
f t •Ibs/s
G,
(s- 1 )
Gt
3595
2970
0.121
705
690
680
870
3480
0.046
0.061
0.075
0.103
1080
1360
1700
2300
530
550
550
560
0.022
0.035
0.054
2920
inches wide, all the calculations had to be based on the
center line depth.
have thus
The curvature of the water surface may
yielded a low
center
line depth
the calculated *G-value to be too high.
reading
None
of
causing
the
other
channel slopes employed produced flows with extreme surface
curvat ur e. '
In looking
at the actual values
for power dissipation
for the j u m p , channel and baffled t a n k , it can be seen that
the power requirements for the hydraulic mixers are signifi­
cantly lower
than
the
power
requirements
for
the
baffled
tank.
Hydraulic Residence Times
.
.
The data collected from the dye trace studies was used
to determine the residence time distributions for the baffled
tank and the hydraulic jumps.
for three
different
impeller
Figure 9 represents the RTDs
speeds
in
the
baffled, tank.
O &
4
A
A
N= 50 RPM
O
N = 100 RPM
O
N = 7 0 0 RPM
A
\
/
>
X
\
——— THEORETICAL
D PO
Bx
O
50
' 100
150
TIME (SEC)
FIGURE 9.
Residence Time Distributions for the Baffled Tank Reactor
42
The RTDs
do
not
impeller speed
and
curve derived
However,
the
seem
to
all
be
the
from a mass
mean
distribution)
for
particularly
follows
the
on
of
residence
the
times
impeller
the
theoretical
balance across the reactor
hydraulic
each
data
dependent
[27] .
(centroid
speeds,
listed
of
in
Table 6 , are
significantly different.
Note that changes
I
in the dye concentration within the baffled tank during the
first ten seconds were not included in the numerical integra­
tion. procedure
Table 6 .
used
to obtain the MHRTs listed
Mean Hydraulic Residence Times
IMPELLER
RPM \
for Baffled T a n k .
MEAN HYDRAULIC RESIDENCE TIME
(Seconds)
.
65.6
56.6
50.3
50
100
700
in Table 6 .
\
Reactor Volume/Flow Rate = 52.0 seconds
Two typical RTDs
Figure 10.
above,
two
for the hydraulic
important
RTDs
differences
are
more
respective MHRTs (plug-flow).
for the hydraulic
nitude less
ju m p s ) .
in
In comparison to the baffled tank RTDs discussed
hydraulic jump
or, MHRT
jumps are shown
than
Thus,
the
the
tank
are
closely
evident.
First,
centered about
the
their
Also, the average mixing time
jumps
MHRTs
macroscopic
is about an
(4-5
second
mixing
in
order
MHRT
the
of mag­
for
all
hydraulic
jumps and the baffled tank used in the experiments are vastly
di f f e r e n t .
DYE CONCENTRATION
( m g /lx IO )
43
SLOPE-
O
IO
I.5 7 F T
20
TIME (SEC)
FIGURE 10.
Residence Time Distributions for Two Hydraulic
Jumps.
44
FILTRATION RESULTS
As outlined
filter data
rea di ng s.
by vthe
consisted
The
level
loss data
in the Experimental Work
turbidity
(1-6)
on the
levels through
of
various
data
will
be
and
headless
referenced
directly
other ha n d , is coded by referring
the
headless in the uppermost
loss
occurred.
layer
of
the total headless would be Hl -6
Figures 11 and
turbidity
from which the sample was t a k e n .
which
Two typical
sect io n, the raw
examples
12.
of
coal
He a d - .
to the
For example,
would
the
be Hl-2 and
(see Figure 8 ).
raw
filter data are
The top graph
in each
shown
in
figure, shows the
turbidity removal as a function of time and the lower graph
illustrates the
corresponding
headlosses.
Note
that
the
headless lines are very linear, which is characteristic of a
stable well designed filter system with good depth removal.
A single plant
added.
This
run was also made in which
no
treatment
run
was
establish a base
for the other plant
the no
run
coagulant
move any
of
the
showed
that
nondestabilized
no coagulant was
performed
the
runs .
filter
material.
in
The
order
to
results of
would not re­
This
also
led
to a zero headless bu i l d u p .
Filter Response to Baffled Tank Mixing
As a first attempt to define the effectiveness of vari­
ous mixing conditions for. the baffled t a n k , the rate of headloss buildup and average effluent turbidity for each run was
45
▲ 2
TURBIDITY (TU)
■ INF
HEADLOSS (IN)
TIME (HRS)
•
A
■
HI - 2
HI - 4
HI - 6 TOTAL
TIM E (HRS)
FIGURE 11.
Typical Filter Data for Baffled Tank Mixing
with Alum Treatment (n = 700 PPM).
46
A 2
o 4
• 6
TURBIDITY
(TU)
■ INF
A l
TIME (HRS)
H I-2
A H I -4
■ H I- 6 TOTAL
HEADLOSS
(IN)
•
TIME (HRS)
FIGURE 12.
Typical Filter Data for Hydraulic Jump Mixing
with Ferric Chloride Treatment (slope = 2.0)
in/ft) .
47
calculated and co m p a r e d .
'
The rate of headless is simply the
slope of the linear headless lines which were illustrated in
Figures 11 and
12.
various plant, runs
headloss d a t a .
in as
the
Rates
of headloss
by performing
were
calculated
linear regressions
for
on the
Average effluent turbidity is defined here­
average
of
all
the
effluent
turbidity
readings
up to and including a specified, breakthrough turbidity (C).
The breakthrough
turbidity
was
used
to
determine
the
run
le n g t h , T , and was set at 1.5 TU for M30 suspensions and 1.0
TU for SB3 25 suspensions.
In other w o r d s , when the effluent
turbidity of a given plant run reached the appropriate break­
through turbidity v a l u e , the run was terminated.
The rates of headloss development and average effluent
turbidity for alum treatment of the M30 suspension are shown
in Figure 13.
The rates of headloss do not vary significantly
above a G of approximately 800 s- 1 , however, a minimum rate
seems to have been reached at a G of 3000 s- -*-.
ly, the
average
turbidity
for alum treatment did
significantly with
velocity
of the
in
data
was most
water.
shown
likely
gradient.
Figure
related
to
13
the
seemed
The
to
ferric
chloride
sulfates
and
the
not
vary
insensitivity
unreasonable,
in
In order to avoid the sulfate problem,
was changed
Corresponding­
the
and
influent
the coagulant
experiments
were
repeated.
Figure 14 illustrates the rates of headloss for ferric
chloride treatment
of
the
M30
suspension.
All
trends
in
•
A
HI-2
HI-4
■
HI- 6
o
TREATMENT —
EFFLU ENT
AVERAGE
TURBIDITY
M 3 0 PARTICLES
TURBIDITY
(TU)
RATE
OF HEADLOSS (IN/HR)
— ALUM
IOOO
2000
MEAN VELOCITY GRADIENT -
3000
4000
G
(s')
FIGURE 13.
Rates of Headloss and Average Effluent Turbidities for Baffled Tank
Mixing with Alum Treatment.
— FeCU
TREATMENT —
AVERAGE
▲ HI- 4
EFFLUENT TURBIDITY
M 3 0 PARTICLES
TURBIDITIY
UJ
4.0
(TU)
b 2.0
1000
2000
3000
4000
MEAN VELOCITY GRADIENT - G
(s')
FIGURE 14.
Rates of Headloss and Average Effluent Turbidities for Baffled Tank
Mixing with Ferric Chloride Treatment.
50
Figure 14 are similar to the trends for alum treat me nt ; how­
ever, note the 1 0 0 % change in the rate of headless and near
300% change in the average turbidity over the G-value range
shown.
As a result of the increased sensitivity of the ferric
chloride system, no further alum treatment experiments were
performed.
The SB325
suspension
was
also
treated
with
ferric
chloride and the resulting headless rates and corresponding
turbidities are shown in Figure 15.
headless changes
Once again, the rate of
significantly over the velocity grandient
range employed, with high G-values producing lower rates of
headless.
The
average ' effluent
turbidities
are
somewhat
less sensitive to changes in mixing for the SB325 suspension
in comparison
to
lower sensitivity
basis
turbidities
was
for the
actually
M30
suspension.
expected,
since
on
The
a mass
(mg/ 1 ), a finer distribution of colloids will produce
higher turbidities (light scattering) than a coarser distri­
bution.
It was evident from. Figures 13 through 15, that no clear
cut optimum condition could be identified because the headloss and turbidity data opposed each other.
necessary to
combine
single filtrability
optimum mixing
headless
index
in
con di t io ns .
turbulent microscale
and
and
the
turbidity
order
Also,
T h u s , it became
to
pinpoint
correlations
colloid
data
into
a
possible
between
distributions
the
could
—
FeCI3 TREATMENT —
■ HI-6
AVERAGE
▲ EFFLUENT TURBIDITY
TURBIDITY
(TU)
RATE OF HEADLOSS (IN /H R )
SB 32 5 PARTICLES
iO
2000
3000
4000
MEAN VELOCITY GRADIENT - G
(s-')
FIGURE 15.
Rates of Headless and Average Effluent Turbidities for Raffled Tank
Mixing with Ferric Chloride Treatment.
probably not
be
identified
unless
a
single
parameter
was
used to describe the filter response.
Filtrability Numbers for Baffled Tank Mixing
Filtrability numbers for various plant runs were calcu­
lated using Equation (9) and plotted against average reactor
velocity gradients.
shown in
Figures
.The filtrability
16
through
numbers
18. - Also
obtained are
plotted
in
Figures
16 through 18 are the turbulent microscales for the impeller
stream zone
(see Appendix).
Figure 16 is for alum treatment
of the
suspension
the
M30.
evident.
Figures
once again
that
and
17 and
the
18,
ferric
expected
insensitivity
on the other hand,
chloride
system
is
illustrate
provides
much
more information in regard to the filter's response to vari­
ous degrees
of
are indicative
and 18
seem
mixing.
Since
of poorer
to
for
filtrability
numbers
filtration conditions,
Figures 17
that
filtration
indicate
conditions exist
larger
both the
relative
M30
minimum
and the
SB325 distribu­
tions.
Looking back
and SB325,
shown
at the
grain
size distributions
in Figure 5, it can be seen that the mean
particle size for M30 and SB325 distributions
(number count)
is approximately 3.0 pm and 6.0 p m respectively.
to the
mixing
interactions
for M3Q
interaction
(filtration
turbulent microscales
are
model
developed
conditions)
twice
the
According
herein,
minimum
should exist when the
colloid
size.
Thus,
— ALUM
TREATMENT —
STREAM MICROSCALE ( >jm)
M 3 0 PARTICLES
O 0 .7 5
z 0 .5 0
H 0 .2 5
D
2000
MEAN VELOCITY
GRADIENT -
3000
4000
G
(s*')
FIGUPF 16.
Filtrahility Numbers for Baffled Tank Mixing with Alum Treatment
—
F e C I, TREATM ENT —
STREAM MICROSCALE
M 3 0 PARTICLES
3 0 .5 0
(/jm )
< 0 .2 5
2000
MEAN VELOCITY GRADIENT -
3000
4000
G
(s')
FIGURF 17.
FiltrahiIity Nunhers for Raffled Tank Mixing with Ferric Chloride
Tre at m en t.
----- FeCI 3 TREATMENT PARTICLES
ILTRABILITY
NUMBER xIO
STREAM MICROSCALE (/jm)
SB 3 2 5
D
MEAN VELOCITY GRADIENT -
FIGURE 18.
3000
2000
G
Filtrability Numbers for Baffled Tank Mixing with Ferric Chloride
Tre at m en t.
4000
56
for the M30 and SB325 distributions, minimum filtration con­
ditions should exist when the microscales zare approximately
6 and 12 pm respectively.
of course,
The preceding statement assumes,
that the colloid distribution can be represented
by its mean
size.
The
to velocity gradients
and
6
12 p m microscales
correspond
of 3000 s- -*- and 810 s--*-.
Inspection
of Figures 17 and 18 show that minimum filtration conditions
for the
experimental
very close to 6
that the
in the
systems
at
stream
microscales
pm for M30 and 12 pm for SB325.
coagulant
was
impeller stream,
tion in the
occur
stream
introduced
into
the
a microscale-colloid
region
would
Recalling
baffled
tank
size correla­
seem the most
reasonable.
A secondary minimum filtration condition is also shown
in Figure
18
likely related
(G=3000).
to
the
This
minimum
of
particles
fraction
SB325 distribution
which
particles have
greatest
the
secondary
are
of
small
size
impact
3.0
on
pm;
is
most
in
these
the
small
the measurement
of
turbidity.
ELECTROPHORETIC MOBILITY DATA
.
Recall that the electrophoretic mobility of the desta­
bilized
(level
I)
suspension
throughout each plant
actuality,
the
thus, between
during a given
run.
average
60
and
run.
was
Each
mobility
90 particle
Table
also
measured
regularly
recorded mobility was,
for
ten
particles,
mobilities
7 lists
the
in
and
were measured
average mobilities
57
obtained from
the
treat the M30
and
SB325
Table 7
the
highest
that
close to,
the
discussed in
data seems
drawn from
various
the
of
In f a c t , statistical
any
at,
or
correlation
points
which
supported
by
which make up the average
analysis
of
very
the mobility
conclusion
is
to
can be seen from
Although
measurements
samples
employed
occur
size
section.
scattered,
mobility
It
mobilities
previous
somewhat
conditions
suspensions.
mic ros cale-colI o id
the
large number
mixing
is
the
va lues.
the mobility data yielded
O
9 5 % confidence
intervals could
the mobility
intervals
be
data
bulent microscale
considered
and
gest the existence
which
a size
so
sm a l l ,
insignificant.
filtrability
of
and
were
numbers
correlation
the mean
that
Thus,
seem
between
the
both
to
sug­
the tur­
size of the colloid distri­
bution.
HYDRAULIC MIXER COMPARISONS
Comparing hydraulic mixing
to baffled tank mixing was
handled by utilizing the velocity gradient c o n c e p t .
Within
the context
comparing mixers
from a
related quest io ns .
First,'
of this
G-value standpoint
investigation,
answered
two
the comparisons gave some insight into whether the proposed
method for
calculating
re as ona bl e.
G-values
for
hydraulic
jumps
was
Secondly, comparisons based on G established the
jumps mixing effectiveness
l
of the baffled t a n k .
in relation to the effectiveness
'
.
It should be noted here that a similar
58
Table 7.
Electrophoretic Mobility Data for Treatment
of M30 and SB325 Colloids
'
ELECTROPHORETIC*
MOBILITY
MEAN + 95% C.I.
(lim* cm/volt •s )
VELOCITY
GRADIENT
(s- 1 )
CONDITIONS
0.90 +
1 .01 +
1.00 +
0.94 +
0.98 +
1.15 +
1.16 +
75
Alum Treatment
of M30 Particles
210
385
810
1640
3000
3800
'
810
1640
2290
3000.
3800
*Non destabilized
1.60
1.52
1.70
1.55
1.44
810
1640
3000
3800
G-value for the
particle
mobility
jump and tank does
value.
In fact,
for the
hydraulic
=
not
2.2
imply
for a given G (600— 900 S- M
jump
are
an
0.03
0.04
1.19 + 0.03
1.27 + 0.02
1.31 + 0.03
1.11 + 0.03 ’
210
Ferric Chloride
Treatment of
SB325 Particles
0 .02
0.02
0 .01
1.22 + 0.02
1 .21 + 0.02
1.09 + 0.01
75
210
Ferric Chloride
Treatment of
M30 Particles
0.02
0.04
order
+
+
+
+
+
0.03
0.04
0 .02
0.04
0.03
ym*cm/volt*s
a
similar Gt
the Gt values
of magnitude
smaller
than than the Gt values for the baffled t a n k .
The rates
and channel)
of
headloss
of
both hydraulic
are shown in Figure 19.
mixers
(jump
The data in Figure 19
indicates that the rates of headloss produced in the filter
-H Y D R A U L IC JUMP
FeCL
TREATMENT -
H I- 4
H I- 6
JUMP
o
CHANNEL
2000
A
D
3000
H I-2
HI - 4
H I- 6
4000
MEAN VELOCITY GRADIENT - G
(•-')
FIGURE 19.
Rates of Headloss for Hydraulic Jump and Open Channel Mixing with Ferric
Chloride Treatment.
60
for both the jump and the channel are higher than the rates
which resulted ,from baffled tank mixing.
At
first g l a n c e ,
Figure 19 might suggest that the G-values for the jump, were
tod large since the hydraulic and mechanical data would align
if the jump G-values were less.
dient calculation
However,
the velocity gra­
for open channel mixing
is well defined,
yet the channel data as w e l l , does not align with the baffled
tank data.
From a turbidity
stan dp oi nt , Figure 20 indicates
the hydraulic
mixers
produced
a
effluent than
the baffled
loss depicted
in Figure 19 was understandable.
tank.
slightly
Thus,
the
higher
that
quality
increased head-
Combining the headless and turbidity data to form,cor­
responding filtrability numbers leads to the comparison il­
lustrated in Figure 21.
port the G-value
The filtrability data tends to sup­
calculation method
arid jump data are in good agreement.
since the baffled tank
Furthermore, Figure 21.
indicates that the hydraulic jump and open channel are equal­
ly as effective as the baffled tank in regard to rapid mixing
of chemical coagulants.
Once again,
the actual power dissipations
it should be noted that
\
in the hydraulic mixers were ..
significantly less than the power dissipations for the baffled tank.
FeCL TREATMENT
• JUMP
o CHANNEL
--R EA C TO R
AVERAGE TURBIDITY
(JTU)
HYDRAULIC JUMP
O
3000
2000
MEAN VELOCITY
GRADIENT -
4000
G
(#*')
FIGURE 20.
Average Effluent Turbidities for Hydraulic Jump and Open Channel Mixing
with Ferric Chloride Treatment.
— HYDRAULIC JUMP
rO
0.75
FeCl3 TREATMENT —
•
JUMP
CHANNEL
— — REACTOR
/
0
CE
UJ
X
Z
Z
Z
CD
\
/
3
\
/
0 .5 0
\
/
\
Z
>
\
\
/
2
\
\
Z
t
Z
\
/
CD
<
CE
b 0 .2 5
' V -
IOOO
2000
MEAN VELOCITY GRADIENT -
_ 3000
G
4000
(s')
FIGURE 21.
Filtrability Numbers for Hydraulic Jump and Open Channel Mixing with
Ferric Chloride Treatment.
63
CHAPTER I
CONCLUSIONS
Based on the results of this s t u d y , the following con­
clusions can be. d r a w n :
1. A
size
and the
correlation
colloid
rapid mixing
between
distribution
process
of
of this correlation was
tration data
the
as
well
turbulent
seemed
to
exist
coagulation.
supported
as
particle
microscale
The
for the
existence
by pilot plant
electrophoretic
fil­
mo­
bility data.
2. The microscale-colloid interaction model proposed here­
in may provide
some
mentioned above.
insight
The
into the size
mixing
model
may
correlation
not
actually
define the physical nature of the rapid mixing p r o c e s s ,
however,
the model seems to predict the minimum inter­
action condition
3. In regard
for
the
mixer
to optimum mixing
used
in
this
study I
con di ti on s, the data col­
lected indicates that the size of the colloid suspension
being treated must be considered before optimum condi­
tions can be defined.
The filtr.ability data indicated
that average velocity gradients from 700 to 1500 s- l or,
possibly gradients
above 4000
s ~ " , were
best
for the
64
particle suspension used in this study.
4. Use
of
the
hydraulic
here resulted
in
jump
G-values
functionally correct.
here was
for
jumps
design procedure
which
were
developed
reasonable
and
Although the procedure developed
occurring
in
sloping
channels,
a
similar methodology could be developed for jumps occur­
ring downstream of a critical flow flume.
5. In the average
s- 1 , the
velocity
mixing
was found
to
gradient
effectiveness
be
equal
Fu rt he rmo re , because
to
of
range
of
that
of 600 to 900
the
of
the
the difference
hydraulic
jump
baffled
tank.
in the size of
the power dissipation volumes of the jump and the t a n k ,
significantly less
above G-values
6 . The
power
of a hydraulic
also very app are nt.
to
gradient, all
range.
to limit
It
is
the
operations.
required
to
produce
in the hydraulic jump .
inflexibility
this study
was
In spite
produce
the
jumps
this
use
jumps
fell
the
'
jump as a mixer
was
of the attempts made in
of
variable
within a
inflexibility
of
the
narrow
that
hydraulic
velocity
will
jump
G-value
continue
in
mixing
65
REFERENCES
1.
H i n z e , J .0.
Tu rb ule nc e.
Hill, 1975.
2.
Bradshaw,?.
An Introduction to Turbulence and its
Me as ur em e nt . New Y o r k :Pergamon Press, 1971.
3.
K o l m og or o ff , A.N. "The Local Structure of Turbulence
in Incompressable Viscous Fluid for Very Large
Re y n o l d s •N u m b e r s ." C.R, Acad. Science U . R .S .S .31
. (1941) :518.
2nd
ed.
New
YorkzMcGraw
)
4.
Ko lm o go r of f, A.N.
"On Degeneration of Isotropic Tur­
bulence in an
Incompressible Viscous
Liquid."
C.R. Acad. Science U . R . S . S . 31 (1941):5 38 .
5.
Cutter, Louis, A.
"Flow and Turbulence in a Stirred
Tank."
American Institute of Chemical Engineers
Journal 12 (January, 1966) :35-4 5 ,
6.
O k o m o t o , Y., Nishikawa, M . and Hashim.oto, K . "Energy
Dissipation Rate Distribution and its Effects on
Liquid-Liquid Dispersion
and
Solid-Liquid Mass
Transfer." International Chemical Engineering 21
(January 1981):88-94.
7.
Camp,
8.
Le en tva ar , J. and Y w e m a , T.S.J.
"Some Dimensionless
Parameters of Impeller Power in Coagulation-Floc­
culation. " Water Research
14 (1.980 ): 135-140 .
9.
Chow,
T.R. and Stein, P.C.
"Velocity Gradients and
Internal Work in Fluid Motion."
Journal of the
Boston Society of Civil Engineers 30
(1943):219237 .
V.T.
Open Channel H y d r au li cs .
Hill, 1953.
New YorkzMcGraw
10.
Wilson, G .E . "Initial Mixing and Turbulent
tion"
Ph.D. dissertation. University
fornia, 1972
Floccula­
of Cali­
11.
S t e n qu i st , R.J. and Kaufman, W.J.
Initial Mixing in
Coagulation Pro c e s s e s . U.S.
Environmental
Pro­
tection A g e n c y , EP A-R2-72-053, (1972).
66
12.
Ami rt ha r aj ah f A. and Mills, K . M .
"Rapid-Mix Design
for Mechanisms
of Alum
Coagulation."
Journal
of the American Water Works Association 74 (April
1982): 210-216.
13.
Am i r t h a r a j a h , A. "Initial Mixing." American Water Works
A s s o c i a t i o n , Seminar Proceedings, Coagulation and
Filtration: Back to the B a s i c s . St.
L o u i s , MO,
1981.
14.
L e v y , A.G . and F i l m s , J.W.
"The Hydraulic Jump as a
Mixing D e v i c e ." Journal of the American Water
Works Association 17 (January 1927):1-23.
15.
Smo luc h ow sk i ,
16.
Saffman, P.G. and Turner, J.S.
Drops in Turbulent C l o u d s ."
chanics 16 (1956):16-30.
17.
Spi el ma n , L .A . "Hydrodynamic Aspects of Flocculation"
The Scientific Basis of Flocculation (K.J.
Ives,
editor) Netherla nd s:Sijthoff and No o r d h o f f , 1978.
18.
Adler, P.M.
"Hetrocoagulation in Shear Flow." Journal
of Colloid and Interface Science, 83 (September
19 8 1 ) :106-115.
19.
Garnet, M.B. and Ra de ma ch er , J.M. . "Measuring Filter
Performance."
Water Works Engineering 112 (1959):
117-149.
20.
C l e a s b y rJ . L .
"Approches to a Filtrability Index fqr
Granular Filters."
Journal of the American Water
Works Association 61 (August 1969):372-381.
21.
Ives,
22.
B i s k e r , C.D. and Young, J.C.
"Two-Stage Filtration of
Secondary Effluent."
Journal of the Water Pollu­
tion Control Federation 69
(February
1977).:319351.
23.
L e k k a s , T.D.
"A Modified Filtrability Index for Gran­
ular Bed Water Filters."
Filtration and Separa­
tion 18 (May/June 1981): 214-216.
M.
Physical Chemistry
92
(1917):129.
"On the Collision of
Journal of Fluid Me­
K.J.
"A New Concept on Filtrability." ' Prog'.
Water Technology 10-' (1978) :123-137.
«
67
24. • J a n s s e n s , J.G., A d a m , C . and B u e k e n s , A.
"Statistical
Analysis of Variables Affecting Direct-Filtration"
European Federation of Chemical Engineers, Pro­
ceedings of the International Symposium on Water
Filtr at ion . Antwerp, Be l g i u m :1982.
25.
Amirth ara j a h ,
26.
Ba u m a n n , E .R .
Chapter
2
"Granular-Media
Deep-Bed
Filtration"
Water Treatment Plant Design (R .L .
Sa n k s , editor) Ann ArbortAnn Arbor Science Pub­
lishers, 1978 .
L e tt er ma n , R . D . and V a n d e r b r o o k , S.G.
"Effect of
Solution Chemistry
on
Coagulation with
Hydro­
lyzed A l ( I I I ) !Significance
of
Sulfate
Ion and
pH."
Water Research 17 (1983): 195-204.
'
27.
A.,
Personal
communication.
May
1982.
28.
Amirtharaj a h , A.
Chapter 28 "Design of GranularMedia Filter
Units."
Water Treatment Plant De­
sign (R.L.
S a n k s , editor)
Ann ArbortAnn Arbor
Science Publishers, 1978.
29.
Weber, W.J.
Physicochemical Process for Water Quality
C o n t r o l . New YorktWiley and S o n s , 1972.
)
(
68
APPENDIX
SAMPLE CALCULATIONS
i
69
Hydraulic jump velocity g r a d i e n t .
Measurements of depth and elevation within the
flume led to the following data for the 1.5 inches
per foot slope.
, Ef = 0.078 ft'
F1
= 3.42
-y
S
= 0.128 ft"
= sin 6 = 0.124
The power dissipated in the jump can then be found
using Equation
(7) where
Y = 62.2 Ibs/ft-3 (80°F)
0 = 2.0 gpm = 4.46xl0_3cfs
P = Y Q(Et )
Using F1 , y
and S in Figure 22 yields a jump
2
length:
L=
Thus,
0.461 ft
the dissipation volume, V,
is approximated by a
triangular wedge of 1.0 inch width or
V = 1/2 Ly 2 (0.0 8 3) ft 3
Finally,
substitution into equation
y = 1.799x10-3
(2) where
lb* s/ft 3
G = 705 s-1
The above G-value being an approximation of the velocity
gradient for. the hydraulic jump generated on the 1.5
inches per foot slope.
70
S = 0.25
IO
FROUDE **- - F 1
FIGURE 22.
Hydraulic Jump L e n g t h , L, for Jumps
Ch a n n e l s .
in Sloping
71
2.
Impeller stream microscale
Taking the case for a velocity gradient of 3000
s ~ l , Table 3 lists an average power dissipation of
37.84 ft'lb/s.
Referring to Figure I, the approximate
power dissipation in the impeller stream would be
P m2 = 5.4
(37.84)
= 204.3 ft'lb/s
Converting P m 2 to the power dissipation per unit mass
is accomplished by realizing that
where
P= 1.934 .slugs/ft"* and V 2 = 0.095V = 0.022 ft^
thus
e = 4792.9 ft 2 / S 3
Substituting
into
Equation
V = 9.3x10-6
n = 2x10-5
(I)
ft2/s
where
(80°F)
ft = 6.2 ym
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