FLUIDIZATION APPLIED TO SEDIMENT TRANSPORT · 2012-08-01 · 2 LIST OF FIGURES Fluidization Applied...
Transcript of FLUIDIZATION APPLIED TO SEDIMENT TRANSPORT · 2012-08-01 · 2 LIST OF FIGURES Fluidization Applied...
FLUIDIZATION APPLIED TO SEDIMENT TRANSPORT
by
John T. Kelley, Sr.
A Research Report
Presented to the Graduate Committee
of Lehigh University
in Candidacy for the Degree of
Master of Science
in
Civil Engineering
Lehigh University
1977
. .. . ~
ACKNOWLEDGEMENTS
This study was supported principally by the United States
Army. Partial funding was also obtained for materials and computer
authorization by the Pennsylvania Science and Engineering Foundation.
The author wishes to express gratitude to Dr. Willard A.
Murray for his guidance and assistance during the research project
and in preparation of the final report.
Also to be acknowledged is Mr. E. G. Dittbrenner for his
technical expertise in the construction of the experimental apparatus •
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1.
TABLE OF CONTENTS
ABSTRACT
INTRODUCTION
1.1 Theory of Fluidization
1.2 Sediment Transport using Fluidization -Proposed Applications
1.3 Approach
1
2
6
10
16
2. THEORETICAL ANALYSIS 20
2.1 Pressure in the Unfluidized State
2 .1.1 2 .1.2
Analytical Model Numerical Model
2.2 Pressure in the Fluidized State
2.3 Results of Analysis
22
23 26
27
32
3. EXPERIMENTAL INVESTIGATION 42
3.1 Apparatus
3.2 Test Procedure
3.3 Experimental Data
3.4 Additional Experimental Tests
42
48
50
50
4. DISCUSSION OF EXPERIMENTAL AND ANALYTICAL RESULTS 60
4.1 Comparison of Experimental and Analytical Flow Rate 60
4.2 Variation of Pressure with Flow Rate 61
4.3 Conditions at Incipient Fluidization 77
4.4 Distribution of Pressure· in the Media 78
4:5 Comparison of Fluidized Channels 84
5. SUMMARY AND CONCLUSIONS 87
APPENDIX A - CONSTRUCTION DRA\HNGS 91
APPENDIX B - ANALYSIS OF SNAD USED IN MODEL 94
BIBLIOGRAPHY 98
VITA 100
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LIST OF TABLES .,, ..
Table Page ---
r 1 Experimental Data Distributor No. 1 51
2 Experimental Data - Distributor No. 2 52 -..
3 Experimental Data - Distributor No. 3 . 53
4 Experimental Data - Distributor No. 5 54
5 Summary of Experimental Data 55
... • f
v
I __ ..
...
Figure
1
2
LIST OF FIGURES
Fluidization Applied to Sediment Transport
Cross Section of Fluidized Channel
3
5
3 Variation of Pressure and Bed Length for Fluidized Beds 7
4 Perspex Model used by Hagyard et al. (5 ) 12
5 Crater-Sink Sand Transport System(]) 15
6 Duct-Flow Fluidization 17
7 Seepage Analysis Flow Situation 21
8 Muskat Analysis of Fluidizing System for Unbounded Flow 24
9 Node Configuration for Finite Element Analysis 28
10 Force Balance to Determine Fluidizing Pressure 30
11 Theoretical Pressure Distribution - Dist. 1 33
12 Theoretical Pressure Distribution - Dist. 2
13 Theoretical Pressure Distribution - Dist. 3
14 Theoretical Pressure Distribution - Dist. 5
15a Fluidized Zones Predicted by Muskat Equation
15b Fluidized Zones Predicted by Numerical Model
16 Experimental Apparatus
17 Locations and Detail of Pressure Taps
18 Distributor Configurations
19a Experimental Results - Visual Studies
19b Experimental Results - Visual Studies
20a Experimental Pressure Variation at Point 1
20b Experimental Pressure Variation at Point 2
20c Experimental Pressure Variation at Point 3
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34
35
36
40
41
43
45
47
57
58
62
63
64
Figure Page ......
20d Experimental Pressure Variation at Point 4 65
r 20e Experimental Pressure Variation at Point 5 66
20f Experimental Pressure Variation at Point 6 67
20g Experimental Pressure Variation at Point 7 68
20h Experimental Pressure Variation at Point 8 69
20i Experimental Pressure Variation at Point 9 70
20j Experimental Pressure Variation at Point 10 71
20k Experimental Pressure Variation at Point 11 72
201 Experimental Pressure Variation at Point 12 73
20m Experimental Pressure Variation at Point 13 74
2la Comparison of Experimental and Numerical Pressure 80 Distributor 1
2lb Comparison of Experimental and Numerical Pressure 81 .; Distributor 2
2lc Comparison of Experimental and Numerical Pressure 82 Distributor 3
2ld Comparison of Experimental and Numerical Pressure 83 Distributor 5
22 Comparison of Experimental and Numerical Fluidized Zones 85
' ..
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LIST OF SYHBOLS AND ABBREVIATIONS
A = area, square inches
d depth of burial, inches
e = porosity
F = force in y-direction y
h = head, inches of H2
0
K = hydraulic conductivity, centimeters per second
t = length, inches
tn = natural logarithm
M = strength of source or sink
P,p = pressure, pounds per square inch
P* = pressure including hydrostatic forces
P = pressure force
Pmf = minimum fluidizing pressure, inches of H2
0
Q = discharge, cubic centimeters per second
q = discharge per unit width
Qmf minimQm fluidizing discharge, cubic centimeters per second
r1,r2 radius, inches
Vmf = minimum fluidizing velocity, centimeters per second
W = weight
ys,yw = specific weight
e1,e
2 = angle from well to point
-. ~ = potential
' .. ~ = streamline flow
~ = differential operation
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. ..
. -.
..
ABSTRACT
Sediment can be transported using fluidization techniques by
injecting water into a channel bed via a buried fluidizing pipe having
outlet orifices. The cross-sectional shape of the resulting fluidized
channel is found to be dependent on the location of the outlet ori-
fices on the fluidizing pipe. Two analytical models are developed to
predict the fluidized channel shape for given system parameters;
discharge through the section, depth of burial of the fluidizing pipe,
and location of the outlet orifices on the pipe.
Experimentally, an optimQ~ fluidizing pipe configuration is
found to have the outlet orifices horizontally opposed. This confi-
guration yields the widest fluidized channel of five configurations
tested. Analysis indicates, however, that the fluidized channel width
results from both direct fluidization and erosion effects. Presently,
the erosion effects are not included in the analytical prediction
and are responsible for non-agreement between analysis and experi-
ments.
Experimental and analytical results show good agreement when
comparing conditions of pressure in the media as a function of flow
rate, up to and including incipient fluidization conditions .
1
1. INTRODUCTION
Channel deterioration can be caused by stream bank erosion or
deposition of the stream's sediment load. The result in either case
is an accretion of sediment on the channel bottom causing a reduction
of channel depth. Eventually the channel deteriorates to the point
where it is no longer useful for navigation.
This deterioration may take years to occur in some cases or
it may be an annual occurrance in others. Small tidal inlets and
harbors are areas representative of the latter case. Subjected to
littoral drift and sediment from storm erosion, these channels may
deteriorate rapidly.
It has been proposed by Hagyard et al. (S) that unwanted sediment
can be removed by using a fluidization technique. Figure 1 shows both
a longitudinal and a cross-sectional view of the system. By injecting
water into a sediment bed, via outlet orifices in a buried pipe, a
zone of fluidized sediment can be developed. The zone will extend to
areas of the bed such that the upward pressure force exerted by the
injected water equals or exceeds the weight of solids and liquid above
the pipe.
In the unfluidized state, the sediment, due to its soil struc
ture, will remain in place as long as the channel slope is less than
the angle of repose of the sediment. Fluidizing the sediment destroys
the soil structure of the sediment bed and reduces the angle of repose.
2
-. ...
Sediment
.. ··· ..
Silted Channel
Bottom
. . .. · ... ~ \ ...
Silted Channel
Original Channel
Outlet Orifice
u~ ~z~ng
Liquid (Inflow)
.. :. : .. .. . ... ..... .. .. . .. . . ....... . . . ... · . · ....
.. ' . .. . .. ' -. .. . .. ..... . : ..
.. . .. , __ :: .. .. . .
I
. ·~Blt,t.h:l.i,"Zirlg.·P:i;pe .· ·. • ·. . . . . ""'. \ . . . '. .
., :· .. .. .. ....... · ..... ·
sz
I : • . ' .
Fluidized Zone Boundary
Fluidizing Pipe
Fig. 1 Fluidization Applied to Sediment T~ansport
3
-. ..
The fluidized sediment behaves as a liquid and "flows" on small
slopes, (S) such as those likely to be encountered in channel bottoms.
The fluidizing liquid need only exert sufficient force to
fluidize the sediment, while actual transport of the sediment is
caused by gravity forces acting down the natural channel bottom slope.
The sediment flows down the slope to an area where shoaling is less
harmful and is then redeposited.
For clearing navigable channels, a prediction of top width,
depth, and cross-sectional shape of the fluidized channel must be
known to determine the feasibility of the planned fluidization system,
Figure 2 shows a typical fluidized channel cross section. The flui-·
dized sediment zone results from a given discharge through the sec
tion. This zone can be described by top width at the original bed
surface; depth at the center of the zone; and zone shape described by
the fluidized boundary.
For a given sediment, it is expected that depth of pipe
burial, outlet orifice configuration and upward flow through the
section all are parameters affecting the top width, depth, and shape
of the fluidized channel.
It is the object of this study to determine how the above
parameters affect the size and shape of the fluidized channel.
4
. . . . . ' ' .
Width
Original Bed Surface
depth
·Fluidized Zone Boundary = f(x,y)
Outlet Orifice
Fig. 2 Cross Section of Fluidized Chinnel
. ,
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1.1 Theory of Fluidization
Several general fluidization literature reviews are available
( 1 13) at present. ' Herein, only those areas of research associated
with the immediate problem, applying fluidization to sediment trans-
port, will be reviewed.
Two phenomena can generally be associated with a fluidized
bed; pressure in the bed and expanded bed length. By examining the
pressure in the media and the length of the bed, the effect the
fluidizing flow rate on the bed can be evaluated. Figures 3a and 3b
show the variation of both pressure and bed length with increasing
fluidizing flow rate.
Wnen water is passed through a porous media, a loss in fluid
pressure results. This loss is characteristic of the flow resistance
encountered during passage through the media. By defining the resis-
tance as a drag force, it can be seen that the resistance is propor
tional to the velocity of flow through the media. (l3 )
In the unfluidized media, an increase in flow rate results in
a linear increase in pressure according to Darcy's Law (Figure 3a).
Eventually the force exerted by the pressure becomes sufficient to
support the weight of the media particles. The velocity of flow
through the media at this point is called minimum fluidizing velocity
If flow is considered as it passes through flow paths in the
media, each path of a given area, the discharge at minimum fluidizing
6
. ,
-.
(a)
(b)
(JJ
H :l Ul Ul (JJ
H P-c
/ /
/ ,
B /
/ /
A
G
/ Not / • Fluidized
/ Fluidized
Qmf Flow Rate
Qmf Flow Rate
Uniformly Graded Ideal Bed
Fig. 3 Variation of Pressure and Bed Length for Fluidized Beds
7
--
velocity can be expressed as the product of the channel areas and
velocity, Vmf· This discharge is shown in Figures 3a and 3b. In
creasing the flow rate beyond Qmf causes expansion of the bed and
enlarges the areas of the flow paths, allowing the velocities in the
flow paths to remain at minimum fluidizing velocity. (3 ) The expan
sion of the bed then enables equilibrium to be maintained between the
drag force and the particle weight of the media.
The behavior of the pressure with increa~ing flow rate can be
illustrated by plotting pressure versus discharge (Figure 3a). With
increasing flow rate through the media, pressure loss in the fluidi
zing liquid increases linearly for the unfluidized bed. After fluidi
zation occurs the pressure loss remains constant with increasing flo•~
rate.
If bed length is examined, the opposite trend occurs (Figure
3b). For the unfluidized bed, the bed length is constant with in-
creasing flow rate. Once fluidization occurs, however, bed length
increases with fluid flow rate.
The point at which bed expansion begins and pressure becomes
constant for increasing flow rate is shown as point A in Figures 3a
and 3b. This point defines the condition of incipient fluidization and
is unique for a sand of given specific gravity and particle diameter.
In a graded sand, a distinct point of incipient fluidization (point
A) does not exist. Instead there exists a range of various minimum
fluidization velocities required for the different soil fractions. (l)
8
This is shown as line BC in Figures 3a and 3b. It can be expected
that the more uniform the gradation of the bed, the more distinct
will be the point of incipient fluidization. For the purposes of
this tehsis, incipient fluidization conditions are defined by point
A. This assumes minimum fluidizing pressure (Pmf*) and flow rate
(Qmf*) of an ideal uniformly graded bed.
Channelling and spouting are phenomena that occur in fluidized
beds at or near incipient fluidization conditions. Channelling is a
self-propagating discontinuity in porosity of the media. (l) Since
the channel develops as a path of least resistance through the media,
a majority of the flow tends to flow through the channel. Although
difficult to explain by analytical predictions, channels can readily
be seen in actual fluidized beds as preferred flow paths through the
.media. Since channelling tends to disrupt the uniformity of fluidi
zation of the media it is an undesi~able phenomenon.
Spouting can be visualized as a special case of channelling such
that the channel extends from the flow source to the bed surface.
Wnile undesirable for the purposes of this study, spouting has been
investigated in detail as it relates to chemical engineering applica
tions. (9 ) It is sufficient to note that spouted bed theory appears
to best describe the behavior of the pressure in the media for the
fluidizing system near incipient fluidization conditions.
9
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1.2 Sediment Transport using Fluidization - Proposed Applications
Although literature pertaining to fluidization techniques and
processes in general is in abundance, the application of fluidization
to sediment transport appears to be only of recent interest. Litera
ture pertaining to the specific problem of sediment transport appears
to originate with the study at Westport, New Zealand by Hagyard et
al. (5)
Sediment deposited by littoral drift causing san bar formation
has long been a problem in the harbor at Westport. In the past,
solutions to the problem have been in the form of extensions to
the harbor moles to intercept the littoral drift. (5 ) Experience has
shown this only to be a temporary solution. As permanent solution,
Hagyard proposed that a portion of the sand bar be flu:Ldized to
create a channel to intercept littoral drift and allo·..v it to "flowtt
out to sea. The proposed system included a fluidizing pipe buried
3.5 feet deep and extending 7000 feet from the breakwater to deep
water. An estimated 49 million cubic yards of sand could be removed
annually by this system while only a constant 96 H.P. would be ex
pended to pump the fluidizing liquid through the pipe.
In addition to providing an initial design for the fluidiza-
tion system, Hagyard et al. demonstrated that fluidized sand will
flow on slopes as small as 1:400, the typical sea bed slopes at
Westport.(5 ) This is important if the system r~lies on gravity as
the sole transport mechanism to move the fluidized sediment.
10
...
..
. -
Hagyard et al. also relied on a self-burying design for the
fluidizing pipe. Proper orientation of the fluidizing orifices would
allmv the pipe to fluidize the sediment immediately belo\v the pipe .
The weight of the pipe would then force it to sink in the fluidized
sand. Experiments in the laboratory showed the self-burying concept
to be a valid method of burying the fluidizing pipe. Ho\vever, actual
pilot studies determined that roots and objects below the surface
hindered the self-burying ability of the pipe. (4 )
Channel cross-sectional studies accomplished by Hagyard et al.
were approached in an empirical, manner. Using a perspex model to
simulate a cross section of the fluidized channel (Figure 4), he in-
jected water into a sand media via a jet oriented vertically down.
Assuming a triangular channel cross section, Hagyard et al.
described channel size by the depth of submergence of the pipe and
the angle of divergence of the channel walls. The angle of divergence
of the walls is defined as half the bottom apex angle of the channel
cross section.
Hagyard et al. further defined a quantity of flow called
"leakage" in his experiments. (S) Leakage referred to liquid that
was released to unfluidized portions of the bed. Hagyard et al
determined by dye studies that leakage occurred along horizontal and
downward paths and theorized that it tended to stabilize the sides
of the channel against fluidization. (5)
11
..
....
\.
' ' ' ' '
' ' Unflui"d~ed
Water
I /
/ / /
/ / ./
/
,---~-.:.....-Angle of /
D. / ~verge}lce
./ /
.............. ...... Sand ---
......
...... -- ------
.... ...... ._. __
/ ....... ----~ ---·
./ ./
/ /
/ / /
/ /
/ 1 ft
- Leakage Path / ..... --- ---.._ -- --- --
./ L---------------~ /
Fig. 4 Perspex Model used by Hagyard et al.(S)
12
Results from experimental efforts indicate that for a constant
burial depth of the fluidizing pipe, leakage decreased rapidly with
increasing flow rates of the fluidizing liquid. (5 ) Further experi-
mentation by Hagyard et al. revealed that as flow rates increased,
the angle of divergence increased. (5 ) Although it appears that flow
rate, leakage, and channel width are interdependent, no continuous
expression has been presented to describe this result.
Wnile Hagyard et al. addressed principally the cross-sectional
development of the fluidized channel, others have investigated longi
tudinal development of the channel. Wilson and Mudie(l6) of Scripp's
Institution of Oceanography applied Hagyard's et al. fluidization
technique to create an open channel between a slough and the Pacific
Ocean. The major problem in their endeavor was maintaining flow of
the fluidized sediment through the channel. One consideration that
had not been anticipated was the formation of a delta at the channel
outlet .. · The resulting blockage in flow necessitated constant removal
of the delta to. allmv passage of the sediment. Of major importance
was the tendency for the~ channel to degenerate into a series of
"boiling holes and dams that migrated in position". By increasing
the depth of burial of the fluidizing pipe, the number of holes
decreased and the spacing of the holes increased. As a result they
defined the concept of the "flo\ving fluidized channel". In order to
develop a "flowing channel" it is sufficient that only the tops of
the holes overlap.
13
-.
- .
Fluidization as proposed by Hagyard et al. has been suggested
to enhance the effectiveness of beach replenishment systems. One
such system is the "Crater-Sink Sand Transfer System". (l) The system
relies on a suction type dredge with its inlet located in a crater on
the ocean floor. As dredging proceeds sediment falls down the walls
into the crater to be transported to the beach. During dredging~
material continues to fall until the crater enlarges such that the
angle of the crater walls coincides with the angle of repose of the
bottom sediment. By fluidizing the area around the crater, the
effective area of the crater can be increased without substantially
increasing the depth of the crater. Figure 5 depicts how fluidiza-
tion techniques might enhance the Crater-Sink Sand Transfer System.
A method of sediment transport using fluidization techniques
has been proposed by Inman and Baillard of Scripp's Institution of
Oceanography. (B) The name of this method, "Duct Flmv Fluidization",
arises from the shape of their fluidized channel. Instead of fluid-
izating sediment above the fluidizing pipe, they propose the creation
of a fluidized channel, or duct, directly below the pipe (Figure 6).
This system is used in conjunction with a crater, or sink, to provide
a location for deposition of the fluidized sediment. In this system,
fluidization is used to suspend sediment in the duct. The transport
of sediment through the duct into the sink is accomplished by the
momentum exchange as the fluidizing liquid impinges on suspended
particles. The sediment above the pipe is eroded and replenishes
sediment transported through the duct. Due to sediment flowing into
the duct, a lowering of the channel bed surface results.
14
~-- • -.- .··· ' -- ~- ---- ,.._...,. .• _ . ._.. • •. --~- 'l:'r .. -':"'"'"'"",._..,_, .... ~--·-:'1""".-~- -- ~ ·.- '-, w,., ~--· -· ..--~. ~ . ,..... . - " ... ~- .•• ":'- -~ ---:.·· :.""::"·:;: .--; ..... .._ •••. • .. -v-- \" ~- . _:--·-. ~ ---·-
PMN
Extended Crater Sink Crater Sink due to Angle of Repose
I
sl~
Crater Sink
Consolidated Soil
.. Fig. 5 Crater-Sink Sand Transport System(l)
15
..
. .
The duct flow method is not fluidization as Hagyard et al.
proposed. Increased energy requirements appear necessary over that
required for Hagyard's et al. system since th~ fluidizing liquid
provides the transport mechanism for the sediment.
While it appears that some research has been accomplished
. (5 16 17) toward applying fluidization to sed~ent transport, ' ' none of
the investigators have satisfactorally addressed the problem of cross-
sectional channel shape. Hagyard et al. provided experimental insight
into cross-sectional shape as it relates·to depth of submergence and
fluidizing flow rate~ However, little has been acco~plished in deter-
mining the effect of outlet orifice direction on channel shape.
Herein, the problems of prediction of the channel cross-sectional
shape will be addressed in order to define the effect of flow rate,
pipe submergence, and outlet orifice configuration on the cross-
sectional shape of the fluidized channel.
Problems concerning the linear discontinuity of the fluidized
channel(ll) will not be addressed. In order to determine why the
discontinuities occur and how to prevent them, the basic mechanisms
of the fluidizing system must be known. It is thought that a detailed
investigation of the cross-sectional development will yield this
information.
1.3 Approach
Since fluidization requires that an upward pressure force be
exerted to support the weight of the particle bed, it seems reasonable
16
Water
Channel B-e-~-.. ---.-.-.------.. ---~-,_~--,l •"" -: .. ·. , . . · .
· .. . .. ; : ...
Orifices . ·- ... Fluidized Sediment
........... . ·
Fluidizing Pipe . . ..... .. . ·: .· ..
.. • • • 0 ... -_, . . .. ~ " .... . .. . ~ . • • •• ....... - • .. •• •• 0 ...... ·-~ • •• . . .
Angle of Pipe
: ·: .. ; : ····.'.J .··• .. : __ :._:
.. Fig. 6 Duct-Flow Fluidization
17
I
·-
to investigate the behavior of pressures in the media as they relate
to fluidized and unfluidized portions of the media. This is accom
plished by both analytical and experimental methods for different
configurations of fluidizing pipe.
Analytically the equations must indicate pressure in the two
conditions; unfluidized and fluidized media. Referring to Figure 3a
a seepage flow approach applying Darcy's Law is used to define the
rising portion of the pressure versus discharge curve. The pressure
of the constant portion of the curve is calculated by a force balance;
equating the pressure force to the weight of solids and liquid in the
fluidized column.
The seepage approach has limitations in that the equations are
valid only for the fixed media. Once reorientation of the particles
begins the equations are invalid in describing pressure distributions
in the media. However, Darcy's Law provides a valid solution method
to conditions in the media prior to particle movement. It is thought
that these conditions may indicate how the media will eventually
fluidize.
The pressure distributions obtained using seepage analysis are
representative of a particular total pressure loss through the media.
By increasing this pressure to a typical fluidizing pressure, a
series of locations in the media are obtained where fluidizing pres
sures are present. Fluidized zones obtained in this manner are then
compared to experimental results.
18
An experimental apparatus is constructed to enable measurement
of pressures in a cross section of the fluidized channel. Pressure
distributions measured in the apparatus are compared to analytical
pressure distributions, and fluidized zones measured in. the apparatus
are compared to an analytically determined fluidized zone. Finally,
an evaluation of the validity of the seepage equation predictions of
cross-sectional channel shape is made.
19
2. THEORETICAL ANALYSIS
The problem to be analyzed is the prediction of a fluidized
zone for particular fluid flmv rates and fluidizing pipe configura-
tions. In predicting the fluidized zone it is the pressure at a
point that determines whether or not the media fluidizes. The cruX
of the problem is then to determine the pressure distrigution in a
media resulting from the flow injected by a fluidizing pipe.
The physical problem is illustrated in Figure 7. The fluidi-
zing pipe is shown buried at a depth, d, in the silted channel. The
known parameters of the flow situation are: flow rate; soil hydraulic
and conductivity, specific gravity, and porosity; depth of burial;
and orientation and configuration of the outlet orifices on the flui-
dizing pipe.
As shown the flmv situation is bounded on three sides by the
original impervious channel walls and bottom. The fourth boundary is
the silted bed surface. In analyzing the boundary conditions then the
flow situation is bounded on three sides by a constant flow along
the boundary and on the fourth side by a constant head determined by
the depth of water above the bed surface.
The solution to the pressure distribution must be divided into
two parts; pressure in the unfluidized bed and pressure in the flui-
dized bed. This is necessary since the development of Figure 3a was
based on the concept that the point of incipient fluidization (point
A) occurred as a discontinuity in the pressure variation for an ideal
20
.. ·:---·.- ~,· .,..,.. ....... - .-,
-..
..
lvater
\ Fluid iz~ Zone
\ \ \ \ 0 \
"- --Impervious Boundaries
/
I
I I I
I I
/
Fig. 7 Seepage Analysis Flow Situation
21
r ••••
.:
""' ""
bed. The pressure variation prior to incipient conditions varies with
flow rate. After incipient fluidization, however, pressure depends
only on the weight of the particle bed.
2.1 Pressure in the Unfluidized State
The pressure in the unfluidized state is determined by a .
seepage analysis of the flow situation shown in Figure 7. The basic
equation for flow through a porous media is Darcy's Law:
where Q = fluid discharge
Q =o -KA dh dt
K =o hydraulic conductivity of soil
A = area of flow path
h = head
t = length of flow path.
Co~bining Darcyts Law and a continuity equation:
V.q=O
where 'V = differential operator
q = discharge vector
~ields the basic equation of motion in a porous media. This equa-
tion can be written as the steady state Laplace Equation in two
dimensions:
22
(1)
(2)
(3)
. .
,.
Two theoretical models are offered which solve the Laplace
Equation subject to boundary conditions in the flow situation
(Figure 7).
2.1.1 Analytical Model
A literature search of two dimensional flow problems yielded
the analytical model to be used. Derived by Muskat, (ll) the analyti-
cal model is a solution to well flow in a porous media.
The flow situation solved by Muskat is shown in Fig. 8. In
the figure a well is being supplied by a stream at a specific distance
from the well. It is assumed that the flow situation is unbounded
except for the stream boundary and that all flow to the well origi-
nates at the stream.
The formulation of the mathematical model requires that the ·
well be replaced by a point source and that the stream boundary be
replaced by an equipotential surface. In Figure 8 a coordinate has
been placed on the physical situation such that the well lies on the
y-axis at (0, -d) and the stream lies on the x-axis.
In order to mathematically develop the x-axis as an equipoten-
tial Muskat locates an image sink at (O,d). Using the method of
images the potential function at any point in the media (x,y) is;
(4)
23
---- ~ 4~---~ - -.. - . ..,. - -~· .............. -. ": .. ..-·:;- ·- --, ·-: _ ..
r
Image Sink (O,d)
Point Source, 0 (0,-d)
Fig. 8 Muskat Analysis of Fluidizing System for Unbounded Flow
24
_I
where ~ = potential
M = strength of source or sink (q/2;)
radius to source
r 2 radius to sink.
The radii, r1
and r2
, can be expressed in terms of x, y, and d:
The equation for the mathematical model can then be written
by combining Equations 4 and 5:
where cp = potential
K= hydraulic
h = head
q = flow per
(x,y) = location
(0,-d) = location
=
rn = -Kh = ....9... Pn fx2 + (y+d )2
T 'V ~~~ + (y-d)Z"" 2n
-Kh
conductivity
unit thickness
of point
of well.
Muskat's model can be easily adapted for the flow situation
(5)
(6)
in Figure 7. The well can be replaced by the fluidizing pipe and the
stream by the bed surface. The only simplifying assumption that must
be made is in fitting the unbounded flow situation of the model to the
bounded conditions of Figure 7.
25
.•
To fit the physical boundary conditions shown in dashed lines
(Figure 8) only the flow between bounding streamlines in the model
will be considered. The bounding streamlines chosen for this study
are those that encompass the full top width of the physical situa
tion .. (AO and BO) as shown in Figure 8. The exact correlation between
the bounded and unbounded situations is developed in Section 4.1.
The mathematical model can detect the effect of variations of
the flow rate, depth of burial of the fluidizing pipe, and hydraulic
conductivity of the soil. There is no parameter, however, to account
for configuration of the outlet orifices on the fluidi~ing pipe. In
using the mathematical model it must be assumed that effects of orifice
configuration are only local to the area around the pipe. In this
respect the mathematical model can only approximate the physical
problem.
2.1.2 NQmerical Model
A numerical model(lO) can be used to directly solve the Laplace
Equation. The numerical model is a computer program designed to solve
the differential equation of motion by a finite element analysis.
The finite element analysis allows the media and fluidizing
pipe to be represented by a grid configuration shown in Figure 9. By
refining the grid in the area of the fluidizing pipe, the shape of
the pipe can be represented by triangular finite elements. The node
points at the pipe surface represent locations of outlet orifices on
the fluidizing pipe.
26
. .__ .... ··- --··· ..
The model has the capability of including more than one media
material in the analysis. Using this feature, the impervious boun
daries of the physical flow situation are simulated by a material of
extremely small permeability. In a like manner, the finite elements
of the pipe are composed of a nearly impervious material.
Initial conditions in the media can be specified as either
a piezometric head or flow at nodes in the finite element grid. For
the problem shmvn in Figure 7, the bed surface boundary condition is
supplied to the model as constant piezometric head at the surface
node points. Boundary conditions at the fluidizing pipe are deter
mined by the outlet orifice configuration. By properly applying
either peizometric head or external flux at points A, B, and C, in
Figure 9, the outlet orifice configurations are varied.
The numerical model allows a more exact representation of the
physical flow situation. Boundaries can be exactly duplicated and,
in fact, must be included for the model's analysis. The fluidizing
pipe also is better represented in the numerical model than in the
analytical model. By choosing the proper grid refinement the size of
the pipe and its interference of the seepage flow paths can be repre
sented. Applying proper boundary conditions the physical outlet
orifice configuration can be duplicated.
2.2 Pressure in the Fluidized State
The pressure in the media after fluidization is determined
by a balance of the forces between the weight of a column of media
27
. -.-""' ; ...
.r
I
Hater P-al 1
Ha terial 2
7)
~ In sert A
Haterial 2
Insert "A"
Fig. 9 Node Configuration for Finite Element Analysis
28
·.
l ' i t
and the upward force exerted by pressure in the media. It is assumed
that contribution of shear forces in the media to the resistance of
motion of the particles is negligible compared to body force (weight).
Referring to Figure 10, the sunullation of forces can be written in the
y-direction as,
where P = pressure force
W = weight.
L:F =P-W y
In the analysis the column is considered to consist of both
(7)
solid particles and the water contained in media voids. The column
of media in Figure 10 has a finite length, 1, and area, 6x~z. The
weight of that column then is the SQ~mation of the volumes of water
and solids in the column multiplied by their respective unit weights.
(8)
where W= weight of column
.t= length of column
t...x~z = area of column
ys = unit weight of solids
y :::z unit weight of water w
e = porosity of media.
Equation 8 can be rearranged to reflect the submerged weight of the
solids and the weight of a water column of dimensions 1 x (~x~z),
W = .t(~x~z)(l-e)(y ""Y ) + y .t(~x~z) s w w (9)
29
y
Bed Surface X
Fluidizing Pipe
Fig •. 10 Force Balance to Determine Fluidizing Pressure
. .
..
The pressure force in Equation 7 can be described as the
pressure in the media acting in an upward direction on the base of
the column. This force can be written as,
P == p (b.xb.z) (10)
where P = pressure force
p == pressure at base of col~~n
b.xb.z = area of column.
In order to fluidize, the summation of forces in Equation 7
must equal zero. Clearly, the pressure force then must equal the
weight of the column. Equating Equations 9 and 10 yields,
P (b.xb.z) -== t(b.xb.z)(l-e)(y -y ) + t(b.xb.z)y s w ·w
Rearranging Equation 11,
where p ·- pressure in media
s = specific gravity of s
yw = unit weight of water
t= length of co lunm
e = porosity.
t(l-e) (S -1) + t s
solid
Equation 12 represents both the pressure due to the weQght
(11)
(12)
of the solids in the column and the pressure due to hydrostatic forces •
The hydrostatic force is constant for a given length of column and,
therefore, can be arbitrarily eliminated from the analysis. This is
31
...
accomplished by defining a pressure, p*, that encompasses both.
contributions to the pressure:
p~: = p + y t w
(13)
The equation for prediction of the fluidizing pressure can be
written by combining Equations 12 and 13,
P* -= t(l-e)(S -1) s
(14)
The fluidized zone in the media extends then to all areas such that
the pressure in the media equals or exceeds P*.
2.3 Results of Analysis
Pressures in the flow situation (Figure 7) were calculated for
unfluidized conditions by the theoretical models. Figures 11 through
14 show the pressure distributions obtained for each of four fluidi-
zing pipe configurations. Shown in dashed lines are the equal pressure
contours obtained by the Muskat model.
Each contour represents a nondimensionalized pressure~ To
formulate the contours, the maximum pressure on the pipe surface is
divided by 10. Each contour then represents an incremental pressure
increase, starting at zero at the bed surface, of 0.1; P being the
m aximun1 pressure.
A comparison of the pressure distributions of models provides
insight to conditions at incipient fluidization. One can note that
32
-- ---,... ....... ··-· -~~-------~-~. ·- ... ~-- .. - .. --:-··.,·.;r<...:r,.•··~-- .... · .• -.o-- ·-"':·-·· .----~··- ·-:·•--':""''' --~ .... ;''""·---;~·---:·· •. •
. . .·
Fig. 11
\
\
I \ \
\ \
\ \
\
\ \
\
\ '\
\ \
\
'\.
I I \ \ \ \
'\.
"'-..
'\.
·" "'
\ \
'\
"' "
/ ---- -
--- ------
-- --
Theoretical Pressure Distribution - Dist. 1
. 33
l I I I I \ I I
I \ I \ I
I \ /
.- / I I \ / \ \ \ /
\ \. /
\ \ \ \ \ ' '-\ \ \ '-... /
\. -\ \ '\.
\ \ ' \
\ \ \ \ \ \ ~
..
......... ........
.........
\ ..........
<lJ --0 --C1j \ -ll-l $.-4 \ ::l Ul
"C) \ ().)
\ r:Q
\ \ \
\ \
\. .
" .
" Fig. 12 Theoretical Pressure Distribution - Dist. 2
. 34
... I \ \ \
•
Fig. 13
T
\ \
\ \ \ \
\
\ \ ~
I \ \
\
\
\
\
\ \
\
~
" "' Theoretical Pressure Distribution - Dist. 3
35
/
/ /
.
--- --
I \
\
~
~ ~ ~ \ \ \ \
\
I \
\ \
\
\
\ \
\
\
\ \
\ \
\
"'
I
/
_..........,. -- - ---
---
Fig. 14 Theoretical Pressure Distribution - Dist. 5
36
.-
..
the tendency of a column to fluidize directly depends on the pressure
gradient through the column. Thus, assuming that pressures throughout
the media are increased uniformly, the vertical sections of the high-
est pressures gradients will be the first sections of the media to
fluidize. It will be these sections that determine incipient fluidi-
zation conditions in the media.
Figures 11 through 14 show that the highest vertical pressure
gradient in the media occurs in the column directly above the fluidi
zing pipe. Examining the pressure gradient in this column of media
will indicate how the incipient conditions vary with the models. The
nQ~erical models all indicate a smaller pressure gradient than that
indicated by the Muskat model in the area above the pipe. It can also
be noted that the pressure gradient in this area increases with pipe
configuration representing distributors 1, 2, 3, and 5 (Figure 18),
respectively.
In the formulation of the models, pressure is determined as it
varies with flow rate. It is expected then that the incipient fluidi
zing flow rate increases as the pressure gradient critical to fluidi-
zation decreases. Conclusions can be made, therefore, about incipient
fluidizing conditions for the theoretical models:
Incipient fluidization velocity varies with outlet orifice
[( configuration
I' '
2. Incipient fluidizing velocity is lowest for the Muskat model
and increases for configurations 5, 3, 2, and 1 in that order.
37
.-
..
An indication of the eventual width of the zone of fluidi-
zation can be obtained by examining the pressure distribution in
the media as a whole. In order to make comparisons between the models,
a "width" of the pressure distribution is defined. This "'vidth'' is
defined as the distance, at the elevation of the pipe, from the center
of the pipe to the contour representing 0.2 P.
It is expected that for a greater "width" of the pressure
distribution, a greater top width of the fluidized zone results.
Comparison of the models on this basis shows the "width" of the pres
sure distribution of configuration number 3 to be similar to the
Muskat solution. In order of decreasing "width" the numerical pressure
distributions can be listed as configuration 5, 3, 2, 1. The respec
tive top widths for the fluidized zones are expected to compare simi
larly to the distribution "widths" for the theoretical models.
Fluidized zones predicted by the Muskat model are shown in
Figure 15. The zones are calculated by an iterative technique that
solves Equations 14 and 6 simultaneously. In the figure, C is a
parameter that varies with applied flow rate:
where C = plotting parameter
Q = flow rate
C = Q/2 ~ K
K ~ hydraulic conductivity •
(15)
Figure 15 shows how the predicted fluidized zone varies with
flow rate. As expected, the fluidized zone becomes larger with
38
increasing flow rate. However, from the figure, it can be seen that
the increase in the top width of the fluidized zone is not a linear
function of C.
· Numerically, fluidized zones were calculated for four fluidi-
zing pipe configurations (Figure 18). Comparison of numerically pre-
dieted pressures at the maximum experimental flow rates (Section 3.3)
were made \vith predictions of P* (Equation 14) to determine the flui-
dized zones. The results of that analysis are shown in Figure lSb.
The top widths of the numerically predicted zones for the different
configurations of fluidizing pipe compare as expected.
From the analysis it is concluded that a fluidized channel
shape can be calculated by the theoretical models.
In addition to predicting channel shape, the models also pre-
dieted pressure in the media and flow required for incipient fluidi-
zation to occur. The analysis model must be altered to account for
outlet orifice effects.
39
c 2
c 2.5
c 3
c =: 4
c 5
c 6
c 7
c Q/2 n K
-.
. • .. Fig. 15a Fluidized Zones Predicted by Muskat Equation
40
.-
0 - Dist. 1
b. - Dist. 2
0 - Dist. 3
0 - Dist. 5
. -··
Fig. 15b Fluidized Zones Predicted by Numerical Hodel
. 41
3. EXPERIMENTAL INVESTIGATION
3.1 Apparatus
To investigate the analytical predictions, a sand model is used.
Shown in Figure 16, the model is a box, 48 inches long, 28 inches
deep and 3 inches thick. It is constructed of ~ inch thick plexi
glass with the joints glued and screwed together. To provide rigidity
to the front and rear faces of the model, one inch steel box supports
span the length of the model at intervals of approximately 9 inches.
Appendix A contains construction details of the sand model.
Discharge of water from the model is controlled by an overflow
weir and an outlet valve (Figure 16). The overflow wier is 2 inches
wide and 4 inches deep with a capacity to discharge in excess of 455
cc/sec (the maximum discharge tested). To enable complete draining
of the model, a valve is located at the base of the model. Both the
weir and the valve discharge to a disposal reservoir, with a 30
gallon capacity for storage.
In order to determine pressures in the media, thirteen pressure
taps are located on the rear face of the sand model. Locations of
the pressure taps (Figure 17) are chosen to provide a refined grid
of taps close to the distributor yet completely cover the flow regime:
A standard spacing of 4" between pressure taps is used in the region
subject to fluidization.
The pressure tap is manufactured from ~ inch hexagonal brass
stock (Figure 17). The exterior- of the tap is machined such that one
42
. '
Disposal Reservoir
Inflow
.......... ,.,..... """" ............
..... ..... ) ·yt ~
Piezometric Columns
Overflow c:r-------------------~~1 Wier II S2.
-,,---~----
13 Pressure Taps "O"
Box Supports
Drain L Valve (r-
on Rear Face II ., II ;·:· . , •.. . . . . . . . . ,. . I 1---:.----.....l•--:-:--:------..-l
'' •, •+.'•' ' ' ' •, ,1''' • '• • • ' . • • .. ' • ·. I • . . .... ""~'· .... +· .. ·· + ....... ~·.:, ...... '·.· !\, .. :.._::·~:: ... :'·~: .~ ...... ::,.,·
.- . ·- ., . . .. ., .... ·, ..... · +' - :. .· ' .. '· II : .. : I • " •• ~-• ~ • • • ... • , •• ~ • 0 • "' ' •
• • • •• - • •• • ... I •• +' .· +"" - Ill ... : . • ·•· ,... ... , : ' ' • • • "' ' I • '• 'f": • ' • . . . . : , . . . .. , . . . "~ I . . . .. •, . . , ~: • ••• :' .... : •• •' • • •• "f. I ·• .. \. I • • • - ""'.· .. ,. ·;.·· .. : .... ~-....
:-. ' .... ... ·+ .·.·. ·. . -. ' ..
' : .,
Distributor
••• '• .... •••• 0 ,.·
l .• • . .. •.
I
. '
I I
end screws into a ~ inch standard thread and the other end accepts
1/8 inch Polyflow Plastic tubing. The tap is screwed into a plexi-
glass block, 1~ x 1~ x 3/8" thich which is, in turn, glued to the face
of the model. The tap, when glued in position is flush with the
inside face of the model wall. The inside diameter of the tap is
1/16" to minimize problems of clogging by particles. During experi-
mentation, however, it was found the taps required cleaning occasion-
ally.
The pressure taps are connected by 1/8 inch Polyflo plastic
tubing to a 6~' manometer board manufactured by Aerolab Supply Com-
pany. This particular manometer board was chosen because it c.ould
easily be converted to provide 20 piezometric columns. Using the
above apparatus allowed simultaneous readings from the thirteen
pressure taps.
Water is injected into the media by a distributor simulating
a portion of the fluidizing pipe (Figure 16). The distributor is
constructed from 1~ inch diameter p.v.c. pipe (Appendix A). It is
3 inches long with the ends capped to fit perpendicular to the viewing
faces of the sand model. A ~ inch tapered thread is tapped into the
distributor to allow it to be screwed onto the inflow pipe. Outlet
orifices drilled into the distributor, are the same size and spacing
as those used by Mudie and Wilson;(ll) 3/32 inch in diameter
at 1 inch intervals.
44
4" 8" 4"
3"
lf3 + 112+ lfl-t
lfl3+ lfoll+ I #6 + f.~s+ 1!4+
I
lf8 + If? + 1112+ lflOt I
if'} +
3"
~· Standard Thread
1/8"
Hole 1/16" Diameter . ,J.
1~ x 1~ x 3/8" Plexiglass
Fig. 17 Locations and Detail of Pressure Taps
45
By varying the location and number of outlet orifices, five
different distributors are constructed (Figure 18). These distri
butors are as follows:
1. Distributor No. 1 - 3 orifices directed downward
2. Distributor No. 2 - 6 orifices directed 45° from horizontal
3. Distributor No. 3 - 6 orifices directed horizontally
4. Distributor No. 4 - 3 orifices directed vertically upward
5. Distributor No. 5 - 9 orifices directed horizontally and
vertically downward.
The bulk of experimentation was conducted with distributors
1, 2, 3, and 5. Distributor No. 4 was used only for the measurements
of incipient fluidizing flow rate and fluidized channel shape.
The main consideration in the design of an inflow pipe is the
prevention of piping in the media at conditions below incipient
fluidization. Actually, piping along the inflow pipe was only a
problem when distributors Nos. 1 and 4 were used. To prevent the
piping in distributors 1 and 4 and to not interfere with outlet
orifices in distributors 2, 3, and 5, two different inflow piping
schemes are used. The inflow scheme used with distributors .No. 1
and No. 4 is shown in Figure 16 by the solid lines. Shm\ln in dashed
lines is the inflow piping scheme used with distributors 2, 3, and 5.
Both. inflow piping schemes provided satisfactory service during exper
imentation with the sand model.
46
3 Holes
3/32 11 @1"
6 Holes
3/32". @1"
Dis t. 1
9 Holes
3/32" @1"
6 Holes
3/32" @1"
3 Holes
. 3/32" @1"
5
Fig. 18 Distribtitor Configurations
47
Flow is supplied to the model via a 3/4" diameter flexible
hose. A pressure regulator is installed in the laboratory supply line ( to insure a constant pressure in the supply line. A shut off valve ~
is located between the flexible base and the inflow pipe to allow
regulation of the flow to the experimental distributor.
Two sands \vere used during experimentation. Initially a fine
sand with a mean grain size of 0.14 mm arid specific gravity of 3.1 was
used. Problems with bulking during fluidization and clogging of the
pressure taps caused abandonment of this sand in preference to a
coarser media. The sand used for the remainder of experimentation has
a mean grain diameter of 0.4 mm.with a uniformity coefficient of 1.37.
A mechanical analysis of the sand is given in Appendix B.
Sand placement in the model is accomplished under saturated
conditions to insure an isotropic media in the model. Rodding of the
media is used only to eliminate large voids that have occurred during
placement. Care is taken that flow channels are not created during
the rodding process. When in place it is assumed the media is at an
uncompacted porosity. Prior to experimentation the media is leveled
to a predetermined depth. Between tests, sand is added to replace
media lost to the disposal reservoir.
3.2 Test Procedure
Itemized below are the steps taken during the fluidizing tests
in the sand model.
48
1. Bury the distributor to the desired desired depth in the media.
Backfilling of the media is conducted as mentioned previously.
.. 2. Open the inflow value slowly to fill the model. Care is taken
not to fluidize to media during this operation. Shut valve off
when model is full.
3. Purge all air bubbles from the tubing and piezometric columns.
Check columns to see that gage "zero" is at the water level in
the model.
4. Open valve slowly to allow flow through the distributor. Read
and record pressures indicated by piezometric column. Also, (
volumentrically measure and record flow from the overflow weir.
5. Make observations as to channel development for the flow rate
and record.
Steps 4 and 5 are repeated as required to give a detailed ·
discription of how the pressures in the media vary with flow rate ..
Steps 1 through 5 are repeated for each distributor to note how the
pressures in the media vary with distributor configurations.
A typical test for a given distributor configuration includes
7 to 13 different flow rates. It is necessary to choose a small
incremental flow increase initially to determine incipient fluidi-
zation conditions. After fluidization fewer readings are required
since the pressure changes are small.
The maximum flow from each distributor, as limited by distri-
butor head loss and inflow capacity, is measured along with the maximum
49
fluidized channel shape. At this point it is assumed that inflow
capacity available to all distributors is the same; distributor con-
figuration alone then limits the maximum flow available for fluidiza-.. tion.
3.3 Experimental Data
Tests were run on each of five distributors for a fixed depth
of burial of 12.75 inches to the center of the distributor.
Data from distributors 1, 2, 3,· and 5 are tabulated in tables
1 through 4. The data in Tables 1 through 4 represent the pressure
in the media for varying flow rates as measured at each of the 13
pressure taps. It should be noted that pressure taps 12 and 13 were
added after the test of distributor No. 1, hence, these pressures are
absent from Table 1.
Table No. 5 summarizes the incipient fluidization
and maximum fluidized channel \vidths for each distributor. Also
listed are areas of maximum cross section of the fluidized channel.
3.4 Additional Experimental Tests
After conclusion of the fluidizing tests additional testing
was conducted to examine the seepage flow patterns generated by each
of four distributors.
50
.. •
Table 1 Experimental Data - Distributor No. 1
Flow Rate Pressure in Media (in H2o)
(cc/sec) 1 2 3 4 5 6 7 8 9 10 11 12 13 t:1
27.48 . 1.3 1.1 0.9 2.2 2.1 1.8 3.3 3.0 5.1 3.1 1.1 Ill rt Ill
39.32 1.7 1.7 1.4 3.3 3.2 2.5 4.9 4.6 7.2 4.5 1.7 H> 0 ~
56.47 2.3 2.4 2.1 5.0 4.9 3.7 7.3 7.0 11.2 6.9 2.5 '"tf 0 !-'• ::s
72.40 2.6 2.6 2.4 6.0 5.9 4. 2 8.9 8.7 14.0 8.7 3.0 rt en
t-'
87. 62'>'c 3.0 2.7 2.2 5.3 5.1 3.7 8.0 7.8 10.5 7.2 3.0 N
Ill ::s
126.67 3.1 3.1 2.6 5.8 5.8 4.2 8.6 8.5 11.1 8.0 3.0 p...
V1 t-' t-' w
255.71 . 3. 0 3.1 3.1 5.7 5.7 4.2 8.5 8.3 10.7 8.7 3.0 z 0 rt
J-:3 *Denotes Incipi~nt Fluidization Ill
"' ro ::s
.I
! .. } .
Table 2 Experimental Data - Distributor No. 2
Flow Rate Pressure in Media (in H20)
(cc/sec) 1 2 3 4 5 6 7 8 9 10 11 12 13 ' ;j
26.11 1.0 1.1 1.0 2.3 2.2 1.8 3.5 3.3 6.5 3.4 1.5 2.2 1.0 '
35.83 1.4 1.3 1.3 3.0 3.0 2.5 4.8 4.6 8.9 4.5 1.7 3.0 1.4
42.86 1.5 1.6 1.5 3.5 3.4 2.9 5.6 5.3 10.4 5.3 2.0 3.4 1.5 i
' (
50.00 1.8 1.9 1.8 4.1 4.0 3.5 6.8 6.4 12.6 6.3 2.4 4.1 1.9 1
64.14 2.2 2.3 2.2 5.2 5.0 4.5 8.8 8.1 15.1 8.0 3.0 5.3 2.2
! 88. 75~'( . 2.7 2.8 2.7 6.6 6.5 5.8 11.1 10.5 15.5 10.8 3.8 6.8 2.9
V1 j N
,! 88.75 3.1 3.0 2.7 6.5 6.4 5.4 9.6 9.4 11.8 9.0 3.5 5.9 2.6 .t .I .'). ' ~ 163.80 3.0 3.2 3.0 6.5 6.4 5.2 9.6 8.7 11.3 9.0 4.1 6.8 3.0 ·i
376.00 3.5 3.5 . 3.5 6.7 6.7 6.7 9.5 9.5 11.0 10.8 4.5 . 7. 6 3.1
*Denotes Incipient Fluidization
..
Table 3
Flow Rate
(cc/sec) 1 2 3 4
9.595 0.5 0.5 0.5 1.2
21.11 0.7 0.7 0.6 1.7
28.24 0.9 0.9 0.8 2.2
35.14 1.1 1.1 1.0 2.6
42.05 1.4 1,.4 1.2 3.3.
64.14 2.2 2.1 2.0 5.1 \.J1 w
74.19 2.6 2.5 2.4 6.2
81. 70-1( 3.0 2.9 2.7 6.8
94.74 3.1 3.1 2.6 6.3
380.95 3.6 3.6 3.6 6.1
*Denotes Incipient Fluidization
Experimental Data - Distributor No.
Pressure in Media (in H20)
5 6 7 8 9
1.0 0.8 2.0 1.7 2.6
1.6 1.3 2.8 2.5 4.0
2.1 1.7 3.8 3.4 5.3
2.5 2.1 4.7 4.2 6.8
3.1 2.6 6.1 5.5 8.8
4.8 4.1 9.3 8.4 12.5
5.9 4.5 10.8 10.1 13.5
6.5 4.7 11.4 10.9 13.6
6.2 4.3 9.0 9.1 10.5
6.1 6.1 9.0 9.1 10.5
3
10 11
1.6 0.7
2.4 1.0
3.1 1.3
3.9 1.5
5.0 1.9
7.7 2.8
9.1 3.3
9.6 3.5
8.2 3.2
8.5 5.0
12
1.2
1.8
2.3
2.8
3.5
5.2
6.0
6.3
5.5
9.0
13
' --~· I
0.5
0.8
1.0
1.3
1.6
2.4
2.7
2.8
2.4
4.0
.. · ..
Table 4 Experimental Data - Distributor No. 5
Flow Rate Pressure in Media (in H2o)
(cc/sec) 1 2 3 4 5 6 7 8 9 10 11 12 13
19.93 0.7 0.6 0.5 2.0 1.8 1.4 3.1 2.7 5.8 2.6 1.0 1.8 0.8
31.69 1:.1-. 1.0 0.8 3.1 2.8 2.2 4.9 4.5 7.8 4.0 1.5 2.8 1.2
36.54 1.3 1.2 1.0 3.4 3.1 2.5 5.6 5.1 8.7 4.5 1.8 3.1 1.3
41.54 1.5 1.4 1.2 3.9 . 3.5 2.9 6.3 5.8 10.2 5.1 2.0 3.5 1.5
51.08 1.8 1.7 1.4 5.0 4.5 3.8 8.1 7.5 12.9 6.6 2.5 4.4 1.9
62.75 2.3 2.1 1.9 6.2 5.7 4. 7 10.1 9.4 15.4 8.2 3.0 5.4 2.3 \J1 +"
75.20* 2.5 2.4 2.0 7.0 6.1 5.1 11.5 10.5 15.9 9.1 3.3 6.0 2.4
85.71 3.2 2.9 2.3 6.4 6.4 4.9 9.6 9.5 12.0 8.1 3.2 5.4 2.4
96.76 3.5 3.5 2.6 6.7 6.7 5.4 9.4 9.6 11.2 8.1 3.2 5.2 2.6
136.30 3.5 3.6 3.1 6.5 6.7 5.8 9.1 9.4 11.0 8.3 3.1 5.4 2.8
163.64 3.5 3.6 3.4 6.5 6.7 6.3 9.2 9.4 11.0 8.2 3.4 6.0 3.0
455.00 3.2 3.2 3.2 5.8 5.8 6.0 8.4 8.1 10.0 9.9 4.4 6.7 3.0
*Denotes Incipient Fluidization
.-
Table 5 Summary of Experimental Data
Incipient Maximum Maximum Fluidizing Haximurn Channel Channel
Distributor Flm-1 Rate Flow Rate Width Area Number (cc/sec) (cc/sec) (in) (in2
)
1 87.62 255.71 11.5 165.75
2 88.75 376.00 20.0 340.00
3 81.70 380.95 28.0 392.00
4 74.17 274.29 11.0 67.10
5 75.20 455.00 21.0 360.00
55
-~.-_,_._ ...... ,. ·-· •- --•,•-,-.,~--, .. --·-·:-• ..,--:......_-- ••• •• • • .• ·····;-r-.o:""'" .. '•~-····. ..... -- .... - ••.• - .... - ~~""!~";'_....- •• -;.·"·--:-:-... -.-~ ... ~- ··-:·---"' .... _ .......
- .·-
This investigation necessitated the use of visual techniques
to trace the flow lines. A series of dye tracing tests provided the
optimum method of studying the flow lines.
The dye chosen for the tests was crystalline Postassium Per
manganate, a strong oxidate leaving a purple trace. Because all
streamlines diverge essentially from a point at the outlet orifice,
it was physically impractical to attempt introducing dye at the dis
tributor and tracing flow lines to the surface. Instead dye crystals
were placed at distinct points on the surface of the media and water
was pumped out of the media via the distributor.
The results of the visual investigations are shown in Figures
19a and 19b. The photographs in the figures show distinct streamlines
resulting from dye tracers. From the photographs, the effect of
distributor configuration on seepage pattern is apparent. The most
promenent effects, as would be expected, are at the distributor sur-·
face. By noting the converging point of the streamlines in distribu
tors 1, 2, and 3, it is easy to determine the outlet orifice location
for the distributor. The streamlines for distributor 5, however, do
not appear to converge at two distinct orifice locations. Instead
the streamlines intersecting the distributor are spaced over the
region between outlet orifices.
A more subtle effect can be seen on the shape of the flow
regime for the different distributors. This effect is most apparent
when comparing the flow regimes of distributors No. 1 and No. 3. The
56
- -· --... ··-:·-::---: .. .-::.-·-::":·--~~~..,..~~- ... ~.-·.•<1' ··-· --,-- .,. ...... ~- ·-·-::·."' -- ....... ,.-.:c·-_,. •.. ·--.. ~-. :"'"'
Fig. 19a Experimental Results - Visual Studies
57
----~------------------------------------
Fig. 19b Experimental Results - Visual Studies
58
flow from distributor 1 is directed downward and it is seen that the
bottom flow line is distorted by the bottom of the model. The same
flow line for distributor 3 is clearly at a higher elevation in the
media. This indicates the same percentage of total flow passes
through a smaller area for distributor 3 than for distributor 1.
Hence, for the same flow rate, velocity through the media would be
higher for distributor 3 than for distributor 1.
The visual tests are useful in the determination of the actual
seepage patterns of the various distributors. Additionally, they give
subjective indications of eventual fluidized zones and are presented
for that purpose. Actual determination of the fluidizing effects on
the media depend on pressure variations in the media and must be
approached by analyzing pressure distributions and not seepage pat-
terns. ()
59
rr- .. -
4. DISCUSSION OF EXPERIMENTAL AND ANALYTICAL RESULTS
4.1 Comparison of Experimental and Analytical Flow Rate
In the study thus far, flow rates for the analytical and
experimental models have not been equated. The Muskat model given
by Equation 6, uses a discharge equal to the flow rate in an unbounded
flow situation of unit width. The numerical analysis and the experi-
mental model require the discharge to be equal to the flow rate in a
bounded flow situation. An adjustment must be made then to equate
the discharges if comparison is to be made between the models.
The first adjustment to be made is the reduction of unbounded
discharge to fit the bounded flow analysis. A simplifying assumption
is made at this point. It is assumed that the boundaries of the
flow situation coincide with a streamline in the unbounded flow
pattern and that only the percentage of the total flow contained
within the bounding streamlines is used in the bounded analysis.
By studying the Muskat stream functions it \vas determined
that the streamline that emerges at the intersection of the bed
surface and confining boundary gives the best physical representation
of both the numerical and experimental boundaries. This streamline
is shown as line AOB in Figure 7. ~
By manipulating the Muskat stream function,
60
·;:,• ~··• ,._ .,_ ~.- • - ~ •• <r-:".T". ,--,._ •' .~. • .__.. -:"'<,. ·~ .• ~··
i""f•v •• • ·-.:- • •
where 'i' flow for streamline
q = total flow
-1 el = tan x/(y-d)
-1 ez ;:::: tan x/(y+d)
d ;:::: depth of burial,
the percentage of the total flow bounded by the streamline is
determined. Using the coordinates of the point at the boundary-bed
surface intersection (point A, Figure 7), the bounded flow adjustment
then is made:
q = 0.32 q b m
where qb ;:::: flow through bounded situation
q = flow in Muskat equation. m
4.2 Variation of Pressure with Flow Rate
The variation of pressure with fluidizing flow rate is dis-
cussed in Section 1.2. In that discussion reference is made to
Figure 3a, representing the pressure variation for an ideal bed. A
series of similar graphs has been developed and is presented to depict
the pressure variations at various locations in the media.
In the course of this study pressure variations in the media
have been investigated both experimentally and theoretically.
Figures 20a through 20m show the results of experimental and mathema-
tic endeavors to determine pressure at a specific location in the
media as it varies ~vith flow rate. The line in the figures shows an
61
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1 1 r II . ' 1 1 -1-i+R=l=R-n-
' .. 0 so 100 150 300 350 400 450
Discharge (cc/sec)
N 6 n
I • .. ,
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c 1-i ro
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0 50 100 150 200 250 300 350 400 450
Discharge (cc/sec)
.
1:?· ·- -..
'±1.:: :: -:_ -- ~- r_ LY -[- L _J_ - -i - rf-f-f·+j±'~ -Itt+ --H++~~--4+~u+l-H- h
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Pressure (in H20)
Fig. 20d Experimental Pressure Variation at Point 4
65
,..... u (!) (/) ....._ u u ~
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I : ~ $ - . . + -f· I l - -- f* __ H±tt ~:LL~~ itrt· N :.:t:ti:t=lttt=:- - -~ ~+-+ , + ct: - --+ - - =r,~-- -n-~ · ~ I -+-W--; · =t- - r - f-t =1= T 1 1 + . + - Trt- ·rtt-H±J-1± -H++ +++++·it:l::t±ttJ±i±t£litl-l1tlitE~!++f-l-+-f.-l I I I I I I
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1 ~ I I i I I I I I I I I I I I 3 ]±-'-
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... ~ 50 100 150 200 250 300 350 400 450
Discharge (cc/sec)
N 0
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0 50 100 150 200 250 300 350 400 450
Discharge (cc/sec)
' .. . \
12 I I
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H- ' I I I i
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1 I
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~:::\~~- ; ~I+_+'· , lr-trl ~-t-:=F - + T -~ ~~-!- I _-~,~- f--+-1 I -- • • 2 f+h+ j~-1 H-t-1 I I I I I I I I ~~ ,-,I I ' ' - + 6 Dist • No • I I
2 j~~ f.--} _j._ H++ u... · D -- Dist. No. 3 1; = -- - £ r : 1+ r+H+ +f+f - 1 + ++-FFFI=·- - --- 0 -- Dist. No. 5 Hu....u-t-t_-H
: = - .It I ' rTH-t--FH-F 1 I ' ' 1 , - -- Theoretical H-++-
lkf=t,:::~tf ~it,-ti~$t •= ttf~ ff+ ,=~~~iilf-f i~;~r- :~tl~ttl~~!t-ft$~t~· 00 0 50 ·---Too 1SO 200 250 300 350 400 450
Discharge (cc/sec)
N 0 1-'•
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~ ti CD
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1 _j_ I t- W- H- ~~~ tiz!--! i : I r,-~-+ -1 I I I I I t+h---!-t-c-H-t--W-: h-;- ~ !--] + RT- H-- i-tt .r- -1\-l +1-!-1!-H-+·++++~• I H+ I -H--l-++++-1-+1++,1++-1+--l-;-1-- h-~!-1--h , .. · ··i I '-,-- I ' Q,-- ~J H- '-rH-H-++--H-l-l-+++-+-H-+-H-t-J_j-_j-jj~~---~~--· ,-i-r-rl-+t-H-l±J-j·r·H+ Tr_,_,_ . + 4-- r-Ai· -H--l I i+- H- r-~-~: j-. I r1TI:Tr:-r-~]T ' I I I I d 1-
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50 100 150 200 250 300 350 400 450
Discharge (cc/sec)
• ' 1 '
12 ! I I +:f± t-C::H- I t I !
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I-' 0 100 150 200 0
250 300 350 400 450
Discharge (cc/sec)
r h
I .j
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1-+11~/++h -t-J , +± ' , 1 1 '1 ,__._. _ _._
1 .....,+ 0 -- Dist. No. 3 'i-1 1.-\-:i~~-~~~--+-l-t-t-++++-t--HI--r , 1 +J-H
1-+.--+-1·1
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0 50 100 150 200 250 300 350 400 450
riischarge (cc/sec)
i :
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50 100 150 200 250 300 350 400 450
Discharge (cc/sec)
.J
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ltl'ltl· tl' tlltf,ll'--tfl-~--~. tfltrll,-1-'"1:/+..:lyl' 11-~-~[rrrr11
llll~-~~-~~~~ 111111111
111 1 LEGEND · ~f--1 ~~ .1~' ' ' ; ' ' ' ' ' ~ :: gi:;: :~: ~ f
lf_ .. -__ ~r.r~~~ +-!+ , , I 1 , 0 -- Dist. No. 3 -. ~]{ I 1-t-+-1--1-iH-+·1--1-I=R+ 1 , '· 0 -- Dist. No. 5 CE,11 J~)t 1-f 1
-- Theoretical E8+ t -~=1-H-+-1--1-1 !-J-J-J-JI---1-H ; ' I =l=j=!=tr,q .. -1 I ' I =~~~-i '~l: I I I -f- 'rf
0 IOO 150 z-oo 250 3'00 350 400 . --'~-·-· --- ..
450
Discharge (cc/sec)
! I ! I ; I r
k.·
• "
J
analytical pressure variation calculated by the Muskat equation
(Equation 6) in the unfluidized state and by Equation 14 in the
fluidized state. Experimental results are plotted as symbols in the
figures •
Agreement between the experimental pressures and the Muskat
equation are good in Figures 20a through 20i. Thus, the Muskat
equation can be used with confidence to predict pressures in the
media at points 1 through 9 (Figure 17) in the experimental model.
At points 10 and 11 (Figure 17) experimental pressures show a small
deviation from predicted pressures. This deviation can be seen in
Figures 20j and 20k. The remaining two points, 12 and 13 (Figure 17),
show even greater deviation of experimental pressures from the pre-
dieted values.
The experimental pressures follow the same general trend,
regardless of the location in the media. While in the unfluidized )
state pressures generally tise along the theoretical line predicted
by Equation 6. A peak in the pressures occurs immediately prior to
the onset of piping in the media. After piping occurs pressures I
exhibit a distinct reduction in magnitude.
With still increasing fluidizing flow rates, the pressure
behaves in one of two modes, depending on whether the location of
the point is in the fluidized or fixed portion of the media. In the
61 . nfluidized portion of the media pressures continue to increase with
flow rate. The rate of pressure increase however, is much less than
75
the rate before piping. This behavior of pressure is evident in
Figures 20k, 1, m, and would indicate that the preponderance of flow
after piping escapes through the fluidized zone in the media.
At locations in the fluidized zone, the pressures after piping
should reduce to the theoretical minimum fluidizing pressure calculated
by Equation 14. As shown in Figures 20d through 20j, the experimental
pressures fall below the theoretical line. The deviation between
calculated and experimental pressures becomes greater as 1) location
gets closer to the fluidizing pipe, and 2) fluidizing velocity
increases.
Maximum disagreement between experimental and theoretical
fluidization pressures occurs at point 9 (Figure 17) in the media
and at the maximum fluidizing flows for each distributor. As noted
in Figure 20i, point 9 was not fluidized during two of the tests,
distributors No. 2 and No. 3. The maximum deviation at point 9 then
is approximately (-)20%, occurring with distributor No. 5.
Error can occur by inaccurate computation of the specific f)'(
gravity o£ porosity of the sand or from inaccurate measurement of the
experimental pressures. Porosity, specific gravity, and depth all
directly affect computation of the theoretical fluidizing pressure.
Clearly, inaccuracies in any of these variables would result in a
noticeable error in comparing theoretical and experimental pressures.
An error.from theoretical calculations would be consistent at a given
depth and would increase linearly with depth. The errors detected in
in the figures, however, do not display a regular pattern.
76
I.
Another possible source of error lies in the experimental appa-
ratus itself~ At high fluidizing velocities the fluidized zone
becomes visibly turbulent. Because of small burrs on the inside of
the pressure tap or because of clogging by sand grains then it might
be expected that turbulent areas can develop at the pressure taps.
In this manner the pressure tap may detect not only the pressure due
to the weight of the ~oil but also that due to small velocity head~.
Because of insufficient time, the source of the error remains unde-
termined.
4.3 Conditions at Incipient Fluidization
Incipient fluidization conditions imply both a flow rate and
a resulting pressure in the media \vhen it first begins to fluidize.
Presented are incipient conditions evaluated both theoretically and
experimentally during the study.
As shown in Figures 20a-m, the experimental pressure at
incipient fluidization is much higher than that predicted by Equa-
tion 14. At locations near the distributor (Figures 20g-j) a dis-
tinct peak in the pressure occurs at incipient fluidization. The
peak is discussed in spouted bed U..terature (9) as a result of local
porosity changes in the media.
By examining changes in the media during experimentation, it
can be concluded that porosity changes do occur. Immediately prior
to fluidization a hole in the media develops at the outlet orifice.
Since the bed has not yet started to expand according to experimental
77
...
observations there must be some way to account for the hole. Intui-
tively, the media particles must be packed more closely than normal
at the walls of the hole. This change in the packing of the parti-
cles causes more resistance to the flow than expected and allows the
incipient fluidization pressure to rise to its peak value prior to
piping.
While the peak pressure is quite different than the theoretical
fluidizing pressure, the incipient fluidizing flow rate agrees well
with predicted values. To provide a location for comparison of
theoretical and experimental values, point 9 (Figure 17) was chosen.
Being the closest experimental location to the distributor, it was
the first location to fluidize during experimentation.
The minimum fluidizing flow rate as calculated by the Muskat
mathematical model is 60.5 cc/sec. The experimental minimQm fluidizipg
flow rates range from 74.11 cc/sec to 88.75 cc/sec. As discussed
earlier (Section 2.3), it is expected that the Muskat solution would -----yield a minimum fluidizing flow rate smaller than the numerical
models. At that time it was concluded that orifice locations account-
ed for small variations in the minimum fluidizing flow rate. The ------·---------prediction of minimum fluidizing flm11 rate by the Muskat equation was
as expected; slightly smaller than the experimental flow rates.
4.4 Distribution of Pressure in the Media
The theoretical model should be capable of simulating the
pressure distribution in the media if it is to satisfactorily predict
78
...
the.eventual fluidized zone. In Figures 2la through 2ld, experimental
pressure in the unfluidized media are shown with numerical model
results.
Since the numerical model contours are based on an arbitrary
pressure at the orifice, a slight adjustment of experimental pressures
is required for the co~parison. Point 9 (Figure 17) was chosen as
the "match point" and the experimental pressure at that point pro-
vided the amount of adjustment required. It should be noted that
the experimental pressures were chosen from data in Tables 1 through
4 such that minimum adjustment was required to fit the numerical
contours.
The experimental pressures, as adjusted, show good agreement
with the numerically calculated results. Exceptions to this occur
in two areas; i~mediately above the distributor and far from the
distributor near the boundary walls. The best fit bet\lleen experi-
mental and numerical results occurred with distributor No. 5. Dis-
tributors Nos. 1, 2, and 3 showed slightly less agreement between
experimental and numerical results.
The Muskat model was not used in the comparison of analytical
and experimental pressure distributions. Since the Muskat model
does not consider either the effects of orifice configurations or the
flow boundaries, it is doubtful that any comparison of experimental
and mathematical pressures would give valid results. In this respect
the Muskat model is best used as a ''standard solution" \llith which to
compare the different numerical models (Section 2.4).
~9
-· • ··- ··•:-.. - ··-.-- .. - .... -·-.- :- ~--- --.·rr-;-;-.';'"' ::--;;.-:-;o-_..--:-•-:-.·· .- ~- ... ,-.--.-.
1.
1.1+
1.0+ 1.9+
+ +
*Denotes Match Pressure
Fig. 2la Comparison of Experimental and Numerical Pressure Distributor 1
80
-·-. ' .~
.. i
•
1 1
1.1
2.6
4.0+
1.1 + 2.6 +
*Denotes Match Pressure
Fig. 2lb Co~parison of Experimental and Numerical Pressure Distributor 2
81"
-. ..
•
1.1
1. +
3.9 +
1.3+ 2.8 +
*Denotes Match Pressure
Fig. 2lc Comparison of E~perimental and Numerical Pressures Distributor 3
.....
•
-..
~----.....-t-----r----'-
1.0 3.1
2.8+
*Denotes Match Pressure
Fig. 2ld Comparison of Experimental and Numerical Pressures Distributor 5
83
4.5 Comparison of Fluid~d Channels
The eventual goal of this study is to be able to predict the
fluidized channel shape given the necessary variables. Throughout
the study it is assumed the following are known:
1. depth of burial of fluidizing pipe
2. outlet orifice configuration
3. hydraulic conductivity of sand
5. specific gravity of the sand.
Knowing the above parameters, fluidized channel shapes have been
calculated using the numerical model for each of the five distributor
configurations. Shown as dashed lines in Figure 22, the fluidized
channel shapes have been calculated using the maximum experimental
rate for each distributor •
... At first glance, there appears to be little or no agreement
bet~11een. the experimentally measured and the. numerically calculated
fluidized zones. However, examining the top widths of the fluidized . ' . ' I .
channels, the numerical model predicted accurately the experiment
top widths of zones resulting from distributors 4 and 5. The numer-
ical model also predicted, ~ith some error, the top width of zones
from distributors 1 and 2 •. There is no agreement, however, in the
top width prediction resulting from distributor No. 3.
It was obser\red experimentally that at high fluidizing flow
-· ... .; rates, a,jet of flow normal to the distributor surface emanated from
the outlet orifice. This jet enlarged the fluidized zone width by
84
-. •
. -
,/ --G------ - -o--;/--
-£)--- -o-
--/
/ ----0 __...[]---- -
Ex~rimental Zones
®- Dist. 1
/A- Dist. 2
m- Dist. 3
X- Dist. 4
@- Dist. 5
Numerical Zones
0- Dist. 1
b.- Dist. 2
0- Dist. 3
X - Dist. 4
0- Dist • 5
I
..........
Fig. 22 Comparison of Experimental and Numerical Fluidized Zones
85
eroding the boundary walls and was particularly prominent in tests
--------------~~----··--- --- . . . with distributors No. 2 and No. 3. The numerical model has no
------~~ mechanism to account for the existence of a jet emanating from the
- .... orifice and cal~ated fluidized zones resulting from distributors
2 and 3 show the zone shape as if no jet erosion occurs. Some ~
adjustment of the calculated fluidized zones then must be made if the
numerical model is to predict the channel shape with any confidence.
---~~-
..... '
86'
.. ..
. .
5. SUMMARY AND CONCLUSION
This study investigates one of the problems involved in
applying fluidization to sediment transport; that of predicting a
fluidized zone shape for a given configuration of fluidizing pipe •
The pressure distributions resulting from different pipe configura-
tions are investigated for the unfluidized state. Fluidized zones
are then developed theoretically from these pressure distributions
and compared to measured experimental fluidized zones.
Theoretically, a seepage analysis is used to predict pressures
in the media and ultimately predict a zone of fluidization. A purely
mathematical model derived by Muskat(ll) provides a "standard solu-
tion" with which to compare numerical and experimental results.
The Muskat model assumes an unbounded flow situation and considers
no effects due to the fluidizing pipe configuration.
A second analytical method is tailored for use on a digital
computer. Solving the Laplace Equation by a method of finite
elements, (lO) it provides a numerical solution to the pressure
distribution in a porous media. By variations of the finite element
grid the effects of the fluidizing pipe configuration .can be modeled.
The numerical method assumes the flow is bounded and can be altered to
fit the physical conditions •
Experimental pressures and fluidized zones are measured in a
sand model. Distributors representing five different fluidizing pipe
configurations are tested. The widest fluidized zone results from a
8.7
distributor having two sets of three orifices horizontally opposed.
The smallest fluidizing zone results from a distributor having three
orifices directed upward. - ....
The seepage analysis yields theoretical predictions of
pressures up to incipient fluidizing conditions. The Muskat model
provides a good approximation of the variation of pressure with flow
rate in the area of the fluidizing pipe. Except at incipient fluidi-
zation, the pressures measured experimentally agree with the Muskat
prediction. At incipient conditions, however, the experimental
pressures reach a peak substantially higher than predicted.
The minimum fluidizing flow rate is analyzed subjectively and
quantatively. As a result of the analytical work, the minimum fluidi-
zing flow rate can be shown to vary for different fluidizing pipe
configurations. Experimentally, minimum fluidizing flow rates varied
from 74.17 cc/sec to 88.75 cc/sec depending on the distributor. The
Muskat equation predicts a minimum fluidizing flow rate of 60.5 cc/sec.
Prediction of the fluidized zones by the Muskat and numerical
models was generally poor. With the exception of distributors 4 and
5, the experimental fluidized zone is altered by erosion. The erosion
caused by the development of jet flow at high fluidizing flow rates
can not be predicted by the numerical or Muskat model.
.. Of the configurations tested it is concluded that the dis-
tributor having two sets of orifices horizontally opposed should be
pursued as the most practical fluidizing pipe configuration. This
88
,.... ··--... .,.. ..... ~. ·.- ~
•.
. . ' . "
•
conclusion, based on observations in the sand model, is justified by
two reasons. First, the distributor does not fluidize the sand
beneath the pipe. When in operation, the fluidizing pipe using this
configuration will not sink, therefore, no elaborate anchoring system
should be required in field tests. Secondly, the above distributor
yields the widest fluidized zone for the given minimum discharge.
It is unfortunate that the theoretical analysis was unable to
predict the fluidized zone for distributor No. 3. The equations
presented provide a basis for analyzing the fluidization problem for
incipient conditions. Additional analysis is required, however, to
evaluate widening effects due to erosion before the theoretical
analysis can successfully predict the eventual fluidized zone shape •
89
. •
APPENDICES
90
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,., 48" -----V
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<: Ill rt !-'• 0 ;:l
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for 8 x 32 x~'' bolt 3" O.C.
·I J .-..-.-.L -.- - .,-- -. - ~ -- - - - ...,. -
Front ----
I
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28-7/16"
Note: (1) Box material is l" p lexiglass, base ma.terial is 3/4'' plywood. (2) All joints butted, glued, and bolted.
... . 2 •
Drill and Tap Hole 3/4" NTP
Side
. -If..._ ..... ... , . l • •
. ..
~ \ t-'• Drill and tap 1,11
2 NPT Cap ends with 1/811 plexiglass ()Q . :X:. I w
1-rj J-' I I I= ------·--..1...--t-'· 0..
/ t-'• N
I 1.0 t-'• w ;:J
()Q I ~ I i-J-'
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' rt 11 - --------t-'· cr c
~ ~ rt 0 11
P.V.C. pipe 1~11 diameter 311
.. . .;
I·
APPENDIX B: ANALYSIS OF SAND USED IN MODEL
A mechanical analysis of the medium sand used in the experi-
mental apparatus was performed by others. To determine properties
of the sand the following standard tests were performed:
1. Dry density test
2. Specific Gravity test
3. Porosity test
4. Sieve analysis
Tables B-1 and B-2 and Figure B-1 show the results of the
mechanical analysis.
94
/".
l
' • .i
- .
Table B-1 · Properties of Medium Sand
Dry Density
(1) Undisturbed -
(2) Compacted -
Specific Gravity
(1) SG -
Porosity
(1) Undisturbed
95.9 lb/ft3
98.3 lb/ft3
2.55
(a) porosity - .464
(b) void ratio - .867
(2) Compacted
(a) porosity - .430
(b) void ratio - .753
95
--':"- -- -_- ... -:· -:--:o -- ----
(' • ~1
"
' ..
Sieve
Number
20
30
40
50
60
80
100
200
Table B-2 Sieve Analysis
Weight (gm) Percent Grain
Gross Tare Net Retained Size (m..rn)
467.9 467.7 0.2 0.0 .841
533.8 508.4 25.4 5.1 .590
853.5 497.2 356.3 71.2 .420
619.6 510.0 109.6 21.9 .297
481.0 474.9 6.1 1.2 .250
438.4 435.7 2.7 0.5 .178
245.9 245.9 o.o 0.0 .149
466.6 466.4 0.2 0.0 .074
DlO - .370
D50 - .480
D90 - .53
Uniformity Coefficient
cu ; D60/Dl0 = 1.37
96
Sieve Number
200 . 140 100 80 60 50 40 30 20 10
Grain Size (mm)
...
Fig. B-1 Grain Size Distribution
- . 97
I I I I ~
r
I I I I
I .• I· ,
I "
I ., 1:
l r I
I
I
I
' ,
BIBLIOGRAPHY
1. Amiratharajah, A., "Expansion of Graded Sand Filters during Backwashing," M.S. Thesis, Ior.va State University, 1970 (unpublished).
2. Amiratharajah, A. and Cleasby, J. L., "Predicting Expansion of Filters during Backwashing," Journal of American \Vater \\forks Association, January 1972, pp. 52-59.
3. Cleasby, J. L., "Water Filtration through a Granular Media," Engineering Research Institute, Iowa State University, May 1969, pp. 33-40.
4. Hagyard, T., "Sand Fluidization Experiments at Westport," September 2, 1970 (unpublished).
5. Hagyard, T., Gilmore, I. A. and Mottram, W. D., "A Proposal to Remove Sand Bars by Fluidization," New Zealand Journal of Science, Vol. 12, December 1969, pp. 851-864.
6. Harr, M. E., "Groundwater and Seepage," McGraw-Hill Book Co., New York, 1962.
7. Inman, D. L. and Harris, R. W., "Crater-Sink transfer System," Proceedings, Twelfth Coastal Engineering Conference, September 1970, pp. 919-933.
8. Inman, D. L. and Bailard, J. A., "Analytical Model of Duct-Flow Fluidization," Symposium on Modeling Techniques, September 1975, Vol. 2, pp. 1402-1421.
9. Mathur, K. B. and Epstein, N., "Spouted Beds," Academic Press, New York, 1974, pp. 1-46.
10. Murray, W. A., "Analysis of Groundwater Flow," Lehigh University, 1973 (unpublished).
11. Muskat, M., "The Flow of Homogeneous Fluids through Porous Media," McGraw-Hill Books Co., New York, 1937, pp. 175-181.
. 12. "Fluidizing Sand to Open the Harbor Bar," New Scientist, March 19, 1970, pp. 556.
13. Othmer, D., "Fluidization," Reinhold Publishing Corporation, New York, 1956, pp. 1-20.
14. Volpicelli, G. et al., "Nonhomogenieties on Solid-Liquid Fluidization," Fluidized Bed Technology, American Institute of Chemical Engineering, 1966, pp. 42-50.
98.
,
f
'
·r
. !I'
15. Watters, G. Z., "Hydronamic Effects of Seepage on Bed Particles," Journal of the Hydraulics Division, ASCE, Vol. HY3, March 1971, pp. 421-439.
16. Wilson, C. R. and Mudie, J. D., Comments on "Removal of Sand Bars by Fluidization," Scripps Institution of Oceanography, Marine Physical Laboratory (unpublished).
17. Wilson, C. R. and Mudie, J. D., "Some Experiments on Fluidization as a Means of Sand Transport," Scripps Institution of Oceanography, Marine Physical Laboratory (unpublished).
99
r
VITA
,.. John T. Kelley was born in Philadelphia, Pennsylvania, on
August 10, 1950. He was raised in Willingboro, New Jersey, where
·he graduated from John F. Kennedy High School in 1968.
The author enrolled at Drexel University \V"b.ere he participated
in the U. S. Army R.O.T.C. program. He graduated in 1973 with a
Bachelor of Science degree in Civil Engineering and was commissioned
as an officer in the U. S. Army. In 1976 he entered a Master of
Science program in the Water Resources Division of the Civil Engi-
neering Department at Lehigh University. In his graduate program he
was supported by a U. S. Army Fellowship.
I 100