flow patterns in cylindrical settling tanks, part i
TRANSCRIPT
Louisiana State University Louisiana State University
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LSU Historical Dissertations and Theses Graduate School
1957
Flow Patterns in Cylindrical Settling Tanks, Part I: Application of Flow Patterns in Cylindrical Settling Tanks, Part I: Application of
Model Analysis Technique to Settling Tank Feed Well Design. Model Analysis Technique to Settling Tank Feed Well Design.
William Leon Barham Louisiana State University and Agricultural & Mechanical College
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Recommended Citation Recommended Citation Barham, William Leon, "Flow Patterns in Cylindrical Settling Tanks, Part I: Application of Model Analysis Technique to Settling Tank Feed Well Design." (1957). LSU Historical Dissertations and Theses. 190. https://digitalcommons.lsu.edu/gradschool_disstheses/190
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FLOW PATTERNS IN CYLINDRICAL SETTLING TANKS
PART I
APPLICATION OF MODEL ANALYSIS TECHNIQUE TO SETTLING TANK FEED WELL DESIGN
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and
Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of
Doctor of Philosophyin
The Department of Chemical Engineering
byWilliam Leon Barham
B.S., Alabama Polytechnic Institute, 19^7B.S., Alabama Polytechnic Institute, 1952M.S., Alabama Polytechnic Institute, 1951*
June, 1957
ACKNOWLEDGMENT
The author desires to express his sincere appreciation to Dr„ Arthur
G. KelleT for his advice and guidance in this investigation., He would
also like to thank Mr. E. E„ Snyder for hie aid in the construction of
much of the apparatus.
He wishes to thank Dr. Norman Bell for hie interest and many helpful
suggestions during the course of this work and Kaiser Aluminum and
Chemical Corporation for the grant which made this research possible.
ii
TABLE CF CONTENTS
Page No.ACKNOWLEDGMENT - - ................................. ±±
LIST OF TABLES - - - -- - ....................................... v
LIST CF FIGURES ----- --------------------------- - - - - vii
LIST CF ILLUSTRATIONS - - - - - - - - - - - - - - - - - - - - ix
ABSTRACT - - --------------------- - - --------------- x
INTRODUCTION........... 1
HYDRAULIC SIMILITUDE AND USE CF MODELS 6
Principle* of Similitude - - - - - - - - - - - - - - - - 6
Geometric - - - - - - - - - - - - - - - - - - - - 7
Kinematic - - - - - - - - - - - - - - - - - - - - - 7
Dynamic 8
Dimensionless Parameters - - - - - - - - - - - - - - - - 8
Viscous forces predominant - - - - - - - - - - - - - 9
Gravitational forces predominant - - - - - - - - - - 9
Surface tension forces predominant - - - - - - - - - 1 0
Application of Similitude and Scale-up to This Research - 10APPARATUS - -- - - - - - - -- - - - - ............. - - - - 12
Model Tray Section and Enclosing Vessel - - - - - - - - - 1 2
Model Feed Well Designs - - - - - - - - - - - - - - - - - 15
Constant Head Feed Tank, Rotameter, Valves and Piping - - 15
i n
Page No.Spheres - - - - - - - - - - - - - - - - - - - - - - - - 15
Sphere Ejector and Accessories - -- -- -- -- -- - ^ 2
Photographic Equipment and Accessories - - - - - - - - 22EXHRBffiNTAL PROCEDURE - -- -- -- -- -- -- -- -- -- 23
QUALITATIVE ANALYSIS OF MODEL FEED WELLS - -- -- -- -- - 26
Qualitative Results - - - - - - - - - - - - - - - - - - 26
Qualitative Discussion of Results - - - - - - - - - - - 26
QUANTITATIVE ANALYSIS OF MCDEL FEED WELLS - -- -- -- -- Lo
Quantitative Results - - - - - - - - - - - - - - - - - Lo
Quantitative Discussion of Results - - - - - - - - - - 50
CONCLUSIONS - -- -- -- -- - ............ 53
BIBLIOGRAPHY ......... 55
APPENDIX I. GLOSSARY QF TER* 6 AND ABBREVIATIONS ----- - 6l
APPENDIX II. THE NUMBER OF SPHERES PER QUADRANT OF MODELTRAY SECTION AND VARIANCE D A T A -------- 64
APPENDIX III SUMMARY QF CALCULATIONS - .................. 77
A. Scale-Down from Prototype to Model - - - - - - - - 77
£>. Application of Statistical Calculations to Data - 80VITA .......................................................84
iv
LIST OF TABLESTable No. Title
I Summary of Model Feed Well Design Data
II Comparison of Model Feed Well PerformanceRatings
III Comparison of Average Model Feed Well Variances
IV Summary of the Analysis of Variance for theEffect of Design and Flow Rate on the Distribution of Spheres per Quadrant for the Model Feed Wells
V Four Point Impingement (Figure 8 ):Summary of the Analysis of Variance for the Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
VI Open Cylinder with Conical Diffuser (Figure 1*0 :Summary of the Analysis of Variance for the Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
VII Open Cylinder (Figure 3)-Summary of the Analysis of Variance for the Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
VIII Two Point Impingement (Figure ' ( ) :Summary of the Analysis of Variance for the Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
DC Cpen Cylinder with Conical Diffuser (Figure 13):Summary of the Analysis of Variance for the Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
X Open Cylinder with Conical Diffuser (Figure 15):Summary of the Analysis of Variance for the Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
Page No.21
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1*8
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4 9
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V
Table No. XI
XII
XIII
XIV
XV
XVI
XVIIXVIII
XIX
XX
XXI
XXII
TitleOpen Cylinder (Figure 3 )*Distribution of Spheres per Quadrant of Model Tray SectionOpen Cylinder, with Conical Diffuser (Small Cone 10" Out of Cylinder, Figure 13): Distribution of Spheres per Quadrant of Model Tray SectionOpen Cylinder, with Conical Diffuser (Large Cone 1 t" Out of Cylinder, Figure lU-): Distribution of Spheres per Quadrant of Model Tray SectionOpen Cylinder, with Conical Diffuser (Large Cone 2 0" Out of Cylinder, Figure 15): Distribution of Spheres per Quadrant of Model Tray SectionTwo Point Implngement (Figure 7 )- Dlstribuiion of Spheres per Quadrant of Model Tray SectionFour Point impingement (Figure 8 )Distribution of Spheres per Quadrant of Model Tray SectionOpen Cylinder. Summary of Variance DataOpen Cvlinder, with Conical Diffuser (Small Cone 1 0 " Out of Cylinder):Summary of ''at lance DataOpen Cylinder, with Conical Dtffueer (Large Cone 1 5" Out of Cylinder)::Summary of Va.r lance DataOpen Cylinder, with Conical Diffuser (large Cone 2.0" Out of Cylinder)::Summary of Variance DataTwo Point Impingement Summary of Variance DataFour Point Impingement: Summary of VarianceData
Page No.
65
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69
70
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72
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7*+
75
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7
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LIST OF FIGURESTitle
Experimental Apparatus
Model Tray Section and Enclosing Vessel
Flow Diagram - Open Cylinder, Single Point Impingement
Flow Diagram - Open Cylinder with Conical Diffuser, Single Point Impingement (large Cone)
Flow Diagram - Open Cylinder with Conical Diffuser, Single Point Impingement (Small Cone)
Flow Diagram - L5° Conical Feed Well, Single Point Impingement
Flow Diagram - 90° Cylindrical Feed Well, Two Point Impingement
Flow Diagram - 90° Cylindrical Feed Well, Four Point ImpingementFlow Diagram - 30° Conical Feed Well, Tangential Feed
Flow Diagram - 60° Conical Feed Well, Tangential Feed
Flow Diagram - 90° Cylindrical Feed Well, Tangential Feed
Flow Diagram - 90° Cylindrical Feed Well, "I" Type Tangential Impingement Feed
Open Cylinder with Conical Diffuser (Small Cone: 1.0" Out of Cylinder)Open Cylinder with Conical Diffuser (Large Cone: 1.5" Out of Cylinder)
Vi I
Figure No. Title15 Open Cylinder with Conical Diffuser (Large Cone
2.0" Out of Cylinder)
viii
LIST CF ILLUSTRATIONS
Plate No. Title
I Open Cylinder, Single Point ImpingementII Open Cylinder with Conical Diffuser, Single
Point Impingement (Large Cone)III Open Cylinder with Conical Diffuser, Single
Point Impingement (Small Cone)IV L5 0 Conical Feed Well, Single Point
ImpingementV 90° Cylindrical Feed Well, Two Point
ImpingementVI 90° Cylindrical Feed Well, Four Point
ImpingementVII 30° Conical Feed Well, Tangential FeedVIII 60° Conical Feed Well, Tangential Feed
DC 90° Cylindrical Feed Well, Tangential FeedX 9 0° Cylindrical Feed Well, "Y" Type
Tangential Impingement Feed
ABSTRACT
Cylindrical settlers have found widespread industrial application
since the turn of the century. In this time, there have been few contributions to the basic understanding of flow patterns in these vessels. Consequently, there is a dearth of knowledge as to how these
patterns affect the distribution of suspended solids for their efficient separation in the settler.
One of the most important factors determining the flow pattern in
a settler is the feed well. These are the distributors of the incoming suspension, and thus are of utmost importance to efficient operation
of the unit.Because of the importance of the feed well and the difficulties
involved in full-scale studies, a simplified method for their evaluation has been devised. By the application of the principles of hydraulic similitude, scaled-down models of ten feed well designs were fabricated for study. The use of these models reduces the expense of experimen
tation to a small fraction of that required for investigation with full size equipment.
The effect of these designs on the flow distribution in a cylindrical vessel was determined by means of small plastic spheres injected into the stream leading to the model feed well. Notion pictures were taken of these spheres and subsequently analyzed to obtain
x
qualitative results. Quantitative results were derived by statistical
treatment of the distribution of spheres.Of the ten designs fabricated, five were considered to be unsatis
factory as a result of the motion picture analysis. The remaining
five designs were examined by statistical procedures. One of the three conical diffuser designs was found to give the most uniform diB-
tirbution of particles, however, a feed well of this type would not be
practical for multitray settlers. The two and four point inplngement
type feed wells were found to produce a very satisfactory flow pattern and particle distribution. These two designs are to be tested in
industrial size mud settlers.
xi
INTRODUCTION
One of the primary functions of a cylindrical settling tank is to separate the suspended solids from the media in which they are carried.To perform this task, the hasic steps involve; introduction of the suspension into the settling tank proper (usually by means of a centrally located feed well), allowing sufficient retention time for the desired removal of suspended matter, and finally withdrawal of the clarified liquid from one or more points around the periphery of the vessel and discharge of the settled solids from the bottom of the tank.
The design of such tanks has emphasized the need for proper retention time to achieve a suitable separation but with a minimum of study given to the hydraulic characteristics of the tank. Since the feed well is the distributor of the incoming suspension, it is a prime factor in determining the flow pattern and disturbing effects within the settler, and thus controls to a large degree the efficiency of the unit.
For the case of long rectangular tanks, the inlet design, resulting flow patterns, and sources of disturbances have been studied extensively. Camp (9 ) pointed out that the flow pattern in such a tank is quite stable because sufficient length Is provided to develop stability through the drag on the walls and the floor of the tank. This long flow path allows
1
vortices end eddies resulting from the turbulence created by the
entering liquid to dissipate, which gives a more uniform velocity dis
tribution. A mathematical expression has been developed by Dobbins (1.3)
showing the concentration changes during settling in an infinitely vide
stream This was applied (8) in determining the rate of particle removal
in a rectangular settling unit. Dobbins and Camp in their derivations
considered flow in two dimensions only, and assumed the mixing coefficient in the vertical direction to be constant. Hiese consider
ations made this problem amenable to mathematical solution. In most
industrial installations, the desirable long length of flow path obtained
In the rectangular settler is not feasible because of the necessity for
economy of space. For this reason many industrial units are cylindrical
in shape with tiers of multiple settling compartments.
A comprehensive survey of the literature indicates that the flow
patterns in cylindrical vessels have not been adequately studied.
Hubbe11 (29) has reported a qualitative technique for the evaluation
of the flow pattern resulting from the feed well design 1 Central up -
flow Type) employed in a single compartment cylindrical settling tank
Numerous authors (7, 9> 2 9 , 57) have stated that conventional cylindrical radial-flow settling tanks have unequal flow in the
different quadrants, and that there is a dearth of qualitative and more
particularly quantitative data for this type vessel. The principal
cause for unequal quadrant flow distribution may be attributed to the feed well, which attests to the need for research aimed at developing
3
both a qualitative and. quantitative approach to its performance
evaluation.
It would be extremely difficult to study several feed well designs
in an industrial size settler from economical and experimental stand
points. Therefore, it was necessary to apply the principles of
hydraulic similitude in the preparation of scaled-down models of a
cylindrical settler tray section and the feed wells to be evaluated.
The application of model results to a much larger unit (prototype)
has been studied extensively (6 , 25, 35* 39, ^3, 51, 52, 60). It has been found possible to predict the behavior of a full scale structure
from model data at a miniwnim cost of time and funds, if the fundamentals
of similitude are obeyed. Some examples of the successful use of model-
prototype scale-up techniques are hydraulic structures, rivers and
harbors, hydraulic machines, airplanes and ships
The successful techniques applied in the case of the rectangular
vessels are not satisfactory for cylindrical vessels. This can be
partially explained by tracing the general flow path of the liquid inI
the two vessels. In the case of the rectangular settling tank, the suspension enters one end and flows in a linear fashion to the effluent
section. In a cylindrical vessel with a central feed well, the flow
suddenly changes direction upon entering the main body of the liquid,
and finally assumes a slightly curved path to the draw-off. These
changes in flow direction in a cylindrical vessel result in the creation
of considerable discontinuity in the flow pattern because of the
1+
development of turbulence and eddy mixing. A detailed mathematical
expression explaining this type of flow would be complex, and difficult,
If not impossible, to solve. The methods employed for tracing flow patterns in rectangular units (1*3, 6 3 ), such as dyes, potassium
permanganate solution, and others, are not applicable here. The
vortices and eddy currents in the cylindrical tank cause a rapid
diffusion of the tracer, and makes it ineffective« For the same
reason, the employment of the special refractive properties of solutions (21*-, M+, 6 3 ) and other methods of indicating flow patterns
in fluids (l6 , 2 7 , 5 0, 5 9 ) would be unsatisfactoryHiese difficulties led to the ultimate development of a technique
for tracing flow patterns and distributions, which was utilized in this work for analyzing feed well design.
In addition to tracing the flow from the feed well, it was necessary to record the observed flow patterns for the purpose of
making a qualitative evaluation of the design.. The successful application of photographic techniques as a means of recording various
phenomena (1 7 , 3 , Ifl, 5 3 ) led to its selection for this Investigation.
Feed well design is a type of problem which is amenable to the
rigorous statistical analytical techniques (15> 2 3 ) developed In recent years. Utilization of these statistical procedures permitted a
quantitative treatment of the experimental data obtained.
During the course of this investigation, a model tray section
was built, and ten model feed wells were fabricated and tested. The
5
results of this work have been susmarized and evaluated in this
dissertation.
HYDRAULIC SIMILITUDE AND USE CF MODELS
It is often advisable to evaluate the performance of a small-scale replica (model) of the full-scale structure (prototype), prior to undertaking an extensive engineering project. These model studies are performed so that costly design errors may be avoided. In addition, these studies provide data, which will aid in the design and operation of the prototype.
In recent years, methods have been developed which, as a result of experiments carried out on a scale model, make it possible to predict the behavior of a full-scale unit or prototype. The principles, on which these procedures are founded, comprise the theory of hydraulic similitude. A review of this theory and the factors pertinent to it follows.Principles of Similitude (25, 35, 39, 3, 51)
The basis for the principles of similitude of fluid motion lies in the assumption that when a force is applied to a fluid mass, the resulting acceleration assumes a direction dependent upon the boundaries of the system and the physical properties of the fluid. The boundaries take Into account the shape, size, and location of all components of the system. Physical properties include the specific gravity, density, viscosity, surface tension, and compressibility.
Similitude between the model and prototype may take the following
6
7
three forms: geometric, kinematic, dynamic.
Geometric. Geometric similarity implies similarity of form. A model
is geometrically similar to the prototype if the ratios of all homologous
lengths in model and prototype are equal.
The quantities involved in geometric similitude are length, area,
and volume. The ratio of homologous lengths, areas and volumes
between model and prototype may be expressed as follows:
Kinematic. The term "kinematic similitude" signifies a similitude
of motions. It also introduces the concept of time as well as length.
Kinematic similarity may be defined by stating that the motions of two
systems are similar, if homologous particles lie at homologous points
at homologous times.
Some oi the kinematic quantities involved in model studies are
linear velocity, acceleration and flow rate. Linear velocity, V, is
expressed in terms of length per unit time:
P OA n * Lin s L p
(1)(2)
AP I*2■3
(3)
00
Where Tp is the time ratio.
Linear acceleration, a, is expressed as length per unit time
8
time squared:
(5)
aP Tr2
The units of flow rate are volume per unit time:
(6)
Dynamic. Two systems are said to be dynamically similar if homologous parts of the system experience similar net forces. This may be expressed by:
The force defined by the equation, F = Ma, has been called inertia force, and equation 8 defines the ratio of homologous inertia forces in model and prototype.Dimensionless Parameters
Several dimensionless parameters have been evolved to describe fluid motion or action. Which should be used depends on whether viscous, gravitational, or surface tension forces are predominant.These three dimensionless groups are Reynolds' number (viscous effect), Froude's number (gravitational effect), and Weber's number (surface
FpFm = F, (7)
Since force equalB mass, M, times acceleration, a,
FR «p ap(8)
tension effect). These three numbers, which have frequently been
the basis for scale-up between model and prototype, are discussed
below.
Viscous farces predo^ n*nt_ if viscous forces Influence the motion
or action of a fluid to so marked a degree that they can be considered
predominant, the force of viscosity as well as the force of inertia governs the motion of any particle. The dimensionless parameter with this force controlling is called Reynolds' number.
A general expression for this dimensionless number is:
Re ; ^ (9)v
where:Dg r Equivalent Diameter, ft.
V = Fluid velocity, ft./sec.2 /v = Kinematic voscosity, ft. /sec.
If the viscous and inertia forces control, similarity between the
model and prototype exists when the Reynolds' number Is the same for
both.
Gravitational forces predn*^ n«nt. Froude's number is the dimensionless parameter, which indicates that the force of inertia
and the force of gravity control the fluid motion. The general expression for this parameter is:
10
where:
V r fluid velocity, ft./sec.
L = linear dimension; for example, the equivalent diameter, ft.
g = gravitational constant, ft./sec.2.
Similarity between model and prototype is obtained, if the gravitational and inertia forces control, when the Froude numbers of both are the same.
Surface tension forceB predominate. Surface tension, denoted by
S', may affect fluid flow,. The dimensionless ratio which expresses the control of the fluid motion when surface tension and inertia forces control is referred to as Weber's number. It may be expressed thus:
W = V2L ^ (1 1)
where:
V = velocity of fluid, ft./sec.L r linear dimension, ft.
f* - density of fluid, lb./ft.^.
& s surface tension, lb./sec.2.
When the forces described above predominate, similarity exists if their Webers' numbers are equal.
Application of Similitude and Scale-up to This ResearchWith this review for a background, it is clear that the possi
bility of the application of experimental data obtained from models is feasible. However, in order for the model data to be significant, the
11
previously discussed principles of similitude must be considered. In
the case of this work, the Reynolds, Froude, and Weber numbers were considered as possible similitude parameters.
The flow in the feed wells of most industrial settlers is predominantly a function of the viscous and inertia forces. Thus, the use of Reynolds' number as the criterion for similitude between the flews in the model and the prototype was justified. Because the remaining two parameters do not contain a viscosity term, they were discarded.
For this investigation, it was possible to scale-down the linear dimensions of the prototype to the model by means of the geometrically similar ratios previously described. Next, the Reynolds number for a feed well of a typical industrial cylindrical settler (multitray) was obtained. It was then necessary to calculate the liquid flow rate, which would satisfy the requirement of similar Reynolds numbers. The flow in the model and prototype might now be considered similar. The experimental data obtained in this manner would be indicative of
conditions in the prototype.
APPARATUS
The apparatus employed In this investigation is shown in Figure 1.A detailed description of the component parts of this apparatus follows.
Model Tray Section and Enclosing Vessel (Figure 2)
A glass cylinder (l6.0" x 10.0") closed at one end was utilized as the model tray section of a settler. Four draw-off openings (spaced 9 0° apart) were placed around the periphery of the vessel and approximately two inches from the top. These draw-offs were connected to a set of four valves by means of copper tubing. This permitted the liquid (water) to be removed from the model tray section by peripheral overflow (all valves closed), or from one to four points.
To prevent optical distortion of the phenomena being photographed, the cylindrical model tray section was enclosed with a square parallelepiped glass vessel (20.0" x 20.0" x 8.0"). This vessel was constructed of four clear glass plate sections, which have the same optical properties as the cylinder. The intervening space between the two vessels of approximately two inches was filled with water, which, in
combination with the curved surface of the cylinder and flat surface of the square vessel, cancels the distortion resulting from the curved
surface alone.
12
13
Figure 1 Experimental Apparatus
<?> ®
Legend:Constant Head Tank2" Steel PipeReducing Tee (2" to 3/*+")3/4" Steel Pipe Globe Valve Union RotameterReducing Union (3/ '* to l/2")1/2" Steel PipeTeeRubber Tubing Connection
—<s><s> d>it,| ' ! ■ ..
©
Bose ClampParticle Dropping Funnr 1 90° EllInlet Pipe (l/2 ")Mud Dovncoaner Model Feed Well Peripheral Overflow Scaled-Down Model Tray Section (Glass: 16" x lO")Square Parelielopiped Enclosing Vessel (2 0 " x 2 0 " x 8 ")
Ik
Figure 2Model Tray Section and Enclosing Vessel
ex.
I________
._L to
ELEVATION
PLANLegend:(§) Scaled-Down Model Tray Section (l6 " x 10")
Square Parallelopiped Enclosing Vessel (20" x 20" x 8 ") Globe Valve Plastic Wedges Liquid Draw-offs
15
Model Feed Well Designs (Plates I - X and Figures 3 - 1 2 )The model feed veils employed were constructed from clear meth
acrylate plastic. To determine the overall dimensions for the Individual designs, the ratio of the diameters of an experimental feed veil (model) and of an industrial settler feed well (prototype) was used for obtaining
the scale-down factor necessary to have geometric similitude between the model and prototype. This ratio was determined to be 0.026, If this ratio had been strictly adhered to, the depth of the feed well would have been too shallow (0 . 2 inches) for a realistic experimental study. Therefore, the vertical dimension was arbitrarily distorted (by a factor of five) much the same as is done in studies of rivers and channels.
A total of ten feed veil, designs were fabricated, which, in conjunction with appropriate dimensions, are enumerated In Table I, Constant Head Feed Tank, Rotameter, Valves and Piping
The constant head tank was constructed of steel, and the inside lined with T ygon primer to minimize rusting. Its dimensions are 3.0' x 2.0' x 2.0', which provides a capacity of ninety-four gallons. To maintain a constant level, a float control valve was used.
A Fisher-Porter rotameter, size 2F, was used to set the liquid flow rate to the model tray section. It has a range of 0.0 to 6.0 gpm.
The location of these two items and the necessary valves and piping are shown in Figure 1.Spheres
A total of forty polystyrene plastic spheres (specific gravity:1.03 1 0.01, diameter: 5-0 mn ♦_ O.l) were employed as tracers. Hie
Plate I, Open Cylinder, Single Point l^plngment
Plate II, 0£en Cylinder with Conical Dtffueer, Single Point linplngeeent (large Cone)
17
Plate 3V, U50 Conical Peed Well,Single Point Impingement
18
Plate V, 90° Cylindrical Feed Well, Two Point Impingement
» * '
- a*
Plate VI, 90° Cylindrical Feed Well,Four Point Impingement
Plate VII, 30° Conical Feed Well, Tangential Feed
T
INCHES
Plate VIII, 60° Conical Feed Well, Tangential Feed
Plate IX, 90° Cylindrical Peed Well,Tangential Peed
Plate X, 90° Cylindrical Feed Well, "3f" Type Tangential Impingement Feed
Table I
Sumary o f Model Feed Well Design Data
f ................................... . T-------“ .----- n_. ■ - __z_ -n_-----:----- . ------
J Diameter of MudiFeed Weil . . . . . . I :hth ; Dovncomer* **
i Design 'i ‘ f . _ ■* . . ■ _ j ■, L . - 1 , ; (O.D. In odes } *
| Open Cylinoer 1 ; - - ; * ...
Open iyU nder v ita jConical D iffu ser 1 - * -
Open Cylinder v ita1i
C onical D iffu ser 3 j - .... .... .. _ _. r
as0 C cni;al - | : •• • '• - - 'j
IVo Point JImpingement } > - ....
Four Point ‘Impingement j " - _ , ' j
30u Conical 7 i _ I _ , Jc ' , '• » 4 -
j 60° Conical ^ . 0
'
P0° C y lin d rica l . '-j •.
"Y" TypeImpingement ’ Q ' 1 : •
i _ . i _ ! . . . . _ .1 . ....... 1 - i .. i
S . P . I . : S in g le Point Impingement *1 i a:::-, t . r Ill nor Axisii
M. P. I.: M ultiple Point Impingement *"*1. .net •:* !1 r w__:
22diameter of the Bpheree was determined by a Bausch & Lcanb Measuring Magnifier (Catalog No. 81-3^-35)* The terminal velocity of the spheres was obtained by noting the time required for each sphere to fall a given
distance in a cylinder of water. This velocity, in conjunction with the sphere diameter and the physical properties of the water, was used to calculate the specific gravity of the spheres by Newton's law (5)-
Sphere Ejector and AccessoriesA conical dropping funnel was connected to a tee section in the
pipe line carrying the liquid feed to the model feed well by means of rubber and glass tubing. Tliis assembly was utilized to eject the plastic spheres into the feed stream. (See Figure 1 for location).
Photographic Equipment and AccessoriesTwo 16 mm Bell and Howell motion picture cameras (2 0 0-TA Auto-
Master, magazine load) were used to record continuously the phenomena
within the model tray section,A 35 mm Praktica FX camera (Tessar 3 5 pre-set lens) was used to
make all "still" photographsThe accessories, which were required in addition to the cameras
to complete the list of photographic equipment, were:a. A Grover Colortran Converterb. A GE exposure meterc. A 16 mm motion picture projectord. Film editing and splicing equipmente . Two tripodsf. Necessary lighting equipment.
EXPERIMENTAL PROCEDURE
The exact procedure employed in investigating each model feed veil is outlined below. To clarify the presentation of this procedure , the letters enclosed in parentheses refer to the schematic drawing of the assembled apparatus as shown in Figure 1 .
Ihe model settler tray section (S) v e b placed in the enclosing vessel (T), and filled with water. The tray section was then adjusted until a uniform peripheral overflow was obtained. (This was done to prevent points of unequal overflow having an effect on the particle distribution from the feed well). The enclosing vessel (T) was then filled with water so that the phenomena within the tray section (s) would not be distorted when photographed. A model feed well (Q) was then securely attached to the inlet pipe (0 ) so that its mid-point was in the center of the tray section (S) and carefully leveled
After this phase bad been completed, the two 16 ran Bell and Howell motion picture cameras were mounted above and to the side of the model assembly. These cameras were employed to record the flow distribution patterns and bottom scour effects.
The rubber tubing connection (K) between the particle dropping funnel (M) and the feed pipe (i) was closed by means of the clamp (l), and the funnel filled with water. The forty polystyrene spheres were then placed in the dropping funnel (M), and aligned in the stem
23
2h
of the funnel to form a column of spheres. The apparatus was now pre
pared to begin a series of experimental determinations.The liquid flow rate from the constant head tank (A) to the model
tray section (S) was established by throttling the globe valve (E)
until the desired reading was obtained on the rotameter (G), Once the flow to the tray section had been stabilized, the plastic spheres were injected into the feed stream by opening clamp (l ) , From the point of injection, the spheres were transported by the liquid into feed well (Q), and then into the tray section proper,
In order to analyze the individual model designs, it was first necessary to evaluate the turbulence and/or bottom scour effects in
the tray section proper. If these effects were excessive, the spheres would be continuously in turbulent motion. Therefore, all
model feed wells were initially investigated at flow rates of 1 , 3 > and
5 gallons per minute visually and photographically to evaluate the
magnitude of both turbulence and bottom scour , In those cases where
these effects were excessive, the designs were not considered for
further study.After this qualitative study, the satisfactory designs were sub
jected to a quantitative evaluation. The procedure employed was as
follows:1. The flow was established, and the spheres injected as
previously described.2 . Once the spheres had exited from the feed well, the flow
25
was stopped, and the spheres allowed to trace the flow pattern or distribution of the liquid from the feed well until they came to rest on the bottom of the tray section.
3 . The number of spheres per quadrant was then noted. (Tofacilitate counting the spheres, a grid, which was subdivided into four quadrants, was placed beneath the enclosing vessel and model tray section prior to beginning a series of determinations)
k, A total of 25 runs at a given flow rate was made.
This procedure was repeated for flow rates of 1, 2, 3, 4, and 5
gallons per minute. The particle count obtained in item (3 ) was utilized as the basic data for evaluating the performance of the design statistically.
QUALITATIVE ANALYSIS OF MODEL FEED WELLS
The ten model feed wells were first evaluated qualitatively. This
evaluation consisted of reviewing motion pictures of each of the model
designs taken from the horizontal and vertical positions. From these pictures, the distribution of spheres (plan view), turbulence in the
tray section (elevation view), and bottom scour effects (elevation view)
were determined In the following discussion, the results of the motion
picture analysis are illustrated in the form of schematic flow diagrams,
and the relative performance of each design 1s evaluated with respect to several flow criteria.
Qualitative Results
The motion picture studies of each feed well design have been
combined into a continuous film A copy of this film, which requires
approximately thirty-five minutes to show,, is submitted with this
dissertation. Schematic flow diagrams of each design, as shown in
Figures 3 through 12, were developed from these motion pictures. The
arrows on these diagrams indicate the flow patterns traced by the
spherical particles. A tabulation of the ratings of each performance
criteria is listed at the bottom of each figure and in Table II. Qualitative Discussion of Results
The Individual model feed well designs were rated with respect to
the following flow characteristics: .l) utilization of tray section area,
£6
27
Figure 3Flow Diagram - Open Cylinder, Single Point Impingement
Flow CharacteristicsUtilization of Tray Section Area : PoorTurbulence in Tray Section Proper : HighBottom Scour : HighFeed Distribution From Well : PoorFlow Direction in Tray Section Proper: Stream directed toward tray section
wall behind the feed inlet.
28
Figure 4Flow Diagram - Open Cylinder with
Conical Diffuser, Single PointImpingement (Large Cone)
/
Flow CharacteristicsUtilization of Tray Section Area Turbulence in Tray Section Proper Bottom ScourFeed Distribution from WellFlow Direction in Tray Section Proper
GoodModerate to HighModerateGoodConical diffuser disperses flow all directions with a slightly higher flow Into two quadrants directly behind feed inlet.
m
Figure SFlow Diagram - Open Cylinder with
Conical Diffuser, Single PointImpingement (Small Cone)
— > n ^ J 51' >
o
1
'i/■* i *> /■ '' \.
/ L \
Flow CharacteristicsUtilization of Tray Section Area Turbulence in Tray Section Proper Bottom ScourFeed Distribution from WellFlow Direction In Tray Section Proper
GoodModerateModerateGoodConical dLffuser disperses flow all directions with a slightly higher flow into two quadrants directly behind feed inlet.
30
Figure oFlow Diagram - ^5° Conical Feed Well,
Single Point Impingement
Flow CharacteristicsUtilization of Tray Section Area : Poor'Turbulence in Tray Section Proper : HighDotton Scour : ExcessiveFeed Distribution iron Weil : PoorFlow Direction in Tray Section Proper: Strikes tray section, and curls
back.
31
Figure 7Flow Diagram - 90° Cylindrical Feed Well,
Two Point Impingement
Flov CharacterietlcsUtilization of Tray Section Area Turbulence in Tray 8ection Proper Bottom ScourFeed Distribution from WellFlow Direction in Tray Section Proper
Fair to Good Moderate Moderate Fair to GoodStream dispersed in two directions with two areas (90° from flow) quiescent-
Figure 0Flow Diagram - 90° Cylindrical Feed Well,
Four Point Impingement
iHtdri!
C
/
Flov CharacteristicsUtilization of Tray Bectlon Area Turbulence In Tray Section Proper Bottom ScourFeed Distribution from WellFlov Direction in Tray Section Proper
Very GoodLowLowVery GoodStream directed toward wall from *+ points with little resulting curl back.
33
Figure 9Flow Diagram - 30° Conical Feed Well,
Tangential Feed
Flow CharacteristicsUtilization of Tray Section Area : PoorTurbulence in Tray Section Proper HighBottom Scour : KxcessiveFeed Distribution from Well : PoorFlow Direction in Tray Section Proper: Strikes tray section wail, and
curls back.
3J+
Figure 10Flow Diagram - 60° Conical Feed. Well,
Tangential Feed
Flow CharacteristicsUtilization of Tray Section Area : GoodTurbulence in Tray Section Proper HighBottom Scour : ExcessiveFeed Distribution from Well : FairFlow Direction in Tray Section Proper: Rotational
Figure 11Flow Diagram - 9O0 Cylindrical Feed Well,
Tangential F-ed
5
Flow CharacteristicsUtilization of Tray Section Area : GoodTurbulence in Tray Section Proper : HighBottom Scour : ExcessiveFeed Distribution from Well : GoodFlow Direction in Tray Section Proper: Rotational
n1 .
36
Figure 12Flow Diagram - 90° Cylindrical Feed Well,
”Y" Type Tangential Impingement Feed
Flow CharacteristicsUtilization of Tray Section Area : PoorTurbulence in Tray Section Proper HighBottom Scour ExcessiveFeed Distribution from Well PoorFlow direction in Tray Section Proper: Rotational, strikes tray section
wall and curls back.
37
2) turbulence in tray section proper, 3 ) bottom scour, and k ) feed dis
tribution from veil. A comparison of the ratings for each design is
presented in Table II. In this tabulation, the order of increasing performance for each of the above flow characteristics is given from left to right.
It is apparent from Table II that those designs, which were fed tangentially, imparted a rotational motion to the entire mass of the liquid in the tray section. Because of this motion, high turbulence
and excessive bottom scour were obtained, which prevented any of the spheres from settling until the rotational energy had been dissipated. Thus, it was not possible for the spheres to assume a true "final position" (based on their point of exit from the feed well), which would be necessary for a valid statistical evaluation. For this reason the following designs were rejected, and not considered In the quantitative evaluation:
1) 30° and 6 0° conical feed wells (Figures 9 and 10),2) 90° cylindrical feed well (Figure 11), and
3 ) 90° cylindrical feed well, ’Y" type Impingement (Figure 12).The ^5° conical feed well (single point impingement, Figure 6)
caused the fluid to short-circuit, and strike the tray section wall
directly opposite the inlet. After hitting the wall, the flow followed the geometry of the tray section, and created points of turbulence on the bottom. T3iis prevented settling of the spheres, and a statistical analysis could not be made. Therefore, this feed well was also discarded, and not considered for further study.
: 1 VTS1?r - .- <
$M8
.1 :«
d
39
The flow characteristics for the remaining feed well designs were
sufficiently good to allow the spheres to assume a true "final
position". These designs (Figures 3> 7 &nd 8) were analyzed
quantitatively by rigorous statistical techniques. The results of this analysis are discussed in the following chapter.
QUANTITATIVE ANALYSIS OF MODEL FEED WELLS
The five model feed wells, which were found to be satisfactory from the qualitative analysis, were evaluated quantitatively. In addition to these five basic designs, a modification of the feed well shown in Figure U (increasing the distance between the bottom of the cylinder and the bottom of the conical diffuser from 1.5" to 2.0") was included in this phase of the investigation. Of the six designs, three employed conical diffusers, and these are illustrated in Figures 13, 1^ and 15.
This analysis was based upon the distribution of the forty spheres in the model tray section, after they had assumed their true "final position". Ilie number of spheres per quadrant at the different flow rates for each design is tabulated in Appendix II. These data were treated statistically to determine the influence of the variables, design and flow rate, on the distribution of spheres in the model tray
section for each feed well design Quantitative Results
The flow distribution characteristics of the model designs were evaluated by comparing them with an ideal feed well. Since a total of forty spheres were admitted to the tray section for each replication, an ideal feed well may be defined as one in which the number of spheres found per quadrant would be ten. Therefore, the value of ten spheres per quadrant was taken as the mean, and the actual count of spheres for
UO
hi
Figure 13Open Cylinder with Conical Diffuser (Small Cone: 1.0" Out of Cylinder)
Scale: 3/4" . l"
Figure 14Open Cylinder with Conical Diffuser (large Cone: 1.5" Out of Cylinder)
Scale: 3/4" = 1"
Figure 15Open Cylinder with Conical Diffuser (l&rge Cone: 2.0" Out of Cylinder)
Scale: 3/^" ■ 1"
1+1+
each replication was converted to a variance, - 10)■» (Xg - 10)2 4 (X^ - 10 )2 where X1, X£ X^ are the count of
spheres per quadrant of model tray section for each replication). The
variance data for each design is tabulated in Appendix II, and summarized
in Table III.
Table III11
Camparison of Average Model Feed Well Variances i
i1Figure Flow Rate (GPM) 1tDesign No. 1.0 2.0 3.0 i I+.O 5.0 ,
Open Cylinder with Conical Diffuser (Large Cone: 2.0" Out of Cylinder) 15 52 38 38 35 39 j
Two Point Impingement 7 33 32 57 1+9 1+1+
Open Cylinder with Conical Diffuser (Large Cone: 1-5" Out of Cylinder) 1 lU 76 M+ 1+7 l+o 1+0 ;
Open Cylinder with Conical Diffuser (Small Cone: 1.0" Out of Cylinder) i 13 58 56 ^3 6 2 1+7 11
Pour Point Impingement 8 6 9 6 5 71 65 ; 6 8
Open Cylinder 3 2ll+ 2 7 2j
J _______ ____
1 9 6 238
j _____
12i+0 11 Ii 1 !
In order to determine the effect of feed well design and flow rate
on the distribution of spheres, a statistical technique was utilized. A two-way analysis of variance was used to examine all of the data obtained
for the six designs and the five flow rates as given in Tables XVII
through XXII in Appendix II. It was found that significant differences
1+1+
each replication vas converted to a variance, N ^ 2 . (N^2 _ - 10 )2* (Xg - 10)2 — - + (Xk - 10)2 where X-, , X2 --- X^ are the count of
spheres per quadrant of model tray section for each replication). The
variance data for each design is tabulated in Appendix IX, and sunmarized
in Table III.
Table III■---'i
Comparison of Average Model Feed Well VariancesI
Figure Flcrw Rate (GFM)Design No. 1.0 2.0 ! 3 - 0 : l+.o
15-0
Open Cylinder with Conical Diffuser (Large Cone: 2.0" Out of Cylinder) 15 52 38
1
38 35 39Two Point Impingement 7 33 32 57 1+9 l+l+
Open Cylinder with Conical Diffuser (large Cone: 1*5" Out of Cylinder) li+ ! 76 l+i+ *7 1+0 1+0
Open Cylinder with Conical Diffuser (Small Cone: 1.0" Out of Cylinder) 13 58 56 « 6 2 1+7
Four Point Impingement 6 ! 691 65 71 65 68
Open Cylinder ! 3 2114- 2 7 2
1
1 9 6 2 3 8
j. ■
, 2l+0!1!
In order to determine the effect of feed well design and flow rate
on the distribution of spheres, a statistical technique was utilized. A
two-way analysis of variance was used to examine all of the data obtained
for the six designs and the five flow rates as given in Ihbles XVII
through XXII in Appendix II. It was found that significant differences
^5
existed between the several designs. Flow rate was found to have no
significant effect on sphere distribution. This may be seen by referring
to Table IV.
Table IVSummary of the Analysis of Variance for the
Effect of Design and Flow Rate on the Distributionof Spheres per Quadrant for the Model Feed
(Method of Calculation, Reference 1 5 )Wells
Source of VariationSum of
SquaresDegrees of Freedom
MeanSquare F
Between Columns (Design) 3A75,858 5 695,172 2 2 2
Between Rows (Flow Rate) 8 ,1 2 6 k 2 ,0 3 2 O .6 5
Residual 2,320,951 7I+O 3,136
Total 5,80^,935 71+9
For 5 and 7* 0 degrees of freedom, an F of 2.23 is significant at the 51 level of confidence and one of 3 - 0 5 at the 1% level.
For 1*- and jUO degrees of freedom, an F of 2 . 3 8 is significant at the 5 t level of confidence and one ox 3 * 3 5 at the 1% level.
46In addition, the data obtained for the individual feed well designs
was subjected to a two-way analysis of variance to determine the effect
of flow rate on the distribution of spheres per quadrant of the model
tray Bection. It was found that for the designs shown in Figures 8 and
14 flow rate had a significant effect on the distribution of spheres.
This may be seen in Tables V and VI.
Table V
Four Point Impinge menx (Figure 8):Summary of the Analysis of Variance for the
Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
(Method of Calculation, Reference 15)
Sum of Degrees of Mean Source of Variation Squares Freedom Square F
Between Columns (Flow Rate) 23,778 4 5 ,9 4 5 3 .9 6 9Between Rows (Replications) 64,391 24 2 ,6 8 3 1 .7 8 0
Residual 143,824 96 1 ,4 9 8
Total 231,993 124
For 4 and 96 degrees of freedom, an F of 2.47 is significant at | the 5^ level of confidence and one of 3*50 at the 1 $ level.
For 24 and 96 degrees of freedom, an I of l.fO is :a:.(at the 5^ level of confidence and one of 2.33 at the 1$ level.
kl
Table VIOpen Cylinder with Conical Diffuser (Figure l4): Sumnary of the Analysis of Variance for the
Effect of Flow Bate on the Distribution of Spheres per Quadrant of Model Tray Section
(Method of Calculation, Reference 15)
Source of VariationSum of Squares
Degrees of Freedom
MeanSquare F
Between Columns (Flow Rate) 23,516 k 5,879 5-130Between Rows (Replications) 29,679 2k 1,237 1.079Residual 109,99^ 96 l,li+6
Total 163,189 12k
For and 96 degrees of freedom, an F of 2.^7 la significant at the 5lt level of confidence and one of 3*5° at the l t level.
For 2k and 96 degrees of freedom, an F of 1.60 is significant at the 5^ level of confidence and one of 2.33 the 1$ level.
For the designs as shown In Figures 3, 7, 13 &nd 15, flow rate was found to have no effect on the distribution of spheres as shown in Tables VII through X.
1*8
Table VII
Open Cylinder (Figure 3)*Summary of the Analysis of Variance for the
Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
(Method of Calculation, Reference 15)
Sum of Degrees of Mean Source of Variation_______ Squares____ Freedom Square_____ F
Between Columns (Flow Rate) 83,3^ 1* 20,835 I .8 9 6Between Rows (Replications) 277,556 21* 11,565 1.053Residual 1,05**,786 96 10,987
Total 1 ,1*1 5 ,6 8 2 121*
For k and 96 degrees of freedom, an F of 2,1*7 is significant atthe 5^ level of confidence and one of 3 50 at the 1$ level.
For 2l* and 96 degrees of freedom, an F of 1.80 is significant atthe 5^ level of confidence and one of 2 .3 3 at the 1^6 level.
Table VIIITwo Point Impingement (Figure 7)-
Summary of the Analysis of Variance for the Effect of Flaw Rate on the Distribution of Spheres
per Quadrant of Model Tray Section (Method of Calculation, Reference 15)
Sum of Degrees of Mean Source of Variation_______ Squares____ Freedom Square_____ F____
Between Columns (Flow Rate) 11,59^ 1+ 2 ,8 9 9 2.000Between Rowb (Replications) 37,305 2k 1,55^ 1.072Residual 139,201 96 1,1*50
Total 188,100 121*
For 1* and 96 degrees of freedom, an F of 2.1*7 is significant atthe 3% level of confidence and one of 3 - 5 0 at the 1$ level.
For 2l+ and 96 degrees of freedom, an F of 1.80 is significant atthe 5 i* level of confidence and one of 2 .3 3 at the lt level.
b9
Table IX
Open Cylinder with Conical Diffuser (Figure 13): Sunmary of the Analysis of Variance for the
Effect of Flow Rate on the Distribution of Spheres per Quadrant of Model Tray Section
(Method of Calculation, Reference 15)
Source of VariationSum of Squares
Degrees of Freedom
MeanSquare
Between Columns (Flow Rate) 6 ,6 1 9 Between Rows (Replications) 33*096 Residual 1 5 8 ,6 1 6
Total 1 9 8,331
k2k96
12 k
1,6551,3791 ,6 5 2
1.0020.835
For 4 and 96 degrees of freedom, an F of 2.^7 is significant at the 5# level of confidence and one of 3 - 5 0 at the 1$ level.
For 2k and 96 degrees of freedom, an F of 1.80 is significant at the 5% level of confidence and one of 2 -3 3 the 1$ level.
-------------------------------------- j
Table X |Open Cylinder with Conical Diffuser (Figure 15)- !Sunmary of the Analysis of Variance for the i
Effect of Flow Rate on the Distribution of Spheres jper Quadrant of Model Tray Section i
(Method of Calculation, Reference 15)
Source of VariationSum of Squares
Degrees of Freedom
MeanSquare F
Between Columns (Flow Rate) 1+,2 M* 1* 1 ,0 6 1 1 .0 0 6Between Rows (Replications) 25,935 2k 1 ,0 8 1 1 .0 2 5Residual 101,283 96 1,055
Total 1 3 1,U6 2 12*1
For U and 96 degrees of freedom, an F of 2.^7 is significant at the yfa level of confidence and one of 3 - 5 0 at the l t level.
For 2k and 96 degrees of freedom, an F of 1.80 is significant at the 3% level of confidence and one of 2 .3 3 the 1£ level.
50
A detailed sample calculation illustrating each of the above
statistical procedures may be found in Appendix III.
Quantitative Discussion of Results
As may be seen in Table III, the open cylinder design (Figure 3) was
found to have the most uneven distribution of spheres in the four quadrants
of the model tray section. This can be explained by referring to the flow
distribution diagram shewn in Figure 3- The two quadrants immediately behind the feed Inlet received the majority of the flow. Thus, these two quadrants would have a high sphere count, which would result in a
very high total variance (Na^) for this design at, all flow rates.
The designs shown in Figures 13, 1 , and 15, using the conical
diffuser, were employed in an attempt to improve the flow distribution.
It is apparent from 'Table I i I that a more uniform distribution vaii
obtained by the three design?, since their variance was lower by a
factor of approximately five from that of the open cylinder (Figure 3)- From a comparison of the three designs alone, it is seen that the feed
wells illustrated in Figures 13 and 14 have & slightly higher variance
than the feed well shown in Figure 15-
In the designs depicted in Figures lU and 15, the distance between
the bottom of the cylinder and the bottom of the conical diffuser was
varied to ascertain the effect of velocity on sphere distribution. It
was found that the lower velocity obtained, by having the conical
diffuser 2.0" below the cylinder Instead of 1.5", gave a more uniform distribution or a lower variance.
From Table III, it can be seen that the two point impingement design
51
(Figure j ) has a variance comparable to that of the conical diffuser
designs. However, it should be noted that the variance increased for the
flow rates of 3* and 5 gpm above that obtained for 1 and 2 gpm. This may be attributed to a more unequal flow distribution resulting at the
higher flow rates because of an increase in the turbulence generated
within the feed well.The four point Impingement design (Figure 8) was devised to improve
the flow distribution found for the two point design by having four inlet streams impinge against each other Instead of two. From Figure 8, it can be seen that a more uniform flow distribution from the feed well was obtained. However, the variance for this design, although it was comparatively uniform at all flow rates, was higher than that for the feed wells shown in Figures 7* 13* 1^ and 1 5 . To explain this higher variance, it was found that the flow from the inlets feeding quadrants 1 and U was unbalanced because of an imperfection in shaping the plastic tubing.
This resulted in quadrant b having a larger number of spheres than quadrant 1, which necessarily gave a higher overall variance for each replication than expected. To substantiate this explanation, it can be seen from Table XVI (Appendix II) that quadrants 2 and 3* which were fed by balanced inlets, have a comparable average distribution of spheres, whereas the average between quadrants 1 and ^ varied considerably.If the imperfection in the inlet feed line to quadrants 1 and U was corrected, it is anticipated that the resulting sphere distribution would be similar to that for quadrants 2 and 3 . Thus, the variance would
52
be reduced considerably, and it is believed that this design would have as low, if not lower, variance than that of the other designs.
CONCLUSIONS
A technique has been developed for the evaluation of feed well
designs by the analysis of scaled models. This technique permits the
study of many designs at a minimum of cost and time. From an analysis of ten models, the following conclusions were drawn.
1. Motion picture analyses of the ten model feed wells showed that the designs illustrated in Figures 6, 9j 10, 11, and 12 were obviously inferior to the others.
2. It was possible to quantitatively analyze the remaining five models (and a modification of one tc give a total of six), as shown in
Figures 3» 7# 8, 13> 1 > and 15, to determine the flow diaIributiun characteristics of each design. It was found that the conical diffuser feed well (Figure 15) has the most uniform distribution. Next, in order of uniformity of flow distribution, are the two point impingement (Figure 7) and the remaining two conical diffuser (Figures ll+ and 13 > designs. The four point impingement design (Figure 8) has a slightly less uniform distribution than the above four (this is believed to be
due to a defective feed inlet). The open cylinder design (Figure 3) vs-6 found to have the poorest distribution of flow.
3. From an analysis of variance of all of the data for the six designs evaluated quantitatively, it was determined that design had a
significant effect on flow distribution and flow rate was insignificant.
53
54
Application of this statistical procedure to the data for the individual
models showed that the effect of flow rate on distribution was significant for two designs (Figures 8 and 14) and insignificant for four
(Figures 3, 7, 13 and 15).From the above conclusions, it can be seen that the evaluation of
feed well design by this technique is feasible. It will permit the selection of one or more designs on the basis of their comparative ratings as a result of the model experiments. The selected designs may then be installed for full scale investigation, thus, eliminating most of the costly changes that would be necessary in the absence of a model study.
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Morrill, A. B., "Sedimentation Basin Research and Design," J. Am. Water Wks . Assoc., 2j+, (1932), 1H42-1H63-
58
43• Murphy, Glenn, Similitude In Engineering. New York: The RonaldPress Co., 1950.
44. Peebles, F. N., et al, "Preliminary Studies of Flow PhenomenonUtilizing a Doubly Refractive Liquid," Proc. Third Midwestern Conference on Fluid Mechs. Minneapolis: University ofMinnesota Press, 1953* pp* 441-454.
45- Perry, J. H., et al, Chemical Engineers1 Handbook. Third Edition. Hew York: McGraw-Hill Book Co., Inc., 1950.
46. Raben, I. A., Flow and Agitation Characteristics of a Self-Contained Water Purification Unit. Louisiana State University Chemical Engineering Department Thesis, 1947*
47- Robinson, C. S., "Some Factors Influencing Sedimentation," Ind. and Eng. Chem., 18, (1926), 8 6 9-8 7 1 .
48. Rouse, H., Fluid Mechanics for Hydraulic Engineers. New York:MeGraw-Hill Book Co., Inc., 1938.
4 9 * Rouse, H., and Howe, J. W ., Basic Mechanics of Fluids . New York: John Wiley 8e Sons, Inc., 1953-
50. Rowe, W. A., "Airfoil Fan Characteristics," Heating, Piping, and Air Conditioning, 8 , (1 9 3 6 ), 112-118.
51- Rushton, J. H., "Application of Fluid Mechanics and Similitude toScale-Up Problems," Chem. Eng. Prog., 48, (1 9 5 2), 33-38,95-102.
52. Rushton, J. H., "The Mechanics of Similitude Applied to Pilot Plant Experimentation," Proc. Midwestern Conference on Fluid pynamlcs. Ann Arbor: Published by J. W. Edwards, 1951*pp. 1 5 6 - 1 7 4 .
53* Santangelo, J. G., and Westwater, J. W., "Photographic Study ofBoiling," Ind. and Eng. Chem., 4£, (1955), l605-l6l0.
54. Schroepfer, G. J., "Factors Affecting the Efficiency of SewageSedimentation," Sewage Wks. J., (1933), 209-232.
55* Slade, J. J., "Sedimentation in Quiescent and Turbulent Basins,"Proc. Am. Soc. Civil Engrs., 6l, (1935), 1435-1451.
5 6 . Slade, J. J., Jr., "The Dynamics of Sedimentation," J. Am. WaterWks. Assoc., 2£, (1937), 1780-1796.
59
57- Stein, P. C., and Perry, L., "Sedimentation and the Design of Settling Tanks," Proc. Am. Soc. Civil Engrs., 71, (1 9I+5 ), 1379-1386.
5 8 . Stewart, R. F., and Roberts, E. J., "The Sedimentation of Fine Particles in Liquids," Trans. Inst. Chem. Engrs. (London),XL, (1 9 3 3 ), 12^-lte.
59* Taylor, M. K., "The Balsa-Dust Method of Air-Flow Visualization and its Application to Model Helicoptor Rotors in Static Thrust," Mech. Eng., 72, (1950), 6 5 8-6 5 9 .
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APPENDIX
I. GLOSSARY OF TERMS
(l) TmpitmpBent Design - A feed veil arrangement In vhlch the incoming feed strlk.es one or more components of the veil Itself. In the case of multiple feed inlets, it may be defined as the collision of the feed streams with each other and/or components of the feed veil.
(2 ) Feed Well - It is that part of the settling tank, vhich is used to distribute the feed within a given tray section.
(3) Short-Circuit - The entering feed tends to flow out from the feed veil in the form of a narrow high velocity stream instead of being uniformly distributed.
(It) Bottom Scour - A resuspension of previously settled material by the Incoming stream.
(5) Multi-Tray Settler - A settler in vhich two or more compartmentsare combined into a single unit.
(6 ) Tray Section - One of the settling canqpartmsnts comprising amulti-tray settler.
(7) True Final Position - The final position assumed by the spherestracing the flow from the feed veil, vhlch position vas not altered by turbulence on the bottom of the tray section.
(8 ) Ideal Feed Mall - One from vhich a perfectly uniform flow distribution is obtained. It may also be defined as one
61
62
with a variance of zero.
(9 ) Conical Diffuser Feed Well Design - The incorporation of a 60° cone, whose apex is directed Into the flow from the inlet, with
a cylindrical feed well to obtain a more uniform distribution of the feed into the tray section.
(10) Model - A small-scale device, which is so related to full-scaleequipment that observations on the model may be employed to predict accurately the behavior of the full-scale equip
ment in the case of the variable studied.(11) Prototype - The full-scale equipment for which predictions are to
be made.
ABBREVIATIONS
Gallons per minute NumberInside diameter Outside diameter Inches FeetAngular degreesMillimetersVolumeSummation
CentlpoisesPoundsCubic feetSquare feetSecond
63
II. THE NUMBER QP SPHERES PER QUADRANT OF MODEL TRAY SECTION AND VARIANCE DATA
Two sets Of data are included in this section™ The first settabulates the number of spheres counted per quadrant of model traysection, and the second summarizes the variances calculated for the individual designs from the above count of spheres.
In order to clarify the position of the quadrants with respect tothe direction of the incoming stream containing the plastic spheres, aplan view of the model tray section is shown below.
The arable numerals refer to the quadrant as given in the data, and the arrow indicates the direction of flow of the feed stream.
6k
J*«pi tr ** l »u*
TotalAyvrtp
1*.U
J I
n
7 *6
I1* 1*7
. b
1 ^
i') 1 1
1 0 «»6
11 1 6
16 1>
ov ufl-y
,u 16-6
ItM* 11Opto <>lind#T (?l|iu« )): Distribution r>f Iphtr** par
o f H a d a l T r a j f t a c t l o n
r
1 ii*.
67 I 5J*li“ ii".
ii i i*.u
.. J
u 1 JO t* uj j 2C y
'I 111i "■ 2 1 i
L \ 26 •i i.
__J 9\ 5 . ... J ; x
in i4f. 2 / sr. 16 p16 1 1 6 1
in * 3I' 16 7 1ll 12 h12 TV <+
:o J 5ri'
3b
► <■ ^ i. ri«
3 ■•17 16r . i*
1« 1 61 r> 1P5 261r' i-
l u 1 117 1 V
inlr.
' 1 * < *1 6 /U 1.
1*» i : 6
IS li. P---
16 1 > 6iV i6 1
1 I .<•Tn
1 3 t «, 1
r< ; u
1 ' r.
16 i,
<■11 *:
20
,16-, L . . .
U07 IN J
131 6 .2
361 lit u
W77 -A iV.l J.n
' 7I'" i ~ 1
- t
-t-
*>A1' 7
12
267 ■11 3 6
66
I fcb U KI1
Qffaa O yU otftar, t l t h C o n i« * l D if fU M r ( t a l l C o n ti 1 . 0 ' Q*t Of C o l l a t o r , F l* u r* 1 ) )
f f l p i r l f t i t l o i a t t f t e n i f * r f tu a d r a n t o f HaAol t ro j r t a c t i o n
F l w A r t* (<7WJ 1 .0 ff.O J .Q _ b 0 5 -0« w A r « j » r M U r i 1 1 \ 1 l 5 1 , X 1 Y X 1 - ft _ . J _ ... - y -
I 9 13 10 Iff 9 13 l i 7 i t 11 ft 9 9 13 Q 9 9 ft U 10
> 1 I f i f k l b lb T A i t 11 7 10 A 19 7 6 10 17 9 n
0 t o a b ft 16 ft ft 6 11 13 10 b 10 13 13 ft IB 119
11l b 5 9 -
ft ft 9 7 19 11 3 13 ft 7 10 11 ft 1
9 7 10 a Iff " n 9 10 3 i o ft 17 6 10 2 16 r 13 12 »
6 Iff 15 ft 9 5 ift 10 7 10 17 e 3 T to ? a a V * 12
7 11 n n ft 6 iff 7 15 A 15 10 9 11 1ft 6 3 i* ft
0 10 17 10 7 13 10 9 ft Iff 13 ft 7 ft 17 a h 12 lb b ft
9 9 16 i t 7 6 i t 19 7 7 19 1L ; 19 lb 4 fc 9 ft 13 10
10 u 10 iffT ’
7 I f 16 9 9 13 9 11 I f 7 10 i t y H 6
LI 10 Iff l i 7 6 ffl 7 6 6 lb u 9 4 16 IP 7 lb 16 U1
ft
Iff 10 10 i t ft 7 lb Iff t e 9 19 ft 5 13 ft lb I f LJ ft 7
U 0 11 13 6 13 ft ft 9 A lb ft >0 IP ;i / 10 9 ft
lb 7 13 iff ft b 19 11 6 Ll n ft ft a 11 lb 7 b i l 9 ib
19 Iff 13 10 5 ft LI i j ft 6 i t 16 <■> 3 17 17 9 lb O 76
16 19 9 ft 10 f 15 10 6 3 16 n ft 7 19 10 b 7 15 ft1
iff
17 6 ffff 7 9 ft 16 « A 7 IT Ll 3 U ft 10 lO 11 iff 9 h
10 11 11 11 7 A 10 13 9 5 n 9 13 10 15 6 9 ft 13 10 9
1* ft 13 Iff 3 3 16 6 13 9 13 Iff 6 9 13 6 l b 9 j I ?
BO U Li Ik 4 15 Iff 10 3 9 u f U il Iff b 13 ip P ft j»-
31 w U Iff U b 13 lb 7 « l? a 7 ft i b 9 9 ) 17 10 10
*1 19 16 9 6 6 Lb l i f 13 7 a iff 6 lb 13 7 16 ft r u
I ) 19 13 7 9 a L6 f 7 10 10 3 15 6 Iff 10 LB ft 1 5 5 Lmtb u 16 • 3 10 lb ft A 7 Iff 10 11 ft 17 i f 7 s ft
H u i f t1 4
IT T 10 6 11 13 10 5 13 7 13 V 16 s
10T o ta l > * 3** B3B 170 1 Roe 3bl « 3 BOB BOb JIB 953 » 1 »05 35 J Bib frfrft 2by 20 r. R-b?
A n n « a » > U 7 1 0 .1 6 . ft . . . 1 3 -6 f . f t a.3 e.t I ff -7 10 .1 9 .0 fl.fr lb.i f t .6 9 1 10 0 \y o ft 2 V p_ J
rirx, fatt tyw). Qi4r*nt JTviyi*rf l e p ] t r o t I o n *
1
I 1 iu
T'ftnlAt» fi r Hf/V'
117lft
l*t
1913 1 7 7
16
1
ffcbla tillOpoi* Cgrllvdar, vttk Cooiofti Dlffuiir (Urfi Ce«i»i l.J" Out of CplLador, Pipur* Ik): PlatrLbutloa of Sphani par uftdmt of Modal froj Metlun
■TP 7Uft iff ;*Oj>.'» 11-V /l fl l
..L A __
ft lr. lo 7 W lO1 o 1 r:i i ‘i k.
n, L't 1 1 7 1 1 1 111 7 7 1 3 1 116 ,, U 6 1 ' 7
- 1 I
J
R10
10 1 3 :. •
1, l 1 -
7 I o i 31 i » 1
10 !.■H i' j
i V
1ft pftft oUft
li.1- > *
10
t J I
in
- l-Q..
\ \ 7 ff R1 1 11 1 1 1
U 8 J*
i.
i
1 I 1 tj L‘': i • io
i*- "
9 i
17 10 rj 710 l* h lo
l'i 31»14 J 1 3
7 0: i
11\) 7 1'
” 11
n n 1 It
11 13
I ,
i l l lO
. _J .. . J t4
t, r \*< -■} \b ft> 1 \o(, Li* I1 1 ft M In i
fb L* Il¥
riTw nit* ^ u n ' i r o r i l K u a l * r
B»pl lnt V vi*I'« 7m r
in \1 I I )1' It
Open Cylinder, vith Conlcnl OlffvMr (Urg» Con* 9.0" Out of Cylinder, r i | m I})-
DlitrlbirtlOH of Sftpni per tiedmit of Itodel lVejr 9i«tlat
I ’"
f f t b l* XV
Two Folut t ^ l n p a e t ( h | u n 7) ; Dlatrlbutioo of Iptaifi par
Q u ^ r u t of Mo4al Troy laetlu
rioa Hrti (gin)Quodrnnl M b « r
I'*
ri
T o t n l
1 .0
7 J T '*3
■> 1110 u1J 7
in i/
in ■> h n
11
Ft o, i f.H *' #
) 'I)•) it* in
J* in i* in' •' ’ 1
*.n 1 I .0 K'.n
9 lo ifl iJ l j
10 1'
n I '11'
1J1J
11
I'l 1 j
-L2- - .r .. ___j...
ii 11
5-if T '
\U11
I
n
r
KVIfour Peim l^l n i n it (Tlffur* 6)'
Miirlfeutlaa of Bpton* p»r 4uft4rut of M U 1 trtjf loetlon
riov Rntc (omjwii!r»nl RuwTfl rftpl Hn»v*
To t<.l Av»!l,lP
1.0 ? 0 Jl P- ._ ij O ; 0I2 .i ■; x ; ~ - _ n j 7 _ { . - * V' ‘ 1 ~T t ~ I ' ' 3
10 U V \ ? 6 n 7 17 1 9 11 j ft i p 16 f, 12 7 1 7
6 10 L 13 15 11 5 9 9 11 9 11 ft 11 16 7 10 u: 1 '
? 1 u U 11, i, 15 10 11 *i 1*. 1** p 7 17 11 l ft- 13 1,
6 6 10 in io in i> 10 it n 1ft 10 6 n 1 L 10 i i n
h 1 1 IP n 16 y 7 3 i f , 12 9 10 fti It 16 t 17 T i l_ , - ---- -------- „----- ’ — ---- ---- ----- . ... --- ---- - ---- - ----- ■ —
\i 10 fi 17 in 1*1 n fl 6 7 15 12 9 U n *. 11 i r . 1J*
5 r 10 IP 'j I P 5 17 3 in 17 7 1 f 11 6 ■? 7 ...
6 i‘ It l? 1 11 i»i ft 7 T it f* 1 \ 1‘ f, j 3 1 •"
7 i i *i P ft f- o r? 0 1 n 9 r. ft) 'J 1 J b r'
7 '■ 1 IP 7 iO in 9 7 1 2 12. '
I 1 1 7 - ■. ’*
it 7 I-5 17 7 11’ 7 ft. A 1 3 13 , t n ft -J ft ■■ ! 1
r r ii, 1 ? - If, iri ft l*. I <• fi ( l 1 ft. 11 1 1, L, 11
V- 'J 17 I P 7 5 ll> O i J Lri . '■ / 1 >i. ft) 9 i r 9 1 * P 16 11 1 * 1 1 I t lo 1 3 1 11 \ P t,
u 1.’ ft > i it 1 i | ’t >4 l i - n /. ■‘ft ' ) l ft , 1
u in 1 is 11 i i IO 1 .i 7 M ft- 1 ' ' ft. :■
.' 1 *1 n > ' 1 1 '■ lo ’ 3 ) n ft 1 ! i ft-i 0 7 s 5 1 1* n , u 20 ? : I, f 1 1 1 ' :■*
o 1. tj l j L 1 / t \ l 12 * I 1ft i<> lo ] J. fi (. 1 i J»
5 ■' 1 15 1. ft 1 ‘l 1 5 7 O S n l«- U l J+ 1 s I '■ It 1
* n li’ 17 I P 1 * 7 n m 1 ■ 1a 3 10 ■■ I v P U P K
i J :■ 17 J 10 * 13 " '# 10 U 10 1/ 1 1 1 1 ?
5 i 17 I P t ■* 10 I-' j* ■i r n ' l , t I ' 1 t
j 7 JO 1*. r If, b 1 3 If. < ft It it I P
i» 0 p IV 10 1 J 7 10 lo n l / i 1 11 ■' 1 **0 .‘20 ?55 777 3-M IT** 317 102 om 21 .n 19 r 27/ « s f 29V i m t'P.’ 4 20
7 !>.? 10 ? u.v n -y lift* 7 0 .2.7 ft 5 : i ft 1 l 2 10-7 7-9 111 ft'. 1 lo r o .n 11 1 i, f. — . .. . - - . . - _
71
Table WIIOpen Cyllnaer:
Bunmary of Variance Data
Tlow Rate (0PM)Replications 1.0 2.0 3-0 4.0 5-0
1 102 146 200 386 200
2 126 360 394 270 146
3 150 66 118 272 178
4 206 216 62 130 204
5 66 286 154 94 700
6 31** 350 162 122 3447 lUo 102 94 342 3506 260 420 240 314 2709 310 190 224 446 2700 100 174 154 356 374
11 302 354 244 206 21412 174 266 170 106 HO
13 146 150 390 150 15814 114 270 ?6o 302 94
15 420 314 174 254 406
16 146 146 136 154 10O
17 446 230 138 210 20018 354 330 170 274 13419 214 250 178 102 11020 110 254 200 214 290
21 152 536 178 210 6622 140 370 492 176 250
23 338 314 86 206 44624 130 250 llO 266 35225 234 374 154 142 150
Total CfcHo?) 5354 6796 4890 5946 599B1 Aye rage 214 272 196 23® 240
72
Table XVIIIOpen Cylinder, with Conical Diffuser (Small Cone: 1.0" Oit of Cylinder):
Summary of Variance Data
Flov Rate (OPReplications 1.0 < __?._0____p____3-0____J„ u-° ----- .2.:° . .
1 3a 20 10 12 lU2 1W 60 lit HO 783 1 U), i*n 26 si* IOU 1*6 31* ll+O 38 305 lU 11* 78 1. s? 76
- - ------- --- ■on 78 171* 1 TO
7 IP si, 1 it ]n6 60P IP H, :u , : K) 80
o 71* M, M. 67 litin 1U 71, 1 /’ ! 1
- - - ■ - (•11 1'* 1-0 7;‘ l''t>2 i V’ sit 7t *1 7 un V. r** ■ * t 11
] 1> 131* 1 3 0
1 5 V 1 M t'i *, !■ ■I
1'- It* a : ■ M i 117 l o t * 1 ,6 t-'.t, p I O
18 2 2 17* 1,-, i+r* 1 i*16 • 8 Hi, 3 0 1 u 1 .(
20 S t . 78 1. *>< > 1 1 1
21 50 86T
n o 2 2
,
r >8
22 86 31* 26 50 5023 68 S O 50 2 '* 6 8
2U 66 ?l* 11* 26 111*25 lUO 7** 26 b8 67
Total ( ? Nffp ) 11*5!* 1398 1060 155** 1170
A v e m y e 58 56 **3 67 <*7
73
Table XDCOpen Cylinder, vlth Conical Piffuaer (Larf.7> Corw* : 1.5" Out of Cylinder):
Hurmnry of Variance Data
Flow Rnte (CPM)Replication* a.o 2.0 3.0 j It.O 5-°
1 110 38 6 ; 30 100
2 13*» 62 5'* 10 50
3 76 30 20 u u 6i* 50 Itlt 2 50 ItO5
j56 72 Ult 70 lU
6 lU 68 38 66
7 U6 38 2 00 208 110 30 62 26 26
9 82 1*6 10*t It 610 36 51* ItO 50 lit
li 72 50 2lt 78 102 I(12 102 7*t 86 26 6
13 23>t 50 9lt 30 38lU 3P 10 it? 5‘t 3615 80 lib 86 Itl. -16 !
il<> 50 LO 1 '.2 t) 0
17 86 8 lt2 30 It?18 110 100 lit 86 66
19 3C 18 38 26 lit20 50 38 Uo lit 7*t
21 70 18 6 26 7022 5U 36 7it 26 3823 U 38 0 BU 22lt 122 56 56 30 6625 98 58 50 1*2 10
Total (SH,£) 1910 1090 110U ioio 99ltAverage 76 Wt 1*7 ItO UO
7U
Table XX
Open Cylinder, vith Conlrnl DlfTufier (larRe Cone: 3.0" Out, of Cylinder):
CuiTwrwry of Variance OnbftI
lonei
- 1 .O 3.0Flovx ..
Hfitn Tr.TM”)7.0_ j _It.O ' 5 .-p..
138 33 18 18 3i»
i? 91* ' I. IO? IO
: 3 79 : 38 18 36
’ u ?6 1 U 11*8 10
5 f-y'* -3 3** 7 8 Mi*- ■•- - - - ■ - - 1 - .... ...... -
r, 1/' 90 , j IO 106
7 70 3B n 38
n <-,0 S7 2 0 78
< j i r,, 8<i 1 It lilt
in *, : ’c 1 3 1 9 s ll* * - .
1 1 yt , p* ■ 0 7'* 7
'IP ..9 si. 7* 1 n
17 1 10 of', 1 0 so
1*j 10 fi (8 ] 1.
•1 u >n
!?u 13 l'*
in 1 7 ] 10 ■ 1 7 c 70
17 nr. 3 0 1 It 70 S3
l8 1: 3 0 ‘■■O 70 ■Mi
1 lo i.0 8 Til 7 0
30 70 ii - 1i
n. 70
21
r ■5** IO V , 36
1 23 7H 1.1. lU 31* 701f
! 231i
9 U lU 120 --- 3s,
i puii 3>+ 50 36 3 86
! pt>i! 68 IB 130 66 76i yy -ir * •• --.3—J-y r- r - rt-i - t--T-. 1. - - -
i .-jTotal (?N^) j I?1*, 9*+? 91. 2 wv, 97b
A v e r n ^ l1 63 38 78 3 r> t ,9L______ _ _ _ 1. _ , _ _________ — _..__ ...____________ l.------
75
Table XXITwo Point Impingement: Surnnary of Variance Data
Rep11c atlonfl 1P
3(t
1.02
20
lh
2o
\h
7)H30
__ Flow Hate "(PPM) "2.0
6
TOBU
14:>4 11*
3 - o
30
10?B30UP
;■ o' ip. 74 7 4
i*.o7656lJ.lt
1H
r
94
-p10
1 410
lo
IO
1 Jh
20 102211822
26
101224 86
8006lfl
lit1090'.0
In.18
'■0
I' >
18
t/,
90
5010
02
3629090
109641+
76
Tfcble XXIIFour Palr»-t Imp 1 nf^mnt.:
Bu m w r y of V«r lsnce Data
Plow Rait« (OPRepllr.atlona 1 .0 2 .0 I 3 -0 U.o 5 -0
1 2U H O U 72 9©
2 26 52 U 62 lU
3 90 62 72 8U H OI* 96 128 10U 3U 6
5 110 50 90 72 8U
6 lU 2U 5U 22 U2
7 98 9U 226 BU 13U
ft Uo 62 30 5U llU
9 30 192 U2 lU 122IO 138 98 18 108 ftft
l i 98 38 38 U2 U6
12 38 62 Uo 3U 5U
13 70 7U 30 66 20li* 06 H O 36 126 Uft
15 56 5U 136 IOU 3U
16 62 22 16 58 6217 38 12 30 3U 7U
18 5U 102 1U6 lU 7U
19 30 20 90 32 '7620 52 102 228 56 IOP
21 106 26 78 70 1622 108 lU 5U 1U6 100
23 eu 6 62 H O 302U 5U 90 6p 10c 10c25 122 18 66 12 20
Tot»l CEH^) 172U 1622 1778 1620 I 690
Average 69 65 71 65 66
5
III. SUMMARY OF CALCULATIONSA. Scale-Dovn from Prototype to Model
Calculations are shown for the determination of the ratio necessary to obtain geometric similitude between the model feed well and the feed well of a typical industrial settling tank. From the dimensions calculated for the model (using this ratio) and the physical properties of the liquid, it was possible to obtain the liquid flow rate, which
would result in identical Reynolds numbers in both units. By having the same Reynolds numbers, the requirement for kinematic similitude is satisfied.
Geometric Similitude
The following dimensions are for the feed well of an industrial
settling tank, which will be designated as the prototype:
a. Inside DiameterJ |,0'
b. Outside Diameter of Mud Downcomer: 3*5'Basis: Experimental feed wells (For example, 2 point Impingement,
Figure 7 ) to be 2.19" I*D.Geometric Ratio:
Dmodel r Dratio = 2.19" = 0 .0 2 6^prototype BfToO11
Mud Downcomer:D Dmodel - ratio^prototype
77
78
Outside diameter of model downcomer • 1+2.0" x 0 026- 1.09" (Use 1 0"
O.Do downcomer)Liquid Flow Rate Required
By using the Reynolds number In the prototype as the similitude factoit it was possible to obtain the flow rate for the model This flow rate resulted in a Reynolds number identical to that In the prototype.
The physical properties of the liquids in the model arxl prototype and a typical flow rate for the prototype, which were necessary to calculate the Reynoldc number (Prototype) and flow rate (model), are as follows:
Model PrototypeLiquidDensity (lbs ./ft.^): Viscosity (cps)Flow Rate
0 81To be determined
62.17
Water Suspension
75-151.1752 00 gpm/tray section
a. Reynolds number in Prototype:Re = V (1)
vwhere: Re is the Reynolds number
De is the equivalent diameter = D2 - D- 1 7 -0 ' - 3 .5 '
: 3-5'V is the fluid velocity = 1 .5 8 x 10 ^ ft./sec.v i6 the Kinematic viscosity = 1.051 ft
sec .Re = 5262
Flew rate In Model:V = Re v
°E
where: Re is 5 2 6 2
Dg is the equivalent diameter in the model
= 2 .1 9 " - 1 .0 ” = 1 .1 9 " = 0 .0 9 9 'V : 8 . 6 7 X 10~^ ft.^/sec.V = 0.U60 ft./sec.
The liquid flow rate corresponding to this velocity
80
B. Application of Statistical Calculations to Data (Details of Methods, References 15 and 23)
The data obtained from the two point impingement design (Figure 7) will be used to illustrate the statistical calculations involved. Variance
The statistic, Variance, may be expressed by the equation,= (X - M )2 (3 )
Npwhere: 6 is the variance
X is the original measurement ( c o u n t o f s p h e r e s
per quadrant).M is the mean (average s p h e r e c o u n t )N is the number of measures in a series (the
number of spheres per replication!.Since forty spheres were used for each replication, t h e v a l u e o f
N remained constant. The value for M in the above equation was taken as ten, which would be the average distribution for an ideal feed well, so that all designs could be evaluated on a coumon basis. Thus, t h e
variance equation was rearranged to simplify the calculation as f o l l o w s .
N i ; (x - 10 ) 2 (1+)
The symbols are the same as given for the original equation.The data tabulated for the 25 replications at 1 gpm, as shown in
Table XV, are utilized to illustrate the application of the modified
81
variance to thla work.. [UO -IO)2 ♦ (10 - 10)2 4 (ll - 10)2 ♦ (9 - io ] 1+
(9 - 10)2 4 (11 - 10)2 4 (13 - 10)2 ♦ (7 - loFj 2 + ...03 - 10)2 * (10 - 10)2 •» (lit - 10)2 ♦ (8 - loFl 25 - 8l8
Die variance for each flcnr rate for the six designs was determined In this manner.Analysis of Variance
This statistical procedure was applied to the data obtained for each design so that the effect of flow rate on the distribution of spheres might be determined. The mathematical proof of the validity of this analysis may be found in any standard text treating statistics (15, 23)/ and therefore is not Included here.
The data employed in the subsequent Illustration were taken from Table XXI and are aumarlzed be lov. The calculations involved are than presented.
Floy Bate (QPH7LI cations 1 2 3 k 5. 2
1 2 6 30 76 3*4 ll*82 20 30 10 56 18 13*43 lit 8 78 ikk 90 33*4k 26 Ik 30 18 56 Ikk5 6 6 k2 5*4 62 1706 Ik 26 218 26 k6 3307 7*4 lU 126 8 38 2608 18 2k 7*4 2k 50 1909 30 Ik 7k 38 6 16210 18 Ik 10 lbO 10 19211 6 30 66 Ok 50 25612 30 96 5*4 k6 26 25213 10 22 Ik k 18 68Ik 166 10 122 30 9*4 k2215 66 lU 7k 8 10 172
(Continued on next page)
82
(Tabulation continued J1 2 3 _ T"" 5
16 26 32 UO 8U 6 18817 1+8 UO 22 10 66 18618 10 78 6 86 20 20019 6 2U 20 2 50 10220 32 17U 36 62 38 3U221 72 26 9U 102 82 37622 5U 2U 3U 18 36 16623 3* 3U U2 26 2 1382h 12 30 10 62 98 21225 2b 18 86 2U 90 2U2
8 1 8 8 0 8 1U3 2 1 2 3 2 IO9 6 5 3 8 6
The total Bum of squares, the sum of squares between columns(flow rate), and the sum of squares between rows (replications) for thesedata may be calculated.
Total = (2)2 * (6 ) 2 * (30)2 .... + (90) 2 - (5386)2 = 188,100125
Between Columns = (8l8 ) 2 ♦ (808)2.■. * (1096) 2 - (5386)2 = 11,5 9U25 25 25 125
Between Rows = (lU8 ) 2 •» (1 3U )2 + (^^U)2 ... ♦ (2U2 ) 2 - (5 3 8 6 )25 5 5 5 125
r 37,305The residual sum of squares may then be obtained as follows:Residual > Total - (Between Columns * Between Rows)
= 1 8 8 ,1 0 0 - (ll,5 9U - 37,305) = 1 3 9 ,2 0 1
The results of the analysis are summarized in the following table.
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Sum ofSource of Variation Squares
Degrees of Freedom*
MeanSquare F
Between Columns (Flow Rate) 11,59** 4 2,899 1.999Between Rows (Replications) 37>305 24 1,55** 1.072Residual 139>201 96 1,1*50
Total 1 8 8 ,100 121+
♦Degrees of freedom:1. Between columns = K - 1
2. Between rows s N - 1
3. Residual = (E - l)(K - l)4. Total = UK - 1where: K - the number of columns or flow rates
N = the number of rows or replications.
For 4 and 9 6 degrees of freedom, an F of 2.1+7 is significant at the5$ level of confidence and one of 3-50 at the 1 6 level. For 24 and 96
degrees of freedom, an F of 1.80 Is significant at the 5$ level of
confidence and one of 2.33 at the 156 level. Thus, It can be seen that neither of the sources of variation (flow rate and replication) had a significant effect on the distribution of spheres when tested with the residual mean square as the denominator of the F ratio.
VITA
William Leon Barham was born at Birmingham, Alabama, on March 5, 1924. He attended the public schools of Birmingham, and graduated from Ramsay High School in May, 1941.
He entered the Alabama Polytechnic Institute in September, 1941.He left the Alabama Polytechnic Institute in April, 1943, and served with the army until March, 1946, attaining the rank of first lieutenant in the Infantry.
He returned to the Alabama Polytechnic Institute in March, 19b6,
and completed the requirements for a B.S, degree in Science and Literature in March, 1947- From this time until he returned to the Alabama Polytechnic Institute in the fall of 1951, he was employed by a trucking firm as superintendent. He campleted the requirements for the B.S. and M.S. degrees in Chemical Engineering in December of 1952 and August of 1954 respectively.
In September of 1954, he entered Louisiana State University to work toward a Doctor of Philosophy Degree in Chemical Engineering.
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EXAMINATION AND THESIS REPORT
Candidate: William L. Barham
Major Field: Chem ical Engineering
Title of Thesis: Flow Patterns in Cylindrical Settling Tanks--Part IApplication of Model Analysis Technique to Settling Tank Feed Well Design
Approved:
M ajor P r o fe s s o r and C h a ir m a n
n ate s r h o o
EXAMINING COMMITTEE:
L
7' s t \
Date of Examination:
May 1957