Experimental investigation and constitutive modeling of cohesive powder flow

MISSING IMAGE

Material Information

Title:
Experimental investigation and constitutive modeling of cohesive powder flow
Physical Description:
vi, 116 leaves : ill. ; 29 cm.
Language:
English
Creator:
Chen, Dongming, 1966-
Publication Date:

Subjects

Subjects / Keywords:
Mechanical Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Mechanical Engineering -- UF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 2001.
Bibliography:
Includes bibliographical references (leaves 112-115).
Statement of Responsibility:
by Dongming Chen.
General Note:
Printout.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 028196616
oclc - 49289646
System ID:
AA00013529:00001


This item is only available as the following downloads:


Full Text










EXPERIMENTAL INVESTIGATION AND CONSTITUTIVE
MODELING OF COHESIVE POWDER FLOW















-By

DONGMING CHEN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2001










ACKNOWLEDGMENTS


I would like to express my sincere gratitude to my advisor, Dr. James F. Klausner,

for introducing me to this field and providing me with valuable academic freedom and

support during the course of this research. Without Dr. Klausner's guidance, encouragement

and support it would have been impossible to bring this work to completion.

I would also like to thank my Ph.D. supervisory committee members, Dr. Renwei

Mei, Dr. Daniel Hanes, Dr. Jocob Chung, and Dr. Gregory Sawyer, for their willingness to

serve on my committee, providing me help whenever needed and for reviewing this

dissertation. I would like to express my special appreciation to Dr. Renwei Mei for the many

discussions I have had with him and for his valuable advice and suggestions.

I am grateful to all my colleagues in the ERC Cohesive Powder Flow Research Group

for their help and support. The financial support provided by the Engineering Research

Center for Particle Science and Technology at the University of Florida is gratefully

acknowledged.

Finally, I should recognize the contribution that my family has made to my academic

success. Their support and patience have played a key role in my achievement.














TABLE OF CONTENTS

Page


ACKNOWLEDGMENTS .............................................. ii

ABSTRACT .......................................................... vi

1. INTRODUCTION ................................ ................. 1

1.1. Background ........................................................ 1
1.2. Literature Review of Current Powder Flow Research ........................ 3
1.2.1. Cohesionless Powders ..................................... 3
1.2.2. Cohesive M materials .................................. .......... 5
1.3. Objectives and Scope ...................................... ......... 8

2. INVESTIGATION OF POWDER RHEOLOGY ............................ 10

2.1. Experimental Apparatus and Procedures ................................. 10
2.1.1. Annular Shear Cell-type Rheometer ................... .......... 10
2.1.1.1. Introduction .................... .... ..... ........... 10
2.1.1.2. Description of Annular Shear Cell-type Rheometer .......... 12
2.1.2. Experimental Procedures ...................................... 15
2.2. Data Reduction ......... .......................................... 16
2.3. Cohesive Powder Properties .......................................... 18
2.4. Extraneous Factors Influencing Experiments .............................. 22
2.5. Powder Rheology .................................................. 26
2.5.1. Cohesionless Powders .................... ........... ... .... 26
2.5.2. Cohesive Powders ......................................... 30
2.6. D discussion ........................................................ 45

3. POWDER PLUG FLOW EXPERIMENT ......................................... 60

3.1. Introduction ..................................................... 60
3.2. Experimental Facility .................................................65
3.3. Calibrations ..... ................................................... 68
3.3.1. Calibration of the Mass Flow Controller ........................ ..68







3.3.2. Calibration of the Validyne Pressure Transducer
3.4. Experimental Protocol ...........................
3.4.1. Procedure ..............................
3.4.2. Data Reduction...........................
3.5. Results and Discussion ...........................
3.5.1. Factors Affecting the Experiment ............
3.5.2. Plug Pressure Drop ......... .............
3.5.3. Plug Velocity ............................


4. MODELING OF COHESIVE POWDER FLOW .........


4.1. Introduction ..............................
4.2. Two-layer M odel ..........................
4.3. Analytical Description of Powder Plug Flow ....
4.3.1. Estimating the Interstitial Fluid Force ..
4.3.1.1. Static Powder Bed Experiment
4.3.1.2. Ergun Equation ... .......
4.3.1.3. Darcy-Forchheimer Law .....
4.3.2. Momentum Equations ..............
4.4. Rheology ................ ...............
4.5. Determination of Shear Layer Velocity Profile and
4.6. Results and Discussion .....................


................... 69
...................70
...................70
...................72
................. 73
.................. 73
...................75
................. 79


.84


.84
.86
.88
.88
.89
.91
.91
.92
.95
.98
103


109


109
111


112


116


.................
.................o

................o
.. ......o........

w...........l....
...t.............
.............o...

Pressure Gradient .

.................


5. SUMMARY AND CONCLUSION ....


5.1. Summary and Conclusion ..........
5.2. Suggestion for Future Studies .......


REFERENCES .....................


BIOGRAPHICAL SKETCH ...........


I


. . . .
..................................










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

EXPERIMENTAL INVESTIGATION AND CONSTITUTIVE
MODELING OF COHESIVE POWDER FLOW

By

Dongming Chen

December 2001


Chairperson: Dr. James Klausner
Major Department: Mechanical Engineering

The flow behavior of cohesive powders in the frictional regime is experimentally

investigated. An annular shear cell type cohesive powder rheometer has been fabricated,

which measures the shear and normal stress states over a range of shear rate and powder

concentration. The theological behavior of both cohesionless powders, 300 p.m silica and

100 atm silica, and cohesive powders, 27 gm silica, 5 pm kaolin and a series of industrial

polymer, has been experimentally investigated in the frictional regime.

It is observed that for cohesionless powders at low shear rates, both the measured

shear stress and the measured normal stress depend on concentration, and are invariant with

shear rate. For cohesive powders in the frictional regime, the shear stress increases with

increasing shear rate until a critical shear rate is reached, at which point the shear stress

decreases with increasing shear rate. The shear to normal ratio for cohesive powders in the






frictional regime is apparently independent of powder concentration, and the data

approximately collapse into a single curve for a given powder. The measured theological

powder properties must be considered in constructing continuum-based powder flow models.

Pneumatic conveying plug flow experiments have been carried through for horizontal

flow of cohesive silica and kaolin powders over a range of gas superficial velocities and plug

lengths. The plug pressure drops and velocities were measured. It is observed that the

pressure drop increases linearly with plug length and increases non-linearly with increasing

gas superficial velocity. A two-layer model is developed to predict the dynamic behavior of

the plug. The inner core of the plug is treated as a Coulomb solid and the shear region of the

plug is treated as a frictional fluid. The powder rheology is an important input to the model.

The predicted pressure drops using the two-layer model agree well with those measured.

Opposed to analyses that treat the entire plug as a Coulomb solid, the two-layer model

correctly predicts the dependence of pressure drop on the gas superficial velocity.










CHAPTER 1
INTRODUCTION


1.1. Background

Based on experimental observations of the bubbling characteristics of different

powders, Geldart [1] has classified powders into four different groups, A, B, C and D, as

shown in Figure 1.1. This classification relates the material's behavior to the influence of

particle mean size and particle density. Corresponding to the Geldart group C classification,

particle assemblies in which surface forces play an important role in the mechanical behavior

of the powder are classified as cohesive. For cohesive powder, the material's behavior is

governed by inter-particle cohesion, friction, and particle collisions. Cohesive powder is

typically associated with small particle size and low density.


-I





U


0.5


\ \___ ILouJ-bte
I --- B-- __S-- -tb








Cohesive.




20 50 100 2 5 1,000

Figure 1.1 Geldart's classification chart

1
2O 50 10 2 5 %OO






2

The utilization of cohesive powders in industrial processing and manufacturing is

extensive. For instance, some cohesive powders are used to coat metallic surfaces on

commercial and industrial products, such as office furniture, garden equipment, automobile

wheels, power tools, and other metal fabricated components. Cohesive powders are used

extensively in pharmaceutical processing, and the manufacturing of paper coatings,

adhesives, fiberglass and insulating glass sealants, among others. The ability of cohesive

powders to flow uniformly and consistently is fundamental to the efficient manufacturing and

processing of powder products.

Although the utilization of cohesive powders is extensive, the bulk flow behavior of

cohesive powders is difficult to predict, and few fundamental studies exist which describe

the dynamic behavior of cohesive powders. One of the main difficulties in mathematically

describing the mechanical behavior of a powder assembly, whether cohesive or cohesionless,

is that its state is not well defined. Powders display a variety of behaviors that are in many

ways different from those of other substances. Due to the unusual behavior associated with

powders, Jaeger and Nagel [2] suggest that a powder cannot be easily classified as either a

solid or liquid. Depending on loading conditions and the forces that dominate the dynamics

of motion, a powder may exhibit two distinct types of behavior: it can flow like a fluid or it

can agglomerate to form clusters and behave like a solid. When the particle size is small,

generally less than 50 gm, in addition to friction and collisions, there exists a multitude of

forces acting on cohesive powder particles such as van der Waals, electrostatic and liquid

bridge among others. Under different conditions, different forces may be dominant in






3

cohesive powders. It has been recognized that a cohesive powder can display both solid-like

and fluid-like behavior, depending on its dynamical state.


1.2. Literature Review of Current Powder Flow Research

1.2.1. Cohesionless Powders

The flow behavior of cohesionless materials has received significant attention,

including both experimental and theoretical investigations. Novosad [3], Scarlett and Todd

[4], and Stephens and Bridgwater [5] have experimentally investigated very low shear rate

granular flows. These authors found very little dependence of the shear stress on shearing

rate. Bridgwater [6], Savage and Sayed [7], Hanes and Inman [8], and Craig et al. [9] have

experimentally investigated granular flow at high shear rate. These authors found that both

the shear and normal stress depend quadratically on the shear rate at constant bed porosity

(or constant particle concentration).

Meanwhile, Savage and Jeffery [10], Jenkins and Savage [II], Lun et al. [12],

Jenkins and Richman [13], and Lun [14] have contributed to the development of constitutive

models for the rapid granular flow of cohesionless materials based on the kinetic theory of

gas-type modeling. The basic idea of kinetic theory of gas-type modeling is that, in the

fluidized regime, particles are assumed to behave like molecules of gas and, except for the

fact that energy is not conserved during the granule collisions, interactions of individual

particles are through binary instantaneous collisions, just like the molecules of gas. Based

on the micro-scale interactions of individual particles, the continuum equations and other

fluid mechanics equations such as balance equations for mass, momentum and energy are





4

derived for the rapid flow regime. This modeling provides a continuum-based Eulerian

description for highly fluidized powders. Under some assumptions, e.g., small contact time

and small inelasticity for particles, these models have been demonstrated to give qualitative

and quantitative predictions in agreement with experimental measurements and discrete

element computer simulations in the rapid shear regime.

Discrete element simulations, which model the dynamics of individual particles, have

been used by Campbell and Brennen [15], Walton and Braun [16], Tsuji et al [17], Campbell

[18], Lun and Brent [19], and Hanes and Walton [20] to study the theological behavior of

granular materials subjected to very high shear rates. The basic idea of the discrete element

simulation is that the bulk behavior of granules is determined by the interactions of

individual particles. While the behavior of bulk granules may not be well understood, the

individual particle interactions, such as surface friction and particle collisions, are described

via analytical expressions. Based on the implementation of inter-particle force models and

the law of particle kinematic trajectories, the bulk behavior of granule assemblies can be

simulated for a small mechanical system of particles. Since it may instantaneously track the

movement of individual particles in the powder assembly, the discrete element simulation

gives detailed information on particle position, stress field, and velocity field for powders

subjected to different shear rates. Discrete element simulations are widely used for

cohesionless powders and have been recently used for cohesive powders under high shear

rates. Hanes and Walton [20] compared the flow behavior of glass beads down an inclined

chute with predictions based on kinetic theory of gas-type modeling and discrete element

simulation modeling. The experimental results compared well with both kinetic theory






5

modeling and discrete element simulation approaches. Mei et al. [21] recently used discrete

element simulations to investigate cohesive powders that are subjected to very high shear

rates.

1.2.2. Cohesive Materials

In contrast to the studies done for cohesionless powder flows, few fundamental

studies exist which describe the dynamic behavior of cohesive powders. Since cohesive

powders may demonstrate a behavior consistent with a rigid plastic continuum, which is

classified as solid-like behavior, an extensive body of literature exists in which powders are

modeled as a plastic continuum using soil mechanics models. Critical state models such as

the Cam-clay model of Puri et al. [22] and the Cam-clay model of Feise and Schwedes [23]

have recently been applied to cohesive powders. Double-surface models, such as the Druker-

Prager/cap elasto-plastic model, have been used by Gethin et al. [24] to describe ceramic

powders. Bauer and Wu [25] extended the incremental nonlinear models of hypo-plastic

type to cohesive powders. A rate-dependent-type model has been proposed by Cazacu et al.

[26] for alumina powder. These models have been demonstrated to be useful in predicting

certain bulk powder behavior prior to flow initiation, but they are not useful in predicting

flow behavior.

It has been noted that a cohesive powder can display both solid-like and fluid-like

behavior, depending on its dynamical state. Typically cohesive powders may be classified

into three different regimes. At very low shear rate, a powder may be in static condition or

time-dependent creep, which is classified as the solid-like regime. With the increase in shear

rate, the powder yield stress will be exceeded, and resistance to flow will be dominated by






6

inter-particle friction and cohesion. This regime is classified as the frictional regime. In the

high shear rate regime, the powder assembly will be highly energized, and resistance to flow

will be dominated by inter-particle collisions. This regime is classified as the fluidized

regime.

At high shear rates, cohesive powder flows have similar features to cohesionless

granular flows since momentum transfer is dominated by inter-particle collisions, just like

cohesionless granular flows. And it is reasonable to describe flow behavior of cohesive

powders at high shear rate by using a kinetic theory of gas-type modeling. When powders

flow in the very high shear rate regime, individual powder particles are sufficiently energized

so that the powder assembly behaves like a freely flowing fluid. Although cohesive powders

have a tendency to flocculate, their kinetic energy is sufficient to overcome the cohesive

binding energy in the fully developed fluidized regime. Mei et al. [21] recently used discrete

element simulations to demonstrate that when cohesive powders are subjected to very high

shear rates, the kinetic theory of gas-type modeling agrees well with the simulated flow

behavior. Mei et al. [21] also defined an energy-based cohesion number that is used to

discriminate between fluid-like behavior and solid-like behavior of the powder assembly.

When a powder assembly is subjected to low or moderate shear rates, its behavior is

neither consistent with that of a solid or a fluid. During slow flow, the overwhelming

interaction between individual particles is considered to be surface friction, during which

particles slide on top of each other during extended contacts. Due to its complexity, the flow

behavior of powders in this regime has received very little attention, and we are not aware






7

of any constitutive continuum models that have been demonstrated to reliably predict such

flow behavior. Yet most industrially relevant powder flow applications occur in this regime.

Tardos [27] recently used a fluid mechanics approach to model slow, frictional flow

of powders. Ancey et al. [28] also suggested that a fluid mechanics treatment of dense

granular flows in the frictional flow regime may be appropriate. The essence of such an

approach is to treat the powder assembly as a compressible continuum and the conservation

of mass and momentum results in

'Pb
conservation of mass: pu = 0 (1-1)



Dpbu
conservation of momentum: DP- pg + V T, (1-2)
Dt


where Pb is the powder bulk density, u is the velocity vector, g is the gravitational vector, and

T is the stress tensor. In order for Equations (1-1) and (1-2) to be useful in describing the

powder velocity and density fields, a constitutive model relating the stress tensor to the rate

of deformation of the powder assembly is required. Tardos [27] and Tardos et al. [29] have

explored the use of different constitutive models relating the stress tensor to the rate of

deformation for frictional flow of powders. A constitutive model relating the stress tensor

to the rate of deformation for an incompressible Coulomb powder was given by Schaeffer

[30] and explored by Tardos [27] and Tardos et al. [29] and is given as:


T p D, (1-3)
,l D')' ^





8

where p is the pressure, 6, the Kroniker delta, and the rate of deformation tensor and the

magnitude of the rate of deformation are respectively expressed as:


aDnd, 1 = + and YD 2 D

For compressible powder flows, Tardos [27] has given a more detailed constitutive equation

1
T = p, + r(p, p) 1 (1-4)
Dov -V.
-2 5



with the specified yield criterion (p, p) = (sin )p a /- with 0.001<3 <0.1.



a and p are two material constants that should be specified.

Tardos [27] and Tardos et al. [29] have used Equations (1-1) and (1-2) to predict

powder behavior in the frictional regime. But only simple limiting cases have been

investigated.


1.3. Objectives and Scope

In this research, attempts are made to investigate the flow behavior of cohesive

powder in the frictional regime. An extensive data base, relating the stress tensor to the rate

of deformation of the powder assembly, has been gathered for different powders. An attempt

is made to use the fluid mechanics approach and the theological behavior of powders to

model cohesive powder flow in the frictional regime. A series of powder plug flow






9

experiments are carried out to validate the modeling approach. To a large extent, this

research aims to:

* Experimentally investigate the theological behavior of cohesive powders in the
frictional flow regime by relating the measured stress components to the imposed
shear rate. An annular shear cell-type cohesive powder rheometer has been
fabricated in which the resulting shear and normal stresses are measured in a simple
Couette type flow;

* Explore the stress-strain rate relationship for different powder compositions, 300 p.m
silica powder, 100 p.m silica powder, 27 pm silica powder, 5 pm kaolin powder and
a series of industrial polymer powders with broad size distributions;

* Set up a simple powder plug flow experiment to investigate the predictive capability
of a continuum flow model based on the macro scale flow behavior of cohesive
powders;

* Use a continuum based conservation of momentum to predict powder pressure drop
based on imposed boundary conditions and system variables;

* Experimentally validate the continuum model predictions with simple powder plug
flow experiments.

Correspondingly, this dissertation consists of three parts. The first part (Chapter 2)

details the annular shear cell-type cohesive powder experiments and results. The second part

(Chapter 3) discusses cohesive powder plug flow experiments and results. The third part

(Chapter 4) describes the development of a continuum-based two-layer plug flow model and

compares predictions with measured experimental results. Ultimately it is demonstrated that

a continuum-based powder flow model that utilizes measured theological powder properties

yields a good predictive capability.










CHAPTER 2
INVESTIGATION OF POWDER RHEOLOGY


2.1. Experimental Apparatus and Procedures

2.1.1. Annular Shear Cell-type Rheometer

2.1.1.1. Introduction

Steady and reliable flow of powders from storage devices or in handling pipelines is

of prime concern in many industries. In order to ensure steady and reliable flow, it is

important to accurately characterize the flow behavior of powders. There are various

experimental apparatus in industry being used to study powder flow behavior. Among them

are funnel flow testers, Jenike Shear Cell, tri-axial testers and annular shear cells.

In a mass flow hopper, the material flows uniformly everywhere. In a funnel flow

hopper, the flow is irregular, with material flowing primarily down a central funnel, leaving

a stagnant region adjacent to the hopper wall. Jenike [31] proposed a specific solution of

steady state equations describing the flow of a cohesionless granular material in a conical

hopper. This solution, the so-called "radial flow solution", predicts stresses and velocities

in a mass flow hopper but does not predict the flow in a funnel flow hopper because the

radial solution breaks down near the transition from mass flow to funnel flow. Funnel flow

testers are simple in design and are mainly used to study powder flow properties such as

internal friction angles, mass flow rate of powder, and the pressure-strength curve, i.e., the

flow function of the powder. Because powders flow by gravity through these devices, arches






11

and rat-holes may form and break repeatedly during the experiments, and thus the

constitutive behavior is difficult to characterize.

Triaxial testers have been widely used to study the mechanical characteristics of

rocks. In recent years, triaxial cell testers have been designed to investigate the properties

of powders at low stresses. A major purpose of triaxial cell testers is to characterize

deformation and strength behavior under applied loading conditions. The triaxial cell tester

is in principle a simple apparatus: an axial stress is applied to a cylindrical sample of powder

which is enclosed within a thin elastic membrane and subjected to a confining pressure.

Among its drawbacks is that each triaxial test requires a large amount of powder and

involves a significant amount of time and labor in sample preparation and testing. Triaxial

cell testers are mainly used to measure the static properties of powders.

The Jenike shear cell tester was specially developed to predict the flow behavior of

powders in a hopper. While a constant vertical load is acting on the sample, a horizontal

force is applied to shear the specimen. The yield locus is given by:

r = a tan + c (2-1)

where T is the applied shear stress, a is the normal stress, 4 is the internal friction angle and

c is the powder cohesion. The Jenike shear cell is useful to measure static properties of

powders such as yield locus and flow function, which has value in establishing a flow or no-

flow criterion. But the Jenike shear cell tester is not useful for measuring the dynamic

properties of powders.

The annular shear cell is a widely used dynamic tester that has been developed to

measure the dynamic properties of powders. By continuously shearing powders at low shear






12

rates, the effect of the rate of strain on stress during shear, may be investigated. Savage and

Sayed [7], Hanes and Inman [8], and Craig et al. [9] used annular shear cells to study the

stress strain-rate relationship ofcohesionless granular materials in the fully mobilized regime

(rapid shearing regime). Recently, Tardos et al. [29] used a rotating cylindrical shear cell to

study the frictional flow of aerated powders over a range of shear rate.

An annular shear cell was fabricated for the present investigation to study the

theological behavior of small diameter cohesive powders. Fabricating a shear cell for small

diameter cohesive powders presents a variety of difficulties that are not encountered with

those used for large granular particles. Since the powder assembly deforms as it undergoes

shearing, dilatancy effects have a tendency to displace the powder from the shear cell.

Therefore, the shear cell must be sealed to confine the powder within the annular shearing

zone. This is a very difficult task because the powders encountered have very small particle

size, and cohesive powders have a very strong tendency to form shearing bands.

2.1.1.2. Description of Annular Shear Cell-type Rheometer

A simplified sketch of the annular shear cell-type powder rheometer is shown in

Figure 2-1. This rheometer satisfies the following requirements:

* Low rotating speed may be obtained in order that powders are tested in the low shear
rate range;

* Powders are not displaced from the shearing zone;

* Powders may be tested under different concentrations; and

* Both the shear stress and the normal stress may be measured at different shear rates
and different powder concentrations.

Generally, this facility consists of three components:









A drive train, which includes a DC motor and a pair of pulleys;

A test cell, which includes a rotating cell, a shearing plate, a shear plate positioner
and seal O-rings;

A data display system, which includes two load cells, a displacement transducer, a
digital tachometer and 3 digital panel meters.

Shear Plate
Positioner



Load Cell
(Normal)

Load Cell
(Tangential) 59Shear Plate
.- ---- 159.32mm -


O-ring Seal

9.40mm
Moment Arm Annular gap

LVDT I --- 44.09mm
Displacement --50.96mm Rotating Cell
Transducer









Pulley DC Motor


Figure 2-1 Annular shear cell-type cohesive powder rheometer

A detailed description of the rheometer is provided as follows. The powder is placed

in the annular gap between the upper shear plate and the bottom rotating cell. The annular

gap has a mean radius of 45.52 mm, a width of 6.87 mm, and a depth of 9.40 mm. The upper






14

shear plate and the bottom rotating cell maintain alignment by means of two linear rotating

bearings. The upper shear plate has an annular protrusion that fits into the annular gap. The

protrusion was cut from stainless steel that is an integral part of the upper shear plate. There

is a groove on each side of the protrusion walls. The powder is sealed within the annular gap

by means of an inner and an outer elastomer O-ring. The upper shear plate remains fixed via

a moment arm with a length of 159.30 mm as shown in the sketch. A load cell is attached

to the moment arm, which is used to measure the mean shear force imposed on the upper

shear plate by the shearing powder. The position of the upper shear plate is controlled via

a threaded shaft that is locked into a fixed position during a shear test. Thus, the volume of

the annular gap remains fixed. In order to assure that the entire particle assembly is in

motion during shearing so that the height of the shear layer corresponds to the powder bed

height, the powder bed height is maintained at approximately 20 particle diameters. A

precision instrument must be used to measure the powder bed height. For 30 mrn powder,

the largest allowable bed height is 600 rtm. Hence the position of the upper shear plate is

measured with an LVDT displacement transducer that is in contact with the moment arm.

The resulting normal stress due to powder compression and dilatancy is measured via a load

cell mounted between the shear plate and the positioner. The bottom rotating cell is driven

by a DC motor, and the rotational speed is controlled with a pulse width modulated power

source. The DC motor is connected to the bottom rotating cell via a pair of pulleys. The

pulley diameter ratio is 1:2.67; in other words, the speed of the rotating cell is 1 rpm when

the speed of the motor is 2.67 rpm. A digital tachometer is used to measure the rotational

speed of the rotating cell. Both load cells are Omega LCF-50 load cells, which have a load






15

range of-222.41N ~ 222.41N (-50 lb ~ +50 Ib). Three Omega DP41-s digital panel meters

are used as indicators and are connected to two load cells and the LVDT displacement

transducer, respectively.

Using this dynamic shear cell facility, extensive work has been done to investigate

the theological behavior of both cohesionless powders and cohesive powders in the frictional

regime by relating the measured stress components to the imposed shear rate. The stress

strain-rate relationship for different powders has been explored.

2.1.2. Experimental Procedures

The experimental procedure includes two major steps: sample preparation and data

collection. Prior to testing, a monolayer of powder is adhered to the shearing surface of the

shear plate using an adhesive. This provides a no-slip boundary condition on the shearing

surface. A very small amount of grease is adhered to both of the inner and outer O-rings, and

then these O-rings are filled into the grooves on the protrusion. The upper shear plate is

lowered until the shear plate is in contact with the bottom of the annular gap and then the

measured bottom bed height, ho, is recorded. The upper shear plate is then lifted in order to

allow powder being poured in the annular gap lately. Loosely packed powder is placed in

a dish and the powder mass, M0, is determined using an Ohaus digital balance. Powder is

added to the dish until the mass is approximately that required to achieve the desired powder

bed height in the annular gap. The powder is loosely and evenly tapped into the annular gap,

and the shear plate is lowered just until contact is made with the powder bed. The rotating

cell is slowly rotated by hand to even out the powder bed. The shear plate is then positioned

so that the desired powder concentration is achieved. The motor controller is used to adjust






16

the shear cell speed so that it is initially set to its lowest achievable steady value (4 rpm or

2 cm/s). When the speed is set below 3 rpm the rotation becomes unsteady.

The shear cell is allowed to rotate until the normal and shear stresses reach a steady

state, which is indicated by a constant normal and tangential load. The measured normal

load, F,, tangential load, F,, bed height, h, and rotational speed of the rotating cell, n, are

recorded. The shear cell speed is then incrementally increased and the raw data are again

recorded when steady state is reached. When sufficient data have been collected, the shear

plate is lifted until it no longer is in contact with the powder bed. The shear cell is again

rotated at various speeds and the resulting normal and tangential loads, f, and f,, are again

recorded. These are the frictional loads associated with the O-rings that seal the powder.

Finally, powder is taken off the annular gap and placed in a dish, and the net powder mass,

M, is determined using the Ohaus digital balance. This procedure is used to collect a locus

of data relating shear and normal stress to shear rate for a fixed powder concentration. This

procedure is repeated when powder is tested under different powder concentrations. In this

investigation, 3 to 4 different concentrations are chosen for each powder.


2.2. Data Reduction

The normal stress depends on the differential of the normal load F. and the no-

loading normal load f,, i.e. the normal stress is computed from:


F,- f n (2-2)
n Ir(),2Ri






17

where Ro = 50.96 mm is the outer annular radius, and R, = 44.09 mm is the inner annular

radius. The torque, T, acting on the shear plate is computed from:

T=(F,- f )L (2-3)

where Lo=159.3 mm is the moment arm length. The relation between the torque and the shear

stress is:


T= J 22rZ (2-4)

and thus the mean shear stress is computed from:


3(F f,)L,
S= (2-5)
2( R_- R3)

The mean shearing velocity is computed from:

2 2 rni R -,
P 0(2-6)

where n is the rotational speed in rpm. The powder concentration, a, is defined as the

fraction of the annular gap volume which is occupied by powder. The powder concentration

is computed from:

M
= P (2-7)
h n(R, R,)

where M is the mass of powder occupying the annular gap, p is the material density of the
powder, and h is the powder bed height. The uncertainties associated with the measurements
of shear stress, normal stress, shear rate, and powder concentration are 0.05 kPa, 0.05 kPa,
20 s-', and 0.005, respectively.








2.3. Cohesive Powder Properties

For this experimental investigation, several different kinds of cohesive powders have

been chosen. The first cohesive powder is a spherical silica powder and the second is an

industrial polymer powder used for powder coating applications, and the third one is a kaolin

powder. The geometrical configuration and surface structure of silica powder and polymer

powder have been explored using a scanning electron microscope (SEM). Figure 2-2 is a

SEM micrograph of typical silica powder particles. In general, the particles are spherical in

shape and are close to mono-disperse. In contrast, the industrial polymer powder shown in

Figure 2-3 is irregular in shape, has sharp edges, and has a wide range of particle size.

Figures 2-4, 2-5 and 2-6 show the particle size distribution for the silica, polymer and kaolin

powders, respectively. The mean particle diameter of the silica powder is 27.5 upm and the

standard deviation is 7.2 jim, while the mean particle diameter and standard deviation of the

polymer powder are 40.5 mun and 17.6 pm. The specific gravity of the silica powder is 2.34

g/cm3 and that of the polymer powder is 1.61 g/cm3. For kaolin powder, the mean particle
diameter is 6.0 pm with a standard deviation of 0.8 gm, and the specific gravity is 1.65g/cm3.

Table 2-1 summarizes the cohesive powder properties.

A Jenike shear cell tester was used to characterize the strength of the silica and

polymer powders. Figure 2-7 shows the yield locus for the two powders. It is seen that the

polymer powder assembly is generally stronger than that of the silica, even though the mean

diameter of the polymer powder is greater than that of the silica powder. The measured angle

of internal friction is 24.50 for the silica powder and 42.20 for the polymer powder. From

Figure 2-7, it is observed that the silica powder is nearly cohesionless, while the cohesion is

0.37 kPa for the polymer powder. These results are also summarized in Table 2-1. For

kaolin powder, a Jenike shear cell test has not been examined.








Table 2-1 Powder properties
Mean Standard Material Internal angle Cohesion E
Diameter Dm Deviation oD Density p of friction 4 (kPa)
Material (jtm) (pm) (g/cm3)
Polymer 40.5 17.6 1.61 42.20 0.37
Silica 27.5 7.2 2.34 24.50 -0
Kaolin 6.0 0.8 1.65 -


Figure 2-2 Silica powder SEM micrograph



































Figure 2-3 Polymer powder SEM micrograph


0 25 50 75 100 125 150 175 200 225 250
Diameter (pm)


Figure 2-4 Particle size distribution of silica powder



































0 25 50 75 100 125 150 175 200 225 250


Diameter (glm)


Figure 2-5 Particle size distribution of polymer powder


0 25 50 75 100 125 150 175 200 225 250

Diameter (pm)


Figure 2-6 Particle size distribution of kaolin powder


5



4



S3

..-
0. 2



1



0












Polymer Powder 4 =42.20
7 --Silica Powder 4 =24.50 A



1./
7/




2 7


07
0 2 4 6 8 10 12 14 16
Normal Stress a (kPa)



Figure 2-4 Yield Locus using Jenike shear cell


2.4. Extraneous Factors Influencing Experiments

Several factors may affect the outcome of experiments and the accuracy of the

measurements. Among them, the temperature and moisture are two main factors,

considering that cohesive powders have a very strong tendency to form shearing bands. Also

at high temperature the properties of powders may be different from those at low

temperature.

Within a moist environment, the sample of powders will have more cohesion. In the

most serious situation during testing, all powder attaches to the surface of the upper plate.

As a result, the shear and normal forces will not change when increasing or decreasing the

speed of the rotating plate. In this case, the experiment is considered a failure and needs to

be redone. Figures 2-8a and 2-8b show an experimental result when the powder is caked

into a solid. The powder tested is silica powder at a 0.49 concentration, with the rotating






23

speed increasing from low to high. To minimize the possibility of the caking and shear

banding, all powders are put in an oven for at least 8 hours before testing. The experiment

is conducted at a dry environment.

Temperature is another important factor that may affect the measurement. At

different temperatures, the physical properties of the powder may change, especially with

polymer powders. When a specimen of powder is sheared for an excessive period of time,

its temperature may significantly increase, and the accuracy of the data is affected.

Experience has shown that when the rotating plate runs continuously for more than 5

minutes, the powder temperature may reach 55 OC (131 OF). To limit the powder temperature

to below 300C (860F), each powder specimen is sheared for no longer than 3 minutes, and

each specimen is tested only one time for a concentration with rotating speed increasing from

low to high and then from high to low. The powder specimen is replaced for each powder

concentration considered.

The sealing efficiency of the O-rings is also a big issue for the success of an

experiment. Since the particle sizes of some powders considered are very small, the powder

particles have a tendency to slip beneath the O-rings. At higher shear rates, this problem can

be more severe due to dilatancy. This problem is clearly reflected in the normal stress

measurement. The normal stress is observed to decrease with increases in shear rate with a

loss of powder. Figures 2-9a and 2-9b show a typical normal and shear stress measurement

for 27 pm silica powder when powder loss occurs in high shear rate regime. When powder

loss occurs, O-rings are replaced and the experiment needs to be redone. Before each test,

the O-rings are carefully checked, cleaned and replaced when necessary. After each test, a

careful check is made to determine whether there is any evidence of powder loss.


































1500

Shear Rate (s')


Figure 2-8a


Normal stress variation with shear rate
when the powder is caked into a solid


500 1000 1500 2000 2500

Shear Rate (s'1)


Figure 2-8b Shear stress variation with shear rate
when the powder is caked into a solid


I I I I I I


Concentration
-e- 0.49


_A -- -- 6 --- -E ---


1000


2000


2500


3000


Concentration
-9- 0.49


4






S3




L
I.-
CO

(0
r2
o)


3000

































500


1000


1500


2000


2500


3000


Shear Rate (s')


Figure 2-9a Normal stress variation with shear rate when powder loss occurs


1000


1500

Shear Rate (s'1)


2000


2500


3000


Figure 2-9b Shear stress variation with shear rate when powder loss occurs


Concentration
- -- 0.52


Concentration
--- 0.52


--~


5



4


0-
3


t, 2
U)
(0


I I I








2.5. Powder Rheology

2.5.1. Cohesionless Powders

The theological properties of cohesionless powders are investigated in this research.

Two different powders, 300 pm silica powder and 100 pm silica powder, are tested. Figures

2-1 Oa and 2-1 0b respectively show the shear stress and normal stress as function of shear rate

at 4 different concentrations for 300 pm silica powder. Figures 2-1 la and 2-1 lb show the

shear stress and normal stress as function of shear rate at 4 different concentrations for 100

pm silica powder, respectively. For the purpose of comparing the results of the 300 uPm and

100 jtm silica powders, both powders are tested at the same set of concentrations. It is

observed that both the shear stress and the normal stress increase slightly with increasing

shear rate in the low shear rate regime, otherwise known as the frictional regime. At the

same concentration, the normal stress for both powders is not too much different in

magnitude while the shear stress of the 100 pm silica powder is slightly lower than that of

300 jm silica powder. The measured shear to normal stress ratio, '/o, is shown in Figure

2-12 as a function of the dimensionless shear rate( y [d, /g] 2) for both 300 pm and 100 pm

silica powders. y is the average shear rate across the shear cell powder bed. It is observed

that the data are collapsed. These results confirm Tardos et al. [29] observations that the

shear to normal stress ratio is invariant with shear rate in the frictional regime. In this

regime, the overwhelming interaction between individual particles is considered to be surface

friction, during which particles slide on top of each other during extend contacts. As the

shear rate increases the inter-particle collisions are stronger and a slight increase in shear

stress with shear rate is observed. With further increase in shear rate, the powder bed is

expected to become fully mobilized, where momentum transfer is dominated by inter-particle

collisions. Shear stress will demonstrate a nonlinear increase with the increase of shear rate.

Due to limitations in the construction of the rheometer, this regime was not achieved for the

cohesionless powders. Savage and Sayed [7], Hanes and Inman [8], and Craig et al. [9]






27

found that for granular flow, both the shear and normal stress depend quadratically on the

shear rate at constant bed porosity (or constant volume) in that regime. Qin [32] also found

that in the high shear rate regime, the shear stress has a quadratic dependence on the shear

rate when she tested 3mm glass beads using an annular parallel-plate shear cell. However

in the low shear rate regime, Qin [32] failed to capture the independence of shear to normal

stress ratio on shear rate. Instead, Qin found the stress ratio increases with the increase of

shear rate in the low shear rate regime. Qin recognized that there is inconsistency between

her experimental result and the theoretical prediction of Savage [33] which also indicates that

the stress ratio is independent of shear rate.


150
Shear Rate (s'1)


Figure 2-10a Shear stress variation with shear rate for 300 pm silica powder


~d/



















Q 4


|3
C)
-i
E
0z
o 2
Z


Shear Rate (s1)


Figure 2-10b Normal stress variation with shear rate for 300 ptm silica powder


0 200


1000 1200


Shear Rate (s")


Figure 2-11 a Normal stress variation with shear rate for 100 gtm silica powder


o


Concentration
-f- 0.502
-- 0.514
0.526
0.538


Concentration
-e- 0.502
-H- 0.514
0.526
0 0.538




/ ,n-.-*


3






2
a..
(g
0


I I I

















(0
C, 4
t(/
O 3

E
z 2
z


0 200


1000


1200


Shear Rate (s1)

Figure 2-11 b Shear stress variation with shear rate for 100 gtm silica powder


1.5 k


S1.0


srl00 vs ts
sr300 vs t/s(300)


A
g~h 4 L


A ZAAA .
Q a LS aA" t.


0.0 '1
0.01 0.1 1


y(d /g)1/2


Figure 2-12 Shear to normal stress ratio as a function of dimensionless shear
rate( y [d /g]'/2) for both 300g m and 100 pm silica powders


Concentration
-E- 0.502
- 0.514
. 0.526
S 0.538


0.5 F


I I


1


........ ..








2.5.2. Cohesive Powders

As mentioned earlier three different cohesive powders are considered: 27 mrn silica

powder, 5 gim kaolin powder, and a series of polymer powders. Figure 2-13a shows the

measured shear stress as a function of shear rate for silica powder at three different

concentrations. At the lowest powder concentration, a= 0.46, the powder bed height is 450

ptm, which corresponds to approximately 16 particle diameters based on the mean diameter.
The variation of shear stress with shear rate follows a similar trend for all three powder

concentrations. At very low shear rate, the powder shear stress increases slightly with

increasing shear rate. As the shear stress is further increased, the shear stress begins to

decline and reaches a minimum. This behavior is characteristic of powder flow in the

frictional regime. As the shear rate is further increased, inter-particle collisions become more

dominant and the shear stress begins to increase with increasing shear rate. This behavior

is characteristic of a mobilized powder. Therefore, the minimum in the shear stress is

indicative of a transition from frictional to mobilized flow. The magnitude of the shear stress

increases with increasing powder concentration. Figure 2-13b shows the behavior of the

normal stress with shear rate at three different silica powder concentrations. In the low shear

rate regime, the normal stress is relatively flat. When the powder flow is in the frictional

regime, it is expected that the normal stress will not significantly increase since inter-particle

collisions are relatively weak. However, when the powder is mobilized, inter-particle

collisions become energetic and the normal stress is expected to increase with increasing

shear rate. Figure 2-13b clearly shows the increase in normal stress with increasing shear

rate in the mobilized regime. Figure 2-13c shows the ratio of the shear to normal stress ratio

with increasing shear rate. The internal angle of friction measured using the Jenike Shear

cell is 24.50 and corresponds to a shear-to-normal stress ratio of 0.46. The shear-to-normal

stress ratio measured using the annular shear cell approaches approximately 0.45 in the zero

shear rate limit. The comparison is good considering the substantial differences in the






31

measuring technique. It is worth noting that in the frictional flow regime at low shear rate,

the shear to normal stress ratio increases with increasing shear rate until a critical shear rate

is reached, at which point the shear to normal stress ratio begins decreasing prior to particle

mobilization. This behavior essentially follows the same trend as the shear stress since the

normal stress does not increase significantly in the frictional regime. In the frictional regime,

the data approximately collapse into a single curve, independent of the powder concentration.

This is very important because the features relating the stress tensor to the rate of

deformation of the powder assembly may be used to formulate a rheology model for a given

powder to be used in a continuum flow prediction model. Figures 2-13d and 2-13e show the

shear and normal stress as a function of shear rate for 27jum silica powder at very low shear

rate range. It is observed that both the normal and shear stress change very slightly in this

regime.

Figures 2-13f, 2-13g and 2-13h show a group of measured data for another 27um

silica powder experiment. It is observed that the variations of normal stress, shear stress and

shear to normal stress ratio with shear rate are consistent with those shown for the previous

experiment. It demonstrates that the experimental results are repeatable. Further more, the

normal and shear stresses are consistent when measured with the rotating speed of the shear

plate increasing from both low to high and then high to low. Figures 2-13i and 2-13j

respectively show the comparisons of shear and normal stress of 27 jIm silica powder at

0.491 concentration, when measured with the rotating speed increasing from low to high and

then decreasing from high to low.




















0.
-d 3
v

U)
r 2
4)


1000


1500


2000


2500


3000


Shear Rate (s"1)


Figure 2-13a Shear stress variation with shear rate for 27 tm silica powder


Concentration
-e- 0.464
7 -+- 0.491
0.522

6


CO


0

S3


2


1
0 500 1000 1500 2000 2500 3000

Shear Rate (s-1)


Figure 2-13b Normal stress variation with shear rate for 27 um silica powder


/'


entration


-e- 0.464
-9- 0.491
- 0.522


-^ Cono
































500 1000 1500 2000 2500
Shear Rate (s1)


3000


Figure 2-13c Shear to normal stress ratio as a function of shear
rate for 27 itm silica powder


4




3




(,

r)

1t--~ -f-





0
0 50 100


Figure 2-13d


150 200 250 300
Shear Rate (s1)


Shear stress variation with shear rate at low
shear rate range for 27 pm silica powder























- 0 0 -
Concentrtation
-e- 0.464
-B- 0.491
S 0.522

0 50 100 150 200 250 300 350
Shear Rate (s-1)


Figure 2-13e



16


14

12 -

10 -


Normal stress variation with shear rate at low shear
rate range for 27 jLm silica powder


1000


1500


2000


2500


3000


Shear Rate V/H (s1)


Figure 2-13f Shear stress variation with shear rate for 27 ptm silica powder


Concentration
-- 0.464
-*- 0.491
A- 0.522


I I 1 i 1 1


~?'


,z













14 -

12 -

10


1000


1500


2000


2500


3000


Shear rate V/H (s-1)


Figure 2-13g Normal stress variation with shear rate for 27 tm silica powder


1.8 -

1.6

1.4


1.2

S1.0


0.8

0.6 -

0.4 -


500 1000 1500 2000 2500 3000
Shear rate V/H (s1)


Figure 2-13h


Shear to normal stress ratio as a function of shear
rate for 27 tim silica powder


Concentrtation
-e- 0.464
-E- 0.491
S 0.522


I I I


Concentration
--- 0.464
-8- 0.491
- 0.522


r1


I


ii'














12 1-


500 100


0 1500 2000
Shear Rate V/H (s1)


2500


3000


Figure 2-13i Comparison of shear stress for 27 mm silica powder when
Measured with speed varying from low to high and high to low


16, i


14

12


1000


1500 2000
Shear rate V/H (s'1)


2500


3000


Figure 2-13j Comparison of normal stress for 27 mm silica powder when
Measured with speed varying from low to high and high to low


Concentration 0.491
-e- Speed from low to high
-e- Speed from high to low








~a,-"ED


Concentration 0.491
-- Speed from low to high
-B- Speed from high to low


I I I I I I


-------- 19





































0 2000 4000 6000 8000

Shear Rate (s"1)


3






2
C,)




I







0


Figure 2-14a Shear stress variation with shear rate for 5 tim kaolin powder


4,-------1 1 1


U)
)3






02
Z


0 2000 4000 6000 8000

Shear Rate (s-1)


10000 12000 14000


Figure 2-14b Normal stress variation with shear rate for 5 pm kaolin powder


10000 12000


Concentration
-e-- 0.243
-- 0.292
0.364
--0- 0.486
-I\ I


1400C


-c .


I











2.0

Concentration
-e- 0.243
1.5 -- 0.292
0.364
0.486

1.0 .



0.5



0.0
0 2000 4000 6000 8000 10000 12000 14000
Shear Rate (s-1)

Figure 2-14c Shear to normal stress ratio as a function of shear
rate for 5 ptm kaolin powder

Figure 2-14a shows the measured shear stress as a fun tion of shear rate for kaolin

powder at four different concentrations. The variation of shear stress with shear rate for

kaolin powder is similar to that for silica powder. At very low shear rate, the powder shear

stress increases slightly with increasing shear rate. As the shear stress is further increased,

the shear stress begins to decline and reaches a minimum. As the shear rate is further

increased, the shear stress begins to increase with increasing shear rate. Considering the

difficulties of the experiment with very thin powder layers in the test bed and the small

particle size of kaolin powder, the kaolin powder base bed is more than 20 particle mean

diameter high. At the lowest powder concentration, a = 0.22, the powder bed height is 400

mrn, which corresponds to approximately 67 particle diameters based on the mean diameter.

Figure 2-14b shows the behavior of the normal stress with shear rate at four different kaolin

powder concentrations. Similar to silica powder, in the low shear rate regime, the normal

stress is relatively flat. When the powder is mobilized, the normal stress increases with






39

increasing shear rate. Due to the smaller particle size, both the normal stress and shear stress

for kaolin powder are a little larger than those of silica powder when compared at the same

concentration and shear rate. Figure 2-14c shows the ratio of the shear to normal stress ratio

with increasing shear rate. In the frictional regime, the data approximately collapse into a

single curve, independent of the powder concentration. The shear-to-normal stress ratio

measured using the annular shear cell approaches approximately 0.65 in the zero shear rate

limit. Kaolin powder is significantly more cohesive than silica powder.

A series of polymer powders have been tested in this investigation. The scanning

electron microscope monograph and the properties of polymer powder #1 is previously

shown as Figure 2-3 and its properties are summarized in Table 1. The measured shear stress

as a function of shear rate is shown in Figure 2-15a for the polymer powder #1. For the shear

rate range covered, the powder remains in the frictional regime. Measurements at higher

shear rates could not be achieved because powder fines could not be sealed within the

annular gap at high shear rate. The variation of shear stress with shear rate follows the same

trend as the silica powder and kaolin powder in the frictional regime. At low shear rates,

the shear stress increases with increasing shear rate until a critical shear rate is reached, at

which point the shear stress decreases with increasing shear rate. The magnitude of shear

stress increases with increasing powder concentration. At the lowest powder concentration,

a=0.40, the powder bed height is 788 ptm, which corresponds to approximately 19 particle

diameters based on the mean diameter. The variation of the normal stress with increasing

shear rate is shown in Figure 2-15b. There is very little increase in normal stress with

increasing shear rate. This behavior is indicative of the fact that the powder flow is in the

fictional regime where the stress contribution from inter-particle collisions is small. The

shear-to-normal stress ratio variation with shear rate is shown in Figure 2-15c. As the shear

rate approaches zero, the shear-to-normal stress ratio approaches approximately 0.80. The

internal angle of friction measured using the Jenike shear cell is 42.20 and corresponds to a






40
shear-to-normal stress ratio of 0.91. The comparison is reasonable. Since the normal stress

does not vary significantly with shear rate, the shear- to-normal stress variation with shear

rate follows the same trend as the shear stress variation. It is also worthwhile noticing that

at a given shear rate and powder concentration, the magnitude of the measured shear stress

is considerably higher for the polymer powder than that for the silica powder. Clearly this

is related to differences in particle shape, size distribution, surface morphology, and possibly

chemical composition.

Figure 2-16a shows the measured shear stress as a function of shear rate at different

concentrations for polymer powder #2. Same as powder #1, for the shear rate range covered,

powder #2 remains in the frictional regime. Figure 2-16b and Figure 2-16c show the

variation of the normal stress with shear rate and the shear to normal stress ratio with

increasing shear rate at different concentrations for polymer powder #2, respectively. The

powder rheology of polymer powder #2 behavior is similar to that of polymer powder #1.

Powder #2 requires larger normal stress and larger shear stress than powder #1 to achieve a

specified concentration and shear rate. Therefore, polymer powder #1 has better flowability

characteristics than powder #2. The shear to normal stress as a function of shear rate is

independent of concentration and nearly identical for both powders. As the shear rate

approaches zero, the shear-to-normal stress ratio approaches approximately 0.80 for powder

#2. Polymer powders #1 has better flowability because it requires less normal stress to

achieve a specified powder concentration; therefore powder flowability is influenced by

powder rheology and consolidation behavior. Figures 2-15d and 2-16d show the particle size

distribution for polymer powder #1 and powder #2, respectively. Compared to powder #2,

powder #1 has a narrower particle size distribution and less fines. The mean diameter of

powder #1 is 40.5 jim and 45.5 pim for powder #2. But the density of powder #1 is 1.61,

only a little higher than that of powder #2, 1.58. Powder #1 has more air pockets between

the particles while powder #2 has more fines.




















a



3
0)
v4
-c


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s )



Figure 2-15a Shear stress variation with shear rate for 40.5 gm polymer powder #1


7 I--- -- i --- i --- ,I i --i--- i ---


6 -


5 -










-0--- 0.3520
S 0.3959




0.4525


-_- 0.35207
SG- 0.3959







Shear Rate (s-1



Figure 2-15b Normal stress variation with shear rate for 40.5 gm polymer powder #1
Figure 2-15b Normal stress variation with shear rate for 40.5 ulm polymer powder #l

























0.5 F


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s-1)


Figure 2-15c Shear to normal stress ratio as a function of shear
rate for 40.5 pmr polymer powder #1


50 100 150 200


Particle Diameter (pm)


Figure2-15d Particle size distribution for 40.5 pim polymer powder #1


Concentration
-e- 0.3520
-e- 0.3959
- 0.4525
- -- 0.5279



















0.
4


5 3
,c
(i)


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s-1)




Figure 2-16a Shear stress variation with shear rate for 45.5 jpm polymer powder #2


G0-- -- 0 -


Concentration
E- 0.3520
-- 0.3959
S 0.4525
- 0.5279


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s-1)


Figure 2-16b Normal stress variation with shear rate for 45.5 pjm polymer powder #2


J


Concentration
-e- 0.3520
-*- 0.3959
- 0.4525
-v- 0.5279





v v v v -v


i- i -- --_ __


~aa~_e~-e,







44


2.0
Concentration
-e- 0.3520
-3- 0.3959
1.5 0.4525
---- 0.5279



1.0




0.5




0.0
0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s-1)

Figure 2-15c Shear to normal stress ratio as a function of shear
rate for 45.5 p.m polymer powder #2


20

18

16

14
.R
12 -

10 -



6

4

2

0
0 50 100 150 200 250


Particle Diameter (pm)


Figure 2-16d Particle size distribution for 45.5 tim polymer powder #2






45

Another 6 polymer powders were chosen to undergo the annular shear cell testing.

Figures 2-17a 2-22d show the experimental results for these powders. These figures are

shown at the end of this chapter. These total 8 polymer powders are divided into 4 groups.

Table 2-2 shows the properties of these powders. The quality of fluidization indicated in

Table 2-2 is based on gas fluidization of the powder in an industrial facility. For each group,

the powder with good fluidization has lower normal stress and shear stress; this is indicative

that the powder rheology is a good indicator of powder flowability.

Table 2-2. Properties of polymer powders
No. Powders No. Fluidization Density(g/cm3) Mean Diameter(upm)
1 IF6855 LOT TCR420 Good 1.61 40.5
2 IF6855 LOT TRR209 Bad 1.58 45.5
3 IF5159 BLUE Okay 1.50 15.7
4 IF5159L(from GE) Bad 1.50 14.5
5 IF2429 lot4780413308 Bad 1.45 32.3
6 IF24290 lot4780729331 Okay 1.45 57.0
7 6593-202E Sample 2 Marginal 1.60 44.0
8 6593-202E Sample 2P Okay 1.60 43.9


2.6. Discussion

The fact that cohesive powders form particle clusters must be accounted for when

considering resistance to flow. Particle clusters existing in cohesive powders is the unique

factor that distinguishes cohesive powder from cohesionless powder. Friction and cluster

break-up are the main mechanisms responsible for resistance to flow in the frictional regime.

Cluster break-up is the main factor distinguishing cohesive and cohesionless powders. When

a powder assembly is sheared, there is a tendency for the powder clusters to break apart. The

shear force must overcome the cohesive binding force keeping the clusters in tact. Shearing

energy is dissipated in the breaking apart of powder clusters. When clusters of powder

particles are initially torn apart at low shear rate, the powder particles have low kinetic

energy, and inter-particle contact is sustained so that the frictional stress remains relatively






46
uniform. As a result, the total shear stress increases with the increase of shear rate. As the

shear rate increases beyond a threshold, powder particles become more energized, inter-

particle contact is not sustained, and powder clusters are not as prevalent. The net result is

a decrease in the total shear stress. This is a possible explanation for the peak in shear to

normal stress ratio variation with shear rate observed for cohesive powders. With further

increases in shear rate, the powder is mobilized, and momentum transfer is dominated by

inter-particle collisions just as with cohesionless powders. In the high shear rate regime,

both cohesive powder flows and cohesionless powder follow the similar flow features.

A constitutive model relating the stress tensor to the rate of deformation for an

incompressible Coulomb powder was given by Schaeffer [30] and explored by Tardos [27]

and Tardos et al. [29] and is given in Equation (1-3).

Applying that constitutive model to the simple Couette flow in the annular shear cell,

results in a = p and t = p sino. It is recognized that the constitutive model given by

Equation (1-3) predicts that the shear stress depends only on the pressure and internal angle

of friction. This might be correct for cohesionless powders in the frictional regime, but

clearly this is not correct for cohesive powders since the shear-to-normal stress ratio

displayed in Figure 2-13c, 2-14c and 2-15c are shear rate-dependent. The predicted shear

stress from Equation (1-3) is in agreement with experiments only in the limiting case where

the shear rate approaches zero.

Two shortcomings associated with the constitutive model given in Equation (1-3) are:

(1) the inter-particle friction is not necessarily constant with shear rate, and (2) in addition

to friction, there exists a mechanism associated with cohesion which provides a resistance

to shearing. Evidence that inter-particle friction is not constant with shear rate is provided

by Ancey et al. [28], where it was found that the measured shear stress decreases with






47

increasing shear rate for frictional flow of glass beads down an incline. Mei et al. [21 ] used

direct element simulations of powder in simple shear flow to demonstrate that the shear

stress decreases with increasing shear rate in the frictional flow regime. Cui [34] obtained

theological measurements for frictional flow of highly concentrated suspensions in a shear

cell device. Similar to the trends observed for the current powder flow measurements, Cui

found that the shear stress initially increases with shear rate prior to fluidization.

Although the constitutive stress-strain rate relationship given in Equation (1-3) has

its flaws, it does suppose that the stress tensor depends on pressure, independent of the

powder concentration. The shear-to-normal stress ratio measurements reported for the

frictional flow regime support this supposition. Although this observation requires further

investigation, it should prove quite useful in constitutive modeling efforts.

It remains a challenge to identify the mechanisms controlling the powder rheology

at the micro-scale so that a constitutive model relating the stress tensor to the rate of

deformation, may be developed that is in agreement with experimental measurements.

































0 200 400 600 800


1000 1200 1400 1600


1 I I


Shear Rate (s"1)



Figure 2-17a Shear stress variation with shear rate for 15.7 tpm polymer powder#3



7,


0 200 400 600 800


1000 1200 1400 160C


Shear Rate (s-1)


Figure 2-17b Normal stress variation with shear rate for 15.7tm polymer powder #3


Concentration
--- 0.3740
-E- 0.4208
0.4809
-V- 0.5610


[
--gV -----"3 ---^ -- -





Concentration
S----~e---- -- 0.3740
-e- 0.4208
0.4809
-v- 0.5610
I I I I I I


.-. 1,1-

























0.5 F


0.0 1 I I
0 200 400 600 800 1000 1200 1400

Shear Rate (s-1)

Figure 2-17c Shear to normal stress ratio as a function of shear
rate for 15.7 gm polymer powder #3


0 50 100 150


200 250 300 350 400 450 500


Particle Diameter (rLm)


Figure 2-17d Particle size distribution for 15.7 mrn polymer powder #3


Concentration
-&- 0.3740
-B- 0.4208
S 0.4809
-- 0.5610


1600


-1 1.0











I I I


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s-1)




Figure 2-18a Shear stress variation with shear rate for 14.5 [tm polymer powder #4




7 1


7 7


0 0 p 0 Ce- ', -


Concentration
-- 0.3740
-*- 0.4208
0.4809
--- 0.5610


0 200 400 600 800 1000 1200 1400

Shear Rate (s"1)


Figure 2-18b Normal stress variation with shear rate for 14.5 itm polymer powder #4


Concentration
-e- 0.3740
-e- 0.4208
0.4809
-- 0.5610


1600


0y

































0 200 400


600 800 1000 1200 1400


Shear Rate (s"')


Figure 2-18c


Shear to normal stress ratio as a function of shear
rate for 14.5 pLm polymer powder #4


20

18

16

14

12

10

2 8
6

4

2

0
0 50 100 150 200 250 300 350 400 450 500


Particle Diameter (pm)


Figure 2-18d Particle size distribution for 14.5 p.m polymer powder #4


Concentration
- 0.3740
. -- 0.4208
-- 0.4809
--- 0.5610


1600


















(-
4


S3
4)


0 200 400 600 800 1000 1200 1400 1600
Shear Rate (s-1)



Figure 2-19a Shear stress variation with shear rate for 32.3 lrm polymer powder #5


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s1)


Figure 2-19b Normal stress variation with shear rate for 32.3 itm polymer powder #5




































600 800 1000

Shear Rate ( sL )


1200 1400 1600


Figure 2-19c


Shear to normal stress ratio as a function of shear
rate for 32.5 tm polymer powder #5


100 200 300 400 500

Particle Size (micron)


Figure 2-19d Particle size distribution for 32.3 pm polymer powder #5


Concentration
--- 0.3743
-e- 0.4211
S 0.4813
0.5615


1.5 F


S1.0 -


0.5 -


0 200


r t I t r r

































0 200 400 600 800 1000 1200 1400


1600


Shear Rate (s-1)



Figure 2-20a Shear stress variation with shear rate for 57.0 gm polymer powder #6


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s1)


Figure 2-20b Normal stress variation with shear rate for 57.0 jim polymer powder #6


Concentration
-e- 0.3743
-*- 0.4211
S 0.4813
---- 0.5615


I I


Concentration
-e- 0.3743
-9- 0.4211
S 0.4813
-V- 0.5615


7


I












4





3





I-


0 200 400 600 800 1000

Shear Rate ( s' )


1200 1400


Figure 2-20c Shear to normal stress ratio as a function of shear
rate for 57.0 itm polymer powder #6


100 200 300 400 500
Particle Size (micron)


Figure 2-20d Particle size distribution for 57.0 ptm polymer powder #6


Concentration
- --- 0.3743
-e- 0.4211
S 0.4813
-A- 0.5615


I I I I




























Concentration
-e- 0.3419
S0.3847
S 0.4396
-- 0.5129

0 200 400 600 800 1000 1200 1400 1600


Shear Rate (s"1)



Figure 2-21 a Shear stress variation with shear rate for 44.0 um polymer powder #7


5

a-
(4


" 3
o
z


- I- [1


C 0 e0 0 0 0


Concentration
-e- 0.3419
-H- 0.3847
0.4396
-V- 0.5129


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s'1)


Figure 2-21 b Normal stress variation with shear rate for 44.0 umr polymer powder #7


I I I I I I i


~t ___g---SL-----ft----_R































400 600 800 1000 1200 1400 1600


Shear Rate ( s- )


Figure 2-21c


Shear to normal stress ratio as a function of shear
rate for 44.0 jpm polymer powder #7


100 200 300 400 500


Particle Size (umn)


Figure 2-21 d Particle size distribution for 44.0 plm polymer powder #7


Concentration
--- 0.3419
-e- 0.3847
- 0.4396
-A- 0.5129


0 200




































0 200 400 600 800 1000 1200 1400


1600


Shear Rate (s1)




Figure 2-22a Shear stress variation with shear rate for 43.9 pm polymer powder #8


0 200 400 600 800 1000 1200 1400 1600

Shear Rate (s1)


Figure 2-22b Normal stress variation with shear rate for 43.9 um polymer powder #8


Concentration
E-- 0.3419
0.3847
0.4396
0.5129


1 1 1 r


I


































800 1000 1200 1400


Shear Rate V/H ( s" )


Figure 2-22c



12


10 -


8-


Shear to normal stress ratio as a function of shear
rate for 43.9 gim polymer powder #8


0 100 200 300 400 500
Particle Size (pm)


Figure 2-22d Particle size distribution for 43.9 pm polymer powder #8


Concentration
-9- 0.3419
-E- 0.3847
--. 0.4396
-A- 0.5129


0 200


1600









CHAPTER 3
POWDER PLUG FLOW EXPERIMENT


3.1. Introduction

Pneumatic conveying involves the use of gas to transport particulate materials

through a pipeline. It has been successfully used for many years in chemical and process

industries for the transport of granular and fine powders such as flour, granular chemicals,

lime and coal. According to the mode of gas-solid flow in the pipeline, pneumatic

conveying may be classified as dilute flow or dense flow. In dilute flow, the majority of the

particles are suspended in the conveying gas. In dense flow, the transport velocity is less

than that necessary to suspend the particles in the conveying gas, and the majority of the

particles are not suspended. Plug flow is one of the distinct modes of dense flow.

Plug flow conveying of solids is a transport technique for delivering a large quantity

of material in a pipeline at low velocity. It is an attractive method of transport due to savings

in pumping energy that can be achieved when successfully deployed. Plug flow transport is

found useful in coal gasification and liquefaction, hydrocarbon catalytic cracking, and oil

shale processing plants[35]. However, due to inter-particle cohesion, the complex nature of

the flow, and limitations in mathematical modeling, there remain many challenges in

developing robust industrial plug flow transport facilities and reliable predictive capabilities.

Experimental investigations and modeling attempts of pneumatically transported plugs have

been reported in the literature. Gu and Klinzing [36], Dhodapkar et al. [37], Plasynski and

Klinzing [38], Laouar and Molodtsof [39], Molerus and Heucke [40], and Konrad [41] have

60






61

extensively studied pressure drop in dense phase pneumatic transport over a range of

velocities with both granular and cohesive type particles for vertical, horizontal, and inclined

flow orientations. Dhodapkar et al. [37] studied the plug flow movement of solids, Aziz and

Klinzing [42] investigated the plug flow transport of cohesive coal for both horizontal and

inclined flow orientations, and Molerus and Heucke [40] also explored the pneumatic

transport of coarse grained particles in horizontal pipes. Meanwhile analytical attempts at

modeling plug flow have been made by Konrad [41], Aziz and Klinzing [42], Borzone and

Klinzing [43], and Levy et al. [44]. The variation of pressure across a moving plug with

varying plug length and gas flow rate have been explored. The experimental data as well as

the analysis show that the pressure drop across powder plugs varies linearly with plug length

for all flow orientations.

Figure 3-1 shows the pressure drop across moving plugs as a function of plug length

in a horizontal pipe for Pittsburgh coal at different gas flow rates as reported by Aziz and

Klinzing [42]. However, the prediction of Aziz and Klinzing [42] shown in Figure 3-1 failed

to capture the pressure drop dependence on air volumetric flow rate. Instead, they suggested

that the pressure drop is independent of the air volumetric flow rate, despite the fact that the

data suggest otherwise. In their research, powder is treated as a Coulomb material and the

wall friction is determined based on the internal angle of friction. In the limiting case of

vertical flow, where the pressure gradient is dominated by gravity, the pressure drop appears

to be independent of gas flow rate. However, in the other limiting case of horizontal flow,

where the gravitational influence is minimal, the pressure gradient is dominated by wall shear

stress, and the pressure gradient shows a substantial dependence on gas flow rate. In this

case, analyses that treat powder as a Coulomb material and determine the wall friction based






62

on the internal angle of friction cannot capture the dependence of pressure gradient on gas

flow rate.


0.25 0.30 0.35 0.40 0.45
PLUG LENGTH (m)


0.50 0.55


Figure 3-1 Variation of pressure drop with plug length for
Pittsburgh coal (data from Aziz and Klinzing[42])

In fact, as a powder plug is conveyed through a pipeline, there exists a layer of

powder material adjacent to the wall in a highly dynamic state, and plastic continuum soil

mechanics type models do not accurately predict the wall shear when the powder is in a

dynamic state. This is clearly demonstrated in the horizontal powder plug flow experiments

of Hirota et al. [45], where the pressure gradient shows a strong dependence on the plug

velocity. In order to account for this dependence, Hirota et al.[45] introduced two dynamic

friction coefficients, the dynamic internal friction coefficient and the dynamic wall friction





63
coefficient, which were measured using a parallel plate shear tester. Figure 3-2 shows a
comparison of experimental and theoretical results from Hirota et al. [45] that shows the
variation of pressure drop with silica powder flow rate. The solid line and the broken line
represent the predictions calculated by using dynamic wall friction coefficient and dynamic
internal friction coefficient, respectively. It is observed that the predictions do not agree well
with the experimental results.


1.5x 104
Silica


I xl10' o
w. O

Tsuji 40




Morikawa

0 2xl03 4x10* 6xl02
(I -E)pg (N/nm)

Figure 3-2 Variation of pressure drop with flow rate of powder
(From Hirota et al.[45])

In this work, a simple powder plug flow experiment has been set up to investigate the
dependence of pressure drop on gas flow rate and plug length for cohesive powder plug
transport through a 19.05 mm (3/4 in.) I.D. horizontal pipe. The transport of two different







64

powders were investigated, 27 pLm silica and 5 p.m kaolin. The pressure drop and powder

plug velocity are measured at different gas flow rates (147-206 cm'/s) and different plug

lengths (3.00-4.50 cm). The main objective of this work is to make bulk flow measurements

and determine whether a continuum flow model, based on theological properties, is sufficient

for predicting bulk flow behavior. Using this facility, packed powder plugs are transported

down the horizontal test section---a circular transparent pipe, with a uniform gas flow rate.

By using a stroboscope and a digital camera, the plug velocity is measured. The pressure

drop across the plug is measured with a differential pressure transducer. It is of interest to

explore the efficacy of the continuum model (to be described in Chapter 4) in predicting the

pressure gradient with a known gas transport velocity and plug length.




Computer & Computer &
Air Compressor Data Acquisition Digital Camera Frame Grabber


Differential
Voltage Source P
..... Pressure
l TTransducer
Pressure Gage -

Valve


LII


Stroboscope


Pressure
Regulator

Desiccant


Mass-flow
Regulator &
Transducer


Figure 3-3 Powder Plug Flow Experiment Facility


19.1mm ID
Flow Tube


Powder
Collection
Chamber








3.2. Experimental Facility

The schematic diagram of the experimental facility used in this work is shown in

Figure 3-3. The facility includes five main components: an air supply system, a test section,

a data acquisition system, a digital imaging system, and a powder collection chamber.

The air supply system includes an air compressor, supply tank, a desiccant drier, a

pressure gage, a pressure regulator, and a mass flow controller. The compressor is a Dayton

Cast Iron Series. Compressed air is stored in a 0.227 m3 (60 gallons) storage tank. From the

storage tank, the air passes through a desiccant drier that removes moisture from the air.

This is important so that the air does not alter the cohesiveness of the powder plug. A valve

is provided to control the pressure of the air entering the pressure regulator. A pressure

regulator is used to maintain a pressure of 552 kPa (80 psi) at the inlet to the mass flow

controller. The varied or controlled pressure, coming out of the pressure regulator can be

checked when the air passes through the pressure gauge. The air pressure can be adjusted

by turning a knob on the pressure regulator. Air flows from the tank to the regulator through

a 12.7 mm (0.5 inch) steel pipe. From the regulator, the air is passed through a 6.4 mm (1/4

inch) diameter polyethylene tube to the inlet of the flow controller. The mass flow controller

is used to control and measure the flow rate of the air passing through the facility. The

Kobold mass flow controller regulates the air flow rate from to 0.064 to 0.378 g/s. The

controller set point is controlled with a 0-5 V DC input that is proportional to the flow rate;

it also has a 0-5 V DC output that indicates the measured mass flow rate. This signal is

output to the digital data acquisition system and is used to monitor the air mass flow rate

with time. In this investigation, the mass flow controller is calibrated by experiment, and the

correlation of the mass flow rate and the output voltage is measured. Using the air density






66

flowing through the test section, the volumetric flow rate (cm3/s) of air is determined, which

ranges from 52 to 309 cm3/s. The mass flow controller was calibrated prior to

experimentation.

The visual test section consists ofa 2 m long transparent tube with an inner diameter

of 19.1 mm (3/4 inch). The tube connects to two fittings and is easily disconnected in order

to place powder plugs in the system. At the end of the tube, the fittings connect with the

powder collection chamber.

The powder collector consists of a large tank with filters covering the exit. This

chamber may be opened to collect powder for recycling purposes.

The data acquisition system includes two subsystems: one is the mass flow controller

that is used measure the gas flow rate; another one is a Validyne differential pressure

transducer. The Validyne differential pressure transducer is used to measure the pressure

drop across the powder plug. Two polyethylene lines connect the transducer to the inlet and

outlet of the test section. The analog signal from the transducer is output to a computer via

an analog and digital I/O card installed in that computer. By running the data acquisition

program written for the current experiments, the pressure drop may be measured, with an

uncertainty of 0.007 kPa. The variation of pressure drop with time is also measured.

The digital imaging system includes a Videk Megaplus camera, a stroboscope with

a frequency range of 110-150000 rpm, a Sony PVM-122 Video monitor, and a computer

which works as a frame grabber. By illuminating the test section with the stroboscope, the

Videk Megaplus Camera is used to capturing multiple plug images when a plug passes

through the tube. The camera works in two modes: a single shooting mode and a continuous

shooting mode, where the camera takes at most 13 pictures consecutively. This camera






67

features a high resolution Charge Coupled Device (CCD) array containing 1320(H) x 1035

(V) light-sensitive picture elements (pixels), and a built-in 8 bit A/D converter that produces

a digital video output signal containing 256 gray levels. This output signal is transferred to

both the computer and the high resolution (40 lines/mm) Sony monitor. The 1320 x 1035

pixel resolution plug images are displayed on the monitor and may be saved in the computer.

A linear scale is mounted beneath the test section in order that the actual distance translated

between successive images may be determined from the digital images. A typical digital

photograph capturing multiple plug images is shown in Figure 3-4 where the air superficial

velocity is 52.0 cm/s and the plug length is 35.0 cm. The uncertainty in the velocity

measurement is 0.5 cm/s.


Figure 3-4


Successive image of kaolin plug conveying through
9.1mm ID duct: U0=52.0cm/s; L=35mm









3.3. Calibrations

3.3.1. Calibration of the Mass Flow Controller

The mass flow controller is calibrated by using air to drive a light plastic piston

through the test section, and using the digital imaging system to measure the piston velocity.

By measuring the mean air velocity driving the piston, the volumetric flow rate may be

computed. These measurements are compared with the output voltage from the mass flow

controller. Figure 3-5 shows the calibration curve of gas volumetric flow rate vs flow

controller voltage. The measured results are from two separate experiments, and the linear

line is the best fit.


0.05 0.10 0.15 0.20 0.25
Voltage (v)


0.30 0.35 0.40


Figure 3-5 Gas volumetric flow rate as a function of voltage


E


0

u.


100

90

80

70

60

50

40

30

20

10

0
0.00


S FlowRate38.73+135oltage
-Flow Rate=38.73+135*Voltage


0.45







69

It is found that the mass flow controller has a threshold flow rate below which there

is no flow. The minimum voltage value is 0.10 Volt, which corresponds to a flow rate of

52.23 cm3/s. Due to this constraint on the mass flow controller, all experiments are done

with a volumetric flow rate larger than 52.23 cm3/s. The uncertainty in the airflow

measurement is 2 cm3/s.

3.3.2. Calibration of the Validvne Pressure Transducer


2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0
0.0


1.2 1.4 1.6 1.8 2.0


Figure 3-6 the Pressure drop variation with the output voltage signal


The Validyne pressure transducer is calibrated against a standard manometer. The

standard manometer is connected to the Validyne pressure transducer and a pressure source.

By measuring the output signal of the Validyne pressure transducer and the corresponding


0.2 0.4 0.6 0.8 1.0
Voltage






70

pressure drop from the manometer, the variation of output signal with the pressure from the

source is obtained. Figure 3-6 shows the calibrated relationship of the output voltage signal

and the pressure drop. This voltage-pressure relationship is used in a data analysis code to

compute the pressure drop from the measured voltage. The uncertainty of the pressure drop

measurement is 0.007 kPa.


3.4. Experimental Protocol

3.4.1. Procedure

Powders are placed into a big dish and are heated continuously in an oven in order

to keep the powders dry. The temperature of the oven is set to about 21.10C (700F). Prior

to testing, powder is placed in a dish and the powder mass, M, is determined using an Ohaus

digital balance. The visual test section is removed and both the outer and the inner surfaces

are cleaned. A long steel rod with a solid plastic plug on one end is placed into the test

section, powder is then poured into the test section via a funnel. Another piston, with a short

steel rod and a solid plastic plug on one end, is used to consolidate the powder sample into

a plug at one end of the test section. When compacting the powder sample, the short piston

is connected to a load cell in order that the applied force, F., may be measured. These two

pistons are then taken out very carefully. The length of the powder plug, L, is measured.

Finally the transparent test section is placed back into the experimental facility.

Next, the air compressor is turned on. The pressure of the compressed air in the

compressor is kept between 827 kPa (120 psi) and 552 kPa (80 psi) automatically. The valve

of the compressor is opened. The pressure gauge in the pipeline is checked to make sure the

pressure in the pipe is maintained at 552 kPa (80 psi). During the experiment, the mass flow






71

controller is used as both a shut-offvalve and an air flow controller. When the input voltage

to the controller is set to zero, it operates as a shut-off valve.

The digital data acquisition facility that collects data on the volumetric flow rate and

pressure drop is initiated and begins recording data. At first, the data acquisition program

is run to measure the zero flow pressure drop offset with the mass flow controller is in the

closed condition ---- the input voltage is zero. Meanwhile the frame grabbing program is

running on the frame grabbing computer. The stroboscope is properly situated in order that

the images of a plug are shown clearly in photographs. The flash frequency of the

stroboscope, f, is properly set so that at least 4 successive images of a plug will appear in a

photograph.

For a certain length powder plug, the volumetric flow rate used to transport the plug

is chosen, and the mass flow controller set point is adjusted using the variable DC power

supply. When the mass flow controller is set to obtain the specified gas volumetric flow rate,

the plug is transported through the test section. Immediately after the mass flow controller

is set and the plug begins to move, both the digital camera and the data acquisition system

are triggered. Since the camera is manually triggered, there involves a degree of trial and

error to capture successive images of the powder plug that are suitable for analysis. Thus the

systematic compilation of a quality set of data is a very tedious process.

In the current experimental investigation five different gas volumetric flow rates i.e.

gas superficial velocities and five different plug lengths were varied independently. So for

each powder, at least 25 experiments should be conducted by following the above procedure.








3.4.2. Data Reduction

To calculate the plug velocity from the digital images, the time interval between two

successive images is determined. The unit of the frequency of the strobe-scope, f, is rpm.

So the time interval in seconds between two continuous images is simply

60 (3-1)
f
Another parameter measured is the actual distance between two successive images.

Photographs are analyzed by using a computer. When the cursor moves from one location

to another, the displacement is determined as a number of pixels. The length corresponding

to each pixel is determined from a scale in the field of view. The velocity is calculated from

the distance traveled between successive images and the time between images.

The powder plug concentration is determined as follows,

M

S(3-2)
AL
where M is the mass of the powder, L is the length of the plug in the test tube after packed,

p is the density of the powder material, and A is the cross-section area of the test section.

The initial consolidating stress is computed from
F
T O= (3-3)

where F, is the normal force applied when compacting the powder plug.

The gas superficial velocity U, is simply calculated from:

U (3-4)
A
where Q is the measured gas volumetric flow rate and A=2.85cm2 is the cross section area

of the pipe.








3.5. Results and Discussion

3.5.1. Factors Affecting the Experiment

During the experiment, the powder plug concentration, the plug length, the air flow

rate, the roughness of the inner wall and the straightness of the test section, will have

influences on the performance of the experiment and the experimental results. The inter-

particle cohesion is the main force that keeps the plug unbroken. The cohesive force is a

combination of adhesion, cohesion, Van der Wall forces, and interfacial forces. The inter-

particle distance, i.e. the concentration of the powder, has significant influence on these

forces. The dynamic wall friction is a function of powder properties, the powder motion and

the wall roughness. Since the powder samples are kept in an oven, the influence of powder

moisture on the stability of the transport of powder plugs is not observed. The influence of

particle size and particle size distribution of powders are not investigated. Using a tube with

a roughened inner wall, the friction acting on the plug will be larger, and the power plug is

easier to break during transport. Experiments are conducted in a smooth wall test section

to avoid additional complications.

When the tube has any kind of bend, the friction between the plug and the wall will

increase when plug pass across the bend point. The direct result of a sudden increase in

friction is that the powder plug will decelerate. When running the experiments it is very

important to maintain a straight test section. Even a large radius of curvature bend will

adversely influence the plug behavior.

Plug flow experiments were carried out with 27utm silica powder and Sm kaolin

powder. These powders were chosen because they are cohesive and their theological






74

properties have been measured with the annular shear cell type powder rheometer previously

described in Chapter 2. The powder properties are again summarized in Table 3-1.

Table 3-1 Powder properties
Material Mean Diameter Dm Standard Deviation GD Material Density p
(po)_ (Plm) (g/cm3)
Kaolin 6.0 0.8 1.65
Silica 27. 5 7. 2 2.34

When the powder concentration is too low, the inter-particle distance is such that the

inter-particle force is not large enough to keep the plug unbroken during an experiment.

Meanwhile the larger the powder plug concentration, the larger the friction between the plug

and the wall of the test section, and thus a larger pressure drop is required to drive the plug.

For a more cohesive powder, this influence is stronger. So in the current investigation, the

concentration for silica powder plugs is about 0.52, while that for kaolin powder plugs is

about 0.29.

The plug length is an important correlating parameter that affects the performance of

the experiment. When the plug length increases, the resistance to gas permeation through

the plug will also increase. On the other hand, if the plug length is too small, for instance,

less than half of the inner diameter of the tube, the plug will collapse before moving through

the tube. And if the plug length is too large, the plug may pass through the test section and

break into several pieces. In this research, the plug length is maintained in the range of

30mm to 45mm.

The air flow rate is an important factor that affects the stability of the plug transport.

The air flow rate should be large enough to drive the plug through the test section. However

large air superficial velocity tends to produce instability in the plug because large air flow






75

rate or large superficial velocity increases the ability of air to permeate the plug, and this

increases the amplitude of perturbations inside the plug and leads to collapse or break-up of

the plug.


Table 3-2 Plug lengths, gas volumetric flow rates and corresponding superficial velocities
S27 ptm Silica Powder 11 5 pm Kaolin Powder I


Plug Gas Gas
Length Volumetric Superficial
(mm) Flow Rate Velocity
(cm3/s) (cm/s)
30.5 166.76 58.5
35.0 176.57 62.0
39.0 186.37 65.4
42.0 196.17 68.8


Plug Gas Gas
Length Volumetric Superficial
(mm) Flow Rate Velocity
(cm3/s) (cm/s)
30.0 147.16 51.6
35.0 156.96 55.1
40.0 166.76 58.5
42.5 176.57 62.0


45.0 205.97 1 72.3 45.0 186.37 65.4


In the current experimental investigation, the pressure drop and the plug velocity are

measured at five different gas flow rates and five different plug lengths for both silica and

kaolin powders. Table 3-2 summarizes the plug lengths, gas volumetric flow rates and

corresponding gas superficial velocities for both powders.

3.5.2. Plug Pressure Drop

Figures 3-7a and 3-8a show the measured pressure drop as a function of gas

volumetric flow rate at 5 different plug lengths for silica powder and kaolin powder,

respectively. Figures 3-7b and 3-8b show the measured pressure drop as a function of plug

length at 5 different gas volumetric flow rates for silica powder and kaolin powder,

respectively. For both powders, the pressure drop is found to increase with increasing of






76

plug length and gas flow rate. The variation of the pressure drop is observed to be linear

with plug length and nonlinear with gas volumetric flow rate. The pressure drop across a

kaolin plug is greater than that of a silica plug with the same plug length and gas volumetric

flow rate. The greater pressure drops are due to greater cohesion of the kaolin powder.

As mentioned earlier, various investigators, especially Klinzing and his co-workers

have extensively studied pressure drop in dense phase pneumatic transport over a range of

velocities with both granular and cohesive type particles. Analytical attempts at modeling

plug flow have also been made. Their work has provided valuable experimental data for

vertical, horizontal, and inclined flows. Their data demonstrate that the pressure drop across

powder plugs varies linearly with plug length for all flow orientations, which is consistent

with their analysis. Figure 3-1 shows experimental data from Aziz and Klinzing [42]. It can

be noted that the results of current investigation are consistent with those of Aziz and

Klinzing [42]. However the analysis of Aziz and Klinzing [42] does not capture the variation

of pressure drop with air flow rate. Their analysis incorrectly suggests the pressure drop is

independent of the gas flow rate, while the result of the current investigation shows the

pressure drop increases with increasing gas volumetric flow rate.

In the case of vertical flow, where the pressure gradient is dominated by gravity, the

pressure drop is approximately independent of gas flow rate. However, in the case of

horizontal flow, where gravitational influence is minimal, the pressure gradient is dominated

by wall shear stress, and the pressure gradient shows a strong dependence on the gas flow

rate, i.e., the gas superficial velocity.

The reason the analysis of Aziz and Klinzing [42] fails to capture the flow rate

dependence of the pressure gradient is that they treat powder as a Coulomb material and







77

determine the wall friction based on the internal angle of friction. Treating the wall shear

stress based on quasi-static measurements of wall shear stress does not reflect a shear rate

dependence. However, the cohesive powder rheology measurements clearly identifies a

dependence of shear stress on shear rate.


7



6


Q.
-5
CL
2



1:.
I4

0C


2
160


170


180 190

Air Volume Flow Rate (cm3/s)


200


210


Figure 3-7a Variation of pressure drop with gas volumetric flow rate for silica powder


Plug Length
(mm)
-e- 30.50
-35.00
-A- 39.00
---- 42.00
45.00
































28 30 32 34 36 38 40 42 44 46
Slug Length (mm)


Figure 3-7b Variation of pressure drop with plug lengths for silica powder


-A--


Plug Length
(mm)
30.0
35.0
400


4;
4!


5



4
140 150 160 170 180 1

Air Volume Flow Rate (cm3/s)


Figure 3-8a Variation of pressure drop with gas volumetric flow rate for kaolin powder


Air Flow Rate
(cm'/s)
--- 166.76
S176.57
-A- 186.37
-- 196.17
205.97


2.5
5.0











9 11-i -I II

Gas Volumetric
Flow Rate (cm3/s)
8 -E- 147.16
156.96
a 166.76
SV 176.57
7 186.37


0-






4 I I I I I I I I
28 30 32 34 36 38 40 42 44 46
Slug Length (mm)


Figure 3-8b Variation of pressure drop with plug lengths for kaolin powder

3.5.3. Plug Velocity

Figures 3-9a and 3-10a show the measured plug velocity as a function of gas

volumetric flow rate for different plug lengths of silica powder and kaolin powder,

respectively. Figures 3-9b and 3-10b show the measured plug velocity as a function of plug

length at different gas volumetric flow rates for silica and kaolin powder, respectively. The

plug velocity increases with increasing plug length and gas flow rate. It is observed that the

variation of the plug velocity with both plug length and gas volumetric flow rate is non-

linear.

It is observed that the plug velocity is lower than the gas superficial velocity. To

illustrate the variation between the gas superficial velocity and the mean plug velocity, the

plug slip velocity is introduced. The plug slip velocity, Op, is defined as P= U(/Ug, where







80

Uo is the plug velocity and Ug is the gas superficial velocity. Figures 3-11 and 3-12 show the

slip velocity as a function of plug length for silica and kaolin powders, respectively. It is

observed that for both silica and kaolin powder, Uo/Ug is in the range of 0.90-0.99. With

larger plug lengths, the plug slip velocity approaches unity. This is because when the plug

length increases, the resistance to gas permeation through the plug will also increase and then

less gas permeates through the plug. In order to conserve mass in the gas-phase, the plug

velocity must increase.

Aziz and Klinzing [42] also showed that the plug velocity varies with the plug length

and is a function of the air velocity in horizontal pipes. The experimental results of the

current investigation are consistent with those of Aziz and Klinzing [42].



75
Slug Length
(mm)
70 30.5
-- 35.0
39.0
.-- 42.0
5 45.0


60
60



55


50
160 170 180 190 200 210
Gas Volumetric Flow Rate (cm3/s)


Figure 3-9a Variation of plug velocity with gas volumetric flow rate for silica powder





















65 F


60 -


28 30 32 34 36 38 40 42 44 46
Slug Length (mm)


Figure 3-9b Variation of plug velocity with plug lengths for silica powder


65 -


60 F


55 k


50 1


45 L
14


0


150 160 170

Gas Volumetric Flow Rate (cm3/s)


Figure 3-10a Variation of plug velocity with gas volumetric flow rate for kaolin powder


Gas Volumetric
Flow Rate (cm3/s)
--- 166.76
176.57
-A- 186.37
- 196.17
205.97












I C


Slug Length
(mm)
-e- 30.0
-- 35.0
S 40.0
-V- 42.5
-<- 45.0


I I I I


























55 F


50 -


45
28 30 32 34 36 38 40 42 44 46

Slug Length (mm)


Figure 3-10b Variation of plug velocity with plug lengths for kaolin powder


1.00


0.98


0.96


0.94


0.92


0.90


0.88


0.86
28


Figure 3-11 Var


Gas Superficial
Velocity (cm/s)
0 58.5
o 61.9
A 65.4
v 68.8
0 72.3


I I


m Silica Powder
27gm Silica Powder


o


I I I


30 32 34 36 38 40 42 44

Plug Length (mm)


nation of slip velocity with plug length for silica powder


Gas Volumetric
Flow Rate (cm3/s)
-e-- 147.16
-E3 156.96
-- 166.76
-- 176.57
-- 186.37


60


., I












5pm Kaolin Powder




0


0.86 I I I I I
28 30 32 34 36 38 40 42 44 46

Plug Length (mm)

Figure 3-12 Variation of slip velocity with plug length for kaolin powder


1.00


0.98 -


0.96 -


Gas Superficial
Velocity (cm/s)
0 51.6
o 55.1
58.5
V 61.9
0 65.4


0.94


0.92


0.90


0.88









CHAPTER 4
MODELING OF COHESIVE POWDER FLOW


4.1. Introduction

In engineering practice, the designer of a pneumatic transport plug flow facility

desires to compute the required pumping power and plug transport velocity for a specified

gas flow rate or gas superficial velocity and plug length. Until now the design of dense

pneumatic transport system remains mostly empirical. The mathematical description of a

translating powder plug is complicated by the fact that a powder can exhibit either a solid-

like or fluid-like state, depending on the local shear rate. In the case of a powder plug, the

core of the plug is typically immobile and demonstrates mechanical behavior consistent with

a solid material. Close to the pipe wall, there exists a shear band that is in a highly dynamic

state and demonstrates mechanical properties consistent with a fluid.

As mentioned before, when a powder assembly is subjected to low or moderate shear

rates, its behavior is neither consistent with that of a solid or a fluid. Tardos [27] used a fluid

mechanics approach to study the dynamic behavior of powder flow in the frictional regime.

Ancey et al. [28] also suggested that a fluid mechanics treatment of dense granular flows in

the frictional regime may be appropriate. Recently Levy [46] used a two-fluid theory to

model plug flow in a horizontal pneumatic conveying pipeline and then solved the

conservation equations for mass and momentum for the gas and solid phases. Such an

approach inherently captures the powder dynamics using conservation of momentum as the

basis for constitutive modeling. In order to use a fluid mechanics approach, a constitutive

84






85

model is required to relate the stress tensor to the rate of deformation of the powder

assembly. Consequently, the powder rheology that relates the stress tensor to the rate of

deformation must be known for a powder. This has been accomplished by using the annular

shear cell powder rheometer.

The main objective of the following analysis is to compute the pressure gradient

across a powder plug for a specified gas superficial velocity, plug length, and specified

theological properties of the powder. In this research, a fluid mechanics approach is applied

to study powder plug flows. The main idea of the present investigation is to use the

rheology data measured for the powder in a dynamic state, coupled with the conservation of

momentum, to deduce the stress field within the powder plug and correctly describe its

dynamic state. Of particular concern is an attempt to correctly describe the fluid-like

behavior within the powder shear band close to the boundary of the pipe wall. Using this

approach the pressure drop can be determined as a function of the gas superficial velocity.

Since the dynamic state of the powder is captured using this approach, it would predict a

pressure drop dependence on the gas flow rate, as reflected in the experimental data.

Compared with the methods used by Ancey et al. [28] and Levy [46], one significant

advantage of this type of approach is that it does not require correlations of dynamic friction

coefficients as inputs. The dynamic wall shear stress is inherently computed as part of the

analysis using measured rheology data.

In investigating powder plug flow, two characteristics mentioned above should be

considered: the core of the plug is typically immobile and demonstrates mechanical behavior

consistent with a solid material; close to the pipe wall, there exists a shear band that is in a

highly dynamic state and demonstrates mechanical properties consistent with a fluid.






86

Therefore, a two-layer model is required to reliably describe the characteristics of a powder

plug and will be used to analyze the powder plug behavior, where the inner layer is treated

as a solid and the outer layer is treated as a fluid. A general description of plug flow for

cohesive powders will be constructed and the predictive capabilities will be compared with

the 27gm silica and 5 um kaolin powder experiments.


4.2. Two-layer Model

Various attempts have been made to model pneumatically transported dense phase

powder flow. Wilson [47] presented a one-dimensional two-layer model for wavy stratified

bed flow. The solids occupy the lower portion of pipe while the transport gas and some

suspended particles move in the upper portion of the pipe. In this model, each layer has a

separate velocity and momentum transfer between layers is governed by shear forces. Levy

[46] used a two-fluid approach to study plug flow in horizontal pneumatic conveying.

Similar to dilute phase flow, Levy assumed that both gas and solids phases occupy any point

of the computational domain each with its own volume fraction. A 3D model was developed

to solve conservation equations for mass and momentum for both the gas and solids phases

using a finite volume numerical approach. Tsuji et al. [48] used the discrete element method

to simulate plug flow in a horizontal pipe. In this work, a two-layer model is used to research

plug flow in horizontal pneumatic conveying. The description of the model is provided as

follows.

An ideal powder plug translating through a pipe of radius R is shown in Figure 4-1.

The powder concentration is typically not uniform across the plug cross-section. The high






87

concentration layer of powder typically occupies the core portion of the pipe, while a

mobilized, looser layer of powder translates near the wall of the pipe.



Px Px+L

-- -- Solid/Shear-layer
Interface ir
/
U.... .U o .... / x
Solid layer


Shear layer :,





Figure 4-1 Simplified diagram of two-layer plug model

In this analysis, it is assumed that the powder concentration is uniform. In order to

mathematically reflect the actual characteristic of the powder plug moving in the pipe, the

moving plug is treated as a combination of two different layers: a solid layer and a mobilized

layer. The solid layer occupies the inner core of the plug. The mobilized layer moves like

a fluid near the pipe wall, and there exists a velocity gradient in the radial direction. The

following assumptions are invoked in order to mathematically simplify the analysis while

maintaining the essential physics in a translating powder plug:

The inner core of the powder plug is immobile and translates with velocity Uo;

There exists a mobilized shear layer close to the pipe wall with thickness 8. In this
layer, friction and collisions dominate momentum transfer;

* The velocity at the edge of the shear layer is the plug velocity, Uo;






88

Steady, fully developed flow with constant density in the shear layer of the cohesive
powder is assumed;

The powder rheology in the shear layer is the same as that measured using a dynamic
shear cell device.

With these assumptions, the velocity Uo and the thickness 8 of the shear layer are

invariant along the length of the plug, and the pressure is radially uniform.


4.3. Analytical Description of Powder Plug Flow

Consider the plug as continuous, steady and axi-symmetric powder flow. The inner

core of the powder plug moves like a solid tube with a constant velocity Uo. The motion of

the mobilized powder in the shear layer is appropriately described with the momentum

equation:


DPbU i
t Pb .VT + Finterstitial (4-1)



where u is the powder velocity vector, g is the gravitational vector, Pb is the powder bulk

density, T is the stress tensor that is comprised of both the pressure and deviatoric

components, and interstitial is the interstitial fluid force per unit volume that arises as a result

of the carrier gas flowing through the plug. Typically, as shown in the experimental results

described in Chapter 3, the difference in velocity between the plug and carrier gas is small,

and it is of interest to deduce the magnitude of this interstitial fluid force.

4.3.1. Estimating the Interstitial Fluid Force

For the purpose of estimating the interstitial fluid force per unit volume across a static

powder plug, it is deduced through a simple integral control volume analysis that










Lnterstitial = (1- ) (4-2)

where Ap is the pressure drop across a static plug, L is the plug length, and c is the powder

porosity. Two equations, those of Ergun and Darcy-Forchheimer, have been used for

estimating the air penetration through a powder plug.

4.3.1.1. Static Powder Bed Experiment

A static powder bed experiment for silica powder plug is conducted to investigate the

air penetration through the plug in this work. Using the present experimental facility for plug

flow, the test section is placed in an inclined position, and a powder plug is placed in the test

section. Using gravity to maintain the plug in a static position, air is driven through the plug

at various superficial velocities, and the pressure drop per unit plug length is measured. By

measuring the pressure drop across a static powder bed and correlating it as a function of the

gas superficial velocity, the relative importance of nterstiial may be gauged.

Figure 4-2 shows the measured pressure drop per plug length variation with the

relative velocity for the static powder bed experiment. (Ap/L), increases with the increase

of the relative velocity V= (Ug-Uo). The plug velocity Uo equals zero in the static bed

experiment. Several plugs with different plug length and same powder concentration are

tested in this experiment. It is noted that all Ap/L follow the same curve when varying the

relative velocity. It appears that the pressure drop is parabolically dependent on the relative

velocity. The values of Ap/L are best fitted with the following function:


S= 6.642x104 V + 4.24x105 V2 (4-3)
L


where the unit of Ap/L is Pa/m, and m/s for V,.











20

*

15


E
10
-J
<3


5 -*



0
0 2 4 6 8 10 12 14 16
Relative Velocity (Ug-Uo) (cm/s)


Figure 4-2 Variant of pressure drop with relative velocity

Depending on the plug length, Uo/Ug typically ranges from 0.9 to 1.0. For all of the

plug flow experiments discussed earlier, the relative velocity, Ug-Uo, is no larger than 7.5

cm/s. That implies that the relative velocity, V,, is usually very small, and the results shown

in Figure 4-2 indicates that the pressure gradient due to interstitial fluid forces are an order

of magnitude less than those for translating powder plugs. Thus, Finterstial may be neglected

in Equation (4-1) for most cases of practical interest. The momentum equation (4-1) may

be written as


Dp b
D Pg + V (4-4)
Dt =








4.3.1.2. Ergun Equation

The Ergun equation is widely used for the predicting the pressure drop through a

porous bed. Using the Ergun equation, the pressure drop through a powder plug may be

described as:


Ap (1-e)2 gVr 1- pgV2
150 + 1.75
L 83 2 d +2 7 3 (pd (4-5)

where d p is the particle diameter (the diameter of a circle with the same area as projection

of particle), cp is the shape factor (the ratio of surface area of a sphere with an equal volume

to surface area of the particle), Vr is the relative velocity, and Vr=Ug-Uo. Uo is the plug mean

velocity, and U, is the gas superficial velocity, t is the viscosity of the fluid, and p, is the

density of the fluid.

With the relative velocity V, is known at the range of 0 7.5 cm/s, i.e., 0 0.075 m/s,

A p/Lcalculated from the Ergun equation is in the range of 0 681 kPa/m. While the

measured total pressure drop across a plug for silica powder is in the range of 0 9 kPa/m.

And the prediction of the Ergun equation does not match with the result of the static powder

bed experiment shown in Figure 4-2. Thus, the Ergun equation is not applicable.

4.3.1.3. Darcv-Forchheimer Law

The Darcy-Forchheimer Law is another equation that is frequently used to investigate

pressure drop in porous bed. The basic idea of Darcy-Forchheimer Law is attempt to

describe the pressure drop as a function of the mean relative velocity of the penetrating fluid.

The pressure drop across a powder plug may be described using the Darcy-Forchheimer Law

as,









Ap gKo(1- I)2 FK ( ) (4-6)
V + p, V (4-6)
L s d, E2 d


where Ko is the permeability factor and F denotes the Forchheimer coefficient. As suggested

by Levy et al. [43], Ko and F must be empirically determined for each powder. For the silica

powder, it is found the resultant Ko and F are determined to be 2.29 and 3.95, respectively.

4.3.2. Momentum Equations

As previously mentioned, the plug flow is considered to be incompressible,

continuous, steady, axisymmetric and fully developed. The x-momentum and the r-

momentum equations for the shear layer are discussed here for a cylindrical coordinate

system.

For steady, axisymmetric plug flow as depicted in Figure 4-1, the gravity component

is zero in the x-direction, and the x-momentum and r-momentum equations for the shear

layer may be written as
Opu opu op aT, 1 a
x-momentum: u +v x+ -r xr,
ax 9r ax ax r ar


r-momentum: apv apv ap 1 ( .
ax ar a r r or xr


In order to nondimensionalize the momentum equation, dimensionless variables are

defined as:
x r U V P P
x r u v p r
x R', r u U, v p- = V
R R Uo Uo 0Cr cro






93

where, R is the radius of the pipe; ro is the initial consolidating stress, where uniaxial

compression is used to prepare the powder plug. The nondimensionalized x-momentum

equation is
.apu 8pu _o op (o atxx 1 ( .)
u + v 2 u r + p 0 -n* 1. (r (4-7)
ox + r U 0x U x r or ax


For a fully developed velocity field in the shear layer, the convective terms on the

left-hand-side are zero, and the x-momentum equation simplifies to


oap 1 a (
x* r x r* )= 0. (4-8)
ox r r "


Similarly considering the momentum equation in the r-direction, we have



x* + r* U2 r r xr (4-9)
Since V=0 everywhere, Equation (4-9) implies that
8p
-0.
or
Thus the pressure p is uniform in the radial direction, i.e.,

p = p*(x). (4-10)

Equation (4-8) may now be written as an ordinary differential equation


dp* 1 d
-* (r-r TJr)= 0
dx r dr x
Integrating the above equation, yields,
r*dp* D
S= xr + -. (4-11)
e2 2dx r
where D is the integration constant.






94
At the edge of the shear layer, r = R-6, the shear stress is that which is just sufficient

to cause failure in the solid-like powder, ro. A force balance on the solid-like portion of the

powder plug, as shown in Figure 4-1 gives:

Apn(R- 5)2 = '27n(R- 6)AL
By defining 6 = 6/R and considering Ap/AL = dp/dx = (dp*/dx' Xao/R), it

follows that

2To
*= 1- /o (4-12)
dx"

To determine the constant D, the boundary condition at r = R-8 is used. At the edge

of the shear layer, r = R-8, the shear stress is ro. Therefore,

xr = o o at r = 1- 6 (4-13)
Substituting Equation (4-13) into Equation (4-11) and combining with Equation (4-

12) yields


To ( 2 dp*o) dp + 2/r0 dp'
o0 2 ao dx* dx o dx


and thus D = 0.

It follows that the x-momentum equation simplifies to:


r r dp
T -r (4-14)
2dx




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID ER8UT0VSP_8CUCPY INGEST_TIME 2013-02-14T16:48:21Z PACKAGE AA00013529_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES