slip velocities in pneumatic transport...dropping distances in air parameter estimates of...
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SLIP VELOCITIES IN PNEUMATIC TRANSPORT
A THESIS SUBMTTED IN FULFILMENT
OF TFIE REQUIREMENT FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
S. Ravi Sankar
M. Tech. (Chem. Ene.)
(I.I.T Madras, India)
MAY 1984
DEPARTMENT OF CHEMICAL ENGINEERING
UNIVERSITY OF ADELAIDE
AUSTRALIA
CERTIFICATE OF CONSENT
i agree that this thesis may be made available for co't,sultation within the
Universìty.
I agree that the thesis may be made available for photo-copying.
I note that my consent is required only to cover the three-year period
following apprwal of rny thesis for the award of my degree.
S. Ravi Sankar
TABLE OF CONTENTS
CERTIFICATE OF CONSENÎ
LIST OF FIGURES
LIST OF TABLES
NOMENCLÀTURE
SUMMARY
DECLARATION
ACKNOWLEDGEMENTS
1. SLIP VELOCITY
1.1 Introduction
1.2 Deûnition
1.8 Signiflcance of Slip VelocitY
1.4 Factore Influencing Slip Velocity
1.4.1 Single ParticÌe Termìnal Velocity
1.1.1.1 Efect ol hee Strcøm Turbulence
1.4.2 Etræ,t of Solid Volumetric Concentrztion
1.1.2.1 Theo¡etical Solutions
1.1.2.2 Semi Th'eoreticøI Solutions
1.1.2.8 Empiñcal Solutions
1.1.2.1 Deaiøtions from Zaki's Co¡relotions
1.4.3 Etræ,t of ?].tbe WaII
1.1.3.1 Hydrodynømic Influence
1.1.3.2 Boundary Influence
1.4.4 Dtrect of Traasport Velocìty
1.1.1.1 SoIid-WalI hiction Factor
ii
Page
t
vl
vlll
tx
xl
xltl
xlv
l-1
l-l
L-2
t-2
t-3
1-4
1-6
1-6
L-7
1-8
l-8
t-12
t-t2
1-13
1-15
t-16
1.1.1.2 Som¿ Othc¡ Dcfinitions of Fhiction Factor
1.1.5 Dtrect of Perticle Sìse and Density
1.6 Concluslons
2. COUNîERCURRENT EXPERIMENTS
2.1 Introductlon
2.2 Countercurrent ÞrPerlments
2.2.1 Apparatus
2.2.2 Experìmenial Results
2.2.3 Range of Vañablæ Invætigated
2.2.1 Anzlysis of Data
2.2.1.1 Conccntmtion'Slþ Velocity Colculotions
2.2.5 Regulte
2.2.5.1 Efrect ol Conccnt¡ation
2.2.5.2 Efrect of Porticlc Proputies
2.2.5.8 Efrect ol I\bc Diameter
2.2.6 Summary of Resulte
2.2.7 Comparison wíth Exìstìng Data
2.2.7.1 Somc Rnmarlæ on thc Cor¡clation
2.2.7.2 Proposcd Co¡¡clation
2.2.8 Theoretical Approaeh
2.2.8.1 Grødicnt Coaguløtion Model
2.3 Sigslflcance of Concentratlon-Sllp Veloclty Relationehlp
2.1 Conclusions
3. CO -:'j- _CURRENT EXPERIMENTS
8.1 Introduction
8.2 Objectives
3.8 Review of Measurement Techniques
3.3.1 Diræ,t Measurements
8.3.1.7 Rodioactivc frøcer Method
iii
t-17
l-19
l-20
2-l
2-l
2-l
2-2
2-2
2-3
2-4
2-6
2-6
2-6
2-7
2-7
2-8
2-lL
2-t2
2-13
2-t1
2-t7
2-19
3-l
3-l
3-l
3-2
3-2
8.3.1.2 Electricol Cøpocity Method
8.3.1.3 Photographic Stroboscopic Method
5.5.1.1 Cine Came¡a Method
3.3.1.5 I-aser Doppler Velocimetry
3.3.2 Indirect Measureme¡ús
3.3.2.1 Isolotion Method
8.3.2.2 Attenuation of Nuclear Radiøtion
3.3.2.3 Optical Method
3.4 Irnpact Meter
3.4.1 Principle of Operation
3.4.2 Description of Impact Meter
3.4.3 Measurement of Thrust on the Impact Plate
3.4.4 Califuation of Thrust due to Air
8.6 Apparatue
3.5.I Solid Fæd Mec-hanisrn
8.5.1.1 Drowback with the Solid Feed Mechønism
3.5.2 Driving Aír Supply
3.5.3 Acceleration Section
3.5.4 Solid Separator
3.5.5 Imapct Plate & I'oad CeII Housíng
3.5.6 Differqtial Pressure ?Ìansducer
8.6 Procedure
3.T Range of Variables Studied
8.8 Analysie of Data
3.8.1 Tå¡ust due to Air
3.8.2 Thrust due to Solids
3.8.3 Solid Velocíty and Concentration
3.8.4 Slip Velocity
3.8.6 Pressu¡e D*p Calculations
3-3
3-3
3-4
3-4
3-5
3-5
3-6
3-6
3-7
3-7
3-8
3-9
3-9
3-10
3-10
3-lI
3-t2
3-12
3-13
3-13
3-13
3-14
3-15
3-15
3-15
3-16
3-16
3-17
3-t7
lv
3.8.5.1 TotoL Pressure DroP
3.8.5.2 Gas-Wo'II Frictional I'ose
3.8.5.8 Pressure Drop due to Solid Holdup
5.8.5.1 SoIid-Wo'II Frictional Loss
3.8.6 Calculation of Solid'Wall F}'iction Factor
3.9 Results and Discuesion
3.9.1 Concentration'SLip Velocity Relationsàip
3.9.2 Slip Velocíty due üo Solid- Wzll tr-riction
3.9.3 Comparison wíth Existíng Data
3.9.3.7 Concentrotion-Slip Velocity Mops
5.9.9.2 Gas-Solid Velocity Relationship
8.9.3.3 Solid'WaJI trlriction Føcto¡
3.lO Prediction of Mininum TÌansport Velocities
g.ll Conclusions
3-17
3-17
3-18
3-18
3-18
3-18
3-19
3-20
3-24
3-25
3-25
3-26
3-28
3-30
BIBLIOGRAPHY
APPENDICES
v
No.
1.1
t.2
LIST OF FIGURDS
DESCRIPTION
Drag Coefficient for Spheres, Disks and Cylinders
Effect of Relative T\rrbulence Intensity on the
Drag Coefficient of Spheres
Terminal Velocity of Silica Spheres, as a F\rnction
of Relative Inteusity of T\rrbulence
Gas-solids Flow Map. Richardson-Zaki Slip Velocity
Gas-Solids Flow Map. Schematic Behaviour for
Ditute Phase Flow when SIip Velocity Increases with
Particle Concentration
Concentration-Slip Velocity Map ( Birchenou gh's Dat a)
Gas-Solid Velocity Map (Bircheuough's Data)
Countercurrent Flow Apparatus
Arrangement for Even Distribution of Solids
Effect of Concentration
Effect of Particle Properties
Effect of Diameter
Concentration-Slip Velocity Map on Log-Log Scale
Correlation of Parameter oAo
Correlation of Parameter uBo
Pictures of 25.4mm Test Section with Sa¡d Flowing
Against Rising Air Stream
Test of Gradient Coagulation Model
Calculated Variation of Solid Volume llaction with
Air Flow in Cocu¡rent Vertical Flow at a Fixed
Velocity of 0.1 m/sec
vi
Page
L-4
1-5
l-51.3
t.4
1.5
l-10
1-10
1.6
t.7
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
l- 15
1-r5
2-L
2-2
2-6
2-6
2-7
2-8
2-9
2-9
2-t4
2.to
2.Lt
2-t6
2- l8
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.1I
3.t2
3.13
3.14
3.15
Impact Plate (Side View)
Pictures of Impact Plate with 644¡r Glass Beads
Ilitting it at an Estimated Speed of 20 m/s
Calibration of load Cell
Calibration of Thrust due to Air
Cocurrent Flow Apparatus
Calibration of Ftictional Pressure Drop due to Air
Concentration-Slip Velocity Map for Cocurrent Tïansport
Effect of Tbansport Velocity on Slip Velocity
Effect of Concentration on Parameter ukn
Egtimation of Parameter 'ko from Momentum Equation
Correlation of Slip Velocity due to Solid-Wall Fliction
Gas-Solid Velocity Maps
Solid-Wall Fliction Factor Versus Concentration
Solid-Wall Fliction Factor Versus Particle FYoude Number
Prediction of Minimum Tlansport Velocities
3-8
3-8
3-9
3-9
3-10
3-18
3-19
3-22
3-23
3-23
3-24
3-25
3-27
3-27
3-29
vu
No.
1.1
t.2
2.1
2.2
2.3
2.4
2.6
3.1
3.2
LIST OF TABLES
DESCRIPTION
Comparison of the Asymptotic Solutions for very Dilute Suspeusions
in Creeping Flow Regiou
Wall Effect on Terminal VelocitY
Details of Materials Used
Test Section Details
Calibration of Mass Flow Rate of Solids
Dropping Distances in Air
Parameter Estimates of Correlation f : ACD
Summary of Vertical Pneumatic Tbansport Investigations
Solid-Wall Friction Correlations
Page
l-6
r-t2
2-3
2-3
2-3
2-3
2-8
3-24
3-27
vllt
A
B
A"
AP
cCo-
C^in
DP
Dt
loI'Íc
Í^r
Íz¡,¡
Fo*
F,
nFs
F,
g
k
NOMDNCLAlURE
Parameter in the comelation 2.8
Parameter in the correlation 2.8
Cross sectional area of transport tube (m2)
Projected area of a particle (m2)
Volume fraction of solids
Drag coefficient in an infinite medium
Lower limit of the concentration
Particle diameter (m)
Tbbe diameter (m)
Doppler shift
Solid-wall friction factor
Gas-wall friction factor
Combined friction factor
Frequency of contact of particles in the test section
Frequency of contacts between clusters of size nN"
Drag force on a particle in an infinite medium (/V)
Froude number
Total thrust on the impact plate (/V)
Thrust due to gas phase (lV)
Thrust due to solid phas" (/V)
Acceleration due to gravity (^t-')Parameter derived from solid phase momentum equation.(/ú is the røtio ol slip
velocity to solid, velocity)
Ratio of observed thrust to theoretical thrust
Equilibrium constant for formation and break up of clusters.
K¡
Kz¡,t
D(
L
Ms
n
Length of test section between pressure taps (m)
Mass flow rate of air through the tube (tgt-t)
Number concentration of particles in the test section
Number concentration of clusters in the test section
Number of particles in a cluster
Total pressure arop (fm-2)
Pressure drop due to static head of g"s (JVm-2)
Pressure drop due to static head of solids (n^-t)
Pressure drop due to gas-wall friction (iV--')
Pressure drop due to solid-wall friction (f--')Loading ratio
T\rbe Reynolds number based on gas velocity
Particle Reynolds number
T\rbe Reynolds number based on terminal velocity
T\¡rbulent intensity
Superficial gas velocity (ms-t)
Mean square fluctuation velocity (-t-t)Volume of a single particle (m3)
Gas velocity (ms-t)
Solid velocity (ms-l)
Terminal velocity of a particte (ms-l)
Slip velocity (ms-t)
Terminal velocity of a particle in the presence of wall (-t-t)Gas viscosity (/Ym-t"-t)
Velocity gradient (t-t)Gas densit¡ (kg^-')Solid deneity (fr9m-3)
Volumetric flux of air in the test section (ms-t)
Volumetric flux of solid in the test section (ms-l)
nN
N
(AP),
(aP)0,
(AP),,
(aP)ro
(aP)r,
R
R"c
R"o
Ret
T
us
u2
up
vs
v,
V¡
v,
Vs
Itg
î
Ps
Pc
os
o,
x
SUMMARY
Slip velocities of solids in a vertical transport line were correlated to solid volume
fraction and the effect of pertinent parameters such as tube diameter, particle properties
and transport velocity were investigated.
Slip velocities were determined from pressure drop measurements in a counter-
current flow apparatus with solid flow against rising air stream. Particles of sand, glass
and steel ranging from l0Omicrons to T00microns in diameter were investigated in 12.7,
lg.l, 25.4 and 38.lmm transport tubes. The ma><imum solid volume fraction realised
in these tests was about 10%. A strong dependence of slip velocity on solid volume
fraction was observed. Its value was larger than single particle terminal velocity and
increased with concentration.
The large slip velocities are due to formation of clusters of particles near the trans-
port walls observed in the tests. A mathematical model based on gradient coagulation
theory is presented to explain these trends. The significance of the concentration-slip
velocity relationship for the definition of uchokingo in forward transport is discussed.
It is established that Richardson & Zaki's correlation which describes the effect
of concentration on slip velocity although appropriate for sedimentation and fluidiza-
tion processes, can not be extended to transport of solids in the tubes where velocity
gradients are considerably large. A correlation of the form V : ACB is proposed
which represents the obtained data with reasonable accuracy. The parameters *Ao and
.Bo are in turn correlated to dimensionless groups involving only system properties
(Dp, Dt,Vt, P", Ps, ttg).
The effect of transport velocity on slip velocity was investigated with cocurrent
transport apparatus. Solid velocity and volume fraction were determined from the
measurement of thrust due to solid phase on an impact plate which deflects the solid
particles right angles to their flight path. This technique proved to be simple and
accurate. Tlansport velocities as high as 15 m/s were investigated.
At large transport velocities solid-wall friction is significant and its effect is to
xl
increase elip velocity. An attempt was made to determine the individual contributions
of transport velocity and solid volume fraction from solid phase momentum balance.
A correlation for the slip velocity due to solid-wall friction is proposed to help esti-
mate the energy losses due to solid-wall friction. Some qualitative understanding of
the mechanism of solid-wall frictioual losses is presented and the need for independent
determination of these losses and the associated mechanisms is stressed.
The usefulness of the concentration-slip velocity relationship obtained from sim-
ple countercurrent flow experiments is demonstrated by comparing the predicted trends
with those obtained from cocurrent transport experiments. The observed minimum
transport velocities are in good agreement with those predicted from concentration-slip
velocity relationship obtained with countercurrent flow arrangemeut.
xll
DECLARATION
This thesig contains no material whicb has been accepted for the
award of any other degree or diploma in any University and, to the begt of
the author's knowledge and belief, the thesis contains no material previ-
ously pubtished or written by another person, except where due reference
is made in the text or where common knowledge is assumed.
S. Ravi Sankar
xlll
ACKNOWLEDGEMENTS
The author wishes to express his appreciation for the assistance that was received
throughout this work.
Special thanks are due to Dr. T.N. Smith, the supervisor of this research, for
his er(cellent guidance and encouragement throughout this work. Thanks a¡e also due
to Dr. B.K. Oneill, for his assistance when Dr. Smith moved to Perth to head the
Chemical Engineering Department at Western Australian Institute of Technology.
The author wishes to thauk the workshop staff, FTed Steer and Brian Inwood in
particular for helping to build and revise the e>cperimental equipment. Thanks are due to
Mechanical Engineering department for lending him the expensive pressure transducer
which made the task of data collection less tedious.
The author wishes to convey his appreciation for the assistance in typing and
proof reading of the document from his wife Padma. The moral support and under-
standing obtained from her throughout his work are also very much appreciated.
The scholarship provided by the Adelaide University which enabled him to pursue
this study is gratefully acknowledged.
xlv
CHAPÎER T
SLIP VELOCITY
l.l Introductlon
Pneumatic transport ¡rs a means of transporting solide is of great interest to
process industries. The use of riser reactors with attractive features, such as better
heat-transfer and continuous operation, placed further emphasis on the need for a better
understanding of fluid-sotid systems. The literature covering this subject is vast and
includes both horizontal and vertical pneumatic conveying. However, the Bcope of the
present work is conûned to vertical pneumatic conveying.
One of the main objectives in the design of solid transporü systems ia to pre-
dict the key parameters, such as pressure drop in the line and slip velocity of solids,
with reasonable confidence. The pressure drop is important in deterurining the power
requirements and in specifying the capacity of driving machinery. The slip velocity is
useful for the estimation of the amount of solids that can be conveyed at a given gas
flow rate. Atso it is important that a lower limit to transport is specified for trouble
free and economic operation.
Some design methods have been proposed (Y*S 1975, Leung & tililes 1976)
based on correlations derived from a large collection of data. Eowever, these procedures
are limited to a certain range of parameters and require caution in their application.
The aim of this work is to investigate the problem of determining the slip velociby and
its dependence on pertinent variables. Delineation of the effects of these variables is of
great value to the desigu of pneumatic transport systems.
1.2 Deflnition
Slip velocity in pneumatic transport is the velocity of the carrying fluid relative
to the moving solid. In the case of fluidization where the net movement of the solids is
1-t
SLTP VELOCITY Chopter I
zero, its value is equal to the gas velocity iu the fluidized bed.
V :Vc -V, (1.1)
where W is the Slip velocity
Vs is the Gas velocity
V, is the Solid velocity
1.8 Stgnlûcance of SltP VeloctüY
Slip velocity is a prime factor in the desigu of transport systems, for it represents
the minimum gaE velocity at which any transport is feasible. ÏVhen a substantially
greater fluid velocity is chosen for practical transport of solid at a specified rate through
a conduit, the slip velocity may be ueed to calculate the solid velocity and the volumetric
concentration in the conduit. These in turn may be used to calculate the contributions
of solid-wall friction and solid weight, to the pressure gradient which must be applied
to the motive fluid in order to maintain transport. While solid weight can simply be
estimated from coucentration and solid density, estimation of solid-wall frictional loss
requires that its dependence on solid velocity is established.
It is therefore, a fundamental requirement for design of transport systems to
identify the factors that influence the slip velocity, and to establish a quantitative rela-
tionship with those factorg.
A systematic analysis of vertical pneumatic system will be presented, accompa-
nied by a review of related works.
1.4 Pactors Influenclng Slip Velocity
Consider the situation of solids transported up against gravity, by a gas in a
vertical conduit . Under steady state conditions, with zero acceleration of solids, the
t-2
SLIP VELOCITY Chop3er I
slip velocity (%) should ideally be equal to the single particle terminal velocity (%)'
if all the particles behaved as if they Ìvere surrounded by an infinite fluid medium.
Therefore:
V,:V (1.2)
where
Vr is the Particle terminal velocity
1.1.1 Single Particle Tæminal Velocity
Terminal velocity is defined as the equilibrium velocity of a particle, moving
under gravity relative to an infinite fluid medium subject to the influence of drag. Its
value is determined by equating the fluid drag (Fp-) on the particle to the net weight
(including buoyancy) of the particle. For a spherical particle of diameter Dp, the force
balance is:
FD* : iolto, - pòe (1.3)
where F¿-is the Fluid drag
Dp is the Particle diameter
po is the Particle density
Ps is the Fluid densitY
g is the Acceleration due to gravity
The hydrodynamic drag experienced by such a particle was first derived theoret-
ically by Stokes (1891), for creeping flow conditions:
FD*:|x¡tsV¡Dp
Itc is the Fluid viscosity
l-3
(r.4)
where
SLIP VELOCITY Cíopter 7
Eowever, it is the turbulent flow case that is of practical interest, and no satisfac'
tory theoretical treatment has been developed because of the complority of representing
the turbulent flow field mathematically; and as a consequence, empirical methods have
been resorted to. Following the dimensional analysis approach, it is established exper-
imentally that the drag coefficient, which is defined as the ratio of drag force per unit
projected area to velocity head, is a unique function of the particle Reynolds number
which characterises the flow field a¡ound the particle. Therefore:
( 1.5)
whereCo*
R"o
AP
is the Drag coefficient in an infinite medium
is the Particle Reynolds number
is the Surface area of the particle
normal to the direction of flow
The above functional relationship is very well established and is presented in
graphical form, tables, or iu the form of empirical equations. The terminal velocity of
a spherical particle can now be expressed as follows (from equations 1.3 and 1.5).
u- 4gDp(p" - pc)
3Cp*pc
Flom the above expression the significance of drag coefficient in determining the
relative motion can be easily seen.
1.1.1.1 Efect ol hee Stream frt¡bùencc
The standard drag coefficient-particle Reynolds number relationship (Fig. 1.1)
is based on the experiments with particlee in a quiescent fluid medium. It has been ac-
cepted practice, in pneumatic transport work, to estimate terminal velocities of particles
using the standard drag curYe.
l-4
(1.6)
loô¡o<ln¡,.-fe.
(,c.g.ca,o
C'oo
oooor ooot o.ot OJ r.o to
Feynolds nrnber N¡. r
tooo ¡o,ooo too,oootoo
!L-sl.
Fig. 1.1 Drag coefficient for spheres, disks and cyl inders.From LappeL and Shepherd (1940).
I
{-
I
I
L
II
I+-
I
II
I
ii!
I
II
I
ind¿rs
\I
TOislsl1
rSgFc.¡¿s
\
-\
\
SLIP VELOCITY Chopler I
However, it was suggested by some workers (Dryden et al 1937, Clamen & Gauvin
1909, Torobin & Gauvin 1961, Clift l¿ Gauvin 1970, Uhlherr & Sinclaìr 1970) that the
free stream turbulence influences the boundary layer around the sphere, thus affecting
the drag coefficient. Consequently, the drag coefficient is a function of not only the
particle Reynolds number but also of the intensity of turbulence. Thus:
CD*: ¡ (Ree,T) lUhlherrl
T
where T is the T\rrbulent intensity
æ is the Mean square fluctuation velocity of gas
Based on experimental results with spheree of diameter ranging from 1.6mm to
lgmm and deneities ranging from 1.0 to 8.0 gm/cc, in water and glycerol solutions of
viscosities up to 4cp, Uhlherr & Sinclair (1970) proposed the following correlations (Fig.
1.2).
-{u2u
for Reo 150 and .05 S f S 0.5
for 50 < Re, < 700 and .05 S f < 0.5
lUhlherrl
In summary, they conclude that the drag coefficient increases with increasing fiee
stream turbulence at higher levels of intensity, while its value is less than the value given
by the standard drag curve at lower levels of intensiüy. The qualitative explanation of
this behaviour was also presented.
Clift E¿ Gauvin (1970) calculated terminal velocities of silica and steel spheres in
air for different levels of free stream turbulence, making use of correlations presented
by Dryden et al (1937), Torobin & Gauvin (1961), and Clamen & Gauvin (f969). The
plot (Fig. 1.3) of terminal velocity versus particle dia¡neter as a function of the relative
turbulent intensity, suggests that the effect of free stream turbulence is predominant
t-5
cD*:{ :ï,f, * #)"u'* o,
tol
Itol
Fig. 1.2
t02 rot to'Re
Effect of relative turbulence intensity on thedrag coefficient of spheres.Fv,om tlhLheru and SincLair (L970).
|J¡¡¡vl
j'. ,o't
E()oJrtt
rotJ
zcr¡il-
ro¡
rolool o.t t.o ro
PARTICLE DIAMETER, D. CM
roo
Terminal velocity of silica spheres, as a function ofrelative intensity of turbulence.From CLift ancl. Gatoín (7970).
TU30 25 20 17o 4tao 35
r5r2lo 7 3 t
!.a
{-+12.t¡
I
aol
llt
2.tt
t
Fig. 1.3
SLIP VELOCITY Chqter I
for larger particles. The smallest particle sise affected at largest turbulent intengitieg
(40%) is about lO00 microns. Consequentln the effect of the free stream turbulence
may safely be ignored for systems with particle size less than 1000 microns. flowever,
it should be borne in mind that the effect of increasing levels of turbulence resultg in a
reduction in the above critical particle size.
Fbom the foregoing discussion, it is clear that the slip velocity between the gas
and solid is the terminal velocity of the particle, the value of which depends on the
properties of the particle and the fluid medium. The slip velocity can therefore be
expressed as follows.
V : Vt, : Í (psr lts, p, Dp) (1.7)
1.4.2 SolÍd Volumetrìc Concentr¿tion
So far the influence of neighbouring particlea on the flow field around a particle
has been ignored. There is ample evidence in the literature to show that such an
influeuce is significant. It is very well established for liquid-solid systems that the slip
velocity is a strong function of solid volumetric concentration which reflects the degree of
proximity of neighbours and their influence on the flow field. Barnea & Mizrahi (1973)
presented an excellent review of the literature on this aspect, for liquid-solid systems.
1.1.2.1 Theorctical Solutio¡æ
Theoretical solutions describing the effect of concentration on slip velocity are
summarized in Table l.l. These solutions are based on the assumptions:
l. creeping flow conditions prenail
2. there is no slip on the solid eurface
3. there are no collieions between particles or formation of clusterg
4. a convenient spatial organisation of the particles may be chosen
t-6
Table (1.1)
Comparison o[ the asymptotic solutions for very dilute suspensions in creeping flow region
(oflet Borneo el. al.,)
' ¡ødded rclerence)
Author Method Equation
Uchida (1e4e) Cell model, cubic cell
Happel (1e58) Cell model, spherical cell, free
surface boundary conditions
Leclair and Hamielee (to6a)
Gal-Or (le7o)
Cell model, spherical cell, zero
vorticity boundary condition
Hasimoto (1959) Point force technique, cubic
arrangement
v,v I1+ 1.?6C
McNown and Lin (1952) Point force technique
Smoluchowski (1911) McNownand Lin (1952) Famularo andHappel (1e65)
Reflections, cubic arrangements v.v Ir.r-#I
Famularo and Happel (1965) Reflections, rhombohedral
arrangements
v,v Il+r.79c
Famularo and Happel (1965) Reflections, raudom arrangement vuBurgers (1942) Random arrangement wv I
T¡'6-T-fõ
'Bachleor (1972) Random allocation V:t-6.5c
SLIP VELOCITY Choptet I
Although almost all the works cited in the Table 1.1 adhered to the hypothesis
of regular arrangement of particles as they settle through the fluid medium, Burgers
(1942) and Batchelor (1972) deduced the effect of concentration on slip velocity assuming
random spatial distribution of eettling particles. In this context it should be mentioned
that the fluidisation experiments with 4mm acrylic spheres in silicone liquid by Smith
(1968), suggest that the spatial dìstribution of settling spheres in a viscous liquid agrees
with a random distribution based on allocation of spheres to space according to binomial
probability distribution at 0.025 concentration level.
However, it is interesting to note that all these solutions indicate that the slip
velocity decreases with increasing solid volumetric concentration. The predicted trend
was confirmed by batch fluidization and sedimentation experiments carried out by sev-
eral workers. These workg were extensively reviewed by Garside & Al-Dibouni (1977)
and Barnea & Misrahi (1973), for liquid-solid systems. Similar trends were also reported
for gas-solicl systems (Mogan et al 1969 & 1971, Rowe et al 1982, Capee & Mcllhinney
1968, Richardson & Davies 1966, Godard & Richardson 1968).
1.1.2.2 Semi Theo¡ctical Solvtions
Ishii & Zuber (1979) and Barnea & Mizrahi (1973) proposed drag similarity cri-
teria based on a mixture viscosity concept, to explain decreasing slip velocity with the
increase in solid volumetric concentration. It was postulated that a particle settling
in a suspension, experiences an increase in effective viscosity of the medium, due to
the interacting momentum exchange mechanism between neighbouring particles. Re-
defining drag coeficient and particle Reynolds number based on mixture viscoeity, it
was postulated that the gtandard drag-Reynolds number relationship could be ottended
to suspensions using a redefined mixture drag co-efrcient as a function of the particle
Reynolds number. Barnea & Mizrahi (1973) and Ishii & Zuber (1979) report that oc-
perimental results from various Bources could be represented successfully by this model,
although different expressions for mixture viscosity were used. The uncertainty in de-
t-7
SLIP VELOCITY Chop3et I
termining the mixture viscosit¡ renders this approach uuattractive. Moreover, viscosity
of a suspensìon is a useful idea only if objects are gettling through it in relative motion
to the body of particles foruring the suspension. If all particles are settling together,
they are moving through the simple fluid.
1.1.2.5 Empirical Solutio¡u
The most wideþ accepted correlation is that due to Richardson and Zaki (1954)'
based on liquid-solid sedimentation and fluidization experiments.
Based on a dimensional analysis, Richardson & Zaki (1954) conclude that the
ratio of slip velocity to single particle terminal velocity is a function of particle Reynolds
number, particle to tube diameter ratio, and solid volumetric concentration.
V,:Vo(l - c)"Do
Vo:V(tO¡-4 l&ichardsonl
n: r RtnDP
,D,
where C is the Volumetric concentration of solids
Dt is the T\rbe diameter
Dp is the Particle diameter
Vs is the Slip velocity corresponding to zero concentration
The above correlation was reported to be applicable to particulate gas-solid sys-
tems by several authors (Capes & Mcllhinney 1968, Richardson & Davies 1966, Godard
& Richardson 1968, Mogan et al 1969), based on fluidization experiments.
1.1.2.1 Deuiations from Zaki's Co¡rcIotion
However, Garside & Al-Dibouni (1977), Addler & Happel (1962), and Capes
(1971) report that the Zaki'e correlation overestimates the slip velocity at solid volu-
metric concentrations leeg thau l%.
l-8
SLIP VELOCITY Chopler 7
Rowe et al (1S82) and Mogan et al (1971) report larger values of exponent (n), for
gas fluidization of ûne solids, which do not give particulate fluidization. Capes (1974)
attributes these larger values of exponents to particle agglomeration, which reduces the
effective voidage and increases the effective particle size. Following the above postu-
lation, and fitting Richardson & Zaki's correlation to Mogan's (1967,1971) data, the
deviations from the calculated and experimental results were reduced.
All the works cited so far report that the slip velocity decrcases with increasing
concentration while its value is always less than the corresponding single particle ter-
minal velocity. This conforms to the theoretical predictions based on the assumPtion
that particulate conditions prevail.
Barfod (1972), based on sedimeutation of fine powders (15-30¡r range) in water,
reports an increase in slip velocity with increasing concentration up to 0.1%. Matsen
(1982) and Barfod (1972) quote the works of Jayaweera et al (1964), Jovanovic (1965),
Johne (1966) and Kaye et al (1962) on sedimentation of solids in liquids reporting an
increase in slip velocity with increasing concentration (up to l%). The slip velocities
were larger than the eingle particle terminal velocity by a factor of l.l to 2.4. The
higher alip velocities are attributed to formation of clouds or clusters of particles whose
effective terminal velocity is higher than the single pa^rticle terminal velocity. The growth
of such clueters is said to increase with increasing concentration thereby increasing the
slip velocity.
Simita¡ trends were reported by Yerushalmi & Cankurt (1979) based on fast
fluidization experiments with ûne particles (33¡r, 49p), in 152mm ({) test section. The
concentration versus slip velocity plot preaented in their work indicates that the slip
velocity is larger than the terminal velocity, and ite value increases with increasing
concentration up to 10%. Eowever, the concentration-elip velocity relatiouship was not
unique, and dependent on the solid flow rate as well. This could be due to non-ideal
effects euch as solid-wall friction and incomplete acceleration of solids.
1-9
SLIP VELOCITY Chopler I
Yousfi & Gau (19?4) reported slip velocities several times larger than the sin-
gle particle terminal velocity, while transporting fine catalyst particles (ZÙtt,183p). It
was obs€rved that large clusters of particles were formed near the walLs. The range of
concentrations gtudied was as high ¿s 22%. Although the concentration and slip veloc-
ity were not preaented o<plicitly, the correlation presented indicatee that slip velocity
increases with increasing concentration.
Capes & Nakamura (f9ß) also report slip velocitiee larger than the single par-
ticle terminal velocity at low gas velocitieg. They attribute this to a particle recir-
culation phenomenon, which is characterised by solid particles gliding down the wall,
then subsequently picked up by the upward moving gas in the core of a transport tube.
Unfortunately, the dependence of slip velocity on concentration was not presented.
Matsen (1982) proposed the following correlation, based on the entrainment data
of lilen & Hashinger (1OOO), and the sedimentation data of Koglin(1971).
v,u
I
1g.ggo.2e3
c < 0.0003
0.0003<c<0.1lMøtsenl
for
for
Their work is signifrcant, as they show that flow maps constnrcted from the
correlations predicting a decrease in slip velocity with increasing concentration (eg.
Richardson & Zaki's conelation), do not predict the experimentally observed voidage
digcontinuity at nchokingl. Their work suggeats such a diecontinuity would be poseible,
only if the slip velocity increases with an increase in the concentration. Fig. 1.4 and Fig.
1.5 are presented to illustrate this point. Unlike Fig. 1.4, Fig. 1.5 exhibits an envelope
which limits the possible solid rate at any given gas velocity. Below this envelope a
slight change in either gas velocity or solid rate results in a corresponding slight change
in voidage. However, as the envelope is reaßhed, in order to accommodate any further
decrease in gas velocity or increase in solid feed rate, the system should change to a
totally different voidage-slip liue. This ¡esults in a sudden iucrease in volume fraction
of solid which is called "choking".
l-10
aa
$('
1.2
¡.0
0.8
0.6
0.4
0.2
00
Fig. 1.4 Gas-solid flo$, map.From Matsen (1982).
¡.0 ¡.2 l.a t.ôu/u¡
Richardson-Zaki sli p veìocity.
lnt fc.Dtt: € ut u
Fiq. 1.5 Gas-solid flow map. Schematic behaviour for dilute phase"J- iiów-wtren slip veiocity increases with particle concentration.From Matsen (1982).
o.2 0.4 0.ö 0.8 t.t
aaù
ur
Lllr¡ ol Co.ì¡t ltSloe¡ -(l -ele I
E¡wrloc¡ Drllnr¡ ll¡¡llurøpr Por¡lu. Wthq¡tAbrupl Gþl¡¡ ln €, . lærc¡¡l¡¡ Solld¡
Concarùat¡il.0rcrcrtil¡Void¡gr
SLIP VELOCITY Choptet I
In summary, the effect of concentration on slip velocity should be considered with
caution. While for particulate conditions in both gas-solid and liquid-solid systems
slip velocity is less than the single particle terminal velocity, and its value decreases
with iucreasing concentration, in aggregative conditions where formation of clusters is
possible, elip velocity exceeds tenninal velocity and its value increases with increase iu
concentration.
At this point, it ehould be noted that the majority of works (Leung 1971, Yang
1973, Capes & Nakamura 1973, Dixon 1976, Arastoopour & Gidaspow 1979) on verti-
cal pneumatic transport assume that Richardson & Zakits correlation can be used to
represent the effect of concentration on the slip velocity. Although extension of batch
fluidization resulte to trausport systems, with the assumption that actual transport
can be represented by moving fluidized bed, was experimentally veriûed for liquid-solid
syatems, so far no euch direct verification is presented for gas-solid systems to the au-
thor's knowledge. Richardson & Zaki (f95a), while discuesing the extension of their
correlation to vertical hydraulic transport, suggeat that the fluid velociüy Fadient and
turbulence could be of considerable importance.
In conclusion, slip velocity is a strong function of concentration. The functional
relationship between these two quantities depends on the flow regime. The above re-
lationship is yet to be established for pneumatic transport of solids. The majority of
studies on pneumatic transport have been done at very low concentrations and high
transport velocities where solid-wall friction governs the slip velocity. Very few studies
(Yerushalmi 1976, Yerushalmi & Cankurt 1979, Yerushaìmi et al 1978, Yousfi & Gau
1974) have extended the results to the situation of low transport velocities and high solid
concentrations. No attempt has been made to correlate concentration to slip velocity.
Clearly, there remains a need to establish this relationship, at low transport
velocities where influence of solid-wall friction on slip velocity is negligible, and the
l-u
SLIP VELOCITY Choplet I
influence of concentration alone is determined
V,: î (U,c) ( 1.8)
1.4.3 Etræt of I\be Wall
1.1.5.1 Hytlrcdynømic Infiuencc
The effect of the tube wall on the single particle terminal velocity was discussed
by Garside & Al-Dibouni (1977) and Barnea & Mizrahi (1973). Theoretical aud em-
pirical solutione quoted by these authors are summarised in Table 1.2. These solutions
indicate that the pr€sence of the wall increases the drag on the particle, thereby reduc-
ing the terminal velocity. The correlation presented by Richardson & Zaki (1954) *.t
derived from extrapolation of slip velocities to zero volumetric concentration of eolids.
Richardson & Zaki's correlation suggests th¿t at any particular solid concentration, the
slip velocity decreases with decreasing tube diameter.
Capes & Nakamura (f9?3) report that the particle terminal velocities determined
from the intercepts of gas-solid velocity plots are less than the calculated terminal
velocity of the particle. The ratio of intercept value to the calculated value decreased
with increasing particle size. This was attributed to a solid-wall effect, and disputed
by Wheeldou & Williams (1980) who maintained that the effect of tube diameter on
terminal velocity was insiguiûcant. They attribute this lower value derived from the
intercept to a failure to use correct gas density in the terurinal velocity calculations.
Yousfi & Gau (1974), based on their investigationg with fine catalyst particles
(ZOtt) in 38mm and 50mm diameter tubes, proposed a correlation which indicates that
the slip velocity decreases with increasing DrlDt ratio.
ft:r+a.zx r0-3(^n,o)o'u (c¡o'zs (+,)-" (i)t-t2
lY ouslil
Toble (1.2) WALL EFFECT ON ?ERMINÁL VELOCITY
Author Correlation Commentg
Eappel and
Brenner (1965) V:t-z.t(fr) +z.re(#)'_0.e6 (+)'
Theoretical solution
(creeping flow)
Ladenburg (1907) n: (t +z.4gi)-'Theoretical solution
(creeping flow)
Flancis (1933) theoretical solution
(creeping flow)
Munroe (1888) V:,- (*)'''T\rrbuleut flow
103 < Ren 13 x 103
Garside and
Al-Dibouni (1977) n: (t + 2.s5#)-1lcReo<120Experimental
Richardson and
Zaki (1e5a) fr: rc-'t0.2<Reo<lO3From extrapolatiouof results tóuero concentration
SLIP VELOCITY Choptet I
where
R", ia the T\be Reynolds number
Van Zuilichem et al (1973) reported that the calculated slip velocities were less
than measured nlues for wheat particles of 4mm size in tubes of diameters 53, 8l and
l30mm. However, this deviation was reported to increase with increasing tube size, con-
tradicting the previously reported trend. Unfortunately the method of determining the
experimental terminal velocities w:rs not presented. It is thue not possible to comment
on the accuracy of their observatione.
f.1.5.2 Boundory Infltence
Apart from the indirect influence of the tube wall on the motion of particles, it is
possible that the effect of actual contact of particles with the tube wall is signiôcant. If
the solid wall friction is sigaificant, and the frequency of contact of the solid particle with
the wall is substantial, then the presence of the wall should result in large slip velocities
in vertical transport. In other words, given that the solid friction is significaut, at a
given solid velocity the frequency of contact between the solid and the tube wall would
be expected to increase with decrease in tube size, reaulting in greater momentum loss,
thus increasing the slip velocity.
Van Zuilichem et al (1973), from their experiments with wheat particles, in tubes
of 51, 80 and l30mm dia,meters, quoted increasing slip velocities with decreasing tube
diameter.
Maeda et al (1974), from their experiments with tube diameters rauging from
8mm to 2Onrmr found that the slip velocity increased with decreasing tube diameter.
Klinzìng & Mathur (1981) quoted the following correlation given by Einkle
(1953), based on high transport velocity experiments.
% : 0.68 (p,)o'u (pù-o'' (Dn)o'nt (D,)-o'u'
l-r3
lKlinzinsl
SLIP VELOCTTY Chopler I
The above correlation also suggests that the slip velocity increas€s with decreas-
ing tube diarneter.
Konno & Satio (1969) maintained that the slip velocity can be approximated
to terminal velocity, based on their experiments with various particle eizes (120-laaOp)
in 26.5 and 46.8rnm tubee. Eowever, closer examination of their results reveals that
for a particular particle size slip velocities in a 26.5mm tube were larger than the slip
velocities in a 46.8mm tube.
Jotaki et al (1978) observed that the influence of tube diameter on slip velocity
was not significant, in their experiments with polyethylene pellets (lmm in diameter
and 2.4m'n thick) in P.V.C. tubeg of diameters 41.2, 52.6, 66.8, 78.3 and l00mm. It
is poasible that this is due to large tube sizes involved in their investigations. Van
Zuilichem et al (1973) also reported significantly smaller changes at large tube sises.
In conclueion, tube diameter appear€ to be a sigaificant factor in determining
slip velocities. Its influence is two fold. Firstly when the solid-wall friction is significant
itg iufluence is to increase the slip velocity. Secondly the indirect influence of the tube
wall, which is imparted through the ûuid medium, ia to decrease the slip velocity by
increasing the hydrodynamic drag on the solid. In batch settling, the particles go down
and the fluid goes up. Drag on the particle is increased by the influence of the wall
on the fluid which is impeding its upward ûow and tending to impart a greater pres-
aure gradient. A eimilar increase in the fluid-eolid drag can be ocpected in the case of
cocurrent transport of solids. However, there is a fundamental difference between these
two processes. In batch settling solids move up the pressure gradient, while in forward
transport they move down the pressure gradient. It is interesting to note that Richard-
son & Zaki (1954) reported that the terminal velocities derived from extrapolation to
zero concentrations agreed well with calculated terminal velocities in the case of sedi-
mentation ocperiments, while lesser values were obtained for fluidization experiments.
In this context, Richardson E¿ Zaki (1954) suggested that the velocity gradient near
l-14
SLIP VELOCITY Chopler I
the wall could be the cause of thie difference. Considering the above observations, the
influence of the transport wall on the slip velocity needq to be investigated for transport
systerns.
V, : I (V,C, Dr) (1.9)
1.1.1 Dfrect of Tlansport VelocitY
There is no doubt that slip velocity increases with transport velocity of the solids.
Fig. 1.6 and Fig. 1.7 present the data published by Birchenough & Mason (f976), but
in a different form. They have plotted the slip velocity againat loading ratio with gas
velocity :rs a parameter. flom their data, the concentration-siip velocity (Fig. 1.6) and
gas-solid velocity (Fig. 1.7) plots we¡e derived, which helped determine the effect of
transport velocity. Their data suggests that at a given volumetric concentration, slip
velocity increases with increasiug mass flow rate of solid. Iu other words, slip velocity
increases with increasing golid velocity. The data further indicateg that for a given mass
flow rate, au increase in solid concentration (thus decreasing the solid velocity) results
in a decrease in the slip velocity. Congidering these trends, and the fact that solid
velocities a¡e aa high as 40 m/sec, the influence of solid velocity on solid-wall friction
is quite clear. The high transport velocity data of Stemerding (1962), Ottjes (1976),
Maeda et at (f974), Mehta et al (1957), Capes & Nakamura (1973) and Reddy & Pei
(1969) suggest a linear relationship between the slip velocity and the transport velocity.
This trcnd is explained simply by greater friction between particles and the wall
with increasing solid velocity. With neglect of auy complication by volume fraction, the
balance of forces on a transporting particle may be written as follows.
(l.lo)
I, is the Solid-wall friction factor
t- l5
DeI"ooo(vo - v")2 : |no,vl +2¿o,o
whe¡e
u
J
CJoJl¡J
fLJvt
20 mrcnon ÊIumtno tn 49.4mm Tubet2
9.6
7.2
{.8
0
0 o. ol 0.02 0.03 0.0{
CONCENTBRT I ON I7I
Fig. 1.6 Concentration-slip veloc'ity map. (Biz'chenough's dnta)
20 mtcnon Hlumlno ln 49.4mm Tube
tl
36
20
20 28 36 {l
GßS VEL0CITf ln/sl
Fig. 1.7 Gas-solid velocity map. (Bit'chenough's dnta)
2.t
60
s2
0. 05
u
E
)-,-(J(]Jtrj
oH
JoU'
28
Ao
ox
+o A
oA x+o o
+o A
ox +
x ++ +
Sol¡d flor lgrlrccl
oo
O- 2{ to 3{a- 39 to {9*- 55 to 65X- ?2 to 82o- 8? to 97
3{t965829?
39
tt
t
ott
B+
2t tototototo
o-A.+-x-o-
5572ø7
5ol¡d flor (9rlrcot
s2 60
SLIP VELOCITY Chopter I
Ae the transport velocity becomes greater, the contribution of the wall friction
term on right hand gide of the equation increases. When it predominates the relationship
between the slip velocity and transport velocity approach direct proportionality. tñrhat
interaction there may be between trausport velocity and volume fraction cannot be
inferred from publiehed data. The difficulty is that small transport velocities are usually
aseociated with small volume fractione. This ie an inevitable reeult of tegte with 6xed
solid feed rate and variable air flow rate.
1.1.1.1 Solid-woll FYiction Focto¡
The frictional loes term in equation (1.10) needs some explanation. The great
majority of workers in the transport field have investigated the energy loss due to the
solid-wall friction. The most common approach to the problem has been to determine
thig loss by subtracting the contributions due to hold up and gas'wall friction from the
total measured preasure loeo.
The frictional loss due to gas phase is generally approximated to that of the value
when gas alone was flowing in the tube. Eowever, Doig & Roper (1967) neport that at
low loading ratios the solid phase causes the air velocities relative to the single phase
profiles to iucrease in the core and decrease at the wall; while at higher loading ratios
the eftect is just the opposite.
\lVhere as, Birchenough & Mason (1976) and Reddy & Pei (1969) report that
the effect of particles on the air velocity profile ie not signiûcant. Reddy & Pei (f969),
quote the works of Soo et al (1964) who also reported similar trends. The influence
of solids on velocity profiles is not very well understood. But the dominant influence
of solid-wall frictional losses at high transport velocities and hold up losses at low
velocities appear to justify the approximation mentioned earlier. Eaving determined
the pressure loss contribution from eolid-wall friction, a solid-wall friction factor (/,) it
defined analogoua to the Íbnning friction factor. The solid phase is approximated to a
continuous phase of density (Cpr), moving at a velocity (%) relative to the solid wall
l- 16
SLIP VELOCITY Chopter I
(D,), where the frictional loss is given by
(AP)¿ : ZI,LC p'v:Dt
( 1.1t)
where
(AP)f, is the Pressure drop due to solid-wall friction
The solid-wall frictiou factor (/r) defined above is generally correlated to either
gas Froude number or solid Floude uumber. l¡t¡hile Maeda et al (1974), Jotaki et al
(1928) and Stemerding (1962) report constant values of friction factor, Reddy & Pei
(1969), Konno & Satio (1969), Swaaij (19?0) and Capes & Nakamura (1973) report
decreasing friction factorg with iucreasing particle Froude number. While Maeda et al
(1974) report decreasing values of friction factor with increasing tube diameter, Jotaki
et al (1978) report quite opposite trends.
Negative friction factorg were algo reported, especially at gas velocities approach-
ing solid terminal velocity, by Yousfi & Gau (1974), Swaaij (1970) and Capes & Naka'
mur¿ (1973). An explanation was proposed based on the obgenred phenomenon of solid
particles moviug downwards along the walls, thus rcsulting in a negative particle-wall
shear.
There are also reports of negative friction factore (Soo 1967) with transport of
fine particlea at higher velocitiee, owing to the damping of the fluid turbulence by the
eolids.
1.1.1.2 Somc othc¡ Dcfinitiotu ol trTiction Focto¡
Jodlowski (1976) and Mehta et al (1957) propose the concept of a combined
friction factor (/-) including gas- wall frictional loesea, with eome variations.
(^P)¡o+ (aP)r,y2
_zl^poLå
l-r7
lJodlowskil
SLIP VELOCITY
where
Chogtcr I
lMchtdl
(^P) ¡o(^P) ¡,l^
is the Pressure drop due to gas-wall friction
is the Preeeure drop due to aolid-wall friction
is the Combined friction factor
is the Correlatiou constaut
:z¡^poLfr(,.(W)")
(AP)¡,:ZlrLCp, (ry)
o
The combined friction factor was correlated to the gas Froude number and volumetric
concentration by Jodlowski, and was found to increase with increasing Fboude number
and solid concentration. Alternatel¡ Mehta et al (f957), attempted to correlate 1^ lo
the gas Reynolda number.
Although introduction of a combined friction factor eliminates the necesEity of
estimating gas- wall frictional losE in the presence of solids, the delineation of the mag-
nitude of individual contributions ig lost.
Barth (1962) introduced another friction factor for solid-wall friction alone.
lBorthl
Eig deñnition is almost similar to the generally accepted definition mentioned
earlier (t.tO), except that inetead of using the square of aolid velocity, the product of
gas and solid velocitieg was used.
The varied definitions of friction factors, range of parameters studied, and the
different dimensionless groups employed to correlate friction factor, make the task of
comparison difficult.
In conclusion, the solid-wall friction is a principal factor in determining the alip
velocity at high transport velocities. Clearly, there is a need to inveatigate the eftect of
transport velocity at high solid concentrations, and the interaction between these two
l-18
SLIP VELOCITY Chaplet 7
quantities in determining the slip velocity.
v: l(v,c,D¡,v,) (1.12)
1.1.5 Etrect oî Pa¡ticle Sise and Densíty
The combined effect of particle size and density is indirectly represented through
particle terminal velocity. Maeda et al (f974), Capes E¿ Nakamura (feæ), and Mehta
et al (f95?) report higher slip velocities for particles with high terminal velocities.
Although one could investigate particles of the sÍune density and of different diameters
or vice versa, it will be impossible to identify the individual influences, aE the terminal
velocities and, therefore, transport velocities will be different as well.
Eowever, the tendency of particles to agglomerate has generally been reported to
increase with decreasing particle size. Consequently, the effect of the size of the particle
on the degree of dependence of slip velocity on volume fraction could be significant.
Yousû & Gau (1974) report ratios of slip velocity to terminal velocity of up to 4
for l38p glass, up to 40 for 55¡r catalyst and up to 300 for ZOp catalyst in the effects of
increasing volume fraction. Similar trends have been reported by Decamps et al (1972),
from their experiments with 55¡r Uranium odde spheres and l65p Aluminium oxide
particles i¡ l0mm glass tube. However, the densities of these particles were slightly
different.
The size and density of particle could be of significance if the number concentra-
tion of pa^rticle is a goveraing factor. The effect of particle ¡ize could also be through
the wall effect (DolD, ratio) mentioned earlier.
v : Í (v,c, Dt,v,, DP, p') (1.13)
Although the above expression iucludes particle sise and density along with the
terminal velocity ¡ra parameters for the reasons mentioned earlier, it ghould be pointed
l-r9
SLIP VDLOCITY Choplct 7
out that tbe te¡¡r¡iual velocity is in fact ¡ur expresgion of the effect of both the ¡ire
and density of the particle on slip velocity. In a given fluid medium heavier ¿nd larger
particles have higher terminal velocities, thus resulting in larger elip velocities. Not
withstanding any complications due to agglomeration, concentration and wall efrects
terminal velocity of single particle and hence the slip velocity can be functionally related
to the particle sise and denaity dependiug on the flow regime-
l.õ Co¡clu¡loa¡
flom the foregoing discusaiou, it is clear that there is a need to
l. investigate the dependence of slip velocity on solid volumetric coucentration for
pneumatic traneport conditions, especially at large concentrations.
2. investigate the effect of key parameters such as tube diameter, particle sice ard
density, ou the above relationship.
3. investigate the significance of transport velocity on solid-wall friction, and the
interaction between transport velocity and concentration iu influencing the slip
velocity.
r-20
CHAPTER 2
COUNTERCURRENT E)(PERIMENTS
2.1 Introduction
Following the conclusione of the previous discussion it was proposed to design
experiments to investigate the effect of concentration ou the slip velocity and also to
study the influence of pertineut variables such as the tube diameter particle size and
density and transport velocity.
The first phase of experimental programme whose prime objective is to determine
slip velocities at large concentrations and low transport velocities is presented in this
chapter.
2.2 Countercurrent Erçeriments
Briefly, experimente were carried out using an apparatus in which solid flows
downward against a rising stream of air. Although this type of arrangement does not
represent actual transport conditions, it corresponds to the region between batch flu-
idization and cocurrent transport. The following factors prompted the choice of such
an arrangement.
l. It is of direct interest for some operations guch as heat transfer,drying and reac-
tion.
2. It facilitates investigation of lower limit to transport velocity.
3. It allows accurate and easy determination of slip velocities, especially at large
solid loadings, which would otherwise be difficult with cocurrent transport.
2.2.1 Apparatus
The apparatus used in the countercurrent experiments is depicted in Fig. 2.1.
2-l
MANOMETERS
1m.
SOTID FEED
SIEVE
ST SECTION3 METRES LONG
souD Ftow
RECEIVER
ROTA METER
AIR SUPPLY
r'-'
1m.
PRESSURETAPPINGS
II
I
I
I
i
I
!-!
I
-l t-
IL
F¡s. 2J NTE URRENT F A TUS
C O U N TERCURREN T EXPERIMEN TS Chopler 2
Solids fall from an open container at a rate fixed by the size of the orifice selected for
the test. To ensure even distribution of solids, arrangement ehown in Fig. 2.2 was used.
The stream of aolide was distributed to a width corresponding to that of test conduit
by a sieve with appropriately chosen mesh. Flom the sieve the solids fall through a
distance calculated to allow them to accelerate to near equilibrium transport velocity
before they pass into the test conduit.
Air from the blower, metered through a rotameter, enters the closed receiving
vessel at the bottom of the tube aud passes through a converging section into the tube.
From the top of the tube the air flow is diffused to the atmosphere through a diverging
section. The test conduit was 3 meters long and was provided with pressure tappings at
half a meter intervals. Two 'Uo tube manometers were connected to pressure tappings
one meter apart, at top and bottom ends of the test section. Butanol (0.808 gm/cc den-
sity) was used as the manometric fluid for good seneitivity. The leads of the manometer
were provided with needle valves to dampen any high frequency oscillations in pressure
differential which are characteristic of two phase flows.
2.2.2 Experimental Procedwe
\ryith a frxed flow of solid through the tube, pressure drop readings at top and
bottom ends of the tube were recorded at several gas flow rates. The air flow was raised
in increments from zero to the point at which solid flow becomes unstable or is arrested,
which is characterised by large fluctuations in pressure drop readings. The procedure
was nepeated with different solid flow ratee fixed by selected orifice sizes at eolid feed
container.
2.2.3 Range of Variables Investþted
Six different materials with mean particle sizes ranging from 96¡r to M4p and
densities ranging from 2.5 gm/cc to 7.6 gm/cc were investigated in four test conduits
with diameters ranging from l2.7mm to 38.1mm. Details of these variables are presented
2-2
F19. 2.2 ARRANGEMENT F0R EVEN DISTRIBUTI0N 0F SOLIDS
..Spl I d FeedContai ner
0ri fi ce
Pipe section havisame diameter oftest sectionSi eve
Convergi ng Cone
Test section
ngthe
Concentri c Pi pesection to containspilling from overflow
JDë
C OU N TERC U RREN T EX PERIMDN TS Chøpter 2
in Tbble (Z.t) and Table (2.2). The volume to gurface mean diameter of particles
presented in the above tables were obtained from the sieve analysis of materials using
standard BSI sieves. Details of the analysis of each material are given in Appendix (A).
The materials chosen were found to be closely sized.
Experiments were conducted at several solid flow rates fixed by the feed container
orifice diameter, which varied from 6mm to 25mm. Calibration of the solid flow rate
with different orifice sizes for all types of solid material used is presented in Tbble (2.3).
The feed rate from the container was found to be constant over the period of the test.
2.2.4 Analysìs of Data
It is desired that solids reach equilibrium velocity by the time they enter the
test section, N steady state conditions are of interest. Details of the minimum fall
distance required to reach equilibrium velocity in etill air, for all particles investigated
are præented in Table (2.4). The calculated dropping distance for the heaviest and
largest particle, used in the tests, to reach its terminal velocity in still air was calculated
to be 9.4 meterg. This requirement is further reduced with increasing upward gas
velocity. F\rrthermore, the distances calculated are overestimated in the sense that it is
impossible to achieve a distribution of particles where particles are dropped individually.
In reality a mærimum dropping distance of about one meter was found to be quite
adequate in the majority of the runs.
The above-mentioned steady state condition is represented by correspondence of
pressure gradients at the top and bottom ends of the test conduit. This was realised in
substantial number of observations. However, at large solid feed rates corresponding to
low gas flows, the pressure drop at the bottom end of the tube was found to be higher
than the pressure drop at the top end of the tube indicating incomplete acceleration of
eolidg. This r€sults from the fall of a group of particles whose dropping distance is larger
than that of an individual particle,. However, such observ¿tions with large differences
in pressure gradients at the top and bottom ends of the tube were excluded from the
2-3
No. Material Density
(gm/sec)
Particle
Diameter
(microns)
Shape
Terminal
Velocity
(*/')I Glass beads 2.47 96 Spherical o.t22
2 Sand 2.63 t73 1.230
3 Glass beads 2.47 644 Spherical 4.740
4 Steel shot 7.62 t7s Spherical 2.786
5 Steel shot 7.62 375 Spherical 5.945
6 Steel shot 7.62 637 Spherical 9.379
Tøble (2.1) DETAILS OF MATERIALS USED
Table (2.2) TEST SECTIOMETAILSNo. T\rbe Diameter (mm) T\rbe Material
I L2.7 Steel
2 19.1 Steel
3 26.4 Steel
4 38.1 Steel
No.
Solid
Material
Mass FIow Rate (gm/sec)
Orifrce Size (mm)
6 I l0 t2 l5 l9 25
I 96¡r Glass 8.74 16.9 27.66 49.39 83.77 139.9 290.47
2 173¡r Sand 7.7r 15.21 24.53 40.6 77.54 r33.4 284.0
3 644¡r Glass 4.76 10.71 18.33 33.87 61.83 ro4.2 227.8
4 179¡r Steel 26.63 53.1 84.47 158.9 27t.4 464.2 974.3
5 375¡r Steel 2t.97 45.44 73.46 t40.7 243.8 425.5 9r9.3
6 637p Steel 18.07 39.18 64.58 r27.1 226.3 386.8 854.7
Table (2.s) CALIBRATION OF MÁSS FLOW RATE OF SOLIDS
Table (2.1 DBOPPING DISTANCES IN .AIR
No.
Particle
Size
(p*)Density
(sm/cmî)
Terminal
Velocity
(*/.)Dropping Distance
(*)1 96 2.47 0.52 0.05
2 173 2.63 t.23 o.2t
3 644 2.47 4.74 2.81
4 179 7.62 2.7s 1.0Í|
5 375 7.62 5.95 4.64
6 637 7.62 9.38 10.01
NB: Dmpping dielonce is the distance travelled by the particle
before reaching 95% of it's equilibrium velocity,
while moving under gravity with zero initial velocity
C OU N TERC URNEN T EXPERIMENTS Chøpter 2
analysis. Observations with pressure drop readings differing by more than l0% at both
ends of the tube were not recorded. A pair of solid volumetric concentration and slip
velocity was obtained at each observation.
2.2.1.1 Concentrøtion - Slip Velocity Calculøtions:
Solid volumetric concentration was derived from the measured Pressure drop.
For steady state conditions the total pressure drop (AP¿) can be expressed as follows.
(AP), : (AP)c, + (AP),, + (AP)/, + (AP)/s (2.r )
where(AP),
(AP)o'
(AP),,
(^P) n(aP)¡,
is the Total pressure drop
is the Pressure drop due to static head of gas
is the Pressure drop due to static head of solid
is the Preesure drop due to gas-wall friction
is the Pressure drop due to solid-wall friction
(aP), : (aP),,
2-4
Since the density of solid is quite large (by a factor of 1000) in comparison with
the density of air, the static head of air (AP)', can be ignored. The solid-wall friction
component (AP)r, can also be iguored as its magnitude is insignificant in comparison
with the solid static head (AP)rr, especially at larger volumetric concentrations. In ad-
dition, solid velocities are always much less than particle terminal velocity due to coun-
tercurrent arrangement. The maximum solid velocity encountered was about 10m/s.
Such low solid velocities justify ueglect of solid-wall friction component. The gas-wall
friction component was experimentally determined with only air flowing through the
test section and was found to be negligible in comparison with the solid static bead
term
Following the above arguments the total pressure drop can be approximated to
solid etatic head component without incurring much error.
(2.2)
C OU N TERC U RREN T EXPERIMEN TS
The pressure drop due to the solid hold up is expressed as follows.
(AP)," : C p,gl
C_ (aP)rP'9L
Chopler 2
where C is the Volumetric concentration of solids
po is the Solid density
L is the Length of test section between pressure taps
Flom equations (2.2) and (2.3), the solid volumetric concentration can now be
derived from the pressure drop as follows.
(2-3)
(2.4\
FTom known quantities of solid and air volumetric fluxes through the tube eolid
and air velocities were derived as follows.
vsos
r-c (2.5)
u":? (2.6)
where Oe is the Volumetric flux of air
O, is the Volumetric flux of solid
The total upward volumetric flux of air (iÞo) through the test section was derived
from the volumetric flow introduced through the blower plus the rate of volume displaced
by the solids collected in the receiving container.
Kuowing air and solid velocities, the slip velocify (Vr) was derived as follows.
V, :Vc *V, (2.7)
The positive sign on the solid velocity is appropriate, since the solids are flowing
in a direction opposite to the air flow.
2-5
C OU N TERCU R REN T EXP ERIMENTS Chopler 2
2.2.5 Results
2.2.5.1 Efect ol Concentrøtion
The concentration and slip velocity nalues derived from the above mentioned pro-
cedure were analysed for a solid material in a test conduit. Slip velocity normalised with
respect to particle terminal velocity was plotted against solid volumetric concentration
to determine the relationship between them. Results of 96¡r glass beads in l2.7mm tube
are presented in Fig. 2.3. The following observations can be made from this plot.
l. Stip velocity is a unique function of solid volumetric concentration. Higher mass
flow rates of solids result iu extension of coucentration ra¡ge.
2. \{hite at very low concentrations slip velocity is almost equal to calculated par-
ticle terminal velocit¡ its value is larger than the terminal velocity at higher
concentrations.
3. The rate of change of alip velocity with concentration decreases with increasing
concentration.
The above trends are typical of the reeults obtained with other particles in all
the tubes investigated. Theee results are presented in Appendix (B).
2.2.5.2 Efrect ol Particlc Propcrties
In order to assegs the influence of particle properties such as density and eize
on the slip velocity-concentration relationship, slip velocitieg normalised with respect
to the coresponding particle terminal velocities were plotted against golid volumetric
concentration. The results of six different solid materials in l2.7mm diameter tube are
presented in Fig. 2.4. The following observations can be made from this graph.
1. At a given concentration smaller particlea have larger dimensionless slip veloci-
ties.
2-6
96 mrcnon Gloss beo'ds ln 12'7 mm T ubel4
11. 4
8.8
6.2
3.6
Solrd Flor¡ lgn/secl- 1.08
2.138.74
- 27.66- 49.39
83.??
Âa*
+
Ac}
2.4
o
X
4.8 7.2
CONCENTBRT I ON (7.)
I+o
A+xo+
1+
utnû)
)
Ê
;
t--
(JÐtll
o-
J(n
++
+o+
o
+
o+
t+to
+ooo
+ o
Xxx
XÀ+q,
+
I9.6 t2
0
Fig. 2.3 Effect of concentration
1,2.7 nn T ube
1uq)
J
Ê,
;
F
CJoJu.J
o-
J(n
t4
ll.4
8.6
6.2
3.6
0 2.4 4.8 7.2 9.6 l2
oo
HqtenrqlO- 961¡ G
^a- 173 l¡ S
+- 644 1¡ G
x- 179U S
o- 375 ¡¡ S
+- 637 l¡ s
oo
Iqss beqdsqndI qssteelteelteel shot
I
oobeqds
shotshot
o
oo
+
ooo
+
sb
++
o
oOO
x
oo
o
À
oo
oo
ooo
À
+
oo
A
oCD
+o
Fi g. 2.4 Eff ect of parti c'le properties
CONCENTRRTIgN V.)
C OU N T E RC U R REN T EX P ERTMEN TS Choptet 2
2. The degree of dependence of slip velocity decreases with increasing particle size.
B. The effect of particle density appears to be of little significance in comparison
with the effect of particle size. This was concluded from the observation that the
results of 173¡r sand (p, :2.6gm/cc) and 179¡r steel shot (p, :7.6gmlcc) tend
to fall on the same line. Similar obseryations can be made with 644p glass beads
and 637p steel shot.
Plots with other tube sizes are presented in Appendix (C). AU these plots show
similar trends.
2.2.5.3 Efect ol I\bc Diomete¡
Ptotting dimeneionless slip velocity against concentration for a given material in
different tube eizee, the following observations can be made. Results of 96p glass beads
in four different tubes are presented in Fig. 2.5.
1. At a given solids concentration dimeneionless slip velocity increases with decreas-
ing tube size.
2. The degree of dependence of slip velocity on concentration decreases with in-
creasing tube diameter.
Simila¡ trends were observed with other particles. These plots are presented in
Appendix (D).
2.2.6 Summary of Results
The trend of entire experimental data from the tests with six different particles
in four different tubes can be summarised as follows.
l. Dimensionless slip velocity is a unique function of concentration for a given solid
material and tube size.
2-7
Uì
h0)
a
=O
(JÐUJ
fL
JU)
t4
11.4
E.8
6.2
3.6
0 2.r
96 mtcnon GloEs becrdE
4.E 7.2
CgNCENTRRT ION ÍI)
9.6 t2
oo
Tubc 0 I ometen
o- 12.?a- 19. I+- 25.4x- 38. I
o
mfn
ñnmnnm
oQoo
oo
oo
+
oo o
osbo
o
+
o
oo
o
o
Aa
+¡X
o
Â
***** I
oo
EooÂ
ooo
A
+
o AÂ
X
o
Â
oo o
Ax
A@
Fig. 2.5 Effect of tube diameter
C OA N TERCURREN T EX PENIMENTS Chopter 2
2. Slip velocity increases with increasing concentration and its maguitude is larger
than the correspondiug single particle terminal velocity.
3. The degree of dependence of slip velocity on concentration decreases with in-
creasing tube sise and particle sise.
4. The influence of particle density on slip velocity-concentration relationship is less
significant thau that of particle size.
2.2.7 Compariæ¡ wiüå Existing Data:
Slip velocities related explicitly to volume fraction are rarely reported in litera-
ture. Iu general data are presented in tenns of loading ratio or gas to solid velocity ratio.
Lack of information about the corresponding solid and ga{t m:ßs flow rates limits the
derivation of the variables of interest. In addition, the majority of the data published
are confined to very low concentrations at high transport velocities, where solid-wall
friction effects dominate.
Matsen ( 1982) presents the following correlation based on the elutriation data of
Wen & Hashinger (1960), with 70¡r glass beads in l00mm diameter bed.
v,V3
I
l0.gCo.2ea
c < 0.0003
c > 0.0003IMatsenl
for
for
The above correlation suggests a linear relationship between the logarithmic val-
ues of volumetric concentration of solids and the dimensionless slip velocity. Analysis of
the present experimental data suggests that such a linear relationship does indeed exist.
Fig. 2.6 represents the data obtained with 96¡r glass beads in 12.7mm tube. Observa-
tions with other particles in all the tubes investigated gave similar results. The values
oI oA' and 'Bo obtained by linear regression are pnesented in T¿ble (2.5) for all the
experiments. The correlation coefficients are also presented. These values are greater
than 0.95 for majority of the data indicating a good fit. However, it is interesting to
note that the values of 'Ao and nB' va¡ied depending on the system properties; that
2-8
t¡u0)
J
e
.
t-(JÐJUJ
fL
(n
oo)o
l. 25
l. 05
0. 85
0. 65
0. 45
-2.2
Iosl0 ( C0NCENTRRTI0N )
.6 Concentration-slip velocity map on 1og-ìog scaìe
96 mrcnon GlosE beo.ds tn 12.7nn Tube
-1.92 -1.64 -1.36 -1.080. 25
-0. I
Fnom Leqst squonc fltSlooe = 0.544T- r ntercePt = l. 59
A
AA ÂA A
AÀ
.A
 A A
AA
A
AA
A
Fis. 2
Toble (2.5) PARAMETER ESTIMATF^S Of' CORRELATION__ ACB
Particle
Size
(¡')
Particle
Density
gmf cms
A B Regression
CoefrcientNo.
T\rbe
Size
(^^)12.7
r9.1
28.4
38.1
12.7
l9.l25.4
38.1
t2.7
l9.l25.4
38.1
t2.7
19.1
25.4
38.1
12.7
l9.l25.4
38.1
t2.7
l9.l25.4
38.1
96
96
96
96
L73
t73173
173
644
644
644
644
179
u9179
179
375
375
375
375
637
637
637
637
2.47
2.47
2.47
2.47
2.63
2.63
2.63
2.63
2.47
2.47
2.47
2.47
7.62
7.62
7.62
7.62
7.62
7.62
7.62
7.62
7.62
7.62
7.62
7.62
39.00
24.O5
13.28
10.45
33.37
t2.64
6.81
7.55
7.38
3.23
1.88
1.29
20.31
r4.60
7.43
5.82
9.07
3.40
3.20
3.31
5.60
2.48
1.76
1.79
0.544
0.482
0.356
0.305
o.642
o.426
o.297
0.320
o.374
0.232
0.160
0.058
0.476
0.467
0.349
0.286
0.359
0.212
0.215
0.231
0.28r
0.164
0.113
0.126
0.988
0.995
0.992
0.974
0.992
0.977
0.989
0.987
0.993
0.967
0.918
o.622
0.992
0.991
0.971
0.980
0.989
0.964
0.959
0.960
0.963
0.953
0.925
0.881
I2
3
4
5
6
7
8
I10
1lt213
L4
l5l6L7
l8l920
2l22
23
24
C OU N TERC URREN T EXPERIMENTS Chopler 2
is tube diameter and particle properties. These two parameters are correlated to three
dimensionless groups which characterise the system namely, particle Reynolds number
(Rrn) particle to tube diameter ratio (DolD¿), particle to air density nlio (p,lp).
The entire data of couutercurrent experiments can be represented by the following
correlation.
v,ACB (2.8)
Vt
A:93.67 (Bro)-o'nnt
B : 1.075 (frro)-o't*
Reo:
The values of the parameters o Ao and oBo predicted by the above correlation
are plotted against the observed values (Fig. 2.7 and Fig. 2.8), and the correspondence
is reasonabty good. For comparison the values of nA" and oB" reported by Matsen
(1982) are also presented. Since Matsents correlation was based on \ilente experiments
with 7lp glass spheres in l0l.lmm tube, the parameters nAo and 'Bo were estimated
from these system properties using the above correlation. The agreement between the
estimated and reported values is good, considering the fact that the tube size used iu
Wen et al's work was larger than the tube diametere used in the present study by at
least a factor of three.
Yerushalmi & Cankurt (1979), report increasing slip velocities with increas-
ing concentration, based on their experiments with 50p spherical catalyst particles in
l52mm tube. However, the concentration-slip velocity relationship was not unique, but
depended on the mass flow rate of solids. The solid volumetric concentration and slip
velocity were inferred from the pressure gradient measur€d over the middle of the test
section. Unfortunatel¡ no mention was made as to whether steady state conditions
prevailed in such measurementg. If the acceleration of solids in the test section is in-
2-9
(+)""(ä)""(#)""(ä)""
50CORRELÊTION OF PRBRMETER 'R'
l0 20 30 {0
OBSERVED VRLUE
CORßELRTION OF PRRRMETER 'B'
++
+ +
0.3 0. {5
30
l¡J:lJG
ol¡Jt-cJ:fLJJG(J
{0
20
0
t0
0 50
+
lr,=JG
ol¡J)-GJ=L)JG(J
Fig. 2.7
0. 75
0.6
0. t5
0.3
0
+
0. 15
+
+
+
+
++
+
*- P¡srcnt dotoX- llot¡cn'¡ doto
++
+x
++
++
+
+
+
+
x+
+++
++
*- Pnc¡cnt dotoIt- Xot¡cn'¡ doto
Fi g. 2.8
0 0. 15
OBSERVEO VÊLUE
0.6 0. 75
C OU N TE NC U R REN T EXP ERIMEN TS Chqtet 2
complete, the concentrations and slip velocities may have been overestimated. This in
tura explains the introduction of solid flux ¡rs a parameter in concentration-slip velocity
relationghip. Also, the size range of catalyst particles ueed in their tests was reported
to be Gl30¡r. Inforrration on the distribution of the particle size was not presented ex-
cept for the mean particle size. The reported slip velocity at negligible solid volumetric
concentration (C < 0.1%) ie about l3 times larger than aingle particle terminal velocity,
when a correspondence is expected instead.
Yousfi & Gau (1974), present the following correlation for slip velocit¡ based on
their experimente with 20¡r and 183¡r particles in 38mm and 50mm diameter tubes. The
range of solid volumetric concentrations reported is 0.5% to 22% .
lYouslil
The above expression suggests that slip velocity is a strong function of concen-
tration and it increases with increasing concentration. Since thie correlation involves
gas Reynolds number as a parameter, comparison with the correlation proposed (2.8)
is not feasible. However, the following observations reported by Yousfi & Gau (1974),
agre€ with the trends observed in the present work.
l. lVhile the dimensionless slip velocity ranged from 8 to 40 in the case of 55p
catalyst particles, its value varied from 40 to 300 in the case of 20¡r particles.
2. For large 290¡r Polystyrene particles dimensionless slip velocity was much smaller
than that observed with fine particles. Its value ranged from I to 4.
3. The effect of concentration on slip velocity was less siguificant in the case of large
particles.
The large magnitudes of dimensionless slip velocity reported in the caee of fine
particles need some attention. In fact, dimensionless slip velocities as high as 300 were
reported while the value predicted by the correlation (2.8) is only about 37 zt the ma:<i-
2-10
ft:r+s.zx ro-3(ß,0)ou "'^ (+)-'" E)
C OU N T E RC U R NEN T EXP ERIMEN TS Chopter 2
mum solid concentration studied (22%).This discrepancy is possibly due to electrostatic
charging and adhesion of particles. These effects are known to be enhanced with de-
creasing particle size. Nevertheless, the correlation proposed [2.8] ig not recommended
for such fine particle systems until additional data are available in support of it.
2.2.?.1 Somc Remø¡ks on the Correløtion
On closer examin tion of the correlation (2.8) the powers on particle Reynolds
number and di'meter ratio groupg are almost same in magnitude, but opposite in sign.
Approximating the magnitudes of the powers to the same value, a new dimensionless
group is realised.
(R"p)(+):ry
This new dimensiouless group is the tube Reynolds number cornesponding to
particle terminal velocity. The siguificance of this group is that it represents ideal
choking flow gas Reynolds number. he correlation presented ea¡lier can be rewritten
in terms of this new group as follows.
l: n"' (2.e)
A: e3.7 (R",)-' (i)"'B: 1.075 (Rrt\
Re¡: DtVtPs
lrs
u r 0.313-0115 li)
The above correlatiou suggests that higher the choking Reynolds number smaller
is the magnitude of dimensionless slip velocity (A) and the degree of dependence on coD,-
centration (B). The contribution of density ratio is also significant. This might appear
to contradict the observed correspondence of slip velocity-concentration relationship
between particles of almost same size with large differences in densities . The following
reasoning should clarify the matter.
2-tt
C OA N TERCURREN T EXPERIMEN TS Chopler 2
Particle Reynolds numbers of the materials used in this study range from 3 to
400 . In this intermediate region the experimental study of Jones et al (1966), suggests
that the terminal velocity of a particle is proportional to
0.66
Vq.DP I)
lJonesl
Since density ratio for gas solid systems is very large the above expression can
be approximated to
DP(2.10)V¿x
Examining the expressiong (2.10) and (2.9), the effect of particle density on
parameters 'Ao and oBn is made clear. Although higher particle density regults in
larger values of density ratio term, a corresponding increase in particle terminal velocity
results in little change in the value of parameter oAn. Similarþ parameter oBo suffers
little change.
Although correlation (2.9) is attractive in terms of fewer dimensionless groups,
correlation (2.8) should be preferred, until additional data are acquired to establish
that the correspondence of powers on particle Reynolds number and tube to particle
size ratio is not fortuitous.
2.2.7.2 Proposed Corrc,Iøtion
The form of the correlation (2.8) to predict slip velocity at any particular solid
concentration, is not completely satisfactory in the sense that it can not be extended to
very low concentrations, and requires the specificatiou of a lower limit to concentration
below which slip velocity is approximately equal to particle terminal velocity. Therefore
other forms of correlations which extend to zero concentration are considered. One of
guch forms is as follows.
Y - r: ACB (2.11)Vt
2-t2
( a_Pe
Pctrog'}ï
2t-Ps
0.66
C O U N TE NC (TN REN T EX PE NIMEN TS Chopter 2
Unfortunately, the degree of fit for the above type of correlation is very poor.
The teft hand term in the above expression is very eensitive at low concentrations and
for large particles where Vrlvl term is closer to unity. As a consequence, the correlation
(2.8) presented earlier is preferred but with the following modification.
wV¿
I
ACE
C 1C^;n
C ) C^¡n(2.r2)
for
for
Cmin: (l¡-r¡awhere
The above correlation suggeets that the lower limit of concentration (C-;o) de-
pends upon the r¡alues of paraneters nAo and 'B' which in turn depend on the system
properties. The orpressions for these parameters euggest that their valuee decrease with
increasing particle size and tube diameter. In other words, for large particles the value
of minimum concentration (C*;"\ is larger than for small particles. This, indeed, is
experimentally observed fact. While for 96¡r glass spheres in l2.7mm tube slip velocity
started to increase even from concentrations as low as O.LTI, its value remained apProx-
imately equal to terminal velocity eveu up to lYo concentration in the case of 644¡r glass
beads in 38.lmm tube.
Atthough the above correlation adequately represents the data of the present
experimental programme as well as some of the data reported in literature, correlations
with gome theoretical background need to be explored. Some of these aspects will be
discussed in the following section.
2.2.8 ThæretícalApprcæh
The significance of the above results, is that it is clearly demonstrated that slip
velocity increases with increasing concentratiou for gas-solid flows in tubes. Remarkably,
in batch fluidization and sedimentation phenomena it is very well established that slip
velocity decreases with increasing concentration. This in turn suggests, a fundamental
2-t3
C OI] N T ENC U R NEN T EX P ERIMEN TS Chopler 2
difference between batch fluidization where there is no net movement of solid phase,
and the flow of gas solid suspensions in tubes.
In order to gain some qualitative understanding of gas-solid flow through tubes,
a section of transparent transport tube of 25.4mm diameter was filmed with 173¡r saud
flowing in it against upward moving air (Fig. 2.9). The photographic study revealed
formation of large clouds of particles which moved downward faster than the individual
particles. Collision a¡d break up of large and fast moving clouds with slow moving
cloude was also observed. This phenomenon was much more pronounced at higher
concentrations corresponding to larger upward gas velocities. When the gas velocity
was sufficieutly high, reverse flow of solid was observed along with extensive backmixing.
Overall the coalescence of particlea into clouds appeared to be more pronounced near
the wall of the tube. Theee observations were limited due to the two dimensional
nature of the pictures and are likely to be subjective. Nevertheless, based on the above
observations, the results obtained in the counter current experiments can be given some
physical basis.
The terminal velocity of a cluster of particles is larger than that of a single
particle. This accounts for the obgerved slip velocities being larger than corresponding
particle terminal velocities. The formation ci such clusters especially near the walls
of the tube suggests that the velocity gradient might be a goveraing factor. In other
words, layers of gas moving at different velocities tend to bring the particles in them
together at least momentarily to form a cluster. This phenomenon, usually known as
gradient coagulation is the basis of the model that will be presented in the following
section, in an attempt to explain the observed trends in slip velocity.
2.2.8.1 Grad.ient Coøgalation Model
Fuchs (1964) presents the gradient coagulation theory proposed by Smolchoweki
(1911). According to this theory, the frequency of contact (f) of a particle by other
particles, moving in a fluid with a velocity gradient transverse to the direction of flow
2-t4
f
o
læ
¡
F
Fig. 2.9 Pictures of 25.4mm test section with sand flowingagainst rising air stream.
C O A NT ERC U N REN T EX PE RIMEN TS
is given by
Chopler 2
(2.13)
where n is the Number concentration of particles
Dp is the Pa¡ticle diameter
r is the Fluid velocitY gradient
The above result has been confirmed by Manley & Mason (1952), based on their
experiments with l73p glass spheres in high viscosity corn syrup, at concentrations
ranging from 0.3 to 2.3% and velocity gradients ranging from 0.4 to 2 sec-1.
Extending the above theory, for gas-solid flow through the tubes, the frequency
of collisiong between the clusters having oNo particles is
lzy q. nNî (2.14)
where iuv is the Number concentration of clusters of size "No
lzn is the trlequency of collisious between clusters
If the total volumetric concentration of solids is C
CnN :
/vup
where
I : f,nnl,
up is the Volume of a particle
JV is the Number of particles in a cluster
lzx q "*h (2.15)
The velocity gradient (r) for pipe flow isVof D6 where Vo zn.d D¿ a;re gas velocity
and tube diameter respectively.
If it is postulated that the frequency of collisiou of these clusters to form a doublet
is in equilibrium with the frequency at which these doublets break up iuto singlets owing
2-t6
C OA NTERC U R REN T EXPENIMEN TS Chopler 2
to what ever destructive forces there are, then the following e)cpression results.
ivI c (2.16)
Kz¡,¡
where
Kz¡r- is tbe Equilibrium constant
This model is extremeþ simplistic because clusters of all sizes will be forming
from collisions of all kinds of groups of particles and there will be some kind of complex
equilibrium. However, the simplified model euggests that the average eize of a clus-
ter (N) is a function of concentration of solids in the tube and the velocity gradient
(VclDr).If the above model adequately describes the gas-solid flow in tubes, the trend
of increasing slip velocity with decreasing tube size at a given volumetric concentration,
can be attributed to higher velocity gradients in smaller tubes. Since slip velocity of a
cluster is a flnction of the size of the cluster(N), plotting slip velocity against product of
solid volumetric concentration and velocity gradient should bring the results of a solid
material in different tubes together.
Plots of dimensionless slip velocity versus product of concentration and velocity
gradient are presented for six differ€nt solid materials in Appendi* (E). For a quick
reference, Fig. 2.f0 represents the results of 179¡r eteel shot in tubes of diameter rang-
ing from l2.7mm to 38.lmm. Although, the correspondence of slip velocities between
different tube sizes is not exact, the general trend is to bring the results of different
tube eizes together.
Some deviations from the general trend were uoticed, especially with the data
corresponding to low gas velocities in small tubes. These deviations are larger for heavy
and large particles (637¡t steel shot). One possible explanation for the deviations at
low gas velocities is that the gradients are not large enough to bring about eubstantial
coagulation. The large deviations for large and heavy particles can be attributed to the
large relarcation times of the particles. In other words, large particles are legs influenced
2-16
vsDt
1?9 mrcr-on SteeI shot7
o
uot)J
e
;
t-(JÐJu.J
o-
JU)
5.8
Tube 0 I qnctcn
O - 12.7 nnA- 19. lmm*- 25.4mmX- 38. 1 nn
4.6
3.4
o o o oo o
2.2 o{oX
X,(-1. 5 -0.8 -0. 1 0.6
Iosl0 ( C x Vs/Dt )
Fi g . 2.LO Test of grad'i ent coagul ati on model
oo
o
^g,Þf*+
oo
oooa
ÐaA
1.3I
2
C O A N T ERC U R NEÌI T EX PERIMETI TS Chopter 2
by intermixing layer of gas. As a result theee particles have lesser tendency to come
into contact with each other, than the tight and small particles which follow the ûuid
stream at a given velocity gradient and particle concentration.
The timitatione of the propoeed model can possibly be overcome by mathemat-
ical representation of the phenomena of coalescence and break up of clusters due to
collisions with each other and the transport tube wall. Lack of understatrding of these
complex mechanisms limits any further analysis. Some quantitative information about
the size distribution of clustere and their densities (solid content) is neceseary before
contemplating modelg with further complications.
2.9 signiflcance of concentration-slip velocity Relationship
Eaving determined the slip velocity and its dependence on concentration, the
significance of such a relationship to the design of forward transport systems will be
congidered here.
If it is presumed that the above relationship obtained from the countercurrent
apparatus, carr be applied to forward transport at velocities which are not too high
(where solid-wall friction component is negligible) the dependence of solid volumetric
concentration on air flow at a ûxed solid throughput may be derived from the coutinuity
and slip velocity statements.
Flom the experimentally determined slip velocity-concentration relationship (eq-
uation 2.8), the solid concentration (C) at a given superficial gas velocity (Oc) for a
fixed solid flux (Or) can be calculated by trial and error procedure using the following
equalities.
Qg: (t - C)Vs (2.17)
lþ,: CV" (2.18)
(for cocurrent transport)
2-t7
V, :Vc -V" (2.1e)
C OA N T E RC A R REN T EX PERIMEN TS Chopter 2
The result of such a calculation using a value of solid flux (O'), of 0.1m/s is given
in Fig. 2.11. What it shows is best appreciated by consideriug a progressive reduction in
air flow (Oo) from an initially large value. The concentration of solid increaseg slowly at
first in accordance with a value of relative velocity not very different from the terminal
velocity of a single sand particle. As this concentration rises, however, its effect on slip
velocity becomes significant. The concentration then rises more steeply with falling air
velocity. The steeper rate of increase leads to a more rapid rate of rise of slip velocity
and the process accelerates. Eventually there is a dramatic increase in concentration
at a particular air velocity. This may be regarded as the nchokingS velocity. The la^rge
concentration of solid presentg a correspondingly large Pl€s8ure gradient to impede
transport.
Other solid loadings produce different curves of concentration against air velocity
and, indeed, different values for the choking velocity. Results for solid fluxes of 0.02
and g.Sm/s are also shown on the Fig. 2.11. At the greater flux the choking velocity is
higher and the rate of onset is more gradual. At the lesser flux the choking velocity is
somewhat lower than for 0.lm/s but the event is catastrophic. The increase in choking
velocity with eolid loading is simply a reflection of the variation of slip velocity with
volume fraction. The different rates of approach to choking can be explained in terms
of the ranges of slip velocity within each of the solid loadings. At the lower solid flux
the range of slip velocity is from the particle terminal velocity to the high value near
the volume fraction of 0.15. For the higher solid flux, range of slip velocity is shorter
because volume fraction is already considerable, so that the slip velocity is substantially
greater than terminal velocity, before the air flow ie reduced.
Alternatively, similar remarks about the family of cun¡es of the concentration
versus gas velocity profiles can be made on the basis of the following mathematical
approach. From equation (2.19)
z9ÞG)-zl¡lozo(J
eJoØ
.15
.10
'05
o
F j g. 2.Il
o10 20
A¡R FLOW (m/s)
Calculated variation of solid volume fraction withai r fl ow i n cocurrent verti cal fl ow at a fi xedsolid flux of 0.1 m/s.
30
III¡
II¡tI
tttt\
___
\
It,,It,tt,t\
IIIIIIIIIIIIIIt
o.o2
\I\r O.5\ \ \
and
C OA N TERC A RRDN T DX PERIMEN TS Chopler 2
For constant solid flux (Or)0v, o, acavo: - cr' av,
ðv, ôv, acðvo ac ïvo
Therefore,AC I
(2.20)ðvo O, AW
æ- ac
The equation (2.20) suggests that the rate of approach of choking (AC lôVs)
depends on the solid flux, concentration and the rate of change of slip velocity with
concentration. What it shows is that the slip velocity concentration relationship governs
the choking phenomenon. At a given concentratiot (ôVrl0C) is frxed. Consequently,
the rate of change of concentration with gas flow rate increases with decreasing solid
volumetric flux. This explains the catastrophic event mentioned earlier at low flux
values. In addition the terln (ôVrlAC) decreases with increasing concentration.
2.1 Conclusion¡
1. The influence of solid volumetric concentration on slip velocity in the case of
gas-solid flowe in pipes is different from that of fluidization and sedimentation phenom-
ena. Use of Zaki's correlation which predicts decreasing slip velocity with concentration
is inappropriate for gas-solid flows in pipes.
2. The experiments carried out covering a wide range of particle and tube sizes,
confirm the conclusion of Matsen (1982) that slip velocity should increase with increasing
concentration in order to explain the choking phenomenon observed in gas-solid flows.
3. An attempt is made to give some physical basis for the observed trends in
slip velocity. The large values of slip velocities are due to the formation of clouds of
particles whose terminal velocities exceed corresponding particle terminal velocity. A
plausible mechanism by which these clouds are formed is gradient coagulation.
2-t9
C OU N T E RC U RREN T EXPERIMEN T S Chagler 2
4. The usefulness of the slip velocity-concentration relationship which was de-
termined with ease and accuracy, in predicting choking flow rates is demonstrated.
Although this was based on the presumption that the above relationship holds good
for forward transport at low transport velocities, the results of subsequent experiments
which will be discussed in the following chapter validate the assumption.
2-20
CHAPTER 3
COCURRENÎ E](PERIMENTS
3.1 Introduction
Having determined the effect of concentration, tube diameter and particle Prop-
erties on elip velocit¡ the effect of transport velocity remains to be investigated. The
cocurrent trausport experiments designed with the following objectives are described in
this chapter.
8.2 Objectives
l. Investig¿te the effect of transport velocity on slip velocity.
2. Determine the significance of solid-wall friction at high transport velocities.
3. Investigate the applicability of the concentration-slip'velocity relationship ob-
tained from cocurrent experiments to forward transport situations.
4. Determine alip velocity with ease and accuracy
3.8 Review of Meaeurement Techniques
To study the effect of transport velocity, accurate determination of the same is
essential. Although solid velocity and concentration valuee were derived from the pres-
sure drop measurements in the countercurrent experiments, similar technique can not
be adopted in the cocurrent transport arrangement since the role of solid-wall friction
can not be ignored, especially at large transport velocities.
Before going into details of the orperimental programme, it is worthwhile review-
ing the various solid velocity measurement techniques. Boothroyd (1971) presented a
good review of instrumentation for use in gas-solid systems. Some of the measure-
3-1
COCURRENT EXPERIMENTS Chopler 3
ment techniques used by several investigators in the flow of gas-solid suspensions will
be reviewed here.
Usually, measurements of mass flow rates of gas and solid a¡e simple and ac-
curate, however determination of solid velocity poses some problems. Basically the
measurement techniques for solid velocity can be classified as direct and indirect mea-
surements.
3.3.1 Direct Measuremenús
3.8.1.1 Radiooctivc høcer Method
Solid velocity is determined by measuring the time required for the radioactively
tagged particles to move from one point to another. This technique was employed by
Ilours & Chen (19?6), Van zuilichem (107e) and Jodlowski (1976). The problem of
contamination of test material by the tagged particles was overcome by making use
of short lived radioactive tracer material for tagging process. The advantages of the
method are as follows.
1. The detectors are external, readily moved to a desired location and do not induce
flow disturbances.
2. The instrument does not require calibration as the pa,rticle velocity is measured
directly.
However, this method is expensive. Brewster & Seader (1980) reported that by
using a phosphorescent material for tagging and photomultiplier detectors for signal
processing the above technique can be made much cheaper. They maintain that the
photomultiplier detectors are far cheaper than the gamma radiation detectors. Except
for the disadvantage of having to have windows at the detector locations their method
offers all the advantages of the radioactive tracer technique.
3-2
COCUNRENT EXPENIMENTS Chopter J
8.8.1.2 Electrical Capøcity Method
This method was developed by Beck et at (1969,1971) and found its application
in industry as a means of mass flow measurement and detection of solids. The velocity
of solid is derived from the transit time of the naturally occurring flow noise pattern
between capacitance transducers at two positions along the transport line. The transit
time is determined by cross correlating the transducer outputs either with an on-line
computer or by collecting the data to be processed later. The adwntage of such a system
is that it does not disturb the flow as the capacitors are fixed on the transport line and
the instrumentation is less involving. Ottjes (1976) used this method to measure the
velocity of eolids in vertical transport. However, one criticism of this technique is that it
assumes that the noise or disturbance travels at the Bame velocity as that of the medium
without being attenuated.
3.3.1.8 Photogrøphic Stroboscopic Methad
In this method, two photographs of the gas-solid stream are superimposed on the
same photographic negative. The velocity of a particle is derived from the displacement
of the particle in the negative. The light Bource is a stroboscope, the interval being
predetermined by a multivibrator. The distance travelled between the flashes should
not coincide with the actual distance between the particles for better contrast. Thig
method was employed by Hitchcock & Jones (1958), Reddy E¿ Pei (1969) and McCarthy
& Olson (1963). Hitchcock & Jones (1958) in their work with particles of size range 2
to 7mm reported difrculty in measurements with emaller particle sizes due to double
images on the photographs. They also observed that the method failed for rotating
particles and dense medium in which resolution was poor. Later works of Reddy &
Pei (1969) and McCarthy & Olson (1968) did not have these difficulties even with
particles as small as 100¡r, probably due to improved techniques such as the use of a
narrow depth of ñeld camera. Although this technique provides the meatrs of obtaining
the local solid velocity, the average solid velocity which ig often of interest can not be
3-3
C O C U R R DN T EX P E R ITI,TX ¡V IS Chopter J
directly determined. Also continuous meisurement of solid velocity is not possible.
5.5.1.1 Cinc Camero Method
Particle velocity is determined by comparing the progress of the coloured gran-
ules, frame by frame against a metered scale. Though the technique is simple it could
not be uged for smaller particles owing to rapid dispersion of coloured particles. Jod-
lowski (1976) employed this technique successfully with large particles (about 3mm in
diameter) and the results were found to be in good agreement with those obtained from
the radioactive tracer technique.
8.3.7.5 Loser Doppler Velocimetry (LDV)
This technique was first introduced in 1964 and was applied to single phase flows.
It is based on the principle that when a particle with some velocity is intercepted by
a laser beam, the scattered beam experiences a shift in its frequency. This frequency
shift is then related to particle velocity. Briefly, the relationship is given as follows.
+I
+I t
where
b:î,Ào
is the Doppler shift
is the Velocity of particle
is the Unit vector along scattered direction
is the Uniü vector along incident direction
lpv,
I,
There are three different modes of operation of LDV, details of which are given
by Durst et al (1972). The advantages of LDV are aÁr follows.
l. The instrument is linear
2. It does not require calibration as all parameters are easily determined
3-4
--+t;
COCURRENT EXPERIMENTS Chopler J
3. Using the dual beam method, a control volume of any dimension can be chosen.
This enables the measrurement of local velocities without disturbance to the flow,
which occurs in probe type instruments.
Studies by Riethmuller & Ginoux (1973) with particles from 100 to 500¡r in
diameter trausported at velocities between 2 and l00m/s showed that LDV can be
successfully used for velocity mea͡urements. Birchenough & Mason (1976) encountered
difrculties at higher solid loading ratios due to a decrease in the signal to noise ratio. It
was reported that the maximum loading that allows measurement is determined by the
point at which processing electronics can no longer deal with the inadequate frequency
information. However, this difficulty was overcome by increasing the intensity of laser
beam. Disadvantages of this technique are that it is expensive and sophisticated.
3.3.2 Indirecú Measure¡nents
The ve;locity of solids can be derived from the knowledge of volumetric concen-
tration of solids in the pipe, and volumetric flux of the solids. The later quantity is
usually easily determined. Some techniques used to determine solid concentration are
described here.
5.3.2.1 Isoløtion Method
The average concentration of solids in a pipe is determined by trapping the solids
in a section of the pipe by simultaneous quick closing valves. Hariu & Molstad (1949),
Gopichand et al (1959) and Capes & Nakamura (1973) used this technique in their
studies. Although this method is simple, its major disadvantage is that it is very time
consuming as the flow needs to be interrupted for each measurement. Another draw
back with this method is the effect of time delay in closing the valves. Although it is
argued that by synchronizing the two valves the effect of time delay is nullified since in
that time the same amount of solids escape from the downstream valve as the amount
of solids that enter through upstream valve, it is hard to account for any disturbances
3-5
COCURRENT EXPERIMENTS Chopler 3
created in the flow by the closing valves. The results of Capes & Nakamura (1973)
are more reliable tha¡ other workers as the hand operated valves were replaced by
elect rically controlled, pneumatically operated valves.
8.8.2.2 Attenvøtion Ol Nucleor Rn'd'iation
A flux of beta particles from a radioactive source can often be attenuated to
a suitably measurable extent by gas-solid suspensions. The degree of attenuation is
a function of the concentration of solids. The beta ray source is of interest because
the attenuation of rays is independent of the atomic number of the material ueed for
calibration and tests. This method gives only average values of concentration and was
used by Jodlowski (1976) in his studies.
8.8.2.3 Optical Method
Arundel et al (1971) used this method to measure the deusity of suspensions. The
instrument used operates on the following principle. Light from a bulb travels along
two identical paths, one beam p:rsses through the suspension while the other passeg
through an optical wedge to simulate the suspension. Fiber optic guides allow the
instrument to traverse without causing the light signal to change. Both the light signals
pass alternately through a chopper to a photo-multiplier. Any fluctuations from the
amplified signal are eliminated by adjusting the optical wedge, the position of which is
calibrated to give suspension density. Unfortunately, this technique requires calibration
prior to use, and may cause damage to the optical surfaces due to particles in the
suspension.
In summary, techniques such as Laser Doppler Velocimetry, Capacitance method
and beta ray adsorption are attractive in the sense that they enable continuous mea-
surement of solid velocity without disturbing the flow. llowever, they suffer from the
disadvantage of being sophisticated and expensive.
3-6
AOCURRENT EXPERIMENTS Chopler I
Consequently, it is felt that the development of a simple technique to measure
solid velocity is necessary. Measurement of the momentum of solid material by impact
on a plate is considered. This technique although not entirely new, has never been used
for solid velocity measurement. Boothroyd (1971) reviewd some of the impact meters
used for mass flow rate measurements.
8.4 frnpact Meter
3.4.1 Príncíple of Operztíon
The total æ<ial momentum of solidg moving in a transport line at a certain
velocity (lzr) can be expressed as follows.
Solid phase a>cial momentum : lÞrprA"V, (3.1)
If this momentum can be accurately measured, then solid velocity (Vr) and con-
centration (C) can be inferred from the knowledge of total volumetric flux of solids (Or),
which is usually determined with ease.
The principle of operation of the impact meter is convension of the axial mo-
mentum to a measurable force. This is achieved by deflecting the solid material at
right angles to their mean travel direction. Ideally, if all the particles lose their a:cial
momentum on impact, the force experienced by the impact plate can be derived from
Newton's second law as follows.
Force : Rate of change of a:<ial momentum
: tÞrprA"V, - QrprA"(o) (B.z)
: lÞrp"A"V,
This force is easily measured with the aid of a load cell. The advantages of such
a system are af¡ fclliows.
3-7
COCURRENT EXPERIMENTS Chøptet 3
l. Measurement of force is simple and accurate
2. Does not disturb the flow, as the impact plate is placed at the exit of the transport
tube.
3. Allows continuous meaÉturement without any interruptions.
3.4.2 Descríption of Impact Meter
The objective in the design of the impact plate is to ensure that the solids lose all
their a>cial momentum on impact. In order to achieve this, the circular impact plate of
l50mm diameter, was machined to have a central cone with a 50mm base radius (Fig.
3.f). The cone w:ur machined to have curvature of 25mm radius. The tip of the coue
was positioned at the exit of the transport line, on the central a:<is of the test section.
The purpose of the curvature on the conical tip is to guide the deflecting solid particles
with successive collisions to a direction normal to their flight path. It is essential that
particles have no vertical velocity component as they leave the impact plate. If the
particles bounce with a velocity component opposite to the general direction of flow,
the measured thrust will be higher and as a consequence solid velocity is overestimated.
On the other hand if the particles leave with some velocity component in the direction
of flow, the thrust will be lower, resulting in underestimation of solid velocity.
In order to ensure proper functioning of the impact plate, flight path of 644u
glass beads travelling at an estimated speed of 20m/s in 25.4mm (/) tube, was filmed
while they were bouncing off the impact plate situated at the exit of the transport
line. Picturee taken at 250 frames per second indicate that the majority of particles
are being deflected by the impact plate normal to the direction of travel. Some of the
frames selected at random are presented (Fig. 3.2). The blurred streaks are the fast
moving glass beads. In general it was observed that the direction of these streaks is at
right angles to the direction of the impinging solid stream.
3-8
Fig. 3.1 Impact P'late (side view)
Fig. 3.2 Pictures of impact plate wÍth 644 micron glass beadshitting it at an estimated speed of 20 m/s.
COCURRENT EXPERIMENTS Choptet I
3.1.3 Meæurement of îàrust on the Impact Plate
The thnrst on the impact plate rvas measured with a oMini Beamo load cell.
The actual load on the load cell was read with the help of a VIP504 Strain-Gauge
input digital process indicator. This device provides the DC excitation necessary for
a four wire strain gauge load cell and gives an easy-to-read display of the transducer
output. The meter also provides an analogue DC signal of one volt corresponding
to the marcimum load specification. This DC signal was fed to a continuous running
uflekidenki" chart recorder. The signal from the transmitter, corresponding to the
weight of the impact plate experienced by the load cell was biased with a variable DC
voltage Bource. ïVhen the impact plate experiences upward thrust due to the impinging
flow of gas-solid suspension, the resulting change in the net force on the load cell beam
is measured by the corresponding change in transmitter output. The transmitter output
was calibrated against the force experienced by the load cell with the help of standard
weights. The response is linear. The gain of the system is 0.485mv lS 1. FiS. 3.3
presents the calibration of the load cell response. In order to dampen the high frequency
fluctuations of the impact due to gas-solid suspension, the signal from the transmitter
was filtered with an RC circuit before feeding it to the chart recorder.
3.4.4 Calibr¿tion of îårust due to Aír
In order to determine the thrust due to the solid phase the contribution of thmst
due to air alone should be determined. It is therefore necessary to calibrate the ac-
tual thrust imparted by air, against the theoretical value obtained by assuming that
the air gives up all of its æcial momentum on impact with the plate. The thrust e><-
perienced by the impact plate was measured at several gas flow rates through all the
tubes investigated. The calculated value was plotted against the observed thrust(Fig.
3.4). A linear relationship is obtained in all experiments. The observed thmsts are in
close agreement with the calculated values in the case of 12.7mm and 19.lmm diameter
tubes. However, with larger tubes, observed thrust is higher than the calculated value
3-9
l 000
Load Cell Gain = 0.485 mv/gmf
600
400
200
o 400 800 1200
nPPLIED L0ßD Ismf)
Fig. 3.3 Calibration of load cell response
800
jl-
=(LFfoJ-Jt¡J(J
oGe
0r 600 2000
5
EIFU)
=E=t-cfttJ
CEIUU)(Do
r2.5
t0
?.5
0
0
Fig.3.a(a)
50
40
0
C or-ne i of ton of T hnuEt due t o R In
2.5 7.5 l0
CRLCULRTED THRUST (smf)
C onne 1o,t r on of T hnuEt due t o fl I n
20
2.5
5 12. s
rÞÉI
F_<l)
=Ê.:trt-oTU
ccU.JU''(DÐ
30
20
l0
0
30
+
Tub¿ 0ronr¿ten! 12.?nn
SL0PE = 1.07
Tube 0romoten: 19. lnm
SLOPE = l.0l
Fig.3.4(b)
l0
CRLCULRTED THRUST ( smf )
40 50
Connelotron of Thnust due to R¡n75
45
;EI
F-a=(t-l-oU.J
ÉlrJanfD0
60
30
t5
Fig.3.a(c)
75
60
30
t5
Tub¿ 0 r oln?t¿F¡ 25. ¡lm¡r
SL0PE = l. t6
0
0 t5 30 45
CRLCULRTED THRUST (gmf )
60
Connelo,tron of Thnust due to Rlr'
?5
45
;EI
l-a=æ-t-oUJ
Ê.trJU)(Do
0
0
Fig.3.4(d)
30 {5
CRLCULRTED THRUST ( smf )
luba 0rolncter.¡ 38. lm¡¡
5L0PE = t.22
l5 60 ?5
COCURRENT EXPERIMENTS Chopler J
by about 20%. This suggests that in the case of large diameter tubes where turbulent
conditions prevail, the air stream is leaving the plate with some negative component of
its axial velocity. Another possible explanation is that the air stream might be flowing
around the edges of the plate resulting in fluid d.ag. Additional thrust could also have
been induced due the pressure difference between the front and rear of the impact plate.
Atl these non-idealities could possibly be associated with a drag coefficient. However,
this f¿ctor should not aftect the accuracy of the measurement of slip velocity, since air
velocity is inferred from total volumetric flow of air and the solid velocity is calculated
from the thrust due to solid phase alone. The thmst due to solid phase is determined
by subtracting the thrust due to air from the total thrust. The underlying assumptiou
in such a procedure is that the thnrst due to air in the presence of solid is same as the
thmst when air alone was flowing at the same velocity.
3.6 Apparatue
The apparatus for cocurrent transport experiments is schematically represented
in Fig. 3.5.
3.5.1 Solid Feed Mechanism
The solid feed mechanism consists of a solid feed tank with provision for inter-
changing orifices at the bottom of the tank. Solid flow through the orifice is controlled
by a tapered plug valve designed to allow gradual opening of the orifice aperture. The
solids leaving the feed tank are introduced into the transport line . The pressure at
this point and the pressure in the feed tank are equalised by a connecting line between
them. This allows smooth flow of solids through the orifice irrespective of the presslure
fluctuations in the system.
The advantages of such a system over the conventional screw feeders, fluidized
stand pipes and venturi feederg (which suffer from the fluctuations in solid feed rate or
limited control over the solid feed rate) are as follows.
3-10
SOLIDLLECTI
.BIN
TORAGEBIN
if ice Plate
id Feed
ug
Load Cellt-I
I
I
SOLIDSEPARATOR
n Air Exhaust
I
oressufeiapþ¡nss
sureTransducer Pen Recorder
DATA COLLECTION
Pre re
Meters
atr
AIR'SUPPLY
air &solid
I
I
I
I
I
I
I
I
I
I
J
t-'
l¡¡z=
TEST SECTION3 Metres Long
Itol¡¡¡¡¡GÐanot¡¡Ec I
iL._.
ACCELERATINGSECTION
Fis.3.5 @trL! FLOW APPARATUS
COCARRENT EXPERTMENTS Chopter J
l. Any golid feed rate can be selected simply by choosing corresponding aperture
Z. Solid feed rate can be accurately predetermined for a given orifice size thus
avoiding insitu arrangemente for solid feed rate measurement euch as continuoue
monitoring of weight of the storage vessel or collection vessel.
3. Does not suffer from fluctuations in eolid feed rate.
4. Simple and trouble free operation.
3.5.1.1 Drowboch with thc Solid Fced Mechanism
Although the solid feed mechanism worked well for coarse particles, runs with
very fine glass beads (96¡r) proved to be problematical. While transporting 96¡r glass
beads the flow of solids through the orifice was found to be oscillatory (stop-start flow).
Thie behaviour was pronounced when the solid bed height above the orifice in the
storage tank was large. This was due to time lag in the pressure equalisation at the
orifice. However, once a certain minimum bed level was reached steady solid flow
was realieed. When solids are introduced into the transport line, the system pressure
increases. For solids to flow through the orifice uninterrupted, a corresponding increase
in pressure above the orifice at the storage vessel should occur. Although the equalising
Iine ensures that there is a corresponding increase in pressure above the solid bed in
the storage vessel, in the case of fine solids the bed resistance is so large, that there
is a time delay in equalising the presr¡ure just above the orifice. As a result solid flow
through the orifice stops. Thus the time delay in pressure equalisation results in stop-
start flow. Once the bed height ie eufficiently low, thus reducing the resistance, emooth
flow is established. No such problem is encountered with a coarse solid bed, since the
resistance offered by a coarse solid bed is small compared with that offered by a fine
solid bed of the same height.
In view of the above difficulty with fine solids, experimental runs with 96¡r glass
3-11
8¡Ze
COCURRENT EXPERIMENTS Choptet 3
beads were carried out with some modifications to the apparatus. Instead of using a
compressor to drive the air through the system, two \¡acuum clea¡ers were ueed down
stream at the solid separator outlet. Thus, air was drawn into the system at atmospheric
conditions, and solids were fed from the feed tank which was oPen to atmospheric pres-
eure. This arr¿ngement ensured smooth flow of solids without any pressure equalising
problems. The air flow was monitored down stream of the apparatus with rotameters
connected at the vacuum cleaner inlet.
3.5.2 Driving.Air Supply
Air from the screw compressor was freed from oil and moisture by a freeze dryer.
The dew point of air leaving the conrpressor was reduced to 4"C. The oil and moisture
free air metered through one of the four rotameters depending on the range of flow
investigated, was fed to the point at which solids are introduced into the transport line.
Air pressure at the rota¡rreter waa measured by a Bourdon type pressure gauge (0 to
l0OKpa) situated downstream.
3.5.3 Acceleration Section
The solids fed from the storage vessel are picked up by the driving air and the
suspension travelg along the gradual 360 degrees bend before commencing its upward
journey through the test section of 3.5meters length. In order to facilitate acceleration
of solids to the equilibrium velocit¡ an acceleration section was provided at the entrance
of the test conduit.
The principle of operation of acceleration section is to reduce the cross sectional
area of the transport line to provide larger air velocities thus increasing the speed of the
solids, over a short dìstance. The acceleration section is shown schematically in Fig. 3.5.
It is designed to facilitate the selection of the desired cross sectional area corresponding
to the degree of acceleration required. The advantage of such a system is that it is not
necessary to provide long test sections in order to realise equilibrium conditions.
3-12
COCURRENT EXPERIMENTS Chopter I
3.5.4 Solid Separator
The solids traversing the test conduit hit the impact plate positioned near the
exit of the conduit, which is situated in the solid sepa^rator. The Purpose of the solid
separator is two fold. (i) To reduce the air velocity so that solids ca¡ be collected from
the gas-solid suspension after impact with the plate. (ii) To provide the housing for the
impact plate and load cell. The solid separator is a cylindrical construction whose cross
sectional area is euch that the air velocity in it corresponding to highest volumetric flow
rate anticipated, ig less than the terminal velocity of smallest particle used in the tests.
This ensured complete separation of solids. The clean air leaves through the two outlets
provided at the sideg of the separator. The separated solids are then collected in the
closed collection vessel situated on the top of the feed tank.
3.5.5 Impæ,t Plate E¿ Load Cell llousing
The load cell was mounted on the top of the solid separator. The extended stem
from the impact plate hung from the beam of the load cell. The beam of the load cell
was provided with an overload protection spring.
The tip of the impact plate was positioned exactly at the centre of the exit of
the test conduit by an air bearing. This prevented displacement of the impact plate in
the lateral direction due to the impinging gas-solid stream, while transmitting the æcial
thnrst without frictional loss.
3.5.6 Differenttal Pressure Tlanducer
Pressure drop along the transport line wzx measured using a high precision MKS
Baratron lype 22OB differential pressure transducer. The transducer is a self contained
unit with the sensor, associated electronics and power supply mounted in a dust proof
box. The sensor ie made up of 3 parts: (f ) a taut metal diaphragm welded to support
rings, (2) a single ceramic-based electrode, and (3) a reference side cover through which
3-13
COCUNNENT EXPERIMENTS Chopter 3
feedthrough terminals make connections to the electrode within the reference cavity.
The P.C. mounted electronic circuitry contains those comPonents necessary to convert
a change in capacitance caused by diaphragm deflection to a linear +10 V DC signal.
The gain of the pressure transducer is lOVolts/lOOTorr. The signal from the
transducer was fed to a uRekidenkin chart recorder. The manometer leads from the
presr¡ure tappings were provided with needle valves which introduce adequate damping
of the pressure signal and provide a steady state average value. Pressure drops along
two sections, each one meter long, downstream of the transport line were measured with
the help of the pressure transducer and a switching station making use of a two way
valves.
3.O Procedure
The solid material being investigated was loaded into the solid feed tank, after
placing the appropriate orifice selected for a desired solid flow rate. Initially, the diame-
ter of the insert in the acceleration section was the same as that of the test conduit. Air
flow from the compressor was established by opening the inlet valve at the rotameters.
Air flow was routed thriugh one of the four rotameters appropriate to the range of air
flows being investigated. With a sufficiently large air velocity established through the
transport line, solids were introduced gradually by withdrawing the tapered plug from
the orifice. Once a solid flow was established, presf¡ure drops at two sections down-
stream of the transport line were recorded. If the pressure drop in the upstream section
was higher than the pressure drop in the downstream section, indicating incomplete
acceleration of solids, then the procedure was repeated (with a smaller insert in the
acceleration section), until the correspondence between the pressure drops was satisfac-
tory. Once this was realised, air pressures at the rotameter and the impact plate in the
solid separator were recorded using Bourdon type pressure gauges. The pressure trans-
ducer signals from the two sectiong of the tube and the signal from the impact meter
were recorded ou the chart recorder. The above procedure was repeated at several gas
3-14
COCURRENT EXPERIMENTS Chryler 3
velocities by reducing the air flow rate progressively until the gas-solid suspension flow
was erratic characterised by large fluctuations in the pressure drop readings. Once all
the material in the feed tank was transported up, the air flow was shut off. Another
solid flow rate was selected by using the appropriate orifice. With the tapered plug
in place the solids in the collection bin were drained back into the solid feed tank by
opening the isolation valve. The procedure was repeated at several solid flow rates.
The experiment was¡ repeated with different solid materials in transport tubes of
different diameters
8.T Range of Variables Studied
Six different solid materials were studied in four different test sections. The golid
materials and the transport tubes used in the study were the same as those used in the
countercurrent experiments. Details of these materials are provided in Chapter 2 [p"gu
31.
The maximum air velocity studied in these experiments was about 2Om/sec. The
range of loading ratios used was about 0 to 60.
8.E Analysis of Data
3.8.I T.hrust Due To Air
Elom the rotameter reading and corresponding air pressure at the rotameter,
inlet mass flow rate of air was determined. The net volumetric flow rate of air introduced
into the test section w¿rs then equal to the total volumetric flow rate less the volume
flow rate of solids introduced into the transport line. With the knowledge of inlet flow
conditions, mass flow rate of air through the conduit was determiued.
Knowing the exit conditions at which the solid velocity was being measured,
3-r5
COCURRENT EXPENIMENTS
contribution of thrust due to air to the total thrust was calculated as follows.
Fc :KÍ (Mù (Vù,,;,
x¡4- (po)r,n/'" (l - c)
where
Chapler I
(3.3)
3.8.2 Tùrust Due To Solids
The total thrust due to the suspension impinging on the impact plate was derived
from the load transducer voltage signal. The thrust due to solid phase was calculated
as follows.
F, : Ft - Fs (3.4)
where
Fs
Ms
C
K¡
Fs
F,
Combinins (3.3), (3.a) and (3.5)
is the Thrust due to air alone
is the Maes flow rate of air
is the Volume fraction of solids
is the Ratio of observed thrust to theoretical thrust
3.8.3 Sofid Yelocíty And Concentration
Knowing that solids loose all their ærial momentum on impact with the plate,
thrust due to the solid phase can be expressed as follows.
is the Thrust due to air alone
is the Thrust due to solid phase
Fr: lþrPrVrA" (3.5)
3-16
(3.6)
COCURRENT EXPERIMENTS Chopler J
(3.7)
and
CV"iD t
Flom equations (3.6) and (3.?) solid velocity and concentration were derived from
the knowledge of the remaining quantities. The above two quantities can be evaluated
either by direct substitution of equation (3.7) in equation (3.6) to yield a quadratic
equation in terms of concentration or solid velocity, or by iterative substitution starting
from an initial guess value.
3.8.4 Slip Velocity
Air velocity was determined from the knowledge of volume flux of gas at exit
conditions and co¡responding voidage. Having determined gas and solid velocities slip
velocity, was calculated as follows.
v¡--vs-vc (3.8)
The negative sign on solid velocity is appropriate since solids a¡e travelling in
the same direction as air flow.
3.8.5 Pressure Drop Calculations
3.8.5.1 Total P¡essure Drop
Flom the pressure transducer voltage signal and the transducer gain, the press¡ure
drop across one meter length of the test eection was derived.
3.8.5.2 Gos-wall hictionol Loss
Pressure drop due to air was determined experimentally with only air flowing
through the test section. The correspondence between orperimental values and the
3-t7
COCARRENT EXPERIMENTS Chopter 3
theoretical values obtained from friction factor-Reynolds number correlation is reason-
ably good. Plots of experimental pressure drop versus calculated values at several air
velocities, for all four test sections used in the study are presented in Fig. 3.6.
The pressure drop due to gas-wall friction in the presence of solids was assumed
to be the same as when air alone was flowing at the same velocity.
8.8.5.8 Pressu¡e Drop Due To Solid Holdup
Pressure drop due to static head of solidg was derived from the knowledge of
solid volumetric concentration.
(AP),, : C p,gL (3.e)
3.8.5.1 Solid-wø,ll hictionol I'oss
Pressure drop due to solid-wall frictional loss was determined by subtracting the
contributions of gas-wall friction and solid static head components from the measured
total pressure drop.
(^P)¡, - (aP), - (AP),, - (^P)rc (3.10)
3.6.6 Calculation Of SoIìd-waII tr'rìction fuctor
Analogous to the Fanning friction factor definition for single phase flow, solid-wall
friction factor defined as follows was calculated from the solid-wall frictional pressure
drop.
(aP)r, :zr,LCp" (H) (4.r1)
3.9 Reeults and Discueeion
As the objective of the cocurrent experiments was to investigate the effect of
transport velocity on slip velocity, and its influence on concentration slip velocity rela-
3-18
1óoud_*
LÐ(Eol¡JrEf(nU)f¡J(ELoUJ
G.lrJU)fDCf
I 000
800
600
{00
200
C or'ne I qt ton of P ne 55ure D nop due t o Ê rn
Tub¿ 0rom¿ten! 12.7nn
Slope = O.94
2oo 400 600 800
CßLCULRTED PRESSUBE DR0P (poscols)
Connelotton of Pressuce Dnop due to f;ln
Slope = 0.96
200 100 600 800
CRLCULRTED PRESSURE DB0P (poscols)
0
0 t 000
Fig. 3.6(a)
I 000
u
douóI(Lo(rolr,(E
=tnU)lrJE(L
olrJ
(Llrltn(D(]
800
600
100
200
0
0
T ube 0 r oñ¿t cl. ! 19. I rnrn
F'ig. 3.6 (b )
I 000
1óol¡ó
_o
(Lo(troU.J(E
=antnLlJ(E(L
oUJ
(trU.Janmo
s00
{00
300
t00
0
Fig. 3.6(c)
ts0
200
Connelqtlon of Pne55ule Dnop due to Êrn
100 200 300 400
CßLCULRTED PBESSURE DR0P (poscols)
Connelqtton of Pnessule Dnop due to Êrn
30 60 90 r20
0
1óouó
_o
Lo(EolrjÊ.
=tnal¡JccLotrl
EUJ<næo
t20
60
90
500
30
0
0
Tub¿ 0roñct¿F: 25.¿lnn
Slope = 1.0
Tub¿ 0ronct¿n¡ 38.lmn
+
+
+
Fig.3.6(d)CRLCULRTE0 PBESSUBE DB0P (poscols)
r50
C O C U RR EN T EX PE RITTE N TS Chøpter 3
tionship obtained from countercurrent experiments, data of both the experiments were
analysed simultaneouslY.
3.9.1 Concentration' SIíp Velocity Relationship
From the procedure presented in the earlier section, slip velocity and concen-
tration values were derived for all the tests with six difterent particles in four different
transport tubes. Data in which the pressure drops at the top and bottom sections
differed by more than l0% were discarded to ensure that only steady state conditions
were analysed. This criterion should ensure that the error in evaluatiou of solid veloc-
ity is much less than l0%. Flom momentum balance the pressure gradient due to the
acceleration effect is proportional to the velocity gradient. Even if one assumes that
the pressure gradient is solely due to acceleration of solids, the corresponding change in
velocity gradient is only f0%. The change in eolid velocity should be much less. More-
over, the adclitional contributions of solid weight and solid-wall friction should further
reduce these errors. Slip velocity normalised with particle terminal velocit¡ was plotted
against solid volumetric concentration for all test runs. For comparisou corresponding
results from countercurrent experiments are also presented on the same plots. These
plots are presented in Appendix (F). For quick reference, results with 375¡r ste€l shot in
12.7mm tube are presented in Fig. 3.7. The following observations can be made from
these plots.
l. Slip velocity is not a unique function of concentration, but also depends on solid
flow rate.
2. At a constant mass flow rate of solids, slip velocity decreases with increasing
concentration.
3. At a given concentration of solids, slip velocity increases with increasing solid
mass flow rate.
4. At low mass flow rates of solids a large reduction in slip velocity results over a
3-19
3?5 mrcnon Steel shot rn 12.7mm Tube{.5
3.8
3. I
2.4
1.7
o
Solrd f lou¡ lgn/seclo- 21.97a- {5.44+- 73. {6x- 140.72*- Countcncunncnt dqtq
x
oA
A+
o A.
Â
o
o
xx
1uo
a
=o
l-(JÐJL¡
(L
Jvt
XX+Aoooo +
A+a+
XX
x
,E )tr+
+
+A
A
oo
A
xX
X
xA
b)*
o++
Ao
Ëx
xxg x
x F lçK
# ;t(
f xx
x
o 0.? l.{ 2'l
CONCENTßßT I ON I'I)
concentration-slip velocity map for cocurrent transport.Countercurrent data is included for comparison.
txxxx
xx&##
#xI
2.8 3.5
Fig. 3.7
COCURRENT EXPERIMENTS Chopter 3
small increase in concentration, while at larger mass flow rates the change in slip
velocity with concentration is gradual.
5. For a fixed solid flow rate, slip velocity decreases as the gas velocity decreases,
and approaches countercurrent experimental data as the lower limit to transport
approaches.
6. At large transport velocities slip velocitieg are much higher than the correspond-
ing values obtained with countercurrent experiments at the same concentration.
Flom the above observations the significance of transport velocity is quite clear.
At large transport velocities solid-wall friction ig dominant. Due to this golid-wall
friction, solid particles hitting the transport line walls loose some of their momentum.
This additional loss in energy should be compensated by larger hydrodynamic drag, in
order to sustain transport. Ilence larger slip velocities.
At a given concentration eolid velocity increases with increasing solid flow rate,
which resulte in larger eolid-wall friction. At a fixed solid flow rate solid velocity de-
creases with increasing concentration. This explains the trends (2) and (3) mentioned
above.
3.9.2 SIìp Velocíty Due To SoIíd-WaII tuìctíon
At large transport velocities, which often are associated with lower volumetric
concentrations, the slip velocity is mainly due to solid-wall frictional loss. In such a
situation momentum balance on solid phase results in the following expression.
!rc, oov? (+) (Ë) : zÍ,c p" (#)
Vr: kV"
t, Pt
Ps
4D¿
8
ã
Then
where k -
3-20
Cp
(3.12)
COCURRENT DXPERIMENTS Chopter t)
Barring any variations in 'ko due to other factors, equation (3.12) suggests that
slip velocity is directly proportional to solid velocity at large transport velocities and
low concentrations.
With the above arguments in mind the observed trends in the concentration-slip
velocity diagrams can be explained mathematically as follows.
At constant solid flux (iÞr) the rate of change of solid velocity with concentration
is given by ôv, o"ac:-æ (3.r3)
writingov, ov, ov,ac ôv, ac
and from eqn. (3.12)ðV,ac-
o,
trYom equation (3.14) it is clear that at a constant solid flux, the rate of decrease
in slip velocity with concentration, increases with decreasing concentration or increase
in solid velocity. This should explain the observation (4) mentioned earlier.
As the transport velocity is reduced, the solid-wall frictiou becomes less signiû-
cant. lilhen the transport velocity approaches almost lower limit, slip velocity is gov-
erned solely by the effect of concentration. This explains the observation (6) mentioned
earlier.
Having determined the effect of transport velocity on slip velocity due to solid-
wall friction qualitatively, it is proposed to investigate the quantitative relationship
between them. Equation (3.12) suggests that slip velocity due to solid-wall friction is
directly proportional to solid velocity. The measured slip velocity in cocurrent trans-
port experiments includes the sum total of the effects of concentration and transport
velocity. Based on momentum balance, if the total energ:f loss is broken into energy
loss due to effect of transport velocity (wall friction), and effect of concentration (solid
(3.14)Cz
3-21
COCARRENT EXPERIMENTS Chopter 3
weight), then the associated squares of slip velocities are additive. This procedure how-
ever, is not rigorous in the sense that even though the energy loss is proportional to
the square of slip velocity the corresponding proportionality constants need not neces-
sarily be the same for all the terms. A rigorous method of determining the individual
eftects of transport velocity and concentration involves solving the solid phase momen-
tum equation after substitution of the concentration-slip velocity relationship from the
countercurrent experiments. Unfortunately, lack of knowledge of drag coefrcient and
solid-wall friction factor values makes the task difficult.
(vl) ¡,r*ro: vl - (v?) "","ent¡ation
(3.15)
The contribution of effect of concentration is already known from the counter-
current low transport velocity data.
Following the above procedure the alip velocity due to eolid-wall friction was
plotted against solid velocity for some of the tests (Appendix G). Results with 375¡r
steel shot in l2.7mm tube are presented in Fig. 3.8. According to equation (3.f2) a
linear relationship between these two is predicted, provided the parameter ok' remains
constant. Fig. 3.8 suggests that the slip velocity due to solid-wall friction indeed
increases with solid velocity. Although the dependence is linear at large solid velocities,
at lower range of solid velocities, rate of change of slip velocity decreases. Also, data with
different mass flow rates, results in different lines. At a given eolid velocity slip velocity
is smaller at higher solid feed rates. lVhat it euggests is that the parameter 'k" which
includes drag coefficient and friction factor is not a constant but varies with other factors.
Flom these obsernations it appears that the value of uko decreases with increasing
conceutration. In other words, at the same solid velocity solid-wall friction component
decreases with increasing concentration. One possible explanation is that the mean free
path of solid particles (which signifies the average distance travelled by a particle before
it comes into contact with another particle) decreases with increasing concentration.
Consequently the relative magnitudes of inter particle collisions to particle wall collisions
3-22
3?5 mrcr'on Steel Ehot rn l2.7mm Tube4
3.2
+t
o
L
L
2.4
1.6
0.8
o 0.4 0.8 l-2 l' 5
Y s/Y tF'ig.3.8 Effect of transport ve'locity on slìp velocìty due to soljd-wall fricti'on
02
oSolrd Flor¡ lgn/secl g A
o- 21.97A - 45.44+- ?3.46x - l 40.72
oo
A+
A + XX
oo
 +A + X
o X
XÂ +
o Â
+
x
+o + X
X
Ao
Ao
Ao
A +
oA
o
COCURRENT EXPERIMENTS Chopter I
increases with concentration. Since interparticle collisions merely result in the transfer
of momentum from fast moving particles to slow moving particles, the net loss in Ðdal
momentum due to such collisions is uegligible. On the other hand particles colliding
with the stationary transport wall suffer considerable momentum loss due to solid-wall
friction. An increase in particle concentration results in fewer solid-wall collisions, hence
smaller slip velocities.
The ratio of slip velocity due to solid-wall frictiou, to solid velocity was plotted
against solid volumetric concentration for 375p steel shot in l2.7mm tube (Fig. 3.9).
The trend clearly indicates that the parameter oko decreases with increasing concen-
tration.
The parameter ok' cau be evaluated (equation 3.12) from knowledge of the eolid-
wall friction factor (tr), drag coefrcient (Cp) and the system properties. The solid-wall
friction factor was derived from the pressure drop ¿nd solid velocity measurements.
The drag coefficient which is a function of particle Reynolds number is derived from the
standard drag coefficient-Reynolds number relationship. uk" values calculated from the
above procedure were compared with the observed values (Fig. 3.10). The scatter about
the correspondence line is large. Of the 900 data points analysed only 300 points are
within lhe *25% confidence limits. One possible source of error is in the evaluation of
drag coefficient. Reddy & Pei (f969), based on the experiments with glass spheres (100
to 300¡r size range) in 100mm diameter tube, indicate that the standard drag coefrcient
is altered by the change in the turbulent flow structure due to the presence of the solids.
Also the friction factor term could be another Bource of error, which is calculated based
on the assumption that the frictional loss due to air is unaffected by the presence of the
solids.
Considering the above obsenr¿tions, it is felt that the slip velocity due to solid-
wall friction is best correlated with the pertinent dimensionless groups such as loading
ratio (R), gas velocity to particle terminal velocity ralio (Vo/V¿), particle to tube diam-
3-23
375 mrcnon steel Ehot rn 12.7nn Tube5.5
4.4
3.3uì
o
¿
:L
qbO¿n
å"r" "
oo
Solrd Flou¡ tgnlseclo- 21.9?a- 45.44+- 73.46x - I 40.72
2.8
o6A
^AAAAA
AA
++++,
I
A+,4 +
I2.2X +
XO
1.1
0.? 1.4 2.1
C0NCENTRÊT I ON I7)
Fig. 3.9 Effect of concentration of parameter "k"
XX
X.X+x
XÂ
03.5
0
t. tr,
It
a.t ," a
'..i t'.:i r.
.t ' . I ¡".¿¿
COCURRENT TRRNSPORT DRTR2.5
2
I 1.5a
Ia
Ot¡J
ElrJa)(Do
I ¡
0.5
a
.'r.J
.¡
a0
0 0.5 t l'5
CRLCULRTED .K.
Fig. 3.10. Estimation of parameter "k" from momentum equation
2 2.5
COCURRENT EXPERIMENTS Chopter I
eter ratio (DplDr) and solid to air density ratio (e,lpr). Experimental dat¿ consisting
of about 1000 observations from 24 tests with six different particles in four test sections
were analysed. The following correlation was obtained from the method of multiple
regression of the va¡iables.
(i) trìctio,,:0'0r(n¡-or
(3)'" (#)"'(ä)"' (s'ro)
where
R is the Loading ratio
The predicted r¡alues of slip velocity were plotted against observed values for all
tests (Fig. 3.ll). The majority of data lies within *'}O% confidence limits. Unfortu-
nately test of correlation(3.16) to the systems beyond the range of variables investigated
in the present work is not feasible, as reported slip velocities include the effect of con-
centratiou. The correlation suggests that the slip velocity due to solid-wall friction
decreases with increasing loading ratio and decreasing gas velocity. At large loading
ratios solid volumetric concentration is higher. Ilence, smaller solid-wall frictional loss.
An increase in gas velocity increases solid velocity thus increasing solid-wall frictional
loss. The correlation also suggests that solid-wall frictional loss increases with decreasing
tube diameter. \ühen the tube diameter is small the number of particle wall collisions
increases thus increasing the energy loss due to such collisions.
3.9.3 Compariæn with Existing Datz
Several investigators have studied the flow of solids in vertical transport lines.
A summary of some of the important works is presented in Table (3.1). The varied
me:xurement techniques, system details, and the range of parameters studied make the
task of comparison difficult. Ilowever, the majority of the works are confined to very
low concentrations ( less than l% ) and high transport velocities, excepting the works of
Yerushalrni & Cankurt (1978, 1979), Yousfi & Gau (1974). While some workers aimed at
correlating gas velocity to solid velocit¡ others have attempted to correlate slip velocity
3-24
TEST OF CORBELRTIONt5
6
PREO i CTEO VRLUE
Fig.3.llCorrelationofslipvelocityduetosolid-wallfrictjon
t2
I
l¡J
=JCE
oU.J
(trl!'a(DÐ
3
0 t5I630r2
.t
-s0la
,.
+50l /
I
I
,
¿a tr.
I/7.
TABLE J.I SUITIIARY OF VERTICAL PIIEUIIAIIC TRAIISP()RT IIIVESTIGAIIOI¡S
Reference
Belden tKassel
( 1e40)
B i rchenough& l{ason
( le76 )
Capes &l{akamu rå
( re73 )
Di xon( le76 )
0oig tRoper
( le67)
ilðteri ål s
used
c¿ta lJst
Al uml napowder
Glass
Steel
Rape Seed
Acralyi cAl kathene
Gl ass
Pa¡t i cl est ze
(u)
1950
20
4'Ì0108017802900
?56535
12002340
¡ 7803400
32803600
Pôrtfc l edens i tJ
( gnlcc )
Temlnalvel oc I ty
(m/s )
Tubedl aneter
Rånge ofgas veìocities
(m/s )
l{ax. ìevel ofconcentràti onstud I ed
(t)("n)
Solid velocity General observationsneàsurenenttechnlque
Soì id velocity lscalcul¿ted frompðrticle terminalvelocity. No di rectmeasuienents weremðde.
Frlction factorstudi es
A combined frictionfactor is correl¿tedto mass velocityrâtlo tern and gasReynoìds number.
963 t?26
8698
00
3.496.55
4-15
l7- 56
l.0r
0.05r LDV Technlque Slip velocitYincreases inllnearly wlth
steeì
3.75 .045 49.4gì ass
3-r8Steel
47.6 tÐ-30
loadiðtavel oc
ng ratiogi ven gasitv
.47
.9
.9
.86
.5t
.85
.7
.7
.08
.91
22
227
777
I0
3.547 .97
tL.2714.944.018.28
14.9322.1
75 3.0
0.6r
Isol atlonlechn i que
Pitot tube
Solid velocityincreases I lnearlywith gas velocity.u-l u- > I for all
t¡Ë' ruìs. The slopeincreases rithincreasing particlete¡rninal veìocity.
Sl ip veìocity atminimu¡n pressuredrop pofnt is derivedto be I.4l timesp¿rticle terminalvelocity
Pð¡ticle and airvel oci tydist¡ibutions werenea su red .
¡legatlve frictlonf¿ctors are reported.Frlctlon fðctor isi nve¡selyproportionaì to solidvel oci ty.
Friction-f¿ctor isreported to be a
constant value (.001)
6. 498.98
l. 180.92 I
10. I9.3s
2.52.5
300750
2.265.45
43Gl ¿ss
5.5 - 10.5 0.6r
TABLE 5.1 (Coxro.) SUllllARY 0F VERTICAL Pt¡EUttATIC ÍRA!¡SP0RI It¡vESTIGAIl0tls
Reference
Gopichandet al
( less )
Harlu &llol stðd
( le4s )
Jodl orsk i( le76)
Jones et al(re67)
l{àterial sused
Cðta I ystSandSilica gel
llheat
Sand
Cataìyst
Polyethyl eneHheàtSandPVC
Gl ¿ssAl uni naZircon Sillc¿Steel Shot
Partl cl eslze
R¿nge200-765
Particledensl tJ
0.893.1I.s51.55.1.44
Range2.5-7.6
Terni n¿lvel oci ty
Tubedi ameter
Steeì
6.7813.5Gl ass
31.649.578.9Steel
Rangeof gas
5-40
5.5-12.5
l0-25
l,l¡r. I eveì ofconcent rat I onstud i ed
(f)
3.75
2.5
0.6
veìocitiesE¡lcc (rnls) (n"¡) (n/s)
Solid veìocity Gene¡al observationsmàsurementtechn i que
Frictlon fðctorstud i es
A frictlon factorbased on totàlpressure drop and¿nd gôs veìocityhead is correlòtedto loading ratio,voidage, 9ås velocityand diameter ratlo.
The order ofmagnitude of frictionfacto¡, aftercorrecting foracceìeration is about-001
t20t9.985.0.07.5
(u)
26t26196392750
ôI23
1.592.t22.8?3.95o.29
7 Isol atlontechn i que
Isoìationtechnique
Cine CaneràandRadlo activet¡¡certechn i ques
Voidð9e ls correlatedto loading ratio ðndgas velocitynoma I I sed vi thterninðl veìocity
Observ¿tlons includeacceleration effects0rå9 co-efficient iscorrelated topartic¡e Reynoldsnumber.
213274357503ll0
36404060105l0o
2.642.64?.642.74
.9'ì7
0.96r.272.581.4
9.6u.8o.620.34
t.7510.2t22.rSteeì
Soìid veìocìty is aI inear functionof gas velocity.
A frlction fàctoris deflned on thebðsis of pressuredrop due to presenceof sollds and g¿svelocf ty heðd.
A friction f¿ctoîbôsed on the coßbinedfriction¿l pressu.edrop cornponents ofgas and soìid phasesand gas velocity headis defined. Thi sfrlction fðctoî iscorrel¿ted to gas Froudenurnber and particìeconcentrati on.
A linear relationshiPbetHeen logarithmicvaìues offriction f¿ctor andloading ratio isobtò i ned.
TABLE l.l (Corro. ) SUllllARY 0F VERTICAL PllEUltATlC TRAIISP0RT ¡llvESTI6AÏl()tS
Reference
Jotðki et aì( le78 )
Kmiec et aì( le78 )
I'laed¿ et ¿l( l914 )
¡lehta et al( lesT )
l,lateri aì sused
PolJetnyl enepel I ets
Turnlp Seed
Silicå gel
Hài ryVetchllillet
Particlesl ze
(u)
3340
22401340u00683
t203?05201050t2027053032501440
Particl edens i tJ
0.8020.8021.154l. ls4
2.5
8.9
Termi nalvel oci ty
Tubed I ¿meter
nangeof gasYeìoclties
(m/s )
l{ax- I evel ofconcentràtl onstudi ed
(t)
4.3r
0.6r
3.0
2.5*
gt¡lcc (m/s) (,tm)
0.57 5.8
Soìld veloclty General observatfons¡Deasutementtechn f que
Frlction factorstudi es
Friction factor isconstðnt for eachtube, and its valueincreases rithincreasing tube
Soìid r¿ll frictionfactor is found todecrease ln theincreasing sol idvel oc i ty.srze.
41.5?.66.78.t00
40
8l0t220
9553l0020
?5598060
0002
8-50
¡-15
PVC tubes
?6.5
46.8 8-20
l-40
t2.7I ron 3-?t
?683
Photographi ctechn i que
Isoì ationlechnique
Photograph i cnethod
Photogrðphl ctechn i que
Isoì¿tiontechn i que
Solid to gasve¡ocity ratio isderlved fromp¡essure dropmeðsurements
l{egatlve frictlonfactors are neportedfor Charnber sectionof the apparatus ¿tìq. gas Yeìocitles
Konno & Satio Glass( le6e )
Copper
6.444.44.673.08
9.810.07. l3
Slip velocities ¿reaìnost equàl tosingle particleternin¿l velocitles.Concentration andsoìid velocltydlstributions àrestud i ed .
Sl ip velocitylncreases Iinearlyrith gas veloclty¿t ì¿rge 9¿svelocities. SI ipvelocities are higherfn snalìer tubes.
So¡id-Í¿ll frictionfactor ls inverselYproportional toparticle Froudenumbe¡.
Friction factor fora glven particìe andtube ls constðnt ðndits value decreasesrith fncreasingtube diðíreter.
Polyethyìene f00Vinyl Chloride 150Gl ass I20Gl ð ss 300
GlassGl¿ss
.35
.44
00
3697
.95
.42
.93
.93
0I22
Acralyi c
2.532.53
0964l
Pðrtlcle velocltylncre¿ses I inearlyrith gðs veloclty.Reproduci bi I i ty :asreported to be poor.
A mixture frictionf¿ctor is deflnedbased on tot¿ìpressure drop òndmixture velocitY.lhe friction factoris correlðted to gasReynolds number.
TABLE 5.t (Co¡ro. ) SUltltARY 0F VERTICAL PI¡EUIiATIC TRAIISP()RT ltvESTIGATl0l{S
Reference
Reddy &Pel
( re6e)
Toili tået al( leso)
Van Zuilichenet ¿l
l{ateri ¿ I sused
Gl ass
Cement
Yheat
Rangeof gnsvelocltfes
(m/s )
7-r4
5-25
2-20
l0-40
t0-30
llax. ìevel ofconcent rat i onstudi ed
(f)
0.F
0.1
l0r
2.3r
3i
Photog raphi ctechnÍ que
Rðdio ôctivetròcer method
100150200?70
2.592.59?.592.59
0.577l.0lt.452.06
P¿rtf cl esrze
(u)
30
4600
4000
Particledens i tJ
glttlcc
Ten¡i nalYel oc i ty
('nls )
Tubedi arneter
('-)
100gl ass
28
Acra ìyi cpipe
5lSteel
Soìld veìocity Geneîal obseryationsmasuremenftechn i que
Frictlon f¿ctorstudi es
A reductlon infrlctlonal pressuredrop ls observed atìorer loading regionfor snall p¿rticles.
Partlcle and gasvelocity profì lesrere studied. 5l ipvelocity iscorrelated to ìoadingrat I o.
lle ratfo of totalpressu¡e drop to gas-rall friction¿ì lossis a linear functionof ìoðding ratio
Shinizu et al Copper( te78)
Steme¡di ng( le62)
catalyst
3.011.681.51.380.75
t7410798926054
46.5
65
8.92
1.39
0.85
0.640.5
1.6 0.19
2.56 0.07
Frcrn pressure dropmeasurernents gas tosolid velocity ratiois reported to beconstant.
Solid-wall frictionf¿ctor is constantfor the systen, andis equ¿¡ to .048.
4t66.8
It ls assumed th¿tthere ls no sìlpbetreen solid andal r. Conpressibìefìo{ condltlons areana lysed.
Slip velocitylncre¿ses I inearlyin the g¿s veìoclty.Slfp velocities òrelarger ln smallertubes.
Frlctlon f¿ctorb¿sed on ôdditfonåìpressure drop andgas veìocitJ head isderived forconpressible fìo;conditions. Frictionf¿ctor decreases rithgas Froude number.
Frlction f¡ctor forpolypropyl ene decreàseswith increðsing gðsFroude nunber. ltsvalue tends to be ð
constdnt rlthlncreaslng n¿ss florrdte of solids.
t3.2
9.5
538l¡30(
t1973)
re80)Polypropy I ene
TABLE l.l (Corro. ) SUITIIARY f)F VERIICAL PIIEUIIATIC TRAI¡SPORT II¡VESTI6ATIO}¡S
Range of l'lax. ìevel of Solid velocity General observðtlonsgas velocltles concentration nEðsurerEnt
(m/s) (1)
Reference
Vogt t fhìte( le4e )
Yausfi &
Gau( le74)
Yerush¿ I rni( re76)
ll¿teri al sused
S¿nd
Stee¡ ShotCìover S€edl¿he¿t
Ratio of totalpressure to Pressuredrop due to ðir iscorrelated to loading
Frlctlon factorstud i es
Frictlon factor iscorreì ated to sol I d
Froude nunber.l{egatiYe frictionfàctors àre reportedat loï Froude numbers.Friction I'actor isabout .0015 at largeFroude ôumbers.
t2.l l5-60
I .7-4. 5
L.7-7.6
Pðrticìesi ze
(u)
20333043472941911704010
Partlcledens i tJ
(S,n/cc )
Tennl nalYel oci ty
(m/s )
Tubedi aneter
(,-)
2.66?.63?.62.567.211.23t.28
l.5l2.593.395.46.355.1lt.77
4r
20
0
ratiand
o, density ratiodiameter ratio.
Gl ass
PoìystyreneC¿ta I yst
Catàlyst 60
4933
0.7650.991.36l. 180.010.078
0.88 0.076 76
?.2.
l.0.0.
u8143183290?o55
?-838
50
7474740686885
Isolatlonnethod
Sìip velocitY isfound to be severalfold higher thônterminaì veìocitY forfine p¿rticles.Fon coarseparticles sìlPvelocities are ofthe order of terminalvel oc I ty.
Sìip velocitYfncreases ¡rithconcent rat I on. Atlarge concentrations¿nd lor solidveìocities solid-wallfriction ls notsignificðnt.
25
l0
veì oci tY '
( rs78) FCC
Di ca lyte1.071.66
0.0780.055
150
Concentrðtion levels ¿re estlm¿ted from loading rdlios'
COCURRENT EXPERIMENTS Chapter J
to gas velocity or loading ratio. Very rarely are concentration values reported. Lack
of information on corresponding solid flow rates makes the derivation of variables of
interest difrcult.
3.9.3.7 Concentrotion'Slip Velocity Meps
Only Birchenough's (1976) data permitted derivation of concentration-slip veloc-
ity map, similar to those presented in this work. Their concentration-slip velocity plot
(f'ig. 1.6) indicates trends similar to those obtained in the present work, significance of
which is already explained.
8.9.8.2 Gos-Solid Velocity Reløtionship
The works of Capes E¿ Nakamura (toZa), Jotaki et al (1978), Jodlowski (1976),
Mehta et al (1957), Konno & Satio (1969) and ïli¡heeldon et al (1980) indicate a linear
¡elationship between gas and solid velocities. Except Konno & Satio (1969), others
report that the rate of change of gas velocity with solid velocity @Vs I AVr) is greater than
one. This clearly indicates that slip velocity increases with transport velocity. Capes 6t
Nakamura (1973) include data corresponding to very low transport velocities, where the
concentration effect dominates. They rightly point out that large slip velocities in this
region are due to particle recirculation, whereas solid-wall friction accounts for large
slip velocities at higher transport velocities.
In order to compare these trends with the present data, solid velocity was plotted
against gas velocity for tests with 644p glass beads in all the four tubes (Fig. 3.12).
Ilom these plots it was observed that in all the four cases the average slope @VslAV'l
is greater than one, and its value decreases with increasing tube diameter.However it
should be noted that in the case of small tubes, at large gas velocities, higher solid flow
rates result in higher eolid velocities. But in the case of large tubes no such dependence
on solid flow rate is observed. Tbis could be explaiued as follows. At a given gas Jelocity,
increasing solid flow rate results in higher concentrations, thus decreasing the eolid-wall
3-25
644 m rcnon G I oEs beqds rn 12.'7 nn T ube{
1lrì[)
J
=o
l-
ooJl¡J
oJE]U'
3.2
2.4
1.6
0.8
0 t. { 5.6
Fig. 3.12(a) Gas-solid velocity map
644 mrcnon GIoss beqdE rn 19.1mm Tube3.5
2.4
2. I
t. {
0.?
0
2.6 4.2
R I R VEL0C I TY (0 ¡ m. Lr-ssl
0 l. | 2.6 4.2
R I R VEL0C I TY (0 r m. lessl
Fig. 3.12(b) Gas-solid velocity map
1
lnult)J
=cl
l-
CJÐJl¡J
oJÐvt
0
1
+o
+x
ôo
oÂ
+
+
o
o
o
1o o
+
o
+X
+
xô
Âo
ô
o
(grn/øcol
+0o
+
"*t *"* x
o++ x ôo óqo^^x+ô
++oGl-O ¿,
Solrd f los,
o- {. ?6a- 10.71+- 18.33x- 33.61o- 6t.83+- l0{.20
o¿l+Xo+
oooX
+Â
+
+1
+I
+ È1
+Â+O
o +¿À
+g
o
+ô
oxo
+
0
x+
oX+Ax
+Â
oxo
Io øo
X
Sol¡d flou (gnleocl- {.?6- 10. ?l- t8.33- 33.8?- 61. 83- 104.20
,+
ooxX
5.6
644 mrcnon Gloss beqds rn 25.4mm Tube
ûu0)
J
=o
)-L)ÐJlJ-t
fl
Jovl
3.5
2.8
2.1
l. {
0.7
ol23
RIR VEL0CITY (0 r m. lesslFig. 3.12(c) Gas-solid velocity map
644 m I cnon Glqss beqds rn 38.1mm Tube3.5
2.6
2. 1
t. {
0.7
0.9 1.8 2.1
R I R VEL0C I TY (D I m. Lessl
Fig. 3.12(d) Gas-solid velocity map
054
1ut¡J
=o
l-(JoJt¡J
oJotn
0
0
00
I Å!
o
o oðo
X
t
ðo
x6*
lgn/ sccl
*<+o¿¡
o-Â-+-x-o-+-
Sol rd f lour
18.3333. 8761. 83
t0{.20227.80363. 40
Âx+
X+
x+
o+x
*+
f
Xfoi
1xE
* )ó
lgn/ sccl
0^ax¡'
#
Solrd f louo- {. ?6a- 10. ?l+- 18.33x- 33.8?o- 6t.83+ - t04. 20x- l6?. t0
3.6 {.5
COCARRENT EXPERTMENTS Chopler 3
frictional loss. This in turu results in lower slip velocities, hence solid velocities are
higher. This effect is predominant in smaller tubes where solid-wall friction is high. In
the case of large tubes this effect is lese significant as solid-wall frictional loes is emall
even at low concentrations. The tube sizes investigated by Capes & Nakamura (1973),
Jotaki et al (1978) and Jodlowski (1976) ranged from 40mm to 100mm in diameter. As
the effect of concentration on slip velocity in these tubes is weak no noticeable change in
solid velocity with mass flow rate is observed. It therefore appears that presentation of
data in terms of concentrations and slip velocities can be mo¡e meaningful in visualieing
the individual effects of concentration and transport velocity rather than in terms of
gas and solid velocities.
8.9.3.8 Solid-woll hiction Factor
Often, the energy loss due to eolid-wall friction ia analysed analogous to single
phase flow situations. However, the definition of friction factors varied. While some
workers defined friction factor based on total pressure drop and mixture velocity head,
other¡ used pressure loss due to frictional loss of both the phases and gas velocity head.
Recent works deûne friction factor as follows.
4 (3.r7)
The above deûnition is arrived at by treating the two phases eeparately. The
frictional pressure drop due to the solid phase is assumed to be same as when a single
phase fluid of density (C p") flows along the tube at mean solid phase velocity. Derivation
of the above friction factor requires the knowledge of total pressure loos, pressure loss
due to static head of solids and gas-wall friction.
A summary of friction factor correlations based on the above definition is pre-
sented in Tbble (3.2). While the works of Konno & Satio (1969), Capes & Nakamura
(1973), Swaaij (9170), Reddy & Pei (1969) indicate that friction factor is an inverse
3-26
l,-Dt
(aP)r,
ïc p,v?
COCARRENT EXPERIMENTS Chopler J
function of solid velocity, works of Stemerding (f962), Hariu & Molstad (1949), Van
Zuilichem (1980) aud Yousfr & Gau (1974) suggest that friction factor is almost con-
stant. While Jotaki et al (1978) based on their experiments with tubes ranging from
40mm to lOûmm in diameter report that friction factor increases with increasing tube
diameter, results of Maeda et al (f974) with tubes ranging from 8mm to 20mm in
diameter show quite opposite trend.
Analysis of data of the present work indicated no significant dependence on solid
velocity. Ilowever, there appears to be some strong dependence on concentration at
very low concentrations. Fig. 3.13 indicates that friction factor decreases rapidly with
increasing concentration up to about 0.5% and levels off to almost a constant value at
larger concentrations. In this region friction factor varied from 0.0005 to 0.0015. These
values are of similar magnitude reported in literature (Table 3.2).
Negative friction factors are also obtained especially corresponding to limiting
gas velocities where solid velocities are very low. FiS. 3.14 represents friction factor
versus particle Floude number data for nrns with all the particles investigated in 12.7mm
tube. lVhat it shows is that the friction factor is negative at very low trÏoude numbers
and its value increases with increasing Froude uumber and levels off at a positive value
at large Floude numbers. A similar trend was reported by Yousfi 6¿ Gau (f974) from
their experiments with 20p and 50¡r catalyst particles in 38 and 50mm tubes. Negative
friction factorg were also reported by Capes & Nakamura (1973), Van Swaaij (1970),
and Yousfi & Gau (1974) corresponding to very low solid velocities.
Considering the scatter and uncertainties in determination of friction factor(/r)
reconciliation of the various forms of correlations presented in literature is difficult. Not
withstanding possible sources of experimental errors, part of the failure to bring out
a unified correlation for solid-wall friction factor similar to single phase flow situations
can be attributed to the following uncertainties.
l. teatment of the solid phase in the transport line as a continuum for the purpose
3-27
ooO)ru
u_
COCURBENT TRRNSPORT DRTR20
16
t2
I
o 0.8 1.6 2- 4
CONCENTRNT I ON (Z )
Fig. 3.13 Deperrdence of solid-wall friction factor on concentration
4
03.2 4
I
I
a a
I .'
a \IL
ltJ. ...it a. !.v\ ot ., ' . ..
TøbIe (s.2)
Solid -\{all Friction Factor Correlations
Reference Correlation
Capes (1974)
Jotaki (1978)
Khan (1e73)
Klinzing (1981)
Kmeic (1978)
Konno (1969)
Maeda (1974)
Reddy (le6e)
Stemerding (1962)
Van Swaaij (1970)
Van Zuillichem (1980)
Yousfi (1974)
r, :2.66co(e) (#);" (3+\'le : o.287uo-r'6tyro'st 11 - c)n .
o.o3o4r,-o.zs¡ -
0.0285Jr - (I|'ro.5),
.f, :0.048Y-t'22
.fe:0.0135(Dr-0.013)
.f' : 0.0015 to 0.003
l, : O.O46V"-\
.f" :0.003
.f, :0.08l/r-l
.f" : 0.001 to 0.002
.f" :0.0015
@)I
;'i'Cl#;'-;it:";rï.i ': ." ". " ' ""r
I
¿l .a-a
a¡ ttìt. a
:r.
a
i
a
12.7nn Tube
-1
-7
-13
-19
-25o sOC l0oo 1500 2000
PRRTICLE FROUDE NUMBER
Fig. 3.14 Dependence of solid-waìl friction factor on particle Froude number
5
ooo:t(o
u-
2500
COCURRENT EXPERIMENTS Chopler 3
of determining the frictional losses analogous to gas phase may not be appropriate
for all the flow regimes.
2. The solid-wall frictional loss is not determined by direct measurements. Instead,
it is inferred from the total frictional loss and gas-wall frictional loss which in
itself is not determined with certainty in the preeence of solid phase'
Considering the above points it might be worthwhile to investigate the mechanism
of solid-wall collisions and the resulting energy losses before attempting to quantify
the solid-wall frictional losses. Direct meas¡urement of particle-wall collision flux and
radial velocity distributions acrose the eection of the pipe, along with Bome qualitative
observations might be helpfut in this regard. Although some preliminary investigations
on these aspects were made by Ottjes (1981) and Ribas et al (f980) for horizontal flow
situations, no such attempts seem to have been made for vertical transport conditions.
Thes€ aspecta however, are beyond the scoPe of this work.
8.lO Prediction of Mi¡lnum Tlansport Velocities
In the previous chapter the significance of concentration-slip velocity relationship
in explaining choking phenomenon is made clear. It is also mentioned in the introductory
remarks of this chapter, that one of the objectives of cocu¡rent transport experiments
is to investigate the extension of countercurrent data to describe forwa¡d transport is
indeed feasible.
One of the observatione from the cocurrent experimental data has been that the
concentration-slip velocity relationship at low transport velocities, approach as that of
the corresponding countercurrent data, although large deviations occur at high trans-
port velocities. What it suggests is that the information derived from simple coun-
tercurrent experimental data is useful in so far as describing the cocurrent transport
at very low transport velocities where solid-wall friction is insignificant. Since "chok-
ing" is a phenomenon associated with low transport velocities and high concentrations,
3-28
COCURRENT EXPERIMENTS Choptet I
the results of countercurrent experiments should prove to be useful in determining the
minimum transport velocities for a given system.
In order to substantiate the above arguments, plots of concentration versus su-
perûcial air velocity for sand particles in all the four tubes investigated are presented
(Fig. B.l5). These plots are similar to the ones predicted from the countercurrent data
(fig. 2.1). The solid lines in Fig. 3.f5 correspond to the predicted behaviour based on
the concentration-slip velocity relationship derived from countercurrent data. The pro-
cedure for calculation of predicted valueg is already described in the previous chapter.
The following observations ca¡ be made from these plots.
f . At large gas velocities the predicted values of concentration are lower than the
experimental values. This deviation is larger in the case of smaller tubes. Thie is
due to the additional effect of solid-wall friction, which comes into play at large
transport velocities.
2. As gas velocity is reduced, the deviation from the predicted curue decreases
gradually. At transport velocities approaching the corresponding lower limit,
observed values agree well with the predicted wlues.
3. The predicted curîes indicate that choking is characterised by a sudden increase
in concentration for small mass flow rates of solids, while at large solid flow rates
the process is gradual. This indeed has been confirmed by the experimental
results.
4. The limiting gas velocity at which "chokingo occurs decreases with decreasing
golid flow rate.
trïom the above observations the sigaificance of countercurrent experiments is
established. These simple experiments provide the valuable concentration-slip velocity
relationship based on which nchokingo velocities can be predicted for a given system. In
the design of pneumatic transport equipment specification of minimum transport veloc-
3-29
Iz.Ðt--CEGF-z.t!CJz-o(J
1,73 mrcr'on Sond rn 12.7mm Tube3.5
2.8
2. 1
1.4
0 6 L2 l8
ÊlR VEL0CITY (Drm. Lessl
Fig. 3.15(a) Prediction of minimum transport velocity
24
I73 mrcr-on Sond rn 19. 1mm Tube
0.7
0
30
2
Solrd Flou Ignlsecl
+
o-A-+-X-0 - 7't.54+- 133. 40
+++
'7.
15.24,46.
7L2L5260
x+
*
z.ÐFCEÉt-z.LTJ(Jz.Ð(J
1.6
0.8
1.2 x
o
X o
+
oo+ ooo
oX
+
X
+
o
+
+ +
0
0 l0 15
R I B VEL0C I TY (0 I m. Less)
Fig. 3.15(b) Prediction of minimum transport veìocity
+0.4 X
+
5
+À
+A
o
o
o
oX
o
o+ o
+ X+ X
++ +o
tà
77 54
Solrd FIou (gmlsec)
o-A-+-X-o-
7.7 L
15. 2124.5246.60
20
Â+
25
t73 mrcnon Sond rn 25.4mm Tube
2.4
1.8
t.2
0
0 l0 l5
RIR VEL0CITY (Drm. Lessl
Fig. 3.15(c) Prediction of minimum trarrsport velocity
I13 mrcr'on Sond rn 38. 1mm Tube
0.8
0.6
0.4
?
ùz.E]
t-CEcc|-z.L¡J(Jz-Ð(J
0.6
5 20 25
ù- ':
z.Ðt-CE(rFz.TL,CJzÐ(J
o.2
0
0 4 I 12
RIR VEL0CITY (Drm. Less)
Fig. 3.15(d) Prediction of minimum transprot velocity
+
o
0+
o+ X
0X
ÂX
+X
+Â +Â
A
+
Ign/sec)Sotrd Flouto- 24.52Â- 46.60+ - 7'7.54x- r33. {0o - 284. 00
+ +- 432.00
A
Ä
À+Xo+
46.77.33.
+
o +
+
o
+o
+oo +
+o
o0 o 0 0
605440
Solrd Floru (gm/secl
15. 2124.52
l6 20
COCARRENT EXPERIMENTS Chaplet 3
ities with confidence is of importance for trouble free operation. AIso, at low transport
velocities knowledge of volumetric concentration of solids is sufficient for epecification of
euergy requirement, as solid hold up is a dominant factor. At large transport velocities,
however, knowledge of the solid-wall friction component is essential in determining the
energ)¡ requirementa.
8.ll Concluelons
l. Derivation of solid velocity from the measured impact of golids on a plate proved
to be simple and accurate.
2. At large transport velocities solid-wall friction is significant. In this region higher
transport velocities result in higher solid-wall frictional loss, thus increasing the
slip velocities.
3. At low transport velocities corresponding to limiting gas velocities, the solid-
wall frictional loss is negligible. Slip velocities in this region are governed by
solid volumetric concentration.
4. The concentration-slip velocity relationship obtained from the countercurrent ex-
periments can be extended to forward transport conditions, provided the trane-
port velocity is small.
5. Limiting gas velocities predicted from countercurrent experimental data agree
with those observed from cocurrent experiments.
6. The additional slip velocity due to solid-wall friction is observed to increase with
increasing solid velocity and decreasing concentration. A correlation to account
for the additional slip velocity due to the solid-wall friction is proposed.
3-30
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Bib-rr
aPPENDD( (A)
DETAILS OF SIZE ANALYSIS OF MATERIALS USED
Appendiz (A)
TøbIe (A-1) SIZE.ANÁ¿ySIS OF e6¡t G¿.ASS BEADS
Surface to volume mean diameter: - 96tt
TøbIe (A-2) SIZE ANA¿ySIS Oî 173p SAND
I
Mesh Size
(microns)
Mean Diameter (Do)
(microns)
Weight fractiou
Ãó,+229
-229+2ll-211+152
-t62+124-124+ru-r04+ 89
- 89+ 76
- 76+ 53
-53
229
220
181.5
138
114
96.5
82.5
64.5
26.5
0.0011
0.0014
0.0
0.2148
0.1615
0.5062
0.0355
0.0568
0.0227
1.0000
Mesh Size
(microns)
Mean Diameter (Dr)(microns)
tileight fraction
AÖ'-295+229
-229+zlr.zLL+LzE
-178+152
-152+104
-r04+ 76
-76
262
220
194.5
165
128
90
38
0.0668
0.332
o.2176
0.2065
o.L427
0.0243
0.01
1.0000
Surface to volume mean diameter: =#
: llStr¿-õ;
A-r
Appendiz (A)
Table (A-s) SIZE.AM¿ySIS OF 6a4¡t G¿ÁSS BEADS
Mesh Size
(microns)
Mean Diameter (Do)
(microns)
Weight fraction
ÃÖ'
+1001
-1001+ 853
- 853+ 7lr- 7ll+ 599
- 599+ 500
- 500+ 422
- 422+ 295
- 295
1001
927
782
655
549.5
461
358.5
t47.5
0.003
0.0077
0.1449
o.6727
0.1549
0.0131
0.0013
0.0024
1.0000
Surface to volume mean diameter : + :644trl'-D;
Table (A-Ð SIZE,{NA¿YSIS OF 17e¡t STEEL SIIOT
Surface to volume mean diameter:I
Mesh Size
(microne)
Mean Diameter (Do)
(microns)
Weight fraction
AÖ'-229+2ll-2ll+178-178+152
-162+124
-124+104
-104+ 76
-76- 295
220
194.5
165
138
114
90
38
147.5
0.0687
0.529
0.306
0.094
0.0
0.002
0.0003
0.0024
1.0000
^-2
t- Aó-¿- -ö;
: l79p
Appendís (A)
Table (A-5) SIZE ^A¡fÁ¿vSIS OF 375¡r STEEL SHOT
Surface to volume mean diameter: : g76tt
TøbIe (A-6) SIZD ANALYSß OF 637¡t STEEL
Surface to volume mean diameter: sr Aó-¿-E:637p
1
I
Mesh Size
(microns)
Mean Diameter (Do)
(microns)
Weight fraction
Aö.,
-599+500
-500+422
-422+3t3
-353+295
-295+2u
549.5
461
387.5
324
253
0.0111
o.2773
o.4457
o.L772
0.0887
r.0000
Mesh Size
(microns)
Mean Diameter
(microns)
Weight fraction
Aö'-1001+
- 853+
- 711+
- 599+
- 500+
- 422+- 353+
853
7tt599
500
422
353
ztL
927
782
655
549.5
461
387.5
282
0.0001
0.0313
0.8157
0.141
0.0077
0.0029
0.0013
1.0000
A-3
APPENDD( (B)
EFFECT OF CONCENTRATION ON SLIP VELOCITY
COUNTERCURRENT TRANSPORÎ DATA
Appendiæ (B)
96 mrcnon GIoEE beqds ln 12.7mm Tube
nuq)
J
=cl
l-(JÐJl¡l
(L
J(n
l{
I t. r
8.8
6.2
3.6
5.8
t.6
3.{
2.t 9.6
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l. E 7.2
CONCENTRRT I ON I7.I
2.8 t.2
CONCENTRNTION 1T,I
l20
7
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:o
t-(JÐJlrl
(L
Jan
70
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A+
I1
1+
III
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+ XX
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o+
+
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2?. 6619. 3983.77
I tt%o
o+
Solrd Flou (grlccolr. 082. 138. ?{
+o
9oxx
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+
+
0
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t5.21
xao
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AÀ
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+¡¡Â>tÀA
o-A-+-x-o-
21.52t6. 60
2.2
t.l
B-1
5.6
I
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644 mrcnon Glqss becrds rn 12.7nn Tube
{ l0
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1?8 mrcnon SteeI Ehot ln 12.?mm Tube
2.9 A.2
CONCENTRNT I ON I7.I
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l-(JoJl¡l
(L
Jan
3.{
2.8
2.2
1.6
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6
5
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3
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Xì ð
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Solrd Floo- 3.05- 7.t8- 26.63- 53. l0- al.1?- 158.9r
l. {
B-2
5.6
I
Appendiæ (B)
3?5 mrcnon Steel shot rn l2.7mm Tube
t.2 2. t 3.6
CONCENTRFT I ON (T.I
t.8
639 mrcnon Steel shot rn 12.?mm Tube
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=o
l-(JoJtrl
(L
J(n
3.t
2.6
2.2
¡.6
2.6
2.2
l. E
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3
nuat
J
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(LH
Ja
60 2.t 3.6
CONCENTBÊTION '7.1
B-3
01
I1
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koIx+Ëõ
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o1
1xo +
( gnl rccl
+
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ox
år^ ^qc^
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ðç
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+
oo
@o
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>fx
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++. .þt* **d
Solrd Flouo- 18.07â- 39. lE+- 6{.58x- l2?. t3o - 226. 33
t. {
t.2 t.E
AppendLæ (B)
96 mrcnon Glqss beo,ds rn 19. lmm TubeE
nur)J
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l-(JÐJlrj
(L
J(n
6.6
5.2
3.E
2. I
1.2
3.{
2.6
t. E
1.6 6.t
L73 mrcnon Sond rn 19.1mm Tube
Sol¡d Flor (gnlcccl
X +
aa+X+xðx
GOA
A¿A
A
3.2 l. E
CONCENTRRT I ON VI
3.2 l. E
CONCENTRRT I gN 1,7,1
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t-(JoJl¡J
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JÉ
+xA
X+A*
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Sol¡d Flor (9rlrccl- 4.74- t6.90- {9.39- 83.17
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t.6
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Appendiæ (B)
644 mrcnon Gloss beqds ln l9.1mm Tube
uuO
J
=_o
t-CJoJlrJ
(L
Jql
2
t.E
t.6
t. {
t.2
t.2
3.{
2.6
t.{ 5.6
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2.6 1.2
CONCENTRRTION (Z)
2. I 3.6
CONCENTRßT I ON I'II
70
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L)oJt¡l
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Jvl
60
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ÂA
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oÂ
x
X
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A+
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Oô'
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+
OaE^
Solrd Flou (gnlecolo- 26.63a- 53. t0+- 158.9{x- 27t.17
t.8
t.2
B-5
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Appendiæ (B)
375 mrcnon SteeI shot rn 19. lmrn Tube2
1lJt0¡
J
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]-(JÐJlrJ
fL
J1n
t.8
1.6
l. {
1.2
t.E
t.6
l. t
oA+xo
60 t.2 t.8
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x *¡x +
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2, I 3.6
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+
+
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t.2
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B-6
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6
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96 mrcnon Glo,ss beods rn 25.4mm Tube
I t0
CONCENTRÊT I ON lTI
1?3 mrcnon Sond ln 25. 4mm Tube
5nu1'J
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f-(JÐJlrJ
(L
Jan
I
3
2
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3
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l-(JoJl¡J
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J(n
2.6
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2.A 1.2
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x
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XX
IXX
x
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+
A
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t+
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+
xx
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4(D
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Sol¡d Flou- 2.06- 7.7t- 15.2t- t6.60- ?7.5{- t33. {0
t. {
t.l
B-7
5.6
Appendiæ (B)
644 mrcnon GlcrsE becrds ¡n 25.4mm Tube
Solrd Flor¡ (grrl¡co)
1.5
t.3
t.2
l. I
2.6
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3
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t. E
t.6
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t.2
1.5
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t.3
t.2
Appendùæ (B)
3?5 mrcnon SteeI shot rn 25.4mm Tube
2
CgNCENTRRT I ON 1'II
639 mrcnon Steel shot ln 25.4mm Tube
3
CONCENTRßT I ON T,T,I
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96 mrcnon Gloss beods rn 38.1mm Tube5
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J
ao
t-(JoJl¡l
(L
Jan
1,2
3. I
2.6
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2.6
2.2
1.8
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3.2 1.8
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2.9 1.2
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644 mrcnon Glqss beods ln 38. lmm Tube
1n0¡J
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t-(JoJtJ.j
(LH
J(n
t. 25
t.2
l. l5
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2
CONCENTBRTION (TI
2.8 1.2
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t. {
t.3
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Appendiæ (B)
3?5 mrcnon Steet shot rn 38. lmm Tube
0.8 1.6 2.1 3.2
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639 mrcnon Steel shot rn 38.1mm Tube
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APPENDD( (c)
EFFECT OF PARTICLE PROPER]TIES ON SLIP VELOCITY
COUNTERCURRENT TRANSPORÎ DATA
uut)J
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t-H
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4.8 7,2
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Appendiæ (C)
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APPENDD( (D)
EFFECT OF TUBE DIAMETER ON SLIP VELOCITY
COUNTERCURRENT TRANSPORT DATA
t{
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8.8
6.2
3.6
5.8
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3.{
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Eotboo o A
2.t
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o
9.6
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o
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L)oJUJ
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3.2 {.8
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2,t 3.6
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aPPENDD( (E)
TEST OF GRADIENT COAGUTATION MODEL
oo
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Appendùæ (E)
t.3
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I1.1
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3.5644 mrcnon Gloss beqdE
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tosl0 ( C x Vs/Dt )
179 m I cnon S t ee I shot
2
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L)ÐJt¡J
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3
2.5
t.5
5.8
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3.1
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0.6
C x Vg./Dt )
2
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A
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1.3
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J(n
3.5
3
2.5
2
t.5
3.5
3
2.5
2
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375 mrcr-on SteeI shot
-0. I 0,6
Iosl0 ( C : VslDt )
637 mrcnon Steel shot
-0.I o.G
tosl0 (CrVg/0t)
AppendLæ (E)
t.3 2
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l-(JÐJlrJ
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Ja
2
oÂ+x
19.25.38.
oo
qf
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APPENDDC (F)
EFFECT OF TRANSPOR:I VELOCITY
ON
CONCENTRATION.SLIP VELOCITY MAPS
COCURRENÎ TRANSPORT DATA
(Countercur¡ent dotø is also prcscntal lor comprison)
7
Appendiæ (F)
96 mrcnon Gloss beods rn t2.7nn Tube
0.5 t.5
CONCENTRßT I gN I7,I
t73 m r cnon S ond r n L2.7 nn T ube
t. { 2. 1
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xo 0x 0x o
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3.{
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F-1
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5
Appendiæ (F)
644 mrcnon Gloss beods rn 12.7nn Tube
0.8 1.6 2. I 3.2
CONCENTRRT I ON I7.I
L18 mrcnon Steei shot rn 12.7nn Tube
1.5
CONCENTBRT I gN (7.1
tJìu[)
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7.5
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1
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3.8
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2.6
2.2
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Appen&Lr (F)
3?5 mrcnon Steel shot ln 12.7mm Tube
0.7 1.4 2.t 2.8
CONCENTRRTION I7I
639 m I CnOn Steet shot rn t2.7nn Tube
1.2 1.8
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Appendiæ (F)
96 mrcnon Gloss beqds rn 19.1mm Tube
0.4 0.8 1.2
CONCENTRRT I ON IX,I
1.6
L73 mrcnon Sond ¡n 19.1mm Tube
20
t¡lna¡
J
=cl
t-(JoJl¡J
fL
Jv,
220 0.8 l.
CONCENTRRT I ON
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++
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A
o
#o
+
Â
oA
+
Âa
A
oa++
:x *x*x
at
*,;*¡+
f ¡|< *x x {'xÂ+*6
+
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ÂooaaÁ aôaÂa
A
ÂÂ
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x
X+o o
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o ++ Io
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+
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+
+
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+
)x
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x x t ll( xtf x xxx
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- 7.7t- 15.2r- 2t.52- 46.60- 77.5t- r33. a0- Countcn¿u¡rant dqto
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644 m r cnon Glq=s beods ln 19.1mm Tube
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a
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(L
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t.5
3.8
3.1
2.t
t.7
6.6
5.2
3.E
0.8 3.2
I78 mrcnon Steel Ehot rn 19. lmm Tube
r.6 2.1
CONCENTRRTION (TI
0.8 1.2
CONCENTRRT I ON 17,1
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I
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(L
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x
+
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xo+ xÂ
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2.2
1.8
Appendïæ (F)
3?5 mrcnon Steel shot rn 19.1mm Tube
0.5 t.5
CONCENTRNT I ON I7.I
639 mrcnon Steet shot rn 19.1mm Tube
20 2.5
3
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=g)-F(JoJt¡J
fL
Jvt
0 2.2 3.3
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ìt(*
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xÀ o
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+ xo¡l xo x o
x oÂo o+ X oXAo x
o
;( x xxfx* *
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x*t(*
*,rf #xx
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+
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Appen&iæ (F)
96 mrcnon GIqss beqds ln 25.4mm Tube
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F-
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(L
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8.2
6.{
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2.ø
5.5
a.6
3.7
2.8
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173 mrcnon Sond
t.2
0.6 0.9
CONCENTRRT I ON IT,I
1.8
CONCENTRRT I ON (7.1
t.2
rn 25.4mm Tube
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)t(*
x;x
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+
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644 mrcnon GIqEE beods ln 25.4mm Tube
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J
ao
t-CJÐJl-¡J
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2
1.8
1.6
t.4
t.2
3.4
2.8 o
2.2
40 0.8 3.2
178 mrcnon Steet çhot rn 25.4mm Tube
r.6 2. A
CONCENTRRT I gN (7.1
+
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x
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CONCENTBR] I ON
2
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x
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375 mrcnon Steel shot rn 25.4mm Tube3
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F
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ts
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t-(JoJUJ
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2.2
t.8
t. {
2
t.8
t.6
l. {
o-Â-+-X-o-+-x-x-
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r40.72?45.83425.50665. 90Count¿ncun¡cnt doto
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x
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127. t3226. 33386. 8062 t. 25Count¿¡cunncnt doto
x
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639 mrcnon Steel shot rn 25.4mm Tube
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Appendiæ (F)
96 m r cnon G I oEE becrds rn 38. l mm T ube
0.2 0.{ 0.6 0.8
CONCENTBHT I ON (7,1
t73 mrcnon Sond rn 38. lmm Tube
0.4 0.6
CONCENTRNT I ON (7)
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2.2
t.8
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3.5
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2.5
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0.8
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644 mrcnon GIoEs beods rn 38. lmm Tube
0.5 t.5
CONCENTRRT I ON l,7,1
178 mrcnon Steel Ehot rn 38.1mm Tube
0. I 0.6
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3?5 mrcnon Steet shot rn 38. tmm Tube2.5
2.2
t.9
1.6
1.3
2
t.8
t.6
l. {
o-A-+-X-o-1-*-
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l{0. ?22{5. 83{25.50Count¿ncunn"nt doto
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639 mrcnon Steel shot In 38. 1'nm Tube
1.5
CONCENTRRT I ON I7.I
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t2?. t3226. 33386. 80Countc¡cr.n¡"nt doto
t,2
0.5
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2.5
APPENDDC (G)
VARIATION OF SLIP VELOCITY DUE TO WAIL FRICTION
WITE
TRANSPORÎ VELOCITY
I
Appen&ir (G)
3?5 mrcnon Steel shot rn 12.?mm Tube
0.4 0.8 1.2 1.6
V s/V t
3?5 mrcnon Steel shot rn 19. lmm Tube
P
IL
t¡-
L
3.2
2.1
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0.8
0
4.5
3.6
2.7
1.8
0.9
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t.2
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o-
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375 mrcnon Steel shot rn 25.4mm Tube
0.4 0.8 t.2 1.6
Vs/Yt
375 mrcnon Steel Ehot rn 38. lmm Tube
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2
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3.3
2.2
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3.6
2.4
Appendiæ (G)
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