EXPERIMENTAL WALL CORRECTION FACTORS OF SINGLE
SOLID SPHERES IN CIRCULAR CYLINDERS
MlTSUGU IWAOKA AND TSUTOMU ISHII
Department of Chemical Engineering, YokohamaNational University, Yokohama240
Introduction
Concerning the wall effect on the slow motion of
single solid spheres in circular cylinders
some
studies1'4"9'13>15)
and
theoretically.
treated
this
methods.
have been made both experimentally
Most
problem
theoretical
with
The experimental
a series
onesl!4'7~9'13>15)
of expansion
ones5>6'14il7)
were based
on measurements of the settling velocities of single
solid spheres in circular cylinders.
However, it cannot be said that there are adequate,
reliable data for the wall correction factors of single
solid spheres in circular cylinders.
In the flow fields of packed beds, particulate fluidization and hindered settling, particles interfere
each other and as a result the drag coefficient
with
of a
particle in a swarm of particles is larger than that of
a single particle in an infinite medium, as the fluid
velocity on the surfaces of all particles is zero and therefore the resistance
force becomes larger for fluid to
pass through such a swarm. The relationship between
the drag coefficient of a particle in a swarm and that
of a single particle in an infinite mediumhas been
expressed by the following void function defined by
Ito11>12), on which previous theoretical
and experi-
mental work is summarized by Ishii10).
CDX=CDs F{e)
(1)
Also, in the case of a single particle settling in a
finite medium surrounded by a solid wall on which the
fluid velocity is zero, the particle interferes with the
solid wall. The drag coefficient of a particle in a
finite mediumis larger than that of a particle in free
settling, if the particle Reynolds number is equal in
both cases. This is because the resistance force becomes larger for a particle to pass through such a
narrow space surrounded by the solid wall. This relationship can be expressed in a similar way by the
CDw = CDsF(r) *
(2)
Here, F(r) is called the wall correction factor.
Received
June 17, 1978.
be addressed to T. Ishii.
VOL
12
NO.
3
1979
Correspondence
circular cylinders as a basic study of the flow fields of
packed beds, particulate
fluidization
and hindered
settling.
This study resulted in the establishment
of
the precise design for a falling ball viscometer, as well
as a basic study on packed beds, particulate
fluidi-
zation and hindered settling.
1.
Experimental
Four glass tubes of about 1 cm I.D. and 100cm in
length were used as experimental apparatus. A
thermometer was inserted into one of the tubes, containing the same liquid as the others, for measuring
the temperature under the same conditions as the
actual experiment in the other tubes. This experi-
ment was done in an air-conditioned room, in which
the room temperature was kept constant at about 20°C.
The tubes for the actual experiment were 0.900, 0.990
and 1.00 cm I.D. and were supported vertically by
making the tubes parallel
weight.
The entrance
to a thread
and end region
suspending
are necessary
a
due to
the unsteady motion of solid spheres influenced by
the upper and lower surface, respectively.
Thus, the distance of 20cm was taken for both
entrance and end region of this experiment and seems
to be adequate, based on fairly many previous studies.
Therefore, the section for measuring the time in which
the spheres fall through was 60 cm in the middk of
the
circular
cylinders.
r.;à"å -. -
Millet jelly aqueous solution was used as a liquid
for this experiment. It is well-known that this is
a highly viscous Newtonian fluids.
Four millet
jelly aqueous solutions of different
concentrations,
designated as A, C, D and E, were prepared by dissolv-
following equation.
051.
Thus the void function and wall correction factor
are proved to have similar properties.
The purpose of this workwas to measure accurately
the wall correction factor of single solid particles in
concerning
this
article
should
M. Iwaoka is now at Japan Steel Works, Muroran
ing given amounts of millet jelly into warmwater.
The viscosities of these solutions were about 2800, 330,
760 and 2000 centipoises respectively, at 20°C.
The viscosities were determined with a Hoeppler
falling ball viscometer at constant temperatures of ap239
Table 1 Previous work
1.
2.
3.
4.
5.
Landenburg13)
Faxen4>
Faxen4)
Lee14>
4.
1
Bohlin1)
7. Haberman
and Sayre8>
Suga17>
10.
Francis6)
ll. Fidleris
l
+9/4*+(9A*)2
2 7r/8)(l
-r)-5/2
F(r)=(l/l
-2.105r+2.087*3)
F(r)=(l
-0.75857r5)/(l
F(r)=(l
-r)-2-25
F(r)={(l-0.475r)/(l-r)}4
and Whitmore5)
r ^0.3
0.7^*^
r^0.3
1
solutions.
Bearing balls of diameters from 1 to 9mmwere
used as solid spheres for this experiment.
2. Results and Discussion
The particle Reynolds number Repw, in which the
settling velocity uw of the sphere influence by the wall
effect was used as a representative velocity, is less than
0.0762. The particle Reynolds number RePF, in
which the settling
velocity
infinite medium calculated
?=0.3
-2.1050r+2.0865*3-1.7068*5+0.72603r6)
rko.5
r^0.832 with glucose
for rotating
spheres with glycerol
r^0.970
Presented graphs
proximately 20, 25, 30, 35 and 40°C.
Distilled water and a standard liquid for correcting
viscometers, prescribed by JIS Z8809, were used as
240
*^0.3
*^0.3
Exact solution of algebraic equations
More exact solution of more than 8 algebraic equations.
Both solutions are almost the same curve in the figure.
F(*)=l/(l
-r)2-1/w-*25
*^0.333
Fig. 1 Previous wall correction factors of single
solid spheres in circular cylinders
standard
*6)
F(*)=l/(l-2.10443*+2.08877r3-6.94813*5-1.372*6+3.87r8-4.19r10)
8.1. Haberman7)
8.2. Paine and Scherr15)
9.
r^O.l
rWl
F(r)=1/(1
-2.104*+2.09*3-0.95*5)
F(*)=(l-V4*)/(l-V+5/2*3-9/4*5+
4.2
F(*)=
4.3
F(?)=(3V
and Byrne9)
Happel
6.
F(r)=l+2Ar
F(r)=1+2.1*
uF of the sphere in an
by Stokes drag formula
Fig. 2 Experimental wall correction factors of single
solid spheres in circular cylinders
was used, is less than 0.748 except that the same
numbers were 3.02 and 2.53 in the case ofDp=0.9 cm.
In such a creeping-flow region the followingrelationship holds for the wall correction factor, which is
calculated from this equation.
F(r) = uF/uw
(3)
All the previous studies are summarized in Table 1
and plotted in Fig. 1, where the correlating equations
are considerably
convergent
in the small r ranges
(solid lines), but they were extrapolated beyond the
applicable ranges (dotted lines).
The plotted data
show large scatter as r increases and it is not known
which correlating
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direction opposite to the direction for rolling along the
wall effect.
Figure 2 shows a semi-log plot of the wall correction
factors of this experiment and dimensionless radius of
the sphere. The data of glucose ofFrancis6} and those
of
Fidleris
and
Whitmore5)
and
McNown, Lee,
McPhersonand Engez14) are also plotted in the same
figure. The data of glycerol of Francis were omitted,
as they concerned rotating spheres
compared with the other theories2'3).
and should be
The data of
this experiment agreed very well with the exact theory
of Haberman7), and Paine and Scherr15).
But the data of McNown,Lee, McPherson and
Engez14), Fidleris and White5\ and Francis6) are lower
for wide ranges of rjR.
Furthermore, although
up to r=0.8 solid spheres
were never rotating, in the case of r approximating
0.9 for this experiment the solid spheres were always
eccentrically rotating as they did when the cylinder was
tilted.
The imperfect roundness of the cross-section,
imperfect verticality
and curvature of the circular
cylinder cause a solid sphere to rotate in part in the
case of very small clearance.
But even though the
necessary precautions were taken, the occurrence of
rotation was unavoidable. According to McNown,
Lee, McPherson and Engez14), at moderately
low
values of the Reynolds number in the creeping-flow
region the sphere settled without rotation in a position
practically concentric with the cylinder, but in this
nearer side of the cylinder wall. The settling velocity
increases monotonically
with lateral position
until
about e=0.98, in which position the sphere settles
approximately
2.1 times faster than for e=0. For
e>0.98 the rate of settling decreases, as does the
angular velocity. Their perturbation solution further
predicts that at some lateral position e, close to unity,
the direction of rotation changes. Although the settling velocity undergoes a two-fold change with lateral
position
remains
The
spheres
in the range 0<e<0.98, the pressure drop
essentially constant over the same range.
data of this experiment except the rotating
was correlated by the following equation
using the least-squares method.
F(r)=l/(1.000000+1.137853p-0.2644559
+0.8474067x
10V+0.1128310X
-0.1003350x
+0.1922832X
+0.5989260X
l0V5+0.1252298
10V-0.6372137-X
10V-0.1967150
x l0-V2
10V4
*
x lOV6
10V8
X 10V°
(4)
In conclusion it turns out that the exact theory of
Haberman7) and the more exact solution of Paine and
Scherr15) can be applied to falling ball viscosity meas-
urements except for very small clearances between a
solid sphere and circular cylinder, and the fluid re-
sistance force for a macromoleculein restricted diffusion can be precisely estimated.
In the case of very small clearance more precise
measurements of rotation speed, rotation direction,
lateral position of a sphere as well as settling velocity
require the use of still and 8-mm low-speed cameras
in future studies.
experiment there was always with rotation in a position
eccentric with the
number range.
cylinder
in the
same Reynolds
Nomenclat ure
Christopherson and Dowson3) applied an elementary
lubrication theory to the settling phenomena of a solid
sphere in a circular cylinder in the case of small clearance and found that the settling velocity of a solid
sphere with rotation in a particular eccentric position
becomes largest and therefore the drag coefficient minisuch an eccentric
settling
with
rotation was predicted as stable from the viewpoint of
minimumviscous dissipation. This theory was based
only on the equation of motion offluid, but to obtain
a conclusion from the viewpoint of minimumviscous
dissipation,
the equation of energy of fluid, must be
considered from the viewpoint of the minimumentropy production rate. The direction of rotation in
the eccentric position in this experiment is opposite to
that reported by christopherson and Dowson3).
Bungay and Brenner2} have established a more precise lubrication
theory, using perturbation
method,
for such a settling problem. In the case of sedimentation of a sphere in a vertical cylinder a concentric
sphere (e=0) settles without rotation. If displaced to
an intermediate
position,
rapidly.
Simultaneously,
VOL.
12
NO.
3
1979
the sphere descends more
it undergoes rotation in a
e
mum.Therefore,
perpendicular distances from sphere center
F*r)
F(e)
R
uw
to cylinders axis
drag coefficient of a single sphere in
an infinite medium
drag coefficient of a single sphere
influenced by the wall effect
drag coefficient of a particle in a swarm
diameter of a solid sphere=2r
eccentricity
bj(R-r)
wall correction factor
void function
radius of a circular cylinder
radius of a solid sphere
= dimensionless radius of a solid sphere=r/R
[-]
[-]
[cm]
[-]
[-]
[-]
[cm]
[cm]
[-]
settling velocity of single solid sphere in
an infinite medium (expressed by Stokes
[cm/sec]
drag formula in this case)
settling velocity of a single solid sphere
[cm/sec]
influenced by the wall effect
= viscosity of solution
= density of solution
[c.p. or poise]
[g/cm3 ]
241
[g/cm3]
= density of a solid sphere
Literature Cited
1) Bohlin, T.: Trans. Roy. Inst. Technol, Stockholm, No. 155
(1960).
2) Bungay, P. M. and J. Brenner: Int. J. Multiphase Flow, 1,
25 (1973).
3) Christopherson,
A251,
550
(1959).
D. G. and D. Dowson: Proc. Roy. Soc,
4) Faxen, H.: Ark. Mat. Astron. Fys., 17, No. 27 (1923).
5) Fidleris, V. and R. L. Whitmore: Brit. J. Appl. Phys., 12,
490
(1961).
6) Francis, A. W.: Physics, 4, 403 (1933).
7) Haberman, W. L.: Ph.D. Thesis, University
(1956).
Basin
Dept.
Report
(1958).
No. 1143,
Washington,
D.C.,
U.S.
Navy
Happel, J. and BJ. Byrne: Ind. Eng. Chem., 46, 1181 (1954).
Ishii, T. : Kagaku Kogaku, 29, 915 (1965).
Ito, S. : "Kagaku-Kogaku-Nyumon",
Kagaku-Kogyo-Sha,
Tokyo
(1957).
Ito, S.: Kagaku Kogaku, 26, 1108 (1962).
Landenburg, R. : Ann. Phys., 23, 447 (1907).
McNown, J. S., M. J. Lee, M.B. McPherson
Engez:
Proc.
p. 17 (1948).
7th
Intern.
Cong.
Appl.
and S. M.
Mech.,
London,
Paine, P. L. and P. Sherr: Biophysical J., 15, 1087 (1948).
Sato, T., I. Taniyama and S. Shimokawa: Kagaku Kogaku,
of Maryland
30,
34 (1966).
Suga, S.: Bull. Inst.
146
8) Haberman, W. L. and R. M. Sayre: David Taylor Model
(1931).
Phys. and Chem. Research,
Tokyo, 10,
NATURAL CONVECTION HEAT TRANSFER FROM
HORIZONTAL CYLINDERS
Jagdish
CHANDand Dharam VIR
Department of Chemical Engineering and Technology, Panjab University,
ponent continuously
Introduction
The problem of natural convection from horizontal
cylinders was treated8>23'25'31)
as one of steady-state conduction in a stationary gas film. However, there is
ample evidence that heat transfer occurs by convection
streams within the so-called film itself.
Analytical
treatment is usually based on the set of equations
representing the conservation of mass, momentumand
energy in a fluid moving under the influence of body
force.
0.25 in the laminar range and 1/3 in the
range. A summary of correlations
of this
type is given in Table 1. However, experimental data
on horizontal
cylinders
collected
by Fishenden
continuously
decreases as (Pr à" Gr) is increased beyond
unity.
For vertical plates, Ostrach29) numerically solved the
equations for 0.01<Pr< 1000 and calculated profiles
for laminar flow to obtain Nu=a(Pr-Gr)0 25. The
computer solution
of Ostrach was shown by Le
Fevre24) to fit the empirical equation
a4-Pr/(2.43478+4.884Pr°-5+4.95283Pr)
Further
calculations
of the same type have been re-
lationship between log Nu and log (Prà" Gr) is essentially non-linear and the experimental slope maydeviate
from 0.25, particularly when Prà"Gr is large. Table 2
shows the values of the coefficients and exponents at
different levels of (Pr-Gr).
It is evident that the exReceived November 2, 1978.
should be addressed to J. Chand.
Correspondence
concerning this
Qualitatively
a similar situation showing that a=
F(Pr) is expected to occur for horizontal cylinders.
This affects the linearity of the relationship
between
log Nu and log (Prà"Gr). Another reason for the nonlinear relationship is the presence of a temperature ratio
term in the relationship by Hermann13).
Nu=F(Pr,
and
Saunders9) and shown in Fig. 1 suggests that the re-
242
increases and that the coefficient
ported10>27'34'35).
The principle of similitude5)6'20)28)
and suitable simplifying assumptions in the boundary layer theory
produce correlating equations of the type Nu=F(Pr,
Gr) and in particular Nu=aPrhGrc with b and c usually
equalling
turbulent
Chandigarh 160014 India
article
Gr,(Tw
- T^)/
TOQ)
Hymanet al.16) correlated natural convection data
with 14 different liquids including liquid metals by the
equation
Nu=0.53
[{Pr1(0.952+Pr)}(Pr
Hausen12)
recommended
à" Gr)]0
the
following
25
empirical
equation :
Nu=0.U(Pr'Gr)1/3+(Pr'Gry
\ lO~7<Pr-Gr<
1012
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