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Suspension of spheres in fully developed pipe flow
The Chemical Engineering Journal, 9 (1975) @ Elsevier Sequoia S.A., Lausanne. Printed
241-249
in the Netherlands
Suspension of Spheres in Fully Developed Pipe Flow B. P. FOSTER*,
A. R. HAIR? and I. D. DOIG
Department of Chemical Engineering, (Received
University of New South Wales, Kensington,
New South Wales 2033
(Australia)
9 July 1974; in final form 6 May 1975)
Abstract
stant sphere-to-fluid density ratio provides several relationships of interest 16:
The suspension of spheres by fully developed flow in a vertical pipe has been investigated for_particle Reynolds numbers between 2 and IO 000, sphere-toconduit diameter ratios between 0.046 and 0.274 and tube Reynolds numbers ranging from 25 to 27 000. Graphs relating drag coefficient, mean sphere position, suspended sphere motion and Strouhal number to Reynolds number and sphere-to-conduit diameter ratio are presented.
Cbt
= tiRept , d/D)
(1)
CDcl
= @@q,l,
(2)
CDt
= WW
d/D) d/D)
blD = @Wep,, d/D)
(4)
b/D = @Wep,l, d/D)
(5)
b/D = @(Ret, d/D)
(6)
Srt
(7)
= tiRep,, d/D)
SrCl = @(Rep,l, d/D) INTRODUCTION
Srt
The hydrodynamic behaviour of freely suspended solid spheres in bounded and non-bounded uniform flow fields both with and without fluid turbulence has been studied experimentally over a wide range of particle Reynolds numbers’ -lo. However, few experimental results are available for the commercially important system in which solid spheres travel within pipes (bounded shear flow field) and most of these relate to spheres travelling with respect to a Poiseuille flow at low particle Reynolds numbers” -14. Studies by Doig15 have shown that solid spheres suspended within a vertical pipe by a fully developed turbulent fluid flow behave quite differently from spheres suspended in the flow fields mentioned above. The study reported here extends this work and covers a particle Reynolds number range of 2 up to 10 000 for pipe Reynolds numbers between 25 and 37 000. Dimensional analysis of the suspension of spheres by fully developed flow in a vertical tube at a con-
* Present address: Australia. t Resent address: Australia.
James Hardie B.H.P. Ltd.,
Ltd., Auburn,
N.S.W.,
140 William Street,
Melbourne,
(3)
= @(Ret, d/D)
(8) (9)
This study explores these quantitative relationships and in addition qualitatively reports the behaviour of spheres suspended in fully developed pipe flow.
EXPERIMENTAL EQUIPMENT Figure 1 shows the arrangement of the experimental equipment. Fuller particulars are given elsewhere 16. and only a brief outline is provided here. Teflon spheres ranging in density from 2.10 to 2.13 g ml-’ and in size from 3.18 to 19 mm were suspended in a test section 50 mm long which formed part of a Perspex vertical tubular riser 5.5 m long with a 69.85 mm bore. The test section was located 0.6 m below the top of the riser. A steady axisymmetrical fluid velocity field was generated in the test section by the use of a constant head tank arrangement, diffuser exit and entry headers, and long radius bends in the fluid supply lines. Fluid flow rate was controlled by a 50 mm ball cock at high flow rates or a 25 mm globe valve at low flow rates. High flow rates were measured with a calibrated orifice plate and low flow rates with a calibrated venturi meter. A calibrated
242
B. P. FOSTER,
Head Tank\]
1
A. R. HAIR,
1. D. DOIG
I
14
I
3
3
4
8
fJ
t-
Fig. 1. Diagram of the experimental
equipment. Fig. 2. Siphtglass device and the method
thermometer
was stationed
for Sucrose solutions
measurement of fluid temperatures. were used as the motive fluid. A 16 mm Bolex tine camera was mounted in the centre of a plane Perspex plate 19 mm thick which formed the top of the exit header during runs 126222. This tine camera was used for measurement of the radial sphere position using the tube wall image reflection technique of Hair and Doig17. During runs I- 125 the sucrose solutions were insufficiently clear and instead of the image reflection method a sightglass device (Fig. 2) was used to observe radial sphere positions. The image reflection method was also found to be inferior for sphere positions remote from the tube wall.
EXPERIMENTAL
of using it.
in the exit header
PROCEDURE
A 61.3% sucrose solution was prepared using Table Grade sugar (99.8% sucrose). The nine spheres under test were placed at the bottom of the vertical riser and all residual air in the system was removed. The flow rate was adjusted to transport the smallest sphere to the test section, after which it was reduced
to hold the sphere suspended in the test section. The pressure drop across the venturi meter, the sphere size and the fluid and room temperatures were recorded. The radial position of the sphere in the test section was estimated using the sightglass device (Fig. 2). Qualitative notes on sphere behaviour during suspension were also recorded. These measurements were repeated for eight larger spheres and at the end of these measurements a sample of the solution was obtained for later analysis. The sucrose solution concentration was then reduced (by the addition of water) and a further set of suspension measurements was carried out. This procedure was used for twenty successive sucrose solution concentrations. Samples of sucrose solutions taken at the completion of each series of suspension measurements were later carefully analysed to determine the sucrose content and used (with the measured fluid temperature) to determine the fluid viscosity and density (for each experimental run) from dataIs. The density of the sucrose’solutions ranged from 1.29 to 1.03 g ml-’ while the viscosity ranged from 60 to 1.1 cP. Occasional sample viscosity measurements were also made to check the tabulated data and ensure that the viscosity did not change with
SUSPENSION
OF SPHERES
243
time. Fuller details of the experimental provided by Foste@.
RESULTS
procedure
are
AND DISCUSSION
Equations (l)-(9) involve two fluid velocities, Vt and I/cl. Vcr was sensitive to errors in the sphere position measurement and yielded less reliable results, especially when introduced into the drag coefficient relationships. Consequently, the macroscale velocity Vr has been emphasized in this paper and relationships involving the local velocity T/cl have been introduced only where they yield additional information. Full details of results based on the local velocity Vcr are provided elsewherer6. Suspended particle motion Observations of sphere behaviour were noted during the experimental runs. These observations were later organized to obtain an overall view of repeating sphere motions in various regimes. Four kinds of repeating motion (or modes) were observed which identified well with Reynolds number regimes. Some overlaps did occur and occasionally some transition behaviour between adjacent modes was noted. The four motion modes have been designated A, B, C and D. In motion mode A the sphere took up a steady position near, but not at, the tube axis and rotated in the direction of the vorticity of the shear flow field. Horizontal, vertical or angular motion was not apparent. In motion mode B the sphere moved radially and axially, but not around the tube axis. A sketch of the motion is given in Fig. 3. Starting from a point near the pipe axis the sphere would rise a short distance and then move radially toward the wall before falling a short distance and moving radially back toward the tube axis. The sphere began to spin at the start of its fall, its angular velocity increasing gradually to a maximum at the bottom of its descent;
at this point spinning abruptly ceased and no spin was observed during its rise. Spin while falling was in the direction of the flow field vorticity. The amplitude of vertical and radial excursions varied, but generally appeared to be in the ranges 0.ti-d and d-3d respectively. Measured data on these amplitudes were not reliable enough to allow any correlation with d/D or Reynolds number. In motion mode C the sphere adopted a metastable position near the conduit wall where it periodically underwent oscillatory radial excursions which decayed to restore the sphere again to its non-oscillatory but metastable position. Suspension in this mode was difficult since the sphere also often travelled erratically in the axial direction. During this axial travel the sphere was observed to spin, the direction being dependent upon the direction of axial movement. Whilst stationary, no spin was observed. At higher Reynolds numbers this periodically damped oscillatory motion changed to a sustained oscillatory motion, motion mode D. In this motion mode, the sphere underwent steady non-spinning radial oscillatory excursions at a constant height. The frequency and amplitude of these oscillations appeared to be reasonably constant. Data on oscillation frequency are presented later in this section. A special motion mode was observed with the largest sphere (d/D = 0.274) which was not observed with any other sphere. In this special case, the sphere took up a steady position near the wall; no vertical, radial or. angular movement or spin was observed. It is thought that this case was a special case of motion mode C in which the tendency to radial excursions was damped by the relatively high inertia of the particle. The relationship between motion mode, Reynolds number (Rept) and sphere-to-conduit diameter ratio is shown in Fig. 4. Note that this relationship is based on the use of the volumetric average fluid velocity Vr in the particle Reynolds number term. When the results were plotted16 using a particle Reynolds number based on the approach velocity Ver of the fluid at the sphere centre, transitions between motion modes were found to occur at the following values of Repcr:
transition transition transition
Fig. 3. Sketch
describing
the sphere
motion
mode
B.
from mode A to mode B, Repcr = lo-20 from mode B to mode C, Repcr = 80-100 from mode C to mode D, Repcr = 200-600
(Note: a range of values of Rep,1 is given for each transition because the transition value of Repcr also
B. P. FOSTER,
244 depends on the sphere-to-conduit diameter ratio (see Fig. 4).) Torobin and Gauvint9 have reviewed literature on the formation of wakes in turbulence-free unbounded uniform flow. They conclude that a stable vortex ring forms behind the downstream hemisphere at particle Reynolds numbers around 10 which grows in size and decreases in stability until it begins to d/D,
I
A. R. HAIR, 1. D. DOIG
oscillate at particle Reynolds numbers of around 130. With increasing particle Reynolds numbers the instability increases, leading to a periodic shedding of vortices at the lower critical Reynolds number. From the conflicting evidence reviewed, Torobin and Gauvin conclude that this shedding probably starts at particle Reynolds numbers around 500. There is an interesting correspondence between the particle Reynolds number values observed at motion mode transitions (A to B, B to C, and C to D) and the values quoted for wake structure changes which suggests that the two phenomena may be related. Drag coefficient and particle Reynolds number results The experimental results for the relationship expressed
Fig. 4. Sphere motion modes grouped by particle Reynolds number and sphere-toconduit diameter ratio. Note: the full line boundaries between the regions are only approximate: the transitions are not sharp and some overlapping occurs. The broken lines denote limits of observations.
in eqn. (1) are plotted in Fig. 5. Smooth curves through these points are presented separately in Fig. 6 and show the effect of increasing the sphereto-tube diameter ratio d/D. Features of these plots are the dependence of the drag coefficient CDr on the diameter ratio d/D, the relatively rapid decrease in CDr in the regime 50 < Rept < 200, the inflections in the Cnt-Repr curves in the regime 200 < Rep, < 800 and the decrease in CDr in the regime 1000 < Rep, < 8000 (shown by broken lines denoting gaps in the data).
Fig. 5. Experimental
Reynolds
I
10"
,,.... lo’
I.,,.,’
‘..,.,I’
lo2
IO'
results plotted
.,,I,,
I 10‘Rep,
as drag coefficient
vs. particle
number
SUSPENSION
245
OF SPHERES
10° Fig. 6. Line plots derived
10'
lo*
lo3
lo4
from Fig. 5. (The lines for d/D ratios 0.114 and 0.159 are omitted
The plot also shows that the value of CDt is generally higher than that given by the standard drag curve with the minor exception of the diameter ratios 0.046 and 0.069 in the particle Reynolds number regimes 130180 and 400-l 500 respectively. The wide divergence from the standard drag curve at values of Rep* below SO results from the use of the velocity Vt in the drag coefficient and particle Reynolds number terms. If the local velocity Vet is used instead, the drag coefficient approaches the standard drag curve as d/D approaches zero 16. The relative decrease in Cot compared with standard drag curve values in the regime 50 < Rep, < 200 coincides with a change in the observed radial sphere position (Fig. 7). Corresponding results plotted on a CDcr-Repcr basis did not exhibit a similar relative decrease16 and this decrease in CDt is therefore attributed to the change in Vt relative to T/,1 introduced by the changes in sphere radial positions occurring in this Reynolds number interval. The decrease in CD~ in the interval 1000 < Rep, < 8000 was also observed in corresponding CD&Rep,, results and therefore cannot be attributed to changes in corresponding I/cl and Vt values. No change in sphere motion was observed in this regime. The regime, however, was incompletely explored and is shown by broken lines
Rep,
for clarity.)
in Fig. 6. The inflections in the course of CDt(d/D) values in the particle Reynolds number interval 180-850 can be associated with the change from sphere motion mode C (periodically unsustained oscillation) to sustained oscillation mode D. Cot values before the inflection in Fig. 6 have Reynolds numbers in the mode C interval in Fig. 4 and are less than Cot values immediately following the inflection which have Reynolds numbers in the mode D interval. (Note that the motion mode boundaries are not sharp: the full lines drawn between regimes in Fig. 4 denote approximately the middle of observed transitions from mode C to mode D which include overlaps. The C-D reg.ime boundary values of Fig. 4 correspond only approximately with the inflection midpoints of Fig. 6.) When the cDt(d/D) values are plotted against the pipe Reynolds number Ret, all of these inflections occur in the range 2300-3000 which is the recognized regime for transition from laminar to turbulent pipe flow. This transition introduces several changes which require consideration. The transition from laminar to turbulent pipe flow changes the local velocity at each position in the flow field and the local velocity gradient, and introduces sustained turbulence. No measurements of turbulence have been made
246
B. P. FOSTER,
A. R. HAIR,
I. D. DOIG
.
loL
10’
Fig. 7. Experimental
showing
10‘
10’
10) results
the observed
mean sphere position
but Pitot tube traverses were made 0.2 m above and 1.9 m below the suspension region of the spheres in the pipe flow Reynolds number regime between 5000 and 32 000. The results showed fully developed axisymmetrical turbulent pipe flow at both positions, an absence of swirl and a close correspondence with the velocity profiles* predicted by Deissler*O. The pressure gradients measured were those predicted for a smooth pipe *l . No spheres were present during these tests and the observations concerning laminar and turbulent pipe flow assume the local turbulence intensity to be a dependent and persistent variable whose presence, or absence, and properties are little affected by the sphere at positions upstream of the sphere. Observations of the sphere position before and after the inflections show that it approximately keeps to its position. Therefore, the laminar-turbulent velocity profile change is likely to change the fluid velocities approaching the sphere. If these local approach velocities in a turbulent velocity profile were higher than those of a Poiseuillian profile (sphere close to the wall), the mean volumetric velocity Vt required to suspend the sphere would decrease and this would be reflected by an increase in CDt. For this argument to account for the observed inflections and increases in Cm values, inflections in the corresponding Rep,,d/&cDc[ relationships should be absent or small. Data relating to the observed inflections are presented in Table 1 and examination shows that the CDct values + The region at the core, however, law=.
followed
a velocity
defect
RePt plotted
against
particle
Reynolds
number.
also increase sharply where the CDt values increase. It follows that something other than changes in local fluid velocities accompanying the laminar-to-turbulent tube flow transition are causing the increases observed in CDr VaheS. Torobin and Gauvin9 have shown that, in unbounded flow systems where turbulence is generated by grids, turbulence generally increases the drag coefficient above values given by the standard drag curve. Clift and Gauvin10 have reviewed more recent work of this kind and report (from Zarin23) that in the particle Reynolds number regime 200 < Rep < 800 the drag coefficient increases sharply with increasing turbulence intensity in the free stream turbulence intensity range 0.004-0.033. It is therefore possible that the increases in CDr during the laminar-toturbulent flow transition interval are due to the introduction of sustained turbulence in the suspending fluid. Figures 5 and 6 also provide comparisons of the present results with those of Eichorn and Smalll*, Fidleris and Whitmoore and DoiglS. Radial position adopted by a suspended sphere The mean radial position adopted by the spheres in
the conduit is presented in Fig. 7. Note that this radial position is expressed as the term b*, which is the ratio of the time averaged distance of the sphere centre from the tube wall divided by the radial distance over which the sphere may move. It allows b* values to range between zero and unity.
SUSPENSION TABLE
OF SPHERES
247
1
Data relating
the points
of inflection
in Fig. 6
Diameter ratio d/D
Tube Reynolds no. Ret
Sphere Reynolds nos. Rept Repel
Drag coefficients CDcl CM
Sphere position ratio b/D
0.214
2820 2820 3050 3430
771 111 834 940
724 690 750 801
0.98 0.92 1.22 1.25
1.11 1.13 1.51 1.42
0.272 0.329 0.323 0.272
0.228
2530 2650 2830 3420
577 603 644 179
516 566 564 594
0.92 0.88 1.16 1.03
1.15 1 .oo 1.50 1.30
0.329 0.272 0.348 0.332
0.183
2320 2560 2580 3010
424 488 471 549
399 395 394 491
0.87 0.75 1.15 1.06
0.98 1.05 1.64 1.33
0.364 0.378 0.386 0.329
0.136
2080 2250 2310 2740
285 308 316 315
99 132 232 335
0.81 0.71 1.03 0.95
6.70 3.89 1.90 1.19
0.443 0.455 0.398 0.329
0.091
2510 3040 3200 3720
233 276 291 331
187 217 228 263
0.71 0.70 0.82 0.77
1.10 1.13 1.33 1.27
0.409 0.421 0.423 0.425
180 < Rept < 850; 2000 Q Ret < 3200.
The most striking aspect of this plot is the sharp change in sphere position in the range 40 < Rep, < 150. Also, for Rept < 150 the distance from the wall increases with the sphere-to-tube diameter ratio d/D, which is consistent with the observation that the larger spheres had the greater oscillational amplitudes. At low particle Reynolds numbers the influence of the diameter ratio d/D on mean position is difficult to discern because of the scatter associated with assessments of the mean position b* when using the sighting device, and a scarcity of data. However, it is noticeable that as Rep, increases from 10 to 40, the value of b* decreases. This decrease can be linked with a change in sphere motion from mode A to mode B. A plot of mean distance b against Ret, presented elsewherem, shows that the laminar-to-turbulent pipe flow transition is not associated with the very pronounced change in radial position from near the tube axis to near the tube wall. (It occurs in the range 320 < Ret < 1000 or 62 < Rep,, < 180.) The results reported in Fig. 7 show that the position of a sphere suspended in fully developed pipe flow is primarily dependent upon the particle Reynolds number, with a discernible secondary d/D effect occurring when the sphere is near the tube wall.
d/D
i
r
7”
/
,046
Fig. 8. Dependence of the Strouhat mode D mean oscillation frequency) diameter ratio.
number (for motion on the sphere-to-tube
248 Radial oscillational frequency
At the higher values of particle Reynolds numbers investigated in this study, the spheres adopted a reasonably steady radial oscillational motion (motion mode D). Dimensional analysis indicates that frequency of oscillation (expressed as a Strouhal number) should be a function of a Reynolds number (Rep,, Rep,, or Ret) and the sphere-to-conduit diameter ratio (eqns. (7), (8) and (9)). When results were plotted, no influence of Reynolds number (Rept , Rep,, or Ret) upon Strouhal number could be discerned (Rep, values ranged from 279 to 10 100). A plot of Strouhal number against sphere-to-conduit diameter ratio only is presented in Fig. 8; the horizontal straight lines indicate Strouhal numbers over the range of Rep, values investigated and the dots are the mean Srt values. The data reported in Fig. 8 were obtained using an image reflection technique17 for measurements over time intervals of 3-5 s.
B. P. FOSTER,
A. R. HAIR,
I. D. DOIG
relative velocity between the sphere and the local fluid velocity approaching the sphere centre at its average position. Relationships employing the former relative velocity should be of use in relating drag coefficient data to macroscopic measurements whilst the relationships based upon the latter will be of use in comparisons with unbounded and bounded quiescent flow systems. For particle Reynolds numbers between 200 and 10 000, spheres suspended in fully developed pipe flow oscillate fairly steadily in a radial direction. The frequency of these oscillations can be expressed as a Strouhal number which appears to be independent of particle Reynolds number but depends upon the sphere-to-pipe diameter ratio. It should be noted that this study concerned suspended spheres at an almost constant sphere-tofluid density ratio. Variation in the sphere-to-fluid density ratio may modify the relationships obtained.
CONCLUSIONS
NOMENCLATURE
Spheres suspended in fully developed pipe flow adopt one of four distinct modes of motion for particle Reynolds numbers between 2 and 10 000. The particular motion mode adopted is primarily dependent upon the particle Reynolds number with a secondary dependency on the sphere-to-pipe diameter ratio. A most interesting observation is that transitions between motion modes coincide approximately with particle Reynolds numbers reported for changes in the wake structure behind a sphere in unbounded systems. Spheres suspended in a fully developed pipe flow adopt a position near the tube axis at particle Reynolds numbers less than 80 but adopt a position near the tube wall at particle Reynolds numbers greater than 200. The mean position adopted by suspended spheres when the spheres are in the vicinity of the conduit wall varies with the sphere-to-conduit diameter ratio. Drag coefficients for spheres suspended in fully developed pipe flow with particle Reynolds numbers between 2 and 10 000 are greater than those reported for spheres travelling through unbounded quiescent fluids and are dependent upon both the particle Reynolds number and the sphere-to-conduit diameter ratio. The particular form of this relationship is dependent upon the velocity used to characterize the relative velocity between the sphere and the fluid. Two characteristic relative velocities were considered in this study-the relative velocity between the sphere and the mean volumetric fluid velocity, and the
b b* CDcl CDi n
d F f
time averaged radial distance between the sphere centre and the conduit axis [L] dimensionless radial sphere position (l-2b)/ (D-4 drag coefficient, 8F/~d2pVc~2 drag coefficient, 8F/nd2p Vt 2 conduit diameter [L] sphere diameter [L] drag force acting on a sphere, .rrd3(ps-p) g/6 [MLT_2] time averaged sphere oscillation frequency
U’-‘I g Repel
Rep, Ret SrCI Srt VCI
vt P P PS
acceleration due to gravity [LTF2] particle Reynolds number, pdVcl/p particle Reynolds number, pd Vt/p Reynolds number for the tube flow, PDT/t/p Strouhal number, fd/Vcl Strouhal number,fd/Vt time averaged “centreline” velocity, being the relative velocity between the sphere velocity and the approach velocity of the fluid at the time averaged position of the sphere centre [LT-‘] volumetric fluid flow rate per unit tube cross sectional area [LT-‘1 fluid viscosity [ML-‘T-l] fluid density [MLe3] density of the solid material of the spheres [ML-3]
SUSPENSION
OF SPHERES
REFERENCES 1 R. A. Castleman, The resistance to steady motion of small spheres in fluids, Nat. Advis. Comm. Aeronaut. Tech. Notes 231,1926. 2 H. Rouse, Nomogram for settling velocity of spheres, Annu. Rept. Comm. Sed., Nat. Res. Council, 1937. 3 C. A. Lapple and C. B. Shepherd, Calculation of particle trajectories, Ind. Eng. Chem., 32 (1940) 605-607. 4 T. Maxworthy, Accurate measurements of sphere drag at low Reynolds numbers, J. Fluid Mech., 23 (1965) 369-312. 5 E. B. Christiansen and D. H. Barker, The effects of shape and density on the free settling of particles at high Reynolds numbers, A.I.Ch.E.J., II (1965) 145-151. 6 J. S. McKnown, H. M. Lee, M. B. McPherson and S. M. Engez, Influence of boundary proximity on the drag of spheres. In Proc. 7th Znt. Congr. for Appl. Mech., Vol. 2, No. 1, 1948, pp. 16-29. deter7 V. Fidleris and R. L. Whitmore, Experimental mination of the wall effect for spheres falling axially in cylindrical vessels, Br. J. Appl. Phys., 12 (1961) 490-494. aspects of 8 L. B. Torobin and W. H. Gauvin, Fundamental solids-gas flow: Part I-Introductory concepts and idealised sphere motion in viscous regime, Can. J. Chem. Eng., 37 (1959) 129-141. 9 L. B. Torobin and W. H. Gauvin, Fundamental aspects of solids-gas flow: Part V-The effects of fluid turbulence on the particle drag coefficient, Can. J. Chem Eng., 38 (1960) 189-200. particles 10 R. Clift and W. H. Gauvin, Motion of entrained in gas streams, Can. J. Chem. Eng., 49 (1971) 439-448. 11 A. M. Fayon and J. Happel, Effect of a cylindrical boundary on a fixed rigid sphere in a moving viscous fluid, A.I.Ch.E.J., 6 (1960) 55-58.
249 12 R. Eichorn and S. Small, Experiments on the lift and drag of spheres suspended in Poiseuille flow, J. Fluid Mech., 20 (1964) 513-517. 13 R. C. Jeffrey and J. R. A. Pearson, Particle motion in laminar vertical tube flow, J. Fluid Mech., 22 (1965) 121-735. on the suspension 14 G. F. Round and J. Kruyer, Experiments of spheres in inclined tubes: I-Suspension by water in turbulent flow, Chem. Eng. Sci., 22 (1967) 1133-1145. 15 I. D. Doig, Suspension of spheres in a vertical tube at high Reynolds numbers, Chem. Eng. Sci., 23 (1968) 794-796. 16 B. P. Foster, Suspension of spheres by a bounded fluid, M.Sc. Thesis, Univ. of New South Wales, Australia, 1973. technique for 17 A. R. Hair and I. D. Doig, A photographic the determination of the position and velocity of particles moving in tubes, Chem. Eng. J., 9 (1975) 175-178. 18 Viscosity of sucrose solutions, in Laboratory Directions, Colonial Sugar Refining Co., Sydney, Australia, 1959, Table 63. aspects of 19 L. B. Torobin and W. H. Gauvin, Fundamental solids-gas flow: Part II-The sphere wake in steady laminar fluids, Can. J. Chem. Eng., 37 (1959) 167-176. 20 R. G. Deissler, Analysis of turbulent heat transfer, mass transfer and friction in smooth tubes at high Prandtl and Schmidt numbers, Nat. Advis. Comm. Aeronaut. Rep. 120,1955. 21 L. F. Moody, Friction factors for pipe flow, Trans. ASME, 66 (1944) 671-678. 22 J. 0. Hinze, Turbulence, McGraw-Hill, New York, 1959. 23 N. A. Zarin, Measurement of non-continuum and turbulence effects on sub-sonic sphere drag, NASA Contract. Rep. CR 1585,197O.
241-249
in the Netherlands
Suspension of Spheres in Fully Developed Pipe Flow B. P. FOSTER*,
A. R. HAIR? and I. D. DOIG
Department of Chemical Engineering, (Received
University of New South Wales, Kensington,
New South Wales 2033
(Australia)
9 July 1974; in final form 6 May 1975)
Abstract
stant sphere-to-fluid density ratio provides several relationships of interest 16:
The suspension of spheres by fully developed flow in a vertical pipe has been investigated for_particle Reynolds numbers between 2 and IO 000, sphere-toconduit diameter ratios between 0.046 and 0.274 and tube Reynolds numbers ranging from 25 to 27 000. Graphs relating drag coefficient, mean sphere position, suspended sphere motion and Strouhal number to Reynolds number and sphere-to-conduit diameter ratio are presented.
Cbt
= tiRept , d/D)
(1)
CDcl
= @@q,l,
(2)
CDt
= WW
d/D) d/D)
blD = @Wep,, d/D)
(4)
b/D = @Wep,l, d/D)
(5)
b/D = @(Ret, d/D)
(6)
Srt
(7)
= tiRep,, d/D)
SrCl = @(Rep,l, d/D) INTRODUCTION
Srt
The hydrodynamic behaviour of freely suspended solid spheres in bounded and non-bounded uniform flow fields both with and without fluid turbulence has been studied experimentally over a wide range of particle Reynolds numbers’ -lo. However, few experimental results are available for the commercially important system in which solid spheres travel within pipes (bounded shear flow field) and most of these relate to spheres travelling with respect to a Poiseuille flow at low particle Reynolds numbers” -14. Studies by Doig15 have shown that solid spheres suspended within a vertical pipe by a fully developed turbulent fluid flow behave quite differently from spheres suspended in the flow fields mentioned above. The study reported here extends this work and covers a particle Reynolds number range of 2 up to 10 000 for pipe Reynolds numbers between 25 and 37 000. Dimensional analysis of the suspension of spheres by fully developed flow in a vertical tube at a con-
* Present address: Australia. t Resent address: Australia.
James Hardie B.H.P. Ltd.,
Ltd., Auburn,
N.S.W.,
140 William Street,
Melbourne,
(3)
= @(Ret, d/D)
(8) (9)
This study explores these quantitative relationships and in addition qualitatively reports the behaviour of spheres suspended in fully developed pipe flow.
EXPERIMENTAL EQUIPMENT Figure 1 shows the arrangement of the experimental equipment. Fuller particulars are given elsewhere 16. and only a brief outline is provided here. Teflon spheres ranging in density from 2.10 to 2.13 g ml-’ and in size from 3.18 to 19 mm were suspended in a test section 50 mm long which formed part of a Perspex vertical tubular riser 5.5 m long with a 69.85 mm bore. The test section was located 0.6 m below the top of the riser. A steady axisymmetrical fluid velocity field was generated in the test section by the use of a constant head tank arrangement, diffuser exit and entry headers, and long radius bends in the fluid supply lines. Fluid flow rate was controlled by a 50 mm ball cock at high flow rates or a 25 mm globe valve at low flow rates. High flow rates were measured with a calibrated orifice plate and low flow rates with a calibrated venturi meter. A calibrated
242
B. P. FOSTER,
Head Tank\]
1
A. R. HAIR,
1. D. DOIG
I
14
I
3
3
4
8
fJ
t-
Fig. 1. Diagram of the experimental
equipment. Fig. 2. Siphtglass device and the method
thermometer
was stationed
for Sucrose solutions
measurement of fluid temperatures. were used as the motive fluid. A 16 mm Bolex tine camera was mounted in the centre of a plane Perspex plate 19 mm thick which formed the top of the exit header during runs 126222. This tine camera was used for measurement of the radial sphere position using the tube wall image reflection technique of Hair and Doig17. During runs I- 125 the sucrose solutions were insufficiently clear and instead of the image reflection method a sightglass device (Fig. 2) was used to observe radial sphere positions. The image reflection method was also found to be inferior for sphere positions remote from the tube wall.
EXPERIMENTAL
of using it.
in the exit header
PROCEDURE
A 61.3% sucrose solution was prepared using Table Grade sugar (99.8% sucrose). The nine spheres under test were placed at the bottom of the vertical riser and all residual air in the system was removed. The flow rate was adjusted to transport the smallest sphere to the test section, after which it was reduced
to hold the sphere suspended in the test section. The pressure drop across the venturi meter, the sphere size and the fluid and room temperatures were recorded. The radial position of the sphere in the test section was estimated using the sightglass device (Fig. 2). Qualitative notes on sphere behaviour during suspension were also recorded. These measurements were repeated for eight larger spheres and at the end of these measurements a sample of the solution was obtained for later analysis. The sucrose solution concentration was then reduced (by the addition of water) and a further set of suspension measurements was carried out. This procedure was used for twenty successive sucrose solution concentrations. Samples of sucrose solutions taken at the completion of each series of suspension measurements were later carefully analysed to determine the sucrose content and used (with the measured fluid temperature) to determine the fluid viscosity and density (for each experimental run) from dataIs. The density of the sucrose’solutions ranged from 1.29 to 1.03 g ml-’ while the viscosity ranged from 60 to 1.1 cP. Occasional sample viscosity measurements were also made to check the tabulated data and ensure that the viscosity did not change with
SUSPENSION
OF SPHERES
243
time. Fuller details of the experimental provided by Foste@.
RESULTS
procedure
are
AND DISCUSSION
Equations (l)-(9) involve two fluid velocities, Vt and I/cl. Vcr was sensitive to errors in the sphere position measurement and yielded less reliable results, especially when introduced into the drag coefficient relationships. Consequently, the macroscale velocity Vr has been emphasized in this paper and relationships involving the local velocity T/cl have been introduced only where they yield additional information. Full details of results based on the local velocity Vcr are provided elsewherer6. Suspended particle motion Observations of sphere behaviour were noted during the experimental runs. These observations were later organized to obtain an overall view of repeating sphere motions in various regimes. Four kinds of repeating motion (or modes) were observed which identified well with Reynolds number regimes. Some overlaps did occur and occasionally some transition behaviour between adjacent modes was noted. The four motion modes have been designated A, B, C and D. In motion mode A the sphere took up a steady position near, but not at, the tube axis and rotated in the direction of the vorticity of the shear flow field. Horizontal, vertical or angular motion was not apparent. In motion mode B the sphere moved radially and axially, but not around the tube axis. A sketch of the motion is given in Fig. 3. Starting from a point near the pipe axis the sphere would rise a short distance and then move radially toward the wall before falling a short distance and moving radially back toward the tube axis. The sphere began to spin at the start of its fall, its angular velocity increasing gradually to a maximum at the bottom of its descent;
at this point spinning abruptly ceased and no spin was observed during its rise. Spin while falling was in the direction of the flow field vorticity. The amplitude of vertical and radial excursions varied, but generally appeared to be in the ranges 0.ti-d and d-3d respectively. Measured data on these amplitudes were not reliable enough to allow any correlation with d/D or Reynolds number. In motion mode C the sphere adopted a metastable position near the conduit wall where it periodically underwent oscillatory radial excursions which decayed to restore the sphere again to its non-oscillatory but metastable position. Suspension in this mode was difficult since the sphere also often travelled erratically in the axial direction. During this axial travel the sphere was observed to spin, the direction being dependent upon the direction of axial movement. Whilst stationary, no spin was observed. At higher Reynolds numbers this periodically damped oscillatory motion changed to a sustained oscillatory motion, motion mode D. In this motion mode, the sphere underwent steady non-spinning radial oscillatory excursions at a constant height. The frequency and amplitude of these oscillations appeared to be reasonably constant. Data on oscillation frequency are presented later in this section. A special motion mode was observed with the largest sphere (d/D = 0.274) which was not observed with any other sphere. In this special case, the sphere took up a steady position near the wall; no vertical, radial or. angular movement or spin was observed. It is thought that this case was a special case of motion mode C in which the tendency to radial excursions was damped by the relatively high inertia of the particle. The relationship between motion mode, Reynolds number (Rept) and sphere-to-conduit diameter ratio is shown in Fig. 4. Note that this relationship is based on the use of the volumetric average fluid velocity Vr in the particle Reynolds number term. When the results were plotted16 using a particle Reynolds number based on the approach velocity Ver of the fluid at the sphere centre, transitions between motion modes were found to occur at the following values of Repcr:
transition transition transition
Fig. 3. Sketch
describing
the sphere
motion
mode
B.
from mode A to mode B, Repcr = lo-20 from mode B to mode C, Repcr = 80-100 from mode C to mode D, Repcr = 200-600
(Note: a range of values of Rep,1 is given for each transition because the transition value of Repcr also
B. P. FOSTER,
244 depends on the sphere-to-conduit diameter ratio (see Fig. 4).) Torobin and Gauvint9 have reviewed literature on the formation of wakes in turbulence-free unbounded uniform flow. They conclude that a stable vortex ring forms behind the downstream hemisphere at particle Reynolds numbers around 10 which grows in size and decreases in stability until it begins to d/D,
I
A. R. HAIR, 1. D. DOIG
oscillate at particle Reynolds numbers of around 130. With increasing particle Reynolds numbers the instability increases, leading to a periodic shedding of vortices at the lower critical Reynolds number. From the conflicting evidence reviewed, Torobin and Gauvin conclude that this shedding probably starts at particle Reynolds numbers around 500. There is an interesting correspondence between the particle Reynolds number values observed at motion mode transitions (A to B, B to C, and C to D) and the values quoted for wake structure changes which suggests that the two phenomena may be related. Drag coefficient and particle Reynolds number results The experimental results for the relationship expressed
Fig. 4. Sphere motion modes grouped by particle Reynolds number and sphere-toconduit diameter ratio. Note: the full line boundaries between the regions are only approximate: the transitions are not sharp and some overlapping occurs. The broken lines denote limits of observations.
in eqn. (1) are plotted in Fig. 5. Smooth curves through these points are presented separately in Fig. 6 and show the effect of increasing the sphereto-tube diameter ratio d/D. Features of these plots are the dependence of the drag coefficient CDr on the diameter ratio d/D, the relatively rapid decrease in CDr in the regime 50 < Rept < 200, the inflections in the Cnt-Repr curves in the regime 200 < Rep, < 800 and the decrease in CDr in the regime 1000 < Rep, < 8000 (shown by broken lines denoting gaps in the data).
Fig. 5. Experimental
Reynolds
I
10"
,,.... lo’
I.,,.,’
‘..,.,I’
lo2
IO'
results plotted
.,,I,,
I 10‘Rep,
as drag coefficient
vs. particle
number
SUSPENSION
245
OF SPHERES
10° Fig. 6. Line plots derived
10'
lo*
lo3
lo4
from Fig. 5. (The lines for d/D ratios 0.114 and 0.159 are omitted
The plot also shows that the value of CDt is generally higher than that given by the standard drag curve with the minor exception of the diameter ratios 0.046 and 0.069 in the particle Reynolds number regimes 130180 and 400-l 500 respectively. The wide divergence from the standard drag curve at values of Rep* below SO results from the use of the velocity Vt in the drag coefficient and particle Reynolds number terms. If the local velocity Vet is used instead, the drag coefficient approaches the standard drag curve as d/D approaches zero 16. The relative decrease in Cot compared with standard drag curve values in the regime 50 < Rep, < 200 coincides with a change in the observed radial sphere position (Fig. 7). Corresponding results plotted on a CDcr-Repcr basis did not exhibit a similar relative decrease16 and this decrease in CDt is therefore attributed to the change in Vt relative to T/,1 introduced by the changes in sphere radial positions occurring in this Reynolds number interval. The decrease in CD~ in the interval 1000 < Rep, < 8000 was also observed in corresponding CD&Rep,, results and therefore cannot be attributed to changes in corresponding I/cl and Vt values. No change in sphere motion was observed in this regime. The regime, however, was incompletely explored and is shown by broken lines
Rep,
for clarity.)
in Fig. 6. The inflections in the course of CDt(d/D) values in the particle Reynolds number interval 180-850 can be associated with the change from sphere motion mode C (periodically unsustained oscillation) to sustained oscillation mode D. Cot values before the inflection in Fig. 6 have Reynolds numbers in the mode C interval in Fig. 4 and are less than Cot values immediately following the inflection which have Reynolds numbers in the mode D interval. (Note that the motion mode boundaries are not sharp: the full lines drawn between regimes in Fig. 4 denote approximately the middle of observed transitions from mode C to mode D which include overlaps. The C-D reg.ime boundary values of Fig. 4 correspond only approximately with the inflection midpoints of Fig. 6.) When the cDt(d/D) values are plotted against the pipe Reynolds number Ret, all of these inflections occur in the range 2300-3000 which is the recognized regime for transition from laminar to turbulent pipe flow. This transition introduces several changes which require consideration. The transition from laminar to turbulent pipe flow changes the local velocity at each position in the flow field and the local velocity gradient, and introduces sustained turbulence. No measurements of turbulence have been made
246
B. P. FOSTER,
A. R. HAIR,
I. D. DOIG
.
loL
10’
Fig. 7. Experimental
showing
10‘
10’
10) results
the observed
mean sphere position
but Pitot tube traverses were made 0.2 m above and 1.9 m below the suspension region of the spheres in the pipe flow Reynolds number regime between 5000 and 32 000. The results showed fully developed axisymmetrical turbulent pipe flow at both positions, an absence of swirl and a close correspondence with the velocity profiles* predicted by Deissler*O. The pressure gradients measured were those predicted for a smooth pipe *l . No spheres were present during these tests and the observations concerning laminar and turbulent pipe flow assume the local turbulence intensity to be a dependent and persistent variable whose presence, or absence, and properties are little affected by the sphere at positions upstream of the sphere. Observations of the sphere position before and after the inflections show that it approximately keeps to its position. Therefore, the laminar-turbulent velocity profile change is likely to change the fluid velocities approaching the sphere. If these local approach velocities in a turbulent velocity profile were higher than those of a Poiseuillian profile (sphere close to the wall), the mean volumetric velocity Vt required to suspend the sphere would decrease and this would be reflected by an increase in CDt. For this argument to account for the observed inflections and increases in Cm values, inflections in the corresponding Rep,,d/&cDc[ relationships should be absent or small. Data relating to the observed inflections are presented in Table 1 and examination shows that the CDct values + The region at the core, however, law=.
followed
a velocity
defect
RePt plotted
against
particle
Reynolds
number.
also increase sharply where the CDt values increase. It follows that something other than changes in local fluid velocities accompanying the laminar-to-turbulent tube flow transition are causing the increases observed in CDr VaheS. Torobin and Gauvin9 have shown that, in unbounded flow systems where turbulence is generated by grids, turbulence generally increases the drag coefficient above values given by the standard drag curve. Clift and Gauvin10 have reviewed more recent work of this kind and report (from Zarin23) that in the particle Reynolds number regime 200 < Rep < 800 the drag coefficient increases sharply with increasing turbulence intensity in the free stream turbulence intensity range 0.004-0.033. It is therefore possible that the increases in CDr during the laminar-toturbulent flow transition interval are due to the introduction of sustained turbulence in the suspending fluid. Figures 5 and 6 also provide comparisons of the present results with those of Eichorn and Smalll*, Fidleris and Whitmoore and DoiglS. Radial position adopted by a suspended sphere The mean radial position adopted by the spheres in
the conduit is presented in Fig. 7. Note that this radial position is expressed as the term b*, which is the ratio of the time averaged distance of the sphere centre from the tube wall divided by the radial distance over which the sphere may move. It allows b* values to range between zero and unity.
SUSPENSION TABLE
OF SPHERES
247
1
Data relating
the points
of inflection
in Fig. 6
Diameter ratio d/D
Tube Reynolds no. Ret
Sphere Reynolds nos. Rept Repel
Drag coefficients CDcl CM
Sphere position ratio b/D
0.214
2820 2820 3050 3430
771 111 834 940
724 690 750 801
0.98 0.92 1.22 1.25
1.11 1.13 1.51 1.42
0.272 0.329 0.323 0.272
0.228
2530 2650 2830 3420
577 603 644 179
516 566 564 594
0.92 0.88 1.16 1.03
1.15 1 .oo 1.50 1.30
0.329 0.272 0.348 0.332
0.183
2320 2560 2580 3010
424 488 471 549
399 395 394 491
0.87 0.75 1.15 1.06
0.98 1.05 1.64 1.33
0.364 0.378 0.386 0.329
0.136
2080 2250 2310 2740
285 308 316 315
99 132 232 335
0.81 0.71 1.03 0.95
6.70 3.89 1.90 1.19
0.443 0.455 0.398 0.329
0.091
2510 3040 3200 3720
233 276 291 331
187 217 228 263
0.71 0.70 0.82 0.77
1.10 1.13 1.33 1.27
0.409 0.421 0.423 0.425
180 < Rept < 850; 2000 Q Ret < 3200.
The most striking aspect of this plot is the sharp change in sphere position in the range 40 < Rep, < 150. Also, for Rept < 150 the distance from the wall increases with the sphere-to-tube diameter ratio d/D, which is consistent with the observation that the larger spheres had the greater oscillational amplitudes. At low particle Reynolds numbers the influence of the diameter ratio d/D on mean position is difficult to discern because of the scatter associated with assessments of the mean position b* when using the sighting device, and a scarcity of data. However, it is noticeable that as Rep, increases from 10 to 40, the value of b* decreases. This decrease can be linked with a change in sphere motion from mode A to mode B. A plot of mean distance b against Ret, presented elsewherem, shows that the laminar-to-turbulent pipe flow transition is not associated with the very pronounced change in radial position from near the tube axis to near the tube wall. (It occurs in the range 320 < Ret < 1000 or 62 < Rep,, < 180.) The results reported in Fig. 7 show that the position of a sphere suspended in fully developed pipe flow is primarily dependent upon the particle Reynolds number, with a discernible secondary d/D effect occurring when the sphere is near the tube wall.
d/D
i
r
7”
/
,046
Fig. 8. Dependence of the Strouhat mode D mean oscillation frequency) diameter ratio.
number (for motion on the sphere-to-tube
248 Radial oscillational frequency
At the higher values of particle Reynolds numbers investigated in this study, the spheres adopted a reasonably steady radial oscillational motion (motion mode D). Dimensional analysis indicates that frequency of oscillation (expressed as a Strouhal number) should be a function of a Reynolds number (Rep,, Rep,, or Ret) and the sphere-to-conduit diameter ratio (eqns. (7), (8) and (9)). When results were plotted, no influence of Reynolds number (Rept , Rep,, or Ret) upon Strouhal number could be discerned (Rep, values ranged from 279 to 10 100). A plot of Strouhal number against sphere-to-conduit diameter ratio only is presented in Fig. 8; the horizontal straight lines indicate Strouhal numbers over the range of Rep, values investigated and the dots are the mean Srt values. The data reported in Fig. 8 were obtained using an image reflection technique17 for measurements over time intervals of 3-5 s.
B. P. FOSTER,
A. R. HAIR,
I. D. DOIG
relative velocity between the sphere and the local fluid velocity approaching the sphere centre at its average position. Relationships employing the former relative velocity should be of use in relating drag coefficient data to macroscopic measurements whilst the relationships based upon the latter will be of use in comparisons with unbounded and bounded quiescent flow systems. For particle Reynolds numbers between 200 and 10 000, spheres suspended in fully developed pipe flow oscillate fairly steadily in a radial direction. The frequency of these oscillations can be expressed as a Strouhal number which appears to be independent of particle Reynolds number but depends upon the sphere-to-pipe diameter ratio. It should be noted that this study concerned suspended spheres at an almost constant sphere-tofluid density ratio. Variation in the sphere-to-fluid density ratio may modify the relationships obtained.
CONCLUSIONS
NOMENCLATURE
Spheres suspended in fully developed pipe flow adopt one of four distinct modes of motion for particle Reynolds numbers between 2 and 10 000. The particular motion mode adopted is primarily dependent upon the particle Reynolds number with a secondary dependency on the sphere-to-pipe diameter ratio. A most interesting observation is that transitions between motion modes coincide approximately with particle Reynolds numbers reported for changes in the wake structure behind a sphere in unbounded systems. Spheres suspended in a fully developed pipe flow adopt a position near the tube axis at particle Reynolds numbers less than 80 but adopt a position near the tube wall at particle Reynolds numbers greater than 200. The mean position adopted by suspended spheres when the spheres are in the vicinity of the conduit wall varies with the sphere-to-conduit diameter ratio. Drag coefficients for spheres suspended in fully developed pipe flow with particle Reynolds numbers between 2 and 10 000 are greater than those reported for spheres travelling through unbounded quiescent fluids and are dependent upon both the particle Reynolds number and the sphere-to-conduit diameter ratio. The particular form of this relationship is dependent upon the velocity used to characterize the relative velocity between the sphere and the fluid. Two characteristic relative velocities were considered in this study-the relative velocity between the sphere and the mean volumetric fluid velocity, and the
b b* CDcl CDi n
d F f
time averaged radial distance between the sphere centre and the conduit axis [L] dimensionless radial sphere position (l-2b)/ (D-4 drag coefficient, 8F/~d2pVc~2 drag coefficient, 8F/nd2p Vt 2 conduit diameter [L] sphere diameter [L] drag force acting on a sphere, .rrd3(ps-p) g/6 [MLT_2] time averaged sphere oscillation frequency
U’-‘I g Repel
Rep, Ret SrCI Srt VCI
vt P P PS
acceleration due to gravity [LTF2] particle Reynolds number, pdVcl/p particle Reynolds number, pd Vt/p Reynolds number for the tube flow, PDT/t/p Strouhal number, fd/Vcl Strouhal number,fd/Vt time averaged “centreline” velocity, being the relative velocity between the sphere velocity and the approach velocity of the fluid at the time averaged position of the sphere centre [LT-‘] volumetric fluid flow rate per unit tube cross sectional area [LT-‘1 fluid viscosity [ML-‘T-l] fluid density [MLe3] density of the solid material of the spheres [ML-3]
SUSPENSION
OF SPHERES
REFERENCES 1 R. A. Castleman, The resistance to steady motion of small spheres in fluids, Nat. Advis. Comm. Aeronaut. Tech. Notes 231,1926. 2 H. Rouse, Nomogram for settling velocity of spheres, Annu. Rept. Comm. Sed., Nat. Res. Council, 1937. 3 C. A. Lapple and C. B. Shepherd, Calculation of particle trajectories, Ind. Eng. Chem., 32 (1940) 605-607. 4 T. Maxworthy, Accurate measurements of sphere drag at low Reynolds numbers, J. Fluid Mech., 23 (1965) 369-312. 5 E. B. Christiansen and D. H. Barker, The effects of shape and density on the free settling of particles at high Reynolds numbers, A.I.Ch.E.J., II (1965) 145-151. 6 J. S. McKnown, H. M. Lee, M. B. McPherson and S. M. Engez, Influence of boundary proximity on the drag of spheres. In Proc. 7th Znt. Congr. for Appl. Mech., Vol. 2, No. 1, 1948, pp. 16-29. deter7 V. Fidleris and R. L. Whitmore, Experimental mination of the wall effect for spheres falling axially in cylindrical vessels, Br. J. Appl. Phys., 12 (1961) 490-494. aspects of 8 L. B. Torobin and W. H. Gauvin, Fundamental solids-gas flow: Part I-Introductory concepts and idealised sphere motion in viscous regime, Can. J. Chem. Eng., 37 (1959) 129-141. 9 L. B. Torobin and W. H. Gauvin, Fundamental aspects of solids-gas flow: Part V-The effects of fluid turbulence on the particle drag coefficient, Can. J. Chem Eng., 38 (1960) 189-200. particles 10 R. Clift and W. H. Gauvin, Motion of entrained in gas streams, Can. J. Chem. Eng., 49 (1971) 439-448. 11 A. M. Fayon and J. Happel, Effect of a cylindrical boundary on a fixed rigid sphere in a moving viscous fluid, A.I.Ch.E.J., 6 (1960) 55-58.
249 12 R. Eichorn and S. Small, Experiments on the lift and drag of spheres suspended in Poiseuille flow, J. Fluid Mech., 20 (1964) 513-517. 13 R. C. Jeffrey and J. R. A. Pearson, Particle motion in laminar vertical tube flow, J. Fluid Mech., 22 (1965) 121-735. on the suspension 14 G. F. Round and J. Kruyer, Experiments of spheres in inclined tubes: I-Suspension by water in turbulent flow, Chem. Eng. Sci., 22 (1967) 1133-1145. 15 I. D. Doig, Suspension of spheres in a vertical tube at high Reynolds numbers, Chem. Eng. Sci., 23 (1968) 794-796. 16 B. P. Foster, Suspension of spheres by a bounded fluid, M.Sc. Thesis, Univ. of New South Wales, Australia, 1973. technique for 17 A. R. Hair and I. D. Doig, A photographic the determination of the position and velocity of particles moving in tubes, Chem. Eng. J., 9 (1975) 175-178. 18 Viscosity of sucrose solutions, in Laboratory Directions, Colonial Sugar Refining Co., Sydney, Australia, 1959, Table 63. aspects of 19 L. B. Torobin and W. H. Gauvin, Fundamental solids-gas flow: Part II-The sphere wake in steady laminar fluids, Can. J. Chem. Eng., 37 (1959) 167-176. 20 R. G. Deissler, Analysis of turbulent heat transfer, mass transfer and friction in smooth tubes at high Prandtl and Schmidt numbers, Nat. Advis. Comm. Aeronaut. Rep. 120,1955. 21 L. F. Moody, Friction factors for pipe flow, Trans. ASME, 66 (1944) 671-678. 22 J. 0. Hinze, Turbulence, McGraw-Hill, New York, 1959. 23 N. A. Zarin, Measurement of non-continuum and turbulence effects on sub-sonic sphere drag, NASA Contract. Rep. CR 1585,197O.