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Forces on objects immersed in fluidized beds
Powder
Technology.-.
19 (19781255 _
0 Elsevier Sequoia S-A.,
- 264
Lausanne -
255
Printed in the Netherlands
Forces on Objects Immersed in Fluidized Beds
T. H. NGUYEN Deportment
(Received
and J. R. GRACE
of Chemical
June 17.1977;
Engineering,
McGill
Untversity.
in revised form September
SUMMARY
The depth of immersion of objects of various sizes and shapes floating at the surface of a fluidized bed was used to evaluate the effective density of the particulate phase. The net buoyancy force under fluidized conditions can be estimated closely by taking the particulate phase density as equal to the bed density at minimum fluidization. Rowever, splashing of particles onto the top of the objects may cause them to sink periodically, even when their density is less than that of the particulate phase_ Local pressure fluctuations caused by the rise of bubbles past horizontal cylindrical tubes were measured in a two-dimensional fluidized bed using a pressure transducer. Integration of the pressure over the surface of the tubes allowed the net vertical force on the tube to be determined as a function of time. The net force uersus time curve is shown to have an approximately sine-wave shape with the maximum upwards thrust corresponding to the arrival of the front of the bubble at the bottom of the tube, and the maximum downwards thrust corresponding approximately to the instant when the rear of the bubble reaches the bottom of the tube.
I?.UTRODUCI’ION In practice,
fluidized
beds
commonly
con-
tubes, diplegs, probes, nozzles, etc. which are immersed in the bed for cooling, heating, addition of solids or fluids, or me asurement purposes_ Objects may be dipped into the bed for coating- In other cases, large lighter objects may float on a fluid&cl bed composed of dense particles, for exampIe in the addition of large pieces of tain
heat
transfer
3480
University
Street.
Montreal
(Canada
H3d
2-4 7)
23. 1977)
coal for combustion to a bed composed mainly of denser ash particles. In all of these cases it is important to know the forces on the immersed bodies. For fixed objects one would like to be able to predict the transient forces which commonly lead to tube vibrations. Some vibrations are probably desirable in promoting favourable heat transfer rates, but escessive vibrations may lead to failure of immersed members. For freely moving objects it is important to know the conditions under which an object will float at the surface- The present paper considers these two questions. Previous work on objects immersed in fluidized beds has been reviewed by Harrison and Grace [ 1, 21.
BUOYANCY Experimental
FORCE
ON
apparatus
FLOATING and
OBJECT
procedrcr-e
it is a well-known prope& of a gas-fluidized bed that objects can be floated on its surface, there appears to have been no systematic work on the conditions under which floating occurs. In the present study, objects of three shapes - spheres, cylinders and recttangular parallelepipeds were floated on the surface of a fluidized bed. The column was 45.7 cm in diameter by 99 cm tall and equipped with a perforated plate distributor. Sand particles with a weight mean diameter of 291 pm and U,, = 7.0 cm/s were used in the experiments. The voidage at minimum fluidization, Emf, was 0.40, and the corresponding bulk density of the bed. D-t = was 1.48 g/cm3. P,U -E,& All of the objects were hollow and could have copper powder added to increase the weight. There were two cylinders, which could be used singly or attached end-to-end to make a longer cylinder_ Similarly, there Although
256 TABLE
1
Object shapes and sizes shape
Dime.?sions
Sphere
3.8 o-d.
Cylinder
3-S o-d., 5.1 long 3.8 0-d.. ‘3.6 long 36 o-d.. 12-7 long 3.8 x 3.8 x 2.5 3.8 x 3.8 x 3.8
Rectangular paraileIepiped
3.8 3.8 3.6 3.8
x x x x
3.8 3.8 3.8 3.8
(cm)
x 6.3 x 7-6 x 10-2 x 11-4
3.8 x 3.8 x 14.0 3-S x 3-8 x 17.8
Orientation
axis vertical axis vertical axis horizontal small side vertical one axis of symmetry vert.
long side vert. or horizlong side vertical long long long long
were fo-ur separate parallelepipeds of the same cross-section which could be immersed separately or attached together in pairs_ The different configurations and orientations used are shown in Table I_ Except for the sphere which was made of plastic, aU the other objects were constructed of balsam. Graduated sloes mere affiied to the outside of the objects to allow the immersion to be measured. When copper powder was added, the top of the copper layer could be fixed by inserting a tightly fitting disc into the cyhnders or parallelepipeds_ The experiments were carried out at superficial velocities of 7-5, S-7 and 10-3 cmis, corresponding to 1_08,1.24 and l-47 X U,, respectively_ Even though these velocities are low, there was sufficient bubbling at the higher two velocities that particles tended to be thrown onto the top of the objects, as observed in somewhat similar experiments by Roche 133 _ When particles lodged on the top of the object, these were swept away before immersed depths were read. This made data taking tedious at the higher velocities, especially for cylinders and parallelepipeds lying with horizontal axes_ Therefore, most of the data correspond to the lowest value of U and to cyhnders and parahelepipeds with their major axes verticalExperimental
results
The volume of dense phase displaced is plotted against the mass of the object in Figs. 1 - 3 for the sphere, cylinders and parallelepipeds respectively_ A straight line,
side side side side
vert_ vert. vert. vert.
or or or or
horiz. horiz. horiz. horiz.
forced through the origin, has been fitted to each set of results using least-squares regression_ It is shown in the Appendix that the drag of the fluidizing gas on the object is negligible provided that the scale of the object is much larger than that of the fluid&d particles. Hence, from Archimedes’ principle, the reciprocal of the slope of each of these lines gives the apparent density of the fluidized bed. The apparent density determined in this manner for the sphere, for cylinders orientated with vertical axis and for parallelepipeds with square cross-section in a horizon&I plane are given in Table 2_ More extensive data are tabulated elsewhere 143 _ Discussion of experimental results It is known that a cushion of gas tends to
form underneath an immersed object in a fluid&d bed, while, for a totahy submerged object, a region of “dead” particles tends to sit on the top surface [5]. Consequently one might expect that the net buoyancy force on an object in a fluid&d bed might differ from what one would predict using Archimedes’ principle and assuming the dense phase voidage to be equal to prnf, the bulk density of the bed at minimum fIuidization_ The effective densities given in Table 2 are ah within 2 5% of pmt = 1.48 g/cm3 for the particles used_ In order to determine whether Peff differs significantIy from pmf, the tdistribution was used to test the significance of differences at the 1% level. The results, given in Table 2, show that the difference is insignificant (i.e. the experimental value of
257
Lin.3 --__--_ _--
OWECT
Fig_ 1. Volume
of fluidized
Symbol
Ultld 1.06 1.24 147
0 &
cl
mnsS.(g)
bed displaced by floating sphere.
i
Fig. 2. Volume
of fluidized
bed displaced by floating cylinders with axis of symmetry
a probability of 0.02 of being exceeded in random sampling) except for the rectangular rate_
When the nine separate means were treated as a single population and a grand mean was formed
(FCff = 1_478),the
compared with tz$g = 2.90). Hence there is strong evidence that prnf canbeusedtocalculate the buoyancy force on floating objects. It is possible, however, that totally submerged objects might be subject to a different net buoyancy force due to the dead region above
~,fwasnotsignificant(It[=0.05
thefstatisticwaslessthanthevalueoftwith
parallelepipedatthelowestgasvdoci~
in the vertical direction.
difference from
200 -
Rectangular ParaIIelePiPedr III
Line ---
0
zca
Fig:. 3_ Volume of fluidized cross-section horizontal_
TABLE
bed
mhnc 1.08
p
1.24
-
3
1.47
200 05JEcT
.
Symbol 0
_--_-
300
MA.ss.Cgl
displaced
by fioating
rectangular
parallelepipeds
of square
cross-section
with
2
Apparent
fluidized
Object
bed
density
calculated
Li
No.
(cm/s)
points
of
uskg
data
Archimedes’
principle
Peff (g/cm?
Erpt.
1.13 l-41
2.76 4.54 3.75
t1-5
ItI
1.55
1.14 2.78 2.40
Vertical cylinder
i-5 s-7 10.3
10 4 4
1.49 1.50 l-49
O-88 0.82 0.43
2.82 4 -4 4.54
Vertical paratlelepiped
7.5 6-7 10.3
36 21 12
I.43 1.52 1.48
9.86
2.44 2.53 2.72
Sphere
5.5
11
8.7 10.3
4 5
the object, but this was not investigated in the present study_ The particles used in the present study correspond to group B in Geldart’s classification [S] , and the finding that peff = pmE should also be used with caution for particles of widely different characteristics. The experiments carried out in this work suggest that there are four situations with respect to floating of objects added to fluidized beds: (i) For very light objects (p, 4 pmf), or for p0 < prnf and shapes such that few particles can collect on the top of the object (e.g. for a tetrahetlron with angle greater than
1.73 0.17
the angle of repose of the particles), the object will at all times stay on the bed surface_ (ii) For objects with p0 < prnf, but such that particles can pile up on the upper surface, the objects should spend periods at the bed surface while splashed particles collect on the top, interspersed with periods of complete immersion. Periodic forces due to bubbles reaching the surface (see next section) will cause disturbances at the bed surface tending to dislodge particles from the top of floating objects. (iii) For objects with p0 = Pmf. the object may circulate entirely within the bed, resting
259
neither at the top surface nor at the distributor plate (iv) For P., Z+ Pmf, the object is expected to go straight to the bottom of the bed and rest on the grid. It seems likely that the transitions between these four classes of behaviour are gradual rather than sharp. Stability
with respect
to tipping
The stability of Boating objects with respect to tipping in real liquids is governed by the relative position of the centre of gravity and the metacentre- Generally speaking, floating objects tend to be unstable when displaced slightly from an equilibrium position if their centre of gravity lies well above the liquid surface and the characteristic dimension of the object in the vertical direction is much greater than that in the horizontal direction. Qualitative studies with the weighted cylinders and parallelepipeds demonstrated that the same general behaviour occurs in a fluidized bed. Some tests were done with bromobenzene, a liquid whose density (1.50 g/cm3) is very close to Pmf for the fiuidized particles used. If an object was unstable or stable in a particular orientation in the flnidized bed, it exhibited the same stability behaviour in the liquid. Hence the analogy between fluidized beds and real liquids not only applies to buoyancy forces, but also appears to extend to the stability of objects floating at the surface under gentle fluidizing conditionsTRANSIENT
FORCES
ON
FLmD
HORIZONT_4L
TUBES
Experimental
apparatus and procedure
The second set of experiments was carried out in order to measure pressure fluctuations on the surface of submerged tubes in order to understand the forces leading to tube vibrations. Reuter [7, 81 and Littman and Homolka [9, lo] have measured local pressure forces with respect to passage of bubbles, but in these cases there was no interaction between the bubbles and submerged objects. In the present case, pressure measurements are nsed to show how the net force on an object varies with timeThe experiments in the present case were carried out in a two-dimensional cohnnn so
that the pressure signal cculd be interpreted with respect to the bubbles giving rise to the pressure variations. The column, 245 cm high X 56 cm wide X 1 cm thick, has been described elsewhere [ 11, 123. The particles were glass microbeads with a weight mean diameter of 177 W, U,, = 6.5 cm/s and E,~ = 0.41. Single bubbles were injected into the column from a pressurized vessel via a solenoid valve through a 0.32 cm port, 12.7 cm above the porous distributor plate. A timer was used to give reproducible settings of the time between bubble injections and the time of solenoid valve opening. A Disa type 51D02-Pu2a low pressure transducer of capacitive type was used to measure local pressures during the passage of single bubbles. The transducer was kept at constant temperature to within +O.Ol “C using a thermostatic bath. Because the pressure changes caused by the passage of a bubble were relatively small, differential pressure measurements were used. This was achieved by blocking one of the two breather holes and connecting the other to a pressured chamber whose pressure was set to match the dense phase pressure at the same level. The output from the transducer assembly was recorded using a Honeywell Visicorder capable of recording signals with less than 2% distortion up to 13 kHz. Three horizontal tubes were used for the esperiments, all hollow with a small orifice in the wall connected to a tube leading to the pressure transducer. The hole was covered by a screen, mounted flush with the outer surface of the tube, to prevent particles from entering. Each of the tubes could be turned manuaily to allow pressures to be measured at any angle to the vertical. In practice, traces were obtained from 0” to 180” inclusive in 30” intervals. The smallest tube, 0.63 cm diam.. was of steel, while the two larger tubes (2.5 and 5.1 cm o-d.) were constructed from plexiglass. Each tube was in turn mounted 101.6 cm above the distributor at right-angles to the front face of the column. Because the span was only 1 cm and the tubes were rigidly supported, tube vibrations were negligible in the present case. In order to interpret the pressure traces, tine photographs were taken at 32 or 64 frames/s using intense back-lighting_ Syn-
260
chronization of the films with the pressure traces was achieved by having a photocell connected 7.6 cm below the centre of the horizontal tube_ When a bubble passed in front of the photocell, a solid-state light, visible in the photographic films, was interrupted and simuhaneously a mark was recorded on the Visicorder output trace. The surface area of single bubbles was determined by weighing paper cut-outs of the bubble contours, traced from projected photographic images_ Bubble velocities were obtained by plotting the nose and rear positions uersus time, and are presented elsewhere [4] _ The background superficial velocities used in the experiments were 30 - 50% in excess of umr. this range being optimal with respect to maintaining stable bubbles. -4 consequence of this background flow rate is that injected bubbles grew somewhat with height at the expense of the dense phase and due to coalescence with smaller bubbles Only one size of bubble was studied in the present work_ This size (580 cm* at 80 cm above the distributor) was chosen to limit both break-up of bubbles as they encountered the tubes and avoidance of the tubes by the bubbles. The bubbles were therefore large enough that side-wall effects were appreciable_ Where bubble splitting did occur or where bubbles veered from a vertical path, the experimental traces were not used. For the largest tube, 51 cm o-d_. it proved to be impossible to produce bubbles which would pass the tube without splitting, so that results are given here only for the two smaller diameter tubes.
Experimental results All pressure traces had qualitatively the same shape. -4 mavimum was observed when the bubble was injected into the bed, as noted by Littman and Homolka [9, lo] _ A second maximum occurred when the front of the bubble arrived at the measuring point and a minimum when the back of the bubble passed this point- Typical traces have been shown by Nguyen [4]. Pressure-time curves, each averaged from five individual recorder traces, are shown in Fig. 4 for the 0.63 cm tube and four angular positions (0”, 60”, 120” and 180” to the vertical). For these curves, r = 0 has been chosen to coincide with a.rrivaI of the bubble nose at the measuring point. Similar curves are shown for the 2.5cm-diam. tube in Fig_ 5_ The frequency of bubble splitting was greater for this larger diameter tube_ Discussion of results The experimental pressure traces for 0” and the 0.63 cm diam. tube were compared with the predicted profties from various theoretical models reviewed by Stewart [ 13]_ Since this tube is much smaher than the approaching bubbles, a reasonable test of the theories is likely for this case_ In front of approaching bubbles, the experimental trace (averaged from our separate traces) lies between values predicted by Colhns [ 141 and Murray [15], more than doubIe the value predicted by Jackson Cl63 and about half that predicted by Davidson 1171. The comparison appears in Fig. 6. Some experimental
Fig_ 4. Pressure-time traces for 0.63 cm ciiaw. tube at four different arrival of the front of the bubble at the meamring point.
ang~kr
positions_Time
0 corresponds
to
261
-7
__---_---
8 60° lZOO 180 0
Fig. 5. Pressure-time traces for 2.5 cm diam. tube at four different a&al of the front of the bubble at the measuring point.
These Experments
(0.63cm
tuoe.8=0
angular positions. Time 0 corresponds
to
1
o Liltman Homdka Experiments
Fig_ 6. Comparison of pressure-time trace for 0.63 cm diam. tube and 8 = 0” with theoretical and with experimental values measured by Littman and Homolka [lo ] _
results due to Littman [IO] which show the same trend, but higher values, are also shown. As with the Littman results, pressure recovery behind the bubble is slower than the build-up of pressure as the bubble approaches. The thecretical models represent our pressure profiles in the wake region better than they do Littman’s results. The instantaneous net upwards pressure force per unit length on each tube at any instant in time is given by F(T) = DT f 0
p(e ,r) cos 0 de
(1)
pressure profiles
w’\pre ~(8 ,T) is the instantaneous pressure at angle 0 measured from the bottom of the tube and DT is the tube diameter. Numerical integration was carried out to give the F(r), and the resulting curves are shown in Fig. 7, with T = 0 corresponding to the instant when the bubble nose reaches the bottom of the tube. It is seen that the tube encounters an up-thrust as the bubble approaches the tube and then a downward force as the bubble rises away from the tube. The curves are roughly sinusoidal in shape, but t&e period of downthrust is considerably longer in duration than the period of up-thrust. The 2.54 cm tube
!6%
Fig_ i. Net nressure force on the 0.63 and 2.5 cm diam. tubes obtained the surface-king eqn_ (1).
encounters considerably larger forces than the 0.63 cm tube, the difference being even greater than the fourfold factor which arises from the appearance of D, in eqn (1). It is clear that this kind of net pressure force can cause vibrations of tubes in large fluidized beds- In freely bubbling beds where many bubbles are present at one time, the amphtude of vibration is likely to depend on such factors as: properties of the bed itself (e-g_ particle properties, bed dimensions, number of tubes present) and operating variables (gas flow rate, bed depth) which determine the bubble size and frequency; orientation, shape, size and length of tubes; the material of construction, wall thickness and means of support of the tubes as well as the mass and viscosity of any fluid contained therein. Some damping due to the viscous nature of a ffuidized bed appears likely. In practice, there are both stochastic and deterministic eIements to the frequency of bubble passage [18], so that bursts of bubbles with different frequencies will occur in freely fluidized beds. The frequency of vibration of the tube will remain similar to the natural frequency of the tube, while the amplitude will vary erratically in response to stimuIi at other
by integrating the pressure profiles over
frequencies [19]_ The worst case will occur if a burst of bubbles arrives with the resonant frequency of the tubes- Because of the stochastic elements in bubble populations, this situation will arise from time to time and so the design must be based on this case_ CONCLUSION
(1) Objects float on the surface of fluidized beds as on the surface of real liquids with the netbuoyancy force determinedasifthedense phase of the fluidized bed had the same density as at minimum fIuidization_ Particles ejected by bubbles bursting at the bed surface may build up on top of floating objects causing periodic complete immersions. (2j Bubbles passing immersed objects cause transient forces- The net pressure force is upwards as the bubble approaches, and downwards with a similar magnitude, but longer duration, as the bubble rises away from the immersed object. The upward and downward thrust can cause severe vibrations of immersed tubes. ACKNOWLEDGEMENTS
The authors are grateful to the National Research Council of Canada for financial
assistance. T. H. Nguyen also wishes to acknowledge a scholarship from the Quebec Ministry of Education.
LIST
C DP DT dP F
F BO F DO
&
x
kf Emf 0
Peff Pi Pmf PO PP 7
OF SYMBOLS
bubble radius drag coefficient defined by eqn_ (A-2) drag coefficient for a single particle in a suspension tube outside diameter particle diameter net force per unit length due as obtained from pressure trace buoyancy force on object drag force on object acceleration due to gravity characteristic height of object = object volume/S, mass of object pressure cross-sectional area of object experimental value of t statistic value of t statistic with a probability of 0.02 of being exceeded in random sampling distance from top of bubble to bottom of tube (see Fig. 7) superficial gas velocity superficial gas velocity at minimum fluidization bed voidage at minimum fluidization angle measured away from lowest position effective density of the dense phase density of fluidizing gas density of gas-fluidized bed at minimum Ruidization = pp(l - E,~) density of object particle density time
APPENDIX
Forces
on floating
objects
For a body fioating at equilibrium on the surface of a gently fluidized bed, the weight of the object must be counterbalanced by buoyancy forces (due to the hydrostatic pressure distribution) and to drag forces contributed by the fluidizing fluid, Le.
Meg = FB~ + FDO
(A-1)
that other forces, suchaselectrical forces, are negligible. We are interested in the relative magnitude of the drag term to the other terms in eqn. (A-l)_ The drag on the object can be expressed as Weassume
F Do
.Pfelf
=
cD,
-53
2
(A-2)
where CD, is a drag coefficient and S, is the cross-sectional area_ The weight of the object is given by fiIl,g = P~H,S&
(A-3)
where HO = object volume/S, is a characteristic height of the object But in a quiescent gas-fluidized bed (pP 5 p,), the weight of each particle must be just balanced by the drag force on that individual particle, Le. (A-4) Combining eqns. (A-2) - (A-4), we ca write the ratio of the drag term to the weight term in eqn. (A-1) as F DO
2cDoPpdp
fifo&Z
~CD,~O%
-=
(A-5)
But pp and PO are expected to be of the same order of magnitude with pp > PO. Drag coefficients for particles in an assembly depend strongly on the local voidage as well as on the particle shape and Reynolds number. The Reynolds number for the object will be many times larger than that for a single particle because of the much larger characteristic dimension; the local voidage may also be larger [5 J _ Hence one would expect Cn, < CDP, but ali that is necessary to the argument here is that CD, be not much greater than Cnpr which is certainly true. Hence, since dp -=Z Ho for any object in question, it is clear that the ratio in eqn. (A-5) must be much less than unity. Equation (-4-l) may therefore be rewritten as fW& = FBO
(A-6)
and sixnpler way of establishing the same result is to imagine an entirely new fhridized bed comprised of objects identical to the one floating on the surface of the much smaller particles. The superficial gas velocity required for the weight of the An
alternative
264 large
objectstobebalancedbydragwouldbe (U,,),,theminim~m fiuidizationvelocity of thenewbed_SinceU,,oftheactualbed~ (U,,),,dragforcesonthe objectinabed of small particles at U at or near U,, must be much less than the weight of the object_
REFERENCES
J_R_GraceandD_ Harrison,Designof fiuidized beds with internal baffies, Chem. Process Eng., 51(1970) 127_ D. Harrison and J_ R_ Grace, Fluidized beds with internal baffles, in J_ F. Davidson and D_ Harrison (Eds.), Fluidiaation, Academic Press, London, 1971_ G. Roche, Etude du comportement d’un milieu fluidi& autour d’un corps immerg& M.Sc_A_ Diss, Ecole Polytecbnique, 1976_ T_ H_ Nguyen, Forces on objects immersed in fhxidized beds, M.Eng. Thesis, McGill Univ_, 1976. D_ H_ Glass and D_ Harrison. Flow patterns near a solid obstacIe in a fluid&d bed, Chem_ Eng. Sci., 19 (1964) lOOl_ D. Geldart, Types of gas fluidization. Powder Technoi., 7 (1973) 285. H_ Reuter, Pressure distribution in a gas-solid fluidized bed, Chem_ Ing_ Tech_, 35 (1963) 98_
8 H. Reuter, On the nature of bubbles in gas and liquid fluidized beds, Cbem- Eng. Rag_ Symp_ Ser., 62 (1966) 92. 9 H. Littman and G. A. J. Homo&a, Bubble rise velocities in two-dimensional gas-fluid&d beds from pressure measurements, Chem. Eng. Prog. Symp_ Ser., 66 (105) (1970) 37. 10 H. Littman and G. A. J_ Homo&a. The nressure fieldaround a two-dimensional gas bubble in a fluidiaed bed, Chem. Eng. Sci., 28 (1973) 2231. 11 C. Chavarie, Chemical reaction and interphase mass transfer in gas-fluidized beds. Ph_D_ Thesis. McGill Univ_. 1973_ 12 C. Chavarie and J. R_ Grace, Performance analysis of a fluidiaed bed reactor, Ind. Eng. Chem. Fundam., 14 (1975) 75. 13 P_ S. B. Stewart. Isolated bubbles in fluidized beds - theory and experiment, Trans. Inst. Chem. Eng.. 46 (1968) 60. 14 R. Collins, An extension of Davidson’s theory of bubbles in fluidized beds, Chem. Eng_ Sci., 20 (1965) 747. 15 J_ D_ M-y. On the mathematics of fluidization. II., J_ Fluid Mech., 22 (1965) 57. 16 R. Jackson. The mechanics of fluid&d beds. II._ n _ Trans. Inst_Chem. Eng., 41 (1963) 22. 17 J. F- Davidson, Symposium discussion. Trans. Inst. Chem. Eng.. 39 (1961) 230. 18 J_ Wertber. Bubble chains m large diameter gasfluidiaed beds, Proc. Int_ Powder and Bulk Solids Handling and Processing Conf., Chicago, 1976. 19 R_ A- Anderson, Fundamentals of Vibrations, Macmillan. New York. 1967.
Technology.-.
19 (19781255 _
0 Elsevier Sequoia S-A.,
- 264
Lausanne -
255
Printed in the Netherlands
Forces on Objects Immersed in Fluidized Beds
T. H. NGUYEN Deportment
(Received
and J. R. GRACE
of Chemical
June 17.1977;
Engineering,
McGill
Untversity.
in revised form September
SUMMARY
The depth of immersion of objects of various sizes and shapes floating at the surface of a fluidized bed was used to evaluate the effective density of the particulate phase. The net buoyancy force under fluidized conditions can be estimated closely by taking the particulate phase density as equal to the bed density at minimum fluidization. Rowever, splashing of particles onto the top of the objects may cause them to sink periodically, even when their density is less than that of the particulate phase_ Local pressure fluctuations caused by the rise of bubbles past horizontal cylindrical tubes were measured in a two-dimensional fluidized bed using a pressure transducer. Integration of the pressure over the surface of the tubes allowed the net vertical force on the tube to be determined as a function of time. The net force uersus time curve is shown to have an approximately sine-wave shape with the maximum upwards thrust corresponding to the arrival of the front of the bubble at the bottom of the tube, and the maximum downwards thrust corresponding approximately to the instant when the rear of the bubble reaches the bottom of the tube.
I?.UTRODUCI’ION In practice,
fluidized
beds
commonly
con-
tubes, diplegs, probes, nozzles, etc. which are immersed in the bed for cooling, heating, addition of solids or fluids, or me asurement purposes_ Objects may be dipped into the bed for coating- In other cases, large lighter objects may float on a fluid&cl bed composed of dense particles, for exampIe in the addition of large pieces of tain
heat
transfer
3480
University
Street.
Montreal
(Canada
H3d
2-4 7)
23. 1977)
coal for combustion to a bed composed mainly of denser ash particles. In all of these cases it is important to know the forces on the immersed bodies. For fixed objects one would like to be able to predict the transient forces which commonly lead to tube vibrations. Some vibrations are probably desirable in promoting favourable heat transfer rates, but escessive vibrations may lead to failure of immersed members. For freely moving objects it is important to know the conditions under which an object will float at the surface- The present paper considers these two questions. Previous work on objects immersed in fluidized beds has been reviewed by Harrison and Grace [ 1, 21.
BUOYANCY Experimental
FORCE
ON
apparatus
FLOATING and
OBJECT
procedrcr-e
it is a well-known prope& of a gas-fluidized bed that objects can be floated on its surface, there appears to have been no systematic work on the conditions under which floating occurs. In the present study, objects of three shapes - spheres, cylinders and recttangular parallelepipeds were floated on the surface of a fluidized bed. The column was 45.7 cm in diameter by 99 cm tall and equipped with a perforated plate distributor. Sand particles with a weight mean diameter of 291 pm and U,, = 7.0 cm/s were used in the experiments. The voidage at minimum fluidization, Emf, was 0.40, and the corresponding bulk density of the bed. D-t = was 1.48 g/cm3. P,U -E,& All of the objects were hollow and could have copper powder added to increase the weight. There were two cylinders, which could be used singly or attached end-to-end to make a longer cylinder_ Similarly, there Although
256 TABLE
1
Object shapes and sizes shape
Dime.?sions
Sphere
3.8 o-d.
Cylinder
3-S o-d., 5.1 long 3.8 0-d.. ‘3.6 long 36 o-d.. 12-7 long 3.8 x 3.8 x 2.5 3.8 x 3.8 x 3.8
Rectangular paraileIepiped
3.8 3.8 3.6 3.8
x x x x
3.8 3.8 3.8 3.8
(cm)
x 6.3 x 7-6 x 10-2 x 11-4
3.8 x 3.8 x 14.0 3-S x 3-8 x 17.8
Orientation
axis vertical axis vertical axis horizontal small side vertical one axis of symmetry vert.
long side vert. or horizlong side vertical long long long long
were fo-ur separate parallelepipeds of the same cross-section which could be immersed separately or attached together in pairs_ The different configurations and orientations used are shown in Table I_ Except for the sphere which was made of plastic, aU the other objects were constructed of balsam. Graduated sloes mere affiied to the outside of the objects to allow the immersion to be measured. When copper powder was added, the top of the copper layer could be fixed by inserting a tightly fitting disc into the cyhnders or parallelepipeds_ The experiments were carried out at superficial velocities of 7-5, S-7 and 10-3 cmis, corresponding to 1_08,1.24 and l-47 X U,, respectively_ Even though these velocities are low, there was sufficient bubbling at the higher two velocities that particles tended to be thrown onto the top of the objects, as observed in somewhat similar experiments by Roche 133 _ When particles lodged on the top of the object, these were swept away before immersed depths were read. This made data taking tedious at the higher velocities, especially for cylinders and parallelepipeds lying with horizontal axes_ Therefore, most of the data correspond to the lowest value of U and to cyhnders and parahelepipeds with their major axes verticalExperimental
results
The volume of dense phase displaced is plotted against the mass of the object in Figs. 1 - 3 for the sphere, cylinders and parallelepipeds respectively_ A straight line,
side side side side
vert_ vert. vert. vert.
or or or or
horiz. horiz. horiz. horiz.
forced through the origin, has been fitted to each set of results using least-squares regression_ It is shown in the Appendix that the drag of the fluidizing gas on the object is negligible provided that the scale of the object is much larger than that of the fluid&d particles. Hence, from Archimedes’ principle, the reciprocal of the slope of each of these lines gives the apparent density of the fluidized bed. The apparent density determined in this manner for the sphere, for cylinders orientated with vertical axis and for parallelepipeds with square cross-section in a horizon&I plane are given in Table 2_ More extensive data are tabulated elsewhere 143 _ Discussion of experimental results It is known that a cushion of gas tends to
form underneath an immersed object in a fluid&d bed, while, for a totahy submerged object, a region of “dead” particles tends to sit on the top surface [5]. Consequently one might expect that the net buoyancy force on an object in a fluid&d bed might differ from what one would predict using Archimedes’ principle and assuming the dense phase voidage to be equal to prnf, the bulk density of the bed at minimum fIuidization_ The effective densities given in Table 2 are ah within 2 5% of pmt = 1.48 g/cm3 for the particles used_ In order to determine whether Peff differs significantIy from pmf, the tdistribution was used to test the significance of differences at the 1% level. The results, given in Table 2, show that the difference is insignificant (i.e. the experimental value of
257
Lin.3 --__--_ _--
OWECT
Fig_ 1. Volume
of fluidized
Symbol
Ultld 1.06 1.24 147
0 &
cl
mnsS.(g)
bed displaced by floating sphere.
i
Fig. 2. Volume
of fluidized
bed displaced by floating cylinders with axis of symmetry
a probability of 0.02 of being exceeded in random sampling) except for the rectangular rate_
When the nine separate means were treated as a single population and a grand mean was formed
(FCff = 1_478),the
compared with tz$g = 2.90). Hence there is strong evidence that prnf canbeusedtocalculate the buoyancy force on floating objects. It is possible, however, that totally submerged objects might be subject to a different net buoyancy force due to the dead region above
~,fwasnotsignificant(It[=0.05
thefstatisticwaslessthanthevalueoftwith
parallelepipedatthelowestgasvdoci~
in the vertical direction.
difference from
200 -
Rectangular ParaIIelePiPedr III
Line ---
0
zca
Fig:. 3_ Volume of fluidized cross-section horizontal_
TABLE
bed
mhnc 1.08
p
1.24
-
3
1.47
200 05JEcT
.
Symbol 0
_--_-
300
MA.ss.Cgl
displaced
by fioating
rectangular
parallelepipeds
of square
cross-section
with
2
Apparent
fluidized
Object
bed
density
calculated
Li
No.
(cm/s)
points
of
uskg
data
Archimedes’
principle
Peff (g/cm?
Erpt.
1.13 l-41
2.76 4.54 3.75
t1-5
ItI
1.55
1.14 2.78 2.40
Vertical cylinder
i-5 s-7 10.3
10 4 4
1.49 1.50 l-49
O-88 0.82 0.43
2.82 4 -4 4.54
Vertical paratlelepiped
7.5 6-7 10.3
36 21 12
I.43 1.52 1.48
9.86
2.44 2.53 2.72
Sphere
5.5
11
8.7 10.3
4 5
the object, but this was not investigated in the present study_ The particles used in the present study correspond to group B in Geldart’s classification [S] , and the finding that peff = pmE should also be used with caution for particles of widely different characteristics. The experiments carried out in this work suggest that there are four situations with respect to floating of objects added to fluidized beds: (i) For very light objects (p, 4 pmf), or for p0 < prnf and shapes such that few particles can collect on the top of the object (e.g. for a tetrahetlron with angle greater than
1.73 0.17
the angle of repose of the particles), the object will at all times stay on the bed surface_ (ii) For objects with p0 < prnf, but such that particles can pile up on the upper surface, the objects should spend periods at the bed surface while splashed particles collect on the top, interspersed with periods of complete immersion. Periodic forces due to bubbles reaching the surface (see next section) will cause disturbances at the bed surface tending to dislodge particles from the top of floating objects. (iii) For objects with p0 = Pmf. the object may circulate entirely within the bed, resting
259
neither at the top surface nor at the distributor plate (iv) For P., Z+ Pmf, the object is expected to go straight to the bottom of the bed and rest on the grid. It seems likely that the transitions between these four classes of behaviour are gradual rather than sharp. Stability
with respect
to tipping
The stability of Boating objects with respect to tipping in real liquids is governed by the relative position of the centre of gravity and the metacentre- Generally speaking, floating objects tend to be unstable when displaced slightly from an equilibrium position if their centre of gravity lies well above the liquid surface and the characteristic dimension of the object in the vertical direction is much greater than that in the horizontal direction. Qualitative studies with the weighted cylinders and parallelepipeds demonstrated that the same general behaviour occurs in a fluidized bed. Some tests were done with bromobenzene, a liquid whose density (1.50 g/cm3) is very close to Pmf for the fiuidized particles used. If an object was unstable or stable in a particular orientation in the flnidized bed, it exhibited the same stability behaviour in the liquid. Hence the analogy between fluidized beds and real liquids not only applies to buoyancy forces, but also appears to extend to the stability of objects floating at the surface under gentle fluidizing conditionsTRANSIENT
FORCES
ON
FLmD
HORIZONT_4L
TUBES
Experimental
apparatus and procedure
The second set of experiments was carried out in order to measure pressure fluctuations on the surface of submerged tubes in order to understand the forces leading to tube vibrations. Reuter [7, 81 and Littman and Homolka [9, lo] have measured local pressure forces with respect to passage of bubbles, but in these cases there was no interaction between the bubbles and submerged objects. In the present case, pressure measurements are nsed to show how the net force on an object varies with timeThe experiments in the present case were carried out in a two-dimensional cohnnn so
that the pressure signal cculd be interpreted with respect to the bubbles giving rise to the pressure variations. The column, 245 cm high X 56 cm wide X 1 cm thick, has been described elsewhere [ 11, 123. The particles were glass microbeads with a weight mean diameter of 177 W, U,, = 6.5 cm/s and E,~ = 0.41. Single bubbles were injected into the column from a pressurized vessel via a solenoid valve through a 0.32 cm port, 12.7 cm above the porous distributor plate. A timer was used to give reproducible settings of the time between bubble injections and the time of solenoid valve opening. A Disa type 51D02-Pu2a low pressure transducer of capacitive type was used to measure local pressures during the passage of single bubbles. The transducer was kept at constant temperature to within +O.Ol “C using a thermostatic bath. Because the pressure changes caused by the passage of a bubble were relatively small, differential pressure measurements were used. This was achieved by blocking one of the two breather holes and connecting the other to a pressured chamber whose pressure was set to match the dense phase pressure at the same level. The output from the transducer assembly was recorded using a Honeywell Visicorder capable of recording signals with less than 2% distortion up to 13 kHz. Three horizontal tubes were used for the esperiments, all hollow with a small orifice in the wall connected to a tube leading to the pressure transducer. The hole was covered by a screen, mounted flush with the outer surface of the tube, to prevent particles from entering. Each of the tubes could be turned manuaily to allow pressures to be measured at any angle to the vertical. In practice, traces were obtained from 0” to 180” inclusive in 30” intervals. The smallest tube, 0.63 cm diam.. was of steel, while the two larger tubes (2.5 and 5.1 cm o-d.) were constructed from plexiglass. Each tube was in turn mounted 101.6 cm above the distributor at right-angles to the front face of the column. Because the span was only 1 cm and the tubes were rigidly supported, tube vibrations were negligible in the present case. In order to interpret the pressure traces, tine photographs were taken at 32 or 64 frames/s using intense back-lighting_ Syn-
260
chronization of the films with the pressure traces was achieved by having a photocell connected 7.6 cm below the centre of the horizontal tube_ When a bubble passed in front of the photocell, a solid-state light, visible in the photographic films, was interrupted and simuhaneously a mark was recorded on the Visicorder output trace. The surface area of single bubbles was determined by weighing paper cut-outs of the bubble contours, traced from projected photographic images_ Bubble velocities were obtained by plotting the nose and rear positions uersus time, and are presented elsewhere [4] _ The background superficial velocities used in the experiments were 30 - 50% in excess of umr. this range being optimal with respect to maintaining stable bubbles. -4 consequence of this background flow rate is that injected bubbles grew somewhat with height at the expense of the dense phase and due to coalescence with smaller bubbles Only one size of bubble was studied in the present work_ This size (580 cm* at 80 cm above the distributor) was chosen to limit both break-up of bubbles as they encountered the tubes and avoidance of the tubes by the bubbles. The bubbles were therefore large enough that side-wall effects were appreciable_ Where bubble splitting did occur or where bubbles veered from a vertical path, the experimental traces were not used. For the largest tube, 51 cm o-d_. it proved to be impossible to produce bubbles which would pass the tube without splitting, so that results are given here only for the two smaller diameter tubes.
Experimental results All pressure traces had qualitatively the same shape. -4 mavimum was observed when the bubble was injected into the bed, as noted by Littman and Homolka [9, lo] _ A second maximum occurred when the front of the bubble arrived at the measuring point and a minimum when the back of the bubble passed this point- Typical traces have been shown by Nguyen [4]. Pressure-time curves, each averaged from five individual recorder traces, are shown in Fig. 4 for the 0.63 cm tube and four angular positions (0”, 60”, 120” and 180” to the vertical). For these curves, r = 0 has been chosen to coincide with a.rrivaI of the bubble nose at the measuring point. Similar curves are shown for the 2.5cm-diam. tube in Fig_ 5_ The frequency of bubble splitting was greater for this larger diameter tube_ Discussion of results The experimental pressure traces for 0” and the 0.63 cm diam. tube were compared with the predicted profties from various theoretical models reviewed by Stewart [ 13]_ Since this tube is much smaher than the approaching bubbles, a reasonable test of the theories is likely for this case_ In front of approaching bubbles, the experimental trace (averaged from our separate traces) lies between values predicted by Colhns [ 141 and Murray [15], more than doubIe the value predicted by Jackson Cl63 and about half that predicted by Davidson 1171. The comparison appears in Fig. 6. Some experimental
Fig_ 4. Pressure-time traces for 0.63 cm ciiaw. tube at four different arrival of the front of the bubble at the meamring point.
ang~kr
positions_Time
0 corresponds
to
261
-7
__---_---
8 60° lZOO 180 0
Fig. 5. Pressure-time traces for 2.5 cm diam. tube at four different a&al of the front of the bubble at the measuring point.
These Experments
(0.63cm
tuoe.8=0
angular positions. Time 0 corresponds
to
1
o Liltman Homdka Experiments
Fig_ 6. Comparison of pressure-time trace for 0.63 cm diam. tube and 8 = 0” with theoretical and with experimental values measured by Littman and Homolka [lo ] _
results due to Littman [IO] which show the same trend, but higher values, are also shown. As with the Littman results, pressure recovery behind the bubble is slower than the build-up of pressure as the bubble approaches. The thecretical models represent our pressure profiles in the wake region better than they do Littman’s results. The instantaneous net upwards pressure force per unit length on each tube at any instant in time is given by F(T) = DT f 0
p(e ,r) cos 0 de
(1)
pressure profiles
w’\pre ~(8 ,T) is the instantaneous pressure at angle 0 measured from the bottom of the tube and DT is the tube diameter. Numerical integration was carried out to give the F(r), and the resulting curves are shown in Fig. 7, with T = 0 corresponding to the instant when the bubble nose reaches the bottom of the tube. It is seen that the tube encounters an up-thrust as the bubble approaches the tube and then a downward force as the bubble rises away from the tube. The curves are roughly sinusoidal in shape, but t&e period of downthrust is considerably longer in duration than the period of up-thrust. The 2.54 cm tube
!6%
Fig_ i. Net nressure force on the 0.63 and 2.5 cm diam. tubes obtained the surface-king eqn_ (1).
encounters considerably larger forces than the 0.63 cm tube, the difference being even greater than the fourfold factor which arises from the appearance of D, in eqn (1). It is clear that this kind of net pressure force can cause vibrations of tubes in large fluidized beds- In freely bubbling beds where many bubbles are present at one time, the amphtude of vibration is likely to depend on such factors as: properties of the bed itself (e-g_ particle properties, bed dimensions, number of tubes present) and operating variables (gas flow rate, bed depth) which determine the bubble size and frequency; orientation, shape, size and length of tubes; the material of construction, wall thickness and means of support of the tubes as well as the mass and viscosity of any fluid contained therein. Some damping due to the viscous nature of a ffuidized bed appears likely. In practice, there are both stochastic and deterministic eIements to the frequency of bubble passage [18], so that bursts of bubbles with different frequencies will occur in freely fluidized beds. The frequency of vibration of the tube will remain similar to the natural frequency of the tube, while the amplitude will vary erratically in response to stimuIi at other
by integrating the pressure profiles over
frequencies [19]_ The worst case will occur if a burst of bubbles arrives with the resonant frequency of the tubes- Because of the stochastic elements in bubble populations, this situation will arise from time to time and so the design must be based on this case_ CONCLUSION
(1) Objects float on the surface of fluidized beds as on the surface of real liquids with the netbuoyancy force determinedasifthedense phase of the fluidized bed had the same density as at minimum fIuidization_ Particles ejected by bubbles bursting at the bed surface may build up on top of floating objects causing periodic complete immersions. (2j Bubbles passing immersed objects cause transient forces- The net pressure force is upwards as the bubble approaches, and downwards with a similar magnitude, but longer duration, as the bubble rises away from the immersed object. The upward and downward thrust can cause severe vibrations of immersed tubes. ACKNOWLEDGEMENTS
The authors are grateful to the National Research Council of Canada for financial
assistance. T. H. Nguyen also wishes to acknowledge a scholarship from the Quebec Ministry of Education.
LIST
C DP DT dP F
F BO F DO
&
x
kf Emf 0
Peff Pi Pmf PO PP 7
OF SYMBOLS
bubble radius drag coefficient defined by eqn_ (A-2) drag coefficient for a single particle in a suspension tube outside diameter particle diameter net force per unit length due as obtained from pressure trace buoyancy force on object drag force on object acceleration due to gravity characteristic height of object = object volume/S, mass of object pressure cross-sectional area of object experimental value of t statistic value of t statistic with a probability of 0.02 of being exceeded in random sampling distance from top of bubble to bottom of tube (see Fig. 7) superficial gas velocity superficial gas velocity at minimum fluidization bed voidage at minimum fluidization angle measured away from lowest position effective density of the dense phase density of fluidizing gas density of gas-fluidized bed at minimum Ruidization = pp(l - E,~) density of object particle density time
APPENDIX
Forces
on floating
objects
For a body fioating at equilibrium on the surface of a gently fluidized bed, the weight of the object must be counterbalanced by buoyancy forces (due to the hydrostatic pressure distribution) and to drag forces contributed by the fluidizing fluid, Le.
Meg = FB~ + FDO
(A-1)
that other forces, suchaselectrical forces, are negligible. We are interested in the relative magnitude of the drag term to the other terms in eqn. (A-l)_ The drag on the object can be expressed as Weassume
F Do
.Pfelf
=
cD,
-53
2
(A-2)
where CD, is a drag coefficient and S, is the cross-sectional area_ The weight of the object is given by fiIl,g = P~H,S&
(A-3)
where HO = object volume/S, is a characteristic height of the object But in a quiescent gas-fluidized bed (pP 5 p,), the weight of each particle must be just balanced by the drag force on that individual particle, Le. (A-4) Combining eqns. (A-2) - (A-4), we ca write the ratio of the drag term to the weight term in eqn. (A-1) as F DO
2cDoPpdp
fifo&Z
~CD,~O%
-=
(A-5)
But pp and PO are expected to be of the same order of magnitude with pp > PO. Drag coefficients for particles in an assembly depend strongly on the local voidage as well as on the particle shape and Reynolds number. The Reynolds number for the object will be many times larger than that for a single particle because of the much larger characteristic dimension; the local voidage may also be larger [5 J _ Hence one would expect Cn, < CDP, but ali that is necessary to the argument here is that CD, be not much greater than Cnpr which is certainly true. Hence, since dp -=Z Ho for any object in question, it is clear that the ratio in eqn. (A-5) must be much less than unity. Equation (-4-l) may therefore be rewritten as fW& = FBO
(A-6)
and sixnpler way of establishing the same result is to imagine an entirely new fhridized bed comprised of objects identical to the one floating on the surface of the much smaller particles. The superficial gas velocity required for the weight of the An
alternative
264 large
objectstobebalancedbydragwouldbe (U,,),,theminim~m fiuidizationvelocity of thenewbed_SinceU,,oftheactualbed~ (U,,),,dragforcesonthe objectinabed of small particles at U at or near U,, must be much less than the weight of the object_
REFERENCES
J_R_GraceandD_ Harrison,Designof fiuidized beds with internal baffies, Chem. Process Eng., 51(1970) 127_ D. Harrison and J_ R_ Grace, Fluidized beds with internal baffles, in J_ F. Davidson and D_ Harrison (Eds.), Fluidiaation, Academic Press, London, 1971_ G. Roche, Etude du comportement d’un milieu fluidi& autour d’un corps immerg& M.Sc_A_ Diss, Ecole Polytecbnique, 1976_ T_ H_ Nguyen, Forces on objects immersed in fhxidized beds, M.Eng. Thesis, McGill Univ_, 1976. D_ H_ Glass and D_ Harrison. Flow patterns near a solid obstacIe in a fluid&d bed, Chem_ Eng. Sci., 19 (1964) lOOl_ D. Geldart, Types of gas fluidization. Powder Technoi., 7 (1973) 285. H_ Reuter, Pressure distribution in a gas-solid fluidized bed, Chem_ Ing_ Tech_, 35 (1963) 98_
8 H. Reuter, On the nature of bubbles in gas and liquid fluidized beds, Cbem- Eng. Rag_ Symp_ Ser., 62 (1966) 92. 9 H. Littman and G. A. J. Homo&a, Bubble rise velocities in two-dimensional gas-fluid&d beds from pressure measurements, Chem. Eng. Prog. Symp_ Ser., 66 (105) (1970) 37. 10 H. Littman and G. A. J_ Homo&a. The nressure fieldaround a two-dimensional gas bubble in a fluidiaed bed, Chem. Eng. Sci., 28 (1973) 2231. 11 C. Chavarie, Chemical reaction and interphase mass transfer in gas-fluidized beds. Ph_D_ Thesis. McGill Univ_. 1973_ 12 C. Chavarie and J. R_ Grace, Performance analysis of a fluidiaed bed reactor, Ind. Eng. Chem. Fundam., 14 (1975) 75. 13 P_ S. B. Stewart. Isolated bubbles in fluidized beds - theory and experiment, Trans. Inst. Chem. Eng.. 46 (1968) 60. 14 R. Collins, An extension of Davidson’s theory of bubbles in fluidized beds, Chem. Eng_ Sci., 20 (1965) 747. 15 J_ D_ M-y. On the mathematics of fluidization. II., J_ Fluid Mech., 22 (1965) 57. 16 R. Jackson. The mechanics of fluid&d beds. II._ n _ Trans. Inst_Chem. Eng., 41 (1963) 22. 17 J. F- Davidson, Symposium discussion. Trans. Inst. Chem. Eng.. 39 (1961) 230. 18 J_ Wertber. Bubble chains m large diameter gasfluidiaed beds, Proc. Int_ Powder and Bulk Solids Handling and Processing Conf., Chicago, 1976. 19 R_ A- Anderson, Fundamentals of Vibrations, Macmillan. New York. 1967.