Department of Chemical Engineering,. The University of ... Abstract. A development of the theory of fluidization to encompass inclined systems is presented. Flow.
Powder
Technology,
169
63 (1990) 169-178
The effect of inclination on fluidized beds D. P. O’Dea, V. Rudolph*, Department
of Chemical
Y. 0. Chong
Engineering,
The University
of Queensland,
St. Lucia,
Qld. 4067 (Austmlia)
L. S. Leung** CSIRO Division of Coal Technology,
P.O. Box 136, North Rude, N.S.W.
2113 (Australia)
(Received October 10, 1989; in revised form May 15, 1990)
Abstract A development of the theory of fluidization to encompass inclined systems is presented. Flow regimes and a transition condition (equivalent to the minimum fluidization condition in vertical fhridized beds) have been identified in experiments with beds inclined at angles between 45 and 90 degrees to the horizontal. Fundamental equations incorporating interparticle friction are derived which estimate the condition for flow regime transition and predict a p&& the pressure-flow-inclination angle relationship recorded in each flow regime. A two-phase description of inclined fluidization is also given.
Introduction
Fluidization phenomena in inclined systems are poorly understood and have received little research attention despite the reality that many flmdized systems are not vertically oriented in practice. In fact, the vertical system may be viewed as a particular case of fluidization in which the drag force and gravitational force are diametrically opposed. There is a need to extend the theory of fluidization to encompass other orientations in the gravitational field. This is pertinent to industrial fluidization processes such as catalytic cracking, coal gasification and iron ore reduction, which often have physical layouts that necessitate the use of angled standpipes or risers. Without fundamental theories, engineers are forced to resort to ‘rules-of-thumb’ or equations for vertical fluidization to design these inclined systems. The lack of even a basic understanding of the effect of inclination on fluidization constitutes an overwhelming obstacle to improving the design, operation or trouble-shooting of inclined systems, and to optimizing the layouts of new solids circulation systems. A survey of *To whom correspondence should be addressed. **Deceased.
0032-5910/90/$3.50
the literature highlights that key research work relates flow in vertical standpipes and risers to flow behaviour in vertical fluidized beds. For example, the expansion equation of Matsen [ 1 ] for vertical gas-solids flow derives from a previous understanding of two-phase bubble flow [2] applied to the vertical fluidized bed model of Davidson and Harrison [ 3 1. It is surmised therefore that flow in inclined systems can be related to flow behaviour in inclined fluidized beds. Research relating to inclined beds is almost non-existent. Apart from Chong et al. [4], the few studies published are not concerned with the overall flow behaviour of the inclined bed but have aimed at describing the slope stability of particulate material influenced by flmd percolation and gravity [5-101. Chong et al. [4], reporting on the observations of Pham Thi [ 111, proposed that three flow regimes can exist in both particulate and aggregative inclined fluidized beds-the tied bed, partially fluidized bed, and fully fluidized bed regimes. The development of a fluidized channel along the upper wall of the inclined bed differentiated the first two flow regimes. An empirical correlation was presented to predict the conditions for channel development. Q Elsevier SequoiaPrInted in The Netherlands
170
No convincing experimental evidence was given for the third regime, which they proposed should occur when the pressure drop across the inclined bed reached its theoretical potential: Ap=(p.-p1)(1
--E)L g sin 0
(I)
Progressing on from this work, we present in this paper a classi&ation of flow regimes for inclined fluidized beds and the derivation of a fundamentally based model which predicts quantitatively the demarcation between flow regimes and the pressure-flow-inclination angle relationship recorded in each flow regime. A comparison with gas-solids downfiow behaviour in an inclined standpipe will be detailed in a succeeding paper soon to be published.
Experimental technique Experiments were conducted in inclined beds pivoted on an A-frame support and set at angles between 45 and 90 degrees to the horizontal as shown in Fig. 1. Perspex columns of two different cross-sections were used-a three-
dimensional cylindrical column (83 mm I.D.) and a two-dimensional thin-slice column (100 mmX 10 mm) for validating two-dimensional mathematical models. Air was uniformly distributed into the beds through sintered bronze distributor plates. Pressures were measured at top, side and bottom tappings around the base of the inclined bed and revealed that negligible pressure variation occurred in the radial direction. Four different powders were used, representing both Geldart class A and class B powders. Table 1 lists the properties of these solids. The internal friction properties of each of these cohesionless powders was characterized by #+, the angle of internal friction, measured using a Jenike Flowfactor Tester [ 121. With the column in the vertical position, the bed of solids was first fluidized vigorously before closing off the air and allowing the bed to settle loosely. This formed the most reproducible packing in terms of average bed voidage. The column was carefully tilted to a selected inclination angle. Air flow was then increased to a required rate at which condition the bed pressure drop was recorded and visual observations made. These measurements were repeated for successively higher rates of air flow. The initial slope of the bed free surface was formed by natural collapse of the solids when the column was tilted to a selected angle. The surface was rarely horizontal as is ideally drawn in Fig. 2(a). However, the slope of the surface was shown to have no significant effect on the inclined bed experiments performed here, unlike previous shallow inclined bed experiments t5-31.
INCLINED FLUIDIZED
Flow regimes and transition condition In each of 61 different inclined powder beds studied, a reproducible, measurable, clearly observable flow regime transition occurred as the gas flow rate was increased. At low gas velocities, the entire bed of powder remained tied as shown in Fig. 2(a). Gas merely percolated through the void spaces between the particles which remained stationary - a packed bed. At a higher velocity, a condition was reached
Fig. 1. Experimental inclinedbed apparatus.
at which the particles at the top of the shortest side of the inclined bed (i.e., at the intersection
171 TABLE 1 Powder properties SOlidS
FCC catalyst
Coarse glass ballotini (CGB)
meglsss balIotini (FGB)
Sand
dp (pm) pa M/m? Geldart classification 4, (degrees) U, (cm/s) 4 & (Pa-s/cm2)
62 1450 A 31 0.33 0.43 250
510 2480 B 30.5 22 0.42 6.6
246 2480 B 26 6.2 0.40 28
228 2620 B 34 4.6 0.40 33
quired to form the channelled bed redirected into the fluidized channel and by-passed the major packed section. This flow regime transition-packed bed to channelled bed -was also clearly marked by a corresponding transition in the relationship of pressure drop to fluid velocity from proportionality to approximate independence. Typical measurements recorded in experimental inclined beds of FCC catalyst and glass ballotini are plotted in Figs. 3(a) and (b) showing this transition. This provides a quantitative flow regime demarcation.
ltj$$$f@ -. . u
I
‘ci
(4
@I
I
"cb (cl
Fig. 2. Flow regime transition in an inclined bed. (a), Packed bed; (b), channel initiation; (c), channelled bed.
between the bed free surface and upper tube wall as shown in Fig. 2(b)) were fluidized - the point of channel initiation (ci). As the velocity was further increased, the channel of fluidized particles and air extended for greater distances down the upper side of the bed, not disturbing the major portion of the bed which remained in its packed state. This phenomenon was also observed by Bridgwater [ 51, who reported that the reproducibility of the depth of the deepest channel for a specified bed inclination and gas velocity was good. A transition of flow regime was very obvious when the channel broke through to the distributor. At this chunnel break-through (cb) condition, a fluidized channel of particles and air now extended from the distributor through to the free surface forming a chunnelled bed (Fig. 2(c)) - a new classification proposed here. This channel provided a less resistive path for air flow than the adjoining packed bed section. Therefore, entering air in excess of that re-
0.2
0 .
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
&J/Qnf (a) 1.2 00 le 00
1.0 0.6
*
hnf
.aa
06.
&Jo0
. 0.4 . 0.2 . 0.0 0.0
@)
90' 75'
4
j',
fiIll
f
1
60” f
f
45”
)*-I r
I . 0.2
0.4
0.6
0.6
1.0
1.2
1.4
Ug/Urnf
Fig. 3. Normaliaed pressure-flow curvesfor inclined 83mm I.D. beds. (a), FCC catalyst; (b), coarse gIass ballotini.
172
Channel break-through condition indepadent of bed dimensirms and _~?uidizing medium During the experimental program, the bed height to diameter ratio (LID) was varied from 2.2 to 8.8. Figure 4 shows that this had no significant influence on the normalized pressure drop recorded at the point of channel breakthrough in several inclined beds. Inclined bed experiments were also conducted in a 2-D column (100 mmx 10 mm). The measured U,,,, of each powder was higher in this thinner bed, possibly due to increased leakage in the wall regions as has been previously postulated [ 131. This increased permeability also existed in the inclined 2-D beds. Therefore their normalized pressure drop vers-us velocity curves did not diverge from those for the corresponding 3-D cylindrical beds (see Pig. 5). Figure 5 also shows that the flow regime transition points for the 2-D and 3-D beds were equal at each angle of inclination. It was concluded that wall effects had negligible influence on the channel break-through phenomenon.
Inclined
fluidization
model
Two distinct flow regimes have been identified for inclined fluidization from this experimental program which included both Geldart class A and Geldart class B powder types:
(i) and
packed bed regime:
U-CUC,
(ii)
channelled bed regime:
U, UC,
where U,, is the fluid velocity at the channel break-through condition.
1 .o
APct,OX. &,,fO,,
Pham Thi [ 111 used both water and air as fluidizing media to study particulate and aggregative fluidization respectively of inclined beds of glass ballotini spheres ranging in size from 420 to 500 pm. Figure 6 shows a close comparison between pressure-flow curves and flow regime transition points for these aggregative and particulate inclined systems. This reproducibility of the channel breakthrough condition in inclined beds irrespective of the bed height, bed cross-section and fluidizing medium highlights its correspondence to the minimum fluidization condition in vertical beds.
60’
~
D
. 45”
0
. .: : .
2
4
8
6
v
6
10
L/D
Fig. 4. Channel breakthrough pressure drop versus length to diameter ratio of various inclined beds.
1.2 ,
The packed bed regime The experimental pressure drop and fluid velocity data for each inclined packed bed were coincident on the same straight line as those recorded for the corresponding vertical packed bed (see Pigs. 3, 5 and 6). This confirms the applicability of the Ergun [ 141 equation or Darcy equation to flow through a fixed bed of solids irrespective of its orientation in the gravitational field. Therefore, the simplified Ergun equation, 1.2
I
1.0 0.6
0 air
. A water .
.a
AP qImf o.6
0.0
0.5
1.0
1.5
2.0
2.5
O.’ .
I
0.2
2-D inclined beds of FCC catalyst.
.
l
a
90.
.
75.
A A
60” 45”
*
A o l 0. A
If 2’
0.0 0.0
3.0
U,Nrnf F’ig. 5. Comparison of pressure-flow
9” *(&a.*
#.OAOA
O...DA
0.5
1.0
1.5
2.0
2.5
U/btlf
curves for 3-D and
Comparison of pressure-flow curves for air-solids and water-soUds inclined beds (after Pham Thi [ 11 I).
Fig.
6.
173
AP
-L = I5@
(1 - E)2 (4$,)2p
u
RepL
(7)
In normalized form, AP~I, =sin AP,
e-tan & cos
8
(8)
Note that eqn. (7) reduces to
correctly describing a vertical fluidized bed. Figures 7(a)-(d) show that the inclusion of internal friction forces in the inclined channelIed bed force balance (eqn. (8)) significantly improves upon the previous estimation of the channelled bed pressure drop given by eqn. (3). However, the assumption that the frictional forces are fully mobilized and equating them to their limiting value (T,,) leads to an over prediction of their infIuence in the force balance.
174 8 (degrees) 45
0.70
6 (degrees)
60
1.0 ~I
0.75
0.60
0.65
sin
75
0.90
0.95
90
1.00
8
0.70
0.76
0.60
0.65
sin
@I
(a)
75
60
90
0.90
0.95
1.00
0.90
0.95
1.00
e
0 (degrees) 60
45
75
90
1.0
0.0
AP,, o.6 Apmfo, 0.2
001 '0.70
0.75
0.80
0.65
sin
0.90
0.95
1.00
6
O%.L,
0.80
0.65
sin
0
eqn. (8)) and liquid head analogy (. . . . . . ., eqn. F’ig. 7. Comparison of simple friction models (-, eqn. (13); ---, (3)) with experimental channelled bed pressure drops. (a), FCC catalyst (t#,,== 23.6”; A= 319; (b), coarse gks ballotid (+,=23”; tj,=30.5”); (c), fine glass ballotini (+,=lS”; t#,=26”); (d), 0, sand (t##,-26.6”; h-343; A, after Brklgwater [5i.
Fig. 8. Notation for force balance at channel/packed bed interface.
It has long been recognized that A, the angle of internal friction of a mass of particles in steady shear (the critical state established in a Jenike shear cell), is greater than the basic physical property of the particles, 4@, the angle of interparticle friction [ 171. The angle of interparticle friction depends on the particle’s
material composition, surface properties, roughness, size and shape. It may be measured by sliding a mass of the particles over a block of the same material with the same surface roughness. Caquot [18] and Horne [19, 201 have studied the behaviour of a shearing assembly of cohesionless particles and derived expressions relating 4, to 6 (also notated in the literature as 4,, the critical angle of shearing resistance at constant volume). The difference between +, and 4, is due to the increased internal frictional losses resulting from the large number of relative movements and contacts between particles in a particle assembly in steady shearing flow, even though the ratio of tangential to normal force at each particle contact nowhere exceeds tan 4p The additional frictional losses can be understood with the aid of a simple model [ 171 which equates this shearing particle flow to two uniform serrated faces in sliding contact (Fig. 9).
175
and S, is solved from the equation (excluding the solution 8, = &): 24 +sin 2&+2 -2s,+sin
cos 261
2&+2
cos 26,
(IOc)
and the stress ratio is equal to
1 (a)
@>
Fig. 9. Simplified model comparison of (a) lnterparticle friction (tan I&J and (b) internal friction (tan +J.
An additional energy loss results from friction on the longer inclined planes as compared with the flat plane, even though the mass as a whole moves as if it were sliding on a flat plane with no volume change. Caquot [ 181 corrected for this additional loss in a shearing assembly of particles by calculating the frictional energy loss over a semicircular length (7r/2 dp) of a hemispherical surface in place of the straight length of a particle diameter (d,) and derived the relation
Horne [ 19, 201 carried out a more rigorous analysis of relative movements and contacts between particles or groups of particles in a shearing assembly and calculated the anisotropy of the assembly in terms of mean projected solid paths. This led to the following complex relationship between 4, and 4, which was derived from a balance of internal and external forces acting in the deforming particle assembly:
Figure 10 shows a comparison between the theoretical curves of Horne [20] and Caquot [ 181 and experimental measurements of 4, and & for a diverse range of particles (covering the range +,= 9” to 39”). Both theoretical predictions are in good agreement with the experimental data, although the simpler analysis of Caquot leads to excessive 4 values for & > 20”. No indefinitely sliding plane of particles (corresponding to a critical state of fully mobilized internal friction) was observed at the channel/ packed bed interface during inclined bed experiments. Therefore, it is reasonable to neglect the additional frictional energy losses associated with relative movements between particles in a shearing assembly and assume that the frictional forces occurring at the interface are only due to interparticle friction. If we equate the stationary element of particles in Fig. 8 to a block of material being held stationary on an inclined plane by surface-surface friction, it is easy to see that only interparticle friction should be included in the interface
_; cos3&-cos3~1
=3 = a, --
a,
$cot
&
1
- ; (sin261+sin
sin 61-sins,
61
+1
sin s, + sin2$)
WW
where 61= f -4,
(IOb)
F’lg. 10. Experhnental measurements of 4 versus +,, compared with the theoretical curves of Home (-, eqn. (10)) and Caquot (- - -, eqn. (9)).0, Crushed glass; V, feldspar; A, A, quartz; 0, zircon; 0, bronze spheres; 0, glass ballotini; I, range of values for steel balls (after Home 1201).
176
force balance. This reduces the frictional forces to a value
#Jjb %
Ts=-
(11)
and the pressure drop across an inclined channelled bed of length L is more correctly given by (combining eqns. (4) (6) and (11))
A~cb= p,(l - E)g(sin 0 -tan qbp cos e-yd
(12)
In normalized form,
A~cb =sin e-tan (bw cos 8 APM
(13)
Equation (12) also reduces to A~cb= APM at 8=90°
= Ps(l - 4gL
describing a vertical fluidized bed. The angle of interparticle friction & of a powder can be measured by sliding a mass of the powder over a block of the same material with the same surface roughness [ 1 I’]. Alternatively, the expression of either Horne (eqn. (10)) or Caquot (eqn. (9)) can be used to determine & if the angle of internal friction &i is known. Table 2 lists the theoretically determined values of 4, for the solids used in the inclined bed experiments. Figures 7(a)-(d) show that equating the frictional force in the inclined channelled bed analysis to the interparticle friction only (eqn. (13)), and not to the complete internal friction, further improves upon the estimation of the channelled bed pressure drop. The theoretical predictions of the simple interparticle friction model are in good agreement with the experimental results. The model now provides a relationship to predict the condition for channel break-through at a given angle of bed inclination and also the resultant pressure drop. TABLE
A two-phase _fluidi.zation
Of:
(i) a particulate phase in which the flow rate is equal to the flow rate required for incipient fluidization (U,) and the voidage is essentially constant at emf, and (ii) a bubble phase which carries any additional flow of fluid. A similar two-phase model of inclined fluidization is proposed here by considering an inclined channelled bed as a two-phase system consisting of (i) a packed bed phase in which the flow rate is equal to the flow rate required for channel break-through (U,,,), and (ii) a channel phase (adjacent to the upper wall of the bed) which carries any additional flow of fluid (U-U,,,). This model is supported by visual observations made of channelled beds and by the maintenance of the simple interparticle friction force balance (eqn. (12)) at fluidizing velocities greater than U,,. F’igure 11 shows the degree of agreement between experimental measurements and the proposed inclined bed model. This two-phase description also highlights two factors which must be remembered in the design or application of any inclined flmdized bed system. Firstly, negligible solids mixing occurs because the major portion of the bed remains in a packed state-only a small proportion of the solids are vigorously agitated in the fluidized channel. Secondly, any fluid 1.2 1.0 .
A
*-o-tTaTa-
-%--~--WV-
4:
31 30.5 26 34
90’ 75.
__4-p_-A-T-
Theoretical & values for experimental solids
FCC catalyst Coarse glass ballothi FYne glass ballotini Sand
of inclined
The two-phase theory of fluidization [3] states that a model of vertical aggregative fluidization may be set up by considering a fluidized bed as a two-phase system consisting
2
solids
description
so0 450
& Eqn. (10)
Eqn. (9)
23.5 23 18 26.5
21 20.5 17 23
11. Comparison of experimental data (FCC, #,, = 23.53 and theoretical inclined bed relationships: (I) packed bed regime (-, eqn. (2)); (ii) channelled bed regime (- --, eqn. (12)). Fig.
177
flow in excess of U,, experiences very inefficient contact with solids in the bed as it by-passes the majority of solids viathe upper wall channel. In the preceding simple friction model of the inclined channelled bed, it was assumed that fluid flow normal to the channel-packed bed interface is negligible compared with flow parallel to the interface and has no effect on the force balance. O’Dea [ 151 presented a twodimensional model which included the effect of fluid flow out through the interface. This model predicted that most flow out through the interface occurs near the distributor and exerts a drag force on the particles which reduces the normal stress and therefore reduces the interparticle friction force acting in the axial direction. The pressure gradient down the channelled bed therefore increases to compensate for the reduction in frictional force and maintain the force balance at the interface. However, this model has not been experimentally verified yet.
(3) The normalized operating condition at which channel break-through occurs (U,,/U,> was reproducible and independent of the bed height, bed cross-section and fluidizing medium. (4) Experiments conllrmed that the simplified Ergun equation, eqn. (2), accurately represents the proportionality between pressure gradient and fluid flow in the packed bed regime irrespective of the angle of inclination of the bed. (5) A model of the inclined channelled bed was developed from a force balance on the stationary particles at the channel/packed bed interface. This model has no fitted parameters. For the limited range of powders tested, it predicts the operating conditions for channel break-through and the pressure drop across the channelled bed to be
=
Conclusions
A fundamental model of fluidization in inclined beds has been developed and is summarized in the following conclusions. As an immediate practical application, this model could provide a basis for analysing gas-solids flow in angled standpipes and risers (the subject of a succeeding paper soon to be published). (1) Experiments with several different inclined beds (representing both Geldart class A and class B powders) identified two distinct flow regimes as the fluidizing velocity was increased- the packed bed regime and the chann.ell.ed bed regime. (2) Flow regime transition occurred at the channel break-through condition. A channel of air and fluidized particles that had initiated at the packed bed free surface and developed adjacent to the upper tube wall broke through to the distributor to form a channelled bed. This was quantitatively demarcated by the corresponding transition in the relationship of pressure drop to fluidizing velocity from proportionality to independence (equivalent to the minimum flmdization condition in vertical beds).
sin e-tan
4, cos 8
(13)
Confirmation of its applicability to powders with a wider range of properties, especially friction properties, is still necessary. (6) By direct analogy with the two-phase theory of fluidization, we propose that an inclined channelled bed of powder may also be modelled as a two-phase system consisting of: (i) a packed bed phase in which the flow rate is equal to that required to form a channel (u&), and (ii) a channel phase which carries any additional flow of fluid (U- UC,).
Acknowledgements
The authors wish to thank the Australian Research Council and CSIRO for their financial support of this work.
List of symbols
4 z AP
volume-surface mean particle diameter, m acceleration due to gravity, m/s2 length of bed, m pressure drop, Pa
178
Apct, A~mf
firl u UCb Kf
channelled bed pressure drop, Pa minimum fluidization pressure drop, Pa particle Reynolds number: ,Nd,ll.L, superficial fluid velocity, m/s channel break-through velocity, m/s minimum fluidization velocity, m/s
Greek symbols E average voidage, minimum fluid&&ion voidage, %f 8 angle of inclination of bed to horizontal, rad fluid viscosity, kg/(m . s) CL fluid density, kg/m3 pi true solids density (apparent particle PS density), kg /m3 u normal stress, Pa major, minor principal stress, Pa ffll a3 shear stress, Pa 7s limiting shear stress, Pa 791 angle of internal friction, rad particle shape factor, 2 angle of interparticle friction, rad 4s
References 1 J. M. Matsen, Puwder Techno& 7 (1973) 93. 2 D. J. Nicldin, chsm. Eng. SC& 17 (1962) 693. 3 J. F. Davidson and D. Harrison, Z%k&.se~ Particles, Cambridge University Press, Cambridge, 1963. 4 Y. 0. Cbong, T. H. Nguyen, L. S. Leung and C. S. Teo, AZChE Sgmp. SIX, 80(241) (1984) 149. 6 J. Bridgwater, Powder Techml., 11 (1976) 199. 6 P. M. Heertjea, G. K. Kboe and M. Moncbeaux, Powder
Technd, I5 (1976) 29. 7 L. M&l, L. Krell, H. J. Kiinne and J. Kliefotb, In&rnational C&m. Eng., 26(-2) (1986) 231. 8 J. Verloop and P. M. Heertjes, Powder TechnoL, 7 (1973) 161. 9 M. R. Judd and P. D. Dixon, !&am. ZChenzE, 57 (1979) 67. 10 R. Jackson and M. R. Judd, Tmns. ZchsmE, 59 (1981) 119. 11 T. H. Pbam Tbi, B.E. Thesis, Univ. of Queensland (1972). 12 A. W. Jenike, Bull. Univ. Utah, 53(X) (1964). 13 P. N. Rowe and D. J. Everett, Tmns Znstn Chem.
Engrs, 50 (1972) 49. 14 S. Ergm, Chem. Eng. Frog., 480) (1962) 89. 15 D. P. O’Dea, Ph.D. Thesis, Univ. of Queensland (1989). 16 R. L. Brown and J. C. Richards, Princ&ks of Powder Mechanics, Pergamon Press, Oxford, 1970. 17 P. W. Rowe, Proc. Roy. Sot., A 269 (1962) 500. 18 A. Caquot, Stab&!&% des Terms pulvM et cohfkentes, Paris, Gautbier Villars, 1934. 19 M. R. Home, Z+oc. Roy. Sot., A 286 (1965) 62. 20 M. R. Home, Proc. Roy. Sot., A 310 (1969) 21.
