Modelling the transport disengagement height in ...

Advanced Powder Technology 22 (2011) 155–161

Contents lists available at ScienceDirect

Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

Original Research Paper

Modelling the transport disengagement height in fluidized beds Anke Brems a,*, Chian W. Chan b, Jonathan P.K. Seville b, David Parker c, Jan Baeyens b a Katholieke Universiteit Leuven, Chemical and Biochemical Process Technology and Control Section, Department of Chemical Engineering, Jan Pieter De Nayerlaan 5, 2860 Sint-Katelijne-Waver, Belgium b University of Warwick, School of Engineering, Coventry CV 4 7 AL, UK c University of Birmingham, School of Physics and Astronomy, Edgbaston, Birmingham B 15 2 TT, UK

a r t i c l e

i n f o

Article history: Received 27 May 2010 Accepted 23 July 2010 Available online 4 August 2010 Keywords: Transport disengagement height (TDH) Positron emission particle tracking (PEPT) Carry-over Modelling

a b s t r a c t Bubbles bursting at the surface of a bubbling gas fluidized bed (BFB) project particles into the freeboard. Coarser particles fall back, the solids loading declines with height in the freeboard, and fines are ultimately carried over. The height of declining solids loading is the transport disengagement height (TDH), measured in this research by positron emission particle tracking, and modelled from a balance of forces on ejected particles. Model predictions and PEPT-data are in good agreement. Almost all of the empirical equations overestimate the TDH. Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

1. Introduction Bubbles bursting at the surface of a BFB bed project particles into the freeboard. Depending on their terminal velocities (Ut) and the gas velocity (U), particles are carried up the freeboard to various heights. The solids loading (kg solids/m3 gas) will decline with height, as illustrated in Fig. 1. The region within which the solids loading decreases, is called transport disengagement height (TDH). Coarse particles, even with Ut > U, are flung upwards by the bursting bubbles and then fall back. Above the height they reach, only fines, i.e. particles with Ut < U, are found. The concentration of fines further decreases with height and eventually reaches a constant value. The projection height of the coarse particles is called the splash height or TDH(C). The height above which the fines concentration remains nearly constant is called TDH(F). The definition of a coarse or fine particle will depend upon its Ut and upon U. Ut can be calculated according to Geldart [1]. The TDH and the solids loading in the freeboard influence the entrainment rate and the rate of internal circulation of solids [2], the heat transfer to surfaces in splash zone and freeboard [3], and the progress of reactions in the freeboard [4]. The TDH has been studied by different methods, including measuring the

* Corresponding author. E-mail addresses: [email protected], [email protected] (A. Brems).

entrainment rate in batch or semi-continuous experiments at various velocities for different positions of the gas offtake [5,6]; by visually observing the height above which no downward moving coarse particles are observed [7]; by particle sampling [8]; or from the curve of the pressure drop profile versus height in the freeboard [9,10]. The present paper reports results of real time particle trajectories as measured for the first time in this work by positron emission particle tracking (PEPT). Although the measurement of the pressure gradient along the height of the freeboard is simple and robust, Smolders and Baeyens [10] illustrate that these gradients are generally too small to accurately predict TDH. Visual observation of downward moving marked particles, of different size, is possible although it is considered a tedious task within the splash zone above the bed and does not provide information about the particle trajectories. The particle size analysis at different locations of the gas offtake can provide a reasonable estimate of TDH, although the offtakes can only be located at specific heights in the freeboard, thus limiting the accuracy of the measurement to the separation distance of the offtake point. Sampling probes are of intrusive nature and known to having an effect on the gas–solid flow, thus also affecting the particle samples. Sampling will also be conducted at specific points in the freeboard only and is difficult to use near the wall region of the freeboard. The PEPT technique is non-intrusive, without interference on the gas–solid flow. It provides real time axial and radial tracer particle occupancies and velocities. The particle ejection trajectory can immediately be followed, even in the complex case of coalescing bubbles.

0921-8831/$ - see front matter Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2010.07.012

156

A. Brems et al. / Advanced Powder Technology 22 (2011) 155–161

Nomenclature C CD D dB dB0 dp dp,max F g h H HS N0 Re t

coarse (particles) drag coefficient () diameter of the bed () (m) bubble size (m) diameter of bubbles at the bed surface (m) particle size (m) size of particle with Ut = UB (m) fine (particles) gravitational constant (m/s2) height above the bed surface (m) height above the distributor (m) settled bed height (m) number of holes in distributor (1/m2) Reynolds number, as defined by Eq. (5b) () time (s)

2. Empirical TDH equations Several empirical equations to predict TDH have been presented in literature and are cited in Table 1 with their determining parameters. The TDH can thus be predicted when calculating dB0 using, e.g. Darton’s equation [20], and UB according to, e.g. Werther [19]. For a 90 lm sand in a 0.5 m deep bed of 1 m I.D. at U = 0.4 m/s, predictions differ significantly between 0.55 m [10] and 0.7 m [17,18], to approximately 1.8 m [13–16], and 2.2 m [12]. Similar differences are predicted for different bed materials and operating conditions. There is an obvious need to better predict TDH-values.

TDH(C) TDH(F) U UB UFB Umf Ut v

vr l qg qp XB XFB

transport disengagement height for coarse particles (m) transport disengagement height for fine particles (m) superficial air velocity (m/s) velocity of erupting bubbles (m/s) velocity of air in the freeboard (m/s) minimum fluidization velocity of particles (m/s) terminal velocity of particles (m/s) velocity of particles (m/s) slip velocity (m/s) gas viscosity (Pa.s) density of air (kg/m3) density of particles (kg/m3) surface of the bed (m2) surface of the freeboard (m2)

Operating air velocity and particle size play important roles. The height of projection increases with increasing superficial air flow rate, and/or with decreasing particle size. The TDH-values within given operating conditions show a wide range of values due to the distribution of bubble sizes and velocities and the possible coalescence of two bubbles prior to erupting at the surface. Data were hence analysed statistically, and given below with a value obtained at 50% and 99% of the cumulative TDH curve respectively. Some of the data are illustrated in Tables 2a and 2b. 4. Modelling approach to predict TDH The fundamental model equations were adapted from Do et al. [23] and transformed to make a distinction between fine and

3. Experimental set-up, procedure and illustration of results PEPT is a technique to track a fast moving particle in opaque vessels. It has been extensively applied for obtaining dynamic information of powder flow in a very wide variety of processes. 18 F is prepared and is incorporated into the tracer by an anion-substitution surface adsorption procedure [21,22]. The tracer is located by triangulation of a number of detected annihilation events, and this every 4 ms, thus providing trajectory and velocity information. The bulk bed material was rounded sand (120 lm, 2260 kg/m3) and rounded sand particles of 150, 250, 330, 390, 460 and 550 lm were labelled, as well as alumina (135 lm) and coal (390 lm), each having a density of 1600 kg/m3. A fluidized bed of 0.16 m I.D. with porous plate distributor (high DP) was used, for static bed heights of 0.25, 0.35 and 0.45 m. The superficial gas velocity was varied between 0.029 and 0.352 m/s, mostly in the freely bubbling regime. The detectors, each 0.47 m wide and 0.59 m high, were positioned between 5 and 65 cm above the static bed height, thus allowing for bed expansion. The principle of the experimental layout is shown in Fig. 2. The tracer particles were introduced with the bed material. The exhaust of the rig was connected to a bag filter. The obtained data, an extensive list of consecutive particle locations in three dimensions, determined the instantaneous velocity as well as the location of the particle in the freeboard. The PEPT-experiments lasted for several hours at each combined operating condition. Tracer particles were seen to move each time with an upward and a downward movement. Occasionally, tracer was seen to be re-entrained during its movement by a second erupting bubble. The tracer trajectories are illustrated in Figs. 3 and 4 for the single and multiple jump registration respectively.

Fig. 1. Solids loading profile in a fluidized bed and its freeboard.

157

A. Brems et al. / Advanced Powder Technology 22 (2011) 155–161 Table 1 Empirical correlations to predict the TDH. Authors

Dominant parameters

Comments

Soroko et al. [11]

Settled bed height (Hs) Particle Re and Ar numbers Superficial gas velocity (U) Superficial gas velocity (U) Diameter of bubble at bed surface (dBo) From entrainment rates Bubble velocity and diameter; gas velocity, density and viscosity; particle diameter and density Diameter of bubble at bed surface (dBo) Graphical representation as function of (U Umb) and dBo Bubble velocity (UB) Diameter of bubble at bed surface (dBo) and excess gas velocity

Only valid for TDH(C) Hs < 0.5 m, dp 0.7–2.5 mm TDH(F) TDH(F), for FCC powder TDH(F) TDH(F), for D < 0.6 m TDH(F)

Amitin et al. [12] Fournol et al. [8] Horio et al. [13] Wen and Chen [14] Pemberton and Davidson [15] Hamdullahpur and MacKay [16] Zenz [17] Baron et al. [18] Smolders and Baeyens [20]

coarse particles. Consider an individual particle of size dp and density qp, projected into the freeboard by a bursting bubble rising with velocity UB. The superficial gas velocity in the freeboard is U. The instantaneous particle velocity is v. In a dilute particle flow, without collisions and associated momentum transfer, a balance of forces yields:

TDH(F) TDH TDH(F), UB calculated from Werther [19] TDH(F)

vr ¼ U v qp qg g qp 3 qg b¼ CD 4 qp dp

a¼

Eq. (1) is reduced to:

qp

p

3 dp

3 dp

2 dp

p p 1 dv q ðU v Þ2 ¼ ðqp qg Þg CD 6 dt 6 4 2 g

ð1Þ

The ‘’ sign applies to the region where v > U, the ‘ + ’ sign to the zone where U > v. Introducing the slip velocity, vr, and group-constants a and b:

dvr ¼ a b v r jv r j dt with

dvr ¼ a þ b v 2r dt

ð2Þ

vr < 0

ð3aÞ

5 3 4

1

2

2 Fig. 3. Illustration of a single jump of a 550 lm tracer at U = 0.2 m/s.

6

to computer

compressed air Fig. 2. Experimental set-up with: (1) BFB, (2) c-ray cameras, (3) vent to (4) filter and (5) atmosphere, (6) tracer.

Fig. 4. Illustration of multiple jumps of a 462.5 lm tracer at U = 0.124 m/s.

158


Table 2a TDH (in m) of alumina and coal: TDH0 = minimum TDH measured at lowest velocity, OR = outside range of camera.

pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:54ðU U mf Þ0:4 ðH þ 4 0:001Þ0:8 dB ¼ g 0:2 UB is calculated from Werther [19]:

Alumina

Coal

dp = 135 lm

dp = 390 lm

pffiffiffiffiffiffiffiffiffiffi U B ¼ ðU U mf Þ þ / g DB

TDH50

TDH99

TDH50

TDH99

0.125 0.166 0.255 0.375 0.394 0.515 TDH0 = 0.035

0.241 0.304 0.477 0.609 0.646 OR

0.037 0.084 0.254 0.306 0.340 0.382 TDH0 = 0.024

0.071 0.154 0.416 0.576 OR OR

Table 2b TDH (in m) of sand particles: TDH0 = minimum TDH measured at lowest velocity, OR = outside range of camera. U (m/s)

dp = 150 lm

dp = 320 lm

dp = 460 lm

dp = 550 lm

TDH50

TDH99

TDH50

TDH99

TDH50

TDH99

TDH50

TDH99

0.023 0.069 0.105 0.182 0.242 0.351 TDH0

0.091 0.152 0.185 0.364 0.382 0.455 0.027

0.156 0.278 0.369 0.582 0.642 OR

0.127 0.145 0.158 0.206 0.235 0.307 0.046

0.186 0.240 0.312 0.440 0.515 0.612

0.131 0.139 0.145 0.185 0.226 0.295 0.034

0.167 0.291 0.312 0.462 0.526 0.611

0.146 0.194 0.162 0.185 0.224 0.266 0.041

0.223 0.275 0.308 0.375 0.452 OR

dvr ¼ a b v 2r dt

vr > 0

ð3bÞ

The equation was solved for an initial boundary condition that corresponds with vr(t = 0) = U UB. Therefore, the height that a particle reaches, can be calculated from:

dh ¼ v ¼ U vr dt

ð4Þ

CD, the drag coefficient, is itself a function of the instantaneous velocity and can be calculated over the whole velocity range by Eq. (5a) below [23]:

CD ¼

24 0:42 ð1 þ 0:15 Re0:687 Þ þ Re 1 þ 42500 Re1:16

with Re ¼

dp ðU v Þ qp

l

ð5aÞ

ð5bÞ

Smolders et al. [10] previously demonstrated that only particles with Ut 6 UB will be ejected, thus fixing the maximum particle size, dp,max, ejected by the bubbles.

Ut ¼ UB

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ðqp qg Þ dp;max g and U t ¼ 3 qg C D

ð6Þ

For reasons of simplicity, it is assumed that the freeboard is not tapered, i.e. that a single value of the superficial gas velocity U can be used throughout the freeboard height and is equal to the superficial velocity in the fluidized bed itself: XB = XFB. The bulk bed sand in the equipment being a Geldart B-powder, the operation was mostly freely bubbling (according to the Yagi and Muchi criterion [24]), except at high air velocities in the 45 cm deep bed. If bubble splitting should occur (mainly A-powders), a stable bubble size is calculated using the method of Geldart as reported by Clift [25]. For freely fluidized beds, and when using a porous plate or multi-orifice distributor with N0 1000/m2, the bubble diameter dB is calculated from Darton [20]:

ð8Þ

where u depends on the type of powder (group A or B) and the diameter of the bed, D:

for B-powders :

for A-powders :

or

ð7Þ

8 D 6 0:1 m > < 0:64 / ¼ 1:6 D0:4 0:1 < D < 1 m > : 1:6 DP1m 8 1 D 6 0:1 m > < / ¼ 2:5 D0:4 0:1 < D < 1 m > : 2:5 DP1m

If slugging occurs (when dB > D/3), the maximum bubble size is then limited to D/3, and the slug velocity of rise should be used [26]. From these equations and expected bubble sizes, it is evident that UB > U. The solution of differential Eqs. (3a) and (3b) with conditions of t = 0 at vr = vr0 and t = ti at vr = vri is as follows [27]:

rffiffiffi ! rffiffiffi ! b b v ri Arctg v r0 a a

vr < 0

1 t i ¼ pffiffiffiffiffiffi Arctg ab

vr > 0

"pffiffi pffiffia # a v ri þ v r0 1 b b pffiffia t i ¼ pffiffiffiffiffiffi ln pffiffia þ v ri v r0 2 ab b b

ð9aÞ

ð9bÞ

Since the drag coefficient CD is a function of vr, a stepwise calculation needs to be used, whereby Dv is calculated in steps of, e.g. 0.01 m/s. The corresponding CD and each corresponding value of ti are calculated. The path length is calculated by:

hi ¼ v i t i ¼ ðU v ri Þ ti

ð10Þ

The particle trajectory is calculated by hi as function of ti. When the particle reaches its final velocity, all hi are summed to calculate the total path covered by the particles. According to the difference between their terminal velocity and respective bubble or superficial gas velocity, we can classify the particles in three groups: – Ut > UB: these particles are not ejected and need not be considered in TDH – U < Ut < UB: As shown in Fig. 5a, particles leave the bed at the bubble velocity UB. At first, the particle velocity exceeds the gas velocity (vr < 0), and the particle is decelerated by gravity and drag. When its velocity becomes smaller than the gas velocity (vr > 0), it is still decelerated by gravity, whilst the gas now tries to accelerate it. Finally, the particle velocity becomes zero (vr = U) and its direction of motion is reversed: the particle falls back into the bed. It is evident that in the deceleration zone, both regimes of vr, will occur since initially v will exceed U, but after deceleration. U will exceed v. Hence both formulae (7a) and (7b) will be used to calculate the total value of time t. – U > Ut: Fig. 5b applies for this case. The particles are also projected at the bubble velocity UB. The particle velocity exceeds the gas velocity (vr < 0) and the particle is decelerated by gravity and drag. When its velocity becomes smaller than the gas velocity (vr > 0), it is still decelerated by gravity, whilst the gas tries to accelerate it. When vr equals the terminal velocity Ut of the particle, the system is in equilibrium and the particle will be carried through the freeboard at this constant velocity (U Ut). According to the definition of TDH(C), one can reasonably assume that this TDH(C) corresponds with the maxi-

159


Height above the bed

Zone of deceleration

Height above the bed

Zone of re-aceceleration

Zone of deceleration

Zone of constant particle velocity (U-Ut)

v

TDHc v

v v

Ugas

v

v

v TDHf

v

Ugas v

v Hbed

Hbed t=0

time

t=0

time

kinetic

drag

vr

drag

vr

(gravity – buoyancy)

vr >0 0

U

(gravity – buoyancy)

vr >0 -v U

-v vr = 0 vr 0 -v 0

U

U -v vr = 0 vr U) and (b) fine particles (Ut 6 U).

mum height reached by those coarse particles with Ut = UB. Similarly, the height at which a particle with Ut = U reaches its final velocity (vr = Ut or v = 0) is called the TDH(F). Do et al. [23] claimed that the coarser the particle, the greater the maximum height the particle can reach, hence they expect TDH(C) to exceed TDH(F). As will be shown hereafter, TDH(F) is not always smaller than TDH(C) and both TDH(F) and TDH(C) depend upon the combination of U and UB. The above treatment is only valid for dilute particle flow, without particle interactions. Since the elutriation rate increases significantly at higher operating air velocities, this assumption is no longer valid at high U, and corrections should be introduced for particle clustering and associated modified drag coefficient, as illustrated by Chan et al. [28] for the acceleration zone of a CFB riser. The reader is hence cautioned not to use the proposed model approach beyond the experimental excess air velocity of 0.35 m/s. Experiments are currently ongoing up to velocities of 1 m/s. Results will be reported in a follow-up paper.

5. Modelling of pept-results The TDH50-values of Table 1 are in fair agreement with empirical predictions of Smolders and Baeyens, as illustrated in Fig. 6. The Smolders and Baeyens equation predicts TDH from [10]:

TDHðFÞ 6½ðU U mf ÞdB0

0:6

ð11Þ

The graphical prediction by Zenz [17] is illustrated in Fig. 7 Clearly for the experimental operating excess velocities of 0.05– 0.35 m/s with predicted bubble eruption diameter of 0.025– 0.125 m, the Zenz predicted TDH-values are between 0.04 and approximately 0.5 m, in line with the PEPT experimental data. Other empirical equations considerably overestimate TDH. Fig. 8 summarises all experimental TDH50 data in comparison with the more fundamental model predictions. A good agreement is obtained, provided the air velocity throughout the particle jetting zone is taken as UB. Since TDH99 is about twice TDH50, probably due to the effect of coalescing bubbles near the bed surface, the TDH99 value is only

160


Fig. 6. Comparison of experimental and predicted TDH using the Smolders and Baeyens correlation [10].

predicted by the model when using a bubble eruption diameter (and associated velocity) corresponding to about 1.5 dB0.

6. Application of the model equation to freely bubbling fluidized beds The model equations can be used to predict TDH(F) and TDH(C) and can also predict the particle trajectories in the freeboard of any

Fig. 8. Comparing TDH model with TDH experimental.

fluidized bed operating at different conditions by using the combined Eqs. (9a and b) and (10). This has by way of example been

10 7 5 3

1 0.7

TDH (m)

0.5 0.3

dB0 (m) 0.025

0.1

0.075

0.07

0.152

0.05

0.305 0.457

0.03

0.610 0.762 0.914

0.01 0.01

0.03

0.050.07 0.1 U-Umb (m/s)

0.3

0.5 0.7

Fig. 7. Empirical prediction of TDH(F) according to Zenz [17].

1


161

results for various velocities. Points marked ‘F’ or ‘C’ correspond, respectively, with model calculated TDH(F) and TDH(C). It can be seen that calculated TDH(F) values exceed TDH(C) at high velocity, whereas the inverse is true at low velocities. Hence either TDH(F) or TDH(C) will determine the lowest position of the bed offtake in function of the velocity applied. Calculations for specific gas velocities (Fig. 10) illustrate how particles of 50 lm achieve their constant velocity from a height of approximately 0.04 m above the bed surface, whereas particles of 212 lm will reach a height of 0.1 m prior to returning to the bed. Particles of 92 lm have a terminal velocity similar to the gas velocity (= 0.5 m/s) and will be held in the gas stream with a particle velocity zero. In this case, TDH(C) will determine the position of the gas off-take. 7. Conclusions

Fig. 9. Predicted TDH(F) and TDH(C) in a 0.4 m I.D. BFB using sand of dp = 40– 800 lm.

PEPT gives a clear picture of the TDH for coarse particles. Model predictions improve understanding the fundamentals of particle projections and the TDH, with a different behaviour of coarse and light particles. A good agreement of experiments and model predictions is obtained, provided the air velocity throughout the bubble eruption/jetting zone is taken as UB. Since TDH99 is about twice TDH50, probably due to the effect of coalescing bubbles near the bed surface, the TDH99 value is only predicted by the model when using a bubble eruption diameter (and associated velocity) corresponding to about 1.5 dB0. References

Fig. 10. Predicted TDH(F) and TDH(C) in a 0.4 m I.D. BFB using sand of dp = 40– 800 lm at U = 0.5 m/s.

performed for a 40–800 lm sand of density 2600 kg/m3, fluidized by air at ambient conditions in a fluidized bed of 0.4 m diameter. Bubble eruption sizes were calculated using Darton’s equation [20] for an expanded bed height of 0.5 m, limited however by the slugging condition (D/3). Freeboard and bed have the same diameter and hence equal gas velocity. Fig. 9 illustrates calculated

[1] D. Geldart, Gas Fluidization Technology, John Wiley and Sons Ltd., N.Y., 1986 (Chapter 6). [2] D. Geldart, N. Broodryk, A. Kerdoncuff, Powder Technol. 76 (1993) 175. [3] S.E. George, J.R. Grace, AIChE J. 28 (1982) 759–765. [4] J. Baeyens, F. Van Puyvelde, J. Hazard. Mater. 37 (1994) 179–190. [5] I.H. Chan, T.M. Knowlton, Paper Presented at Ann. Meeting of AIChE, 1984. [6] H.J.A. Schuurmans, 8th Int. Symp. on Chem. React. Eng., I. Chem. E. Symp. Ser., Vol. 87, 1984, p. 495. [7] M. Sciazko, J. Bandrowski, J. Raczek, Powder Technol. 66 (1991) 33. [8] A.B. Fournol, M.A. Bergougnou, C.G.J. Baker, Can. J. Chem. Eng. 51 (1973) 401. [9] D. Geldart, Y. Xue, H.Y. Xie, AIChE Symp. Ser. 91 (308) (1995) 93–101. [10] .K. Smolders, J. Baeyens, Powder Handling Process. 9 (2) (1997) 123. [11] V. Soroko, M. Mikhalev, I. Muklenov, Int. Chem. Eng. 9 (1969) 280. [12] A.V. Amitin, I.G. Martyushin, D.A. Gurevich, Khim. Tk. Top. Mas. 3 (1986) 20. [13] M. Horio, A. Taki, Y.S. Hsieh, I. Muchi, Fluidization, in: J.R. Grace, J.M. Matsen (Eds.), Plenum Press, 1980, p. 509. [14] C.Y. Wen, L.H. Chen, AIChE J. 28 (1982) 117. [15] S.J. Pemberton, J.F. Davidson, Chem. Eng. Sci. 41 (1986) 253. [16] F. Hamdullahpur, G.D.M. MacKay, AIChE J. 32 (12) (1986) 2047. [17] F.A. Zenz, Chem. Eng. (Nov) 28 (1983) 61–67. [18] T. Baron, C.L. Briens, M.A. Bergougnou, Can. J. Chem. Eng. 66 (1988) 749. [19] J. Werther, Fluidization IV, Japan, Paper 1–12, 1983. [20] R.C. Darton, Trans. Inst. Chem. Eng. 57 (1979) 134. [21] X. Fan, D.J. Parker, M.D. Smith, Nucl. Instrum. Method A 558 (2006) 542–546. [22] X. Fan, D.J. Parker, M.D. Smith, Nucl. Instrum. Method A 562 (2006) 345–350. [23] H.T. Do, J.R. Grace, R. Clift, Powder Technol. 6 (1972) 195. [24] S. Yagi, I. Muchi, Chem. Eng. (Japan) 16 (1952) 307. [25] R. Clift, Gas Fluidization Technology, John Wiley and Sons Ltd., New York, 1986 (Chapter 4). [26] J. Baeyens, D. Geldart, Chem. Eng. Sci. 29 (1) (1974) 255–265. [27] C.R.C. Standard Mathematical Tables, 13th ed., Chemical Rubber Corporation, 1964, p. 294. [28] C.W. Chan, J.P.K. Seville, Z. Yang, J. Baeyens, Powder Technol. 3 (2009) 318– 325.