Effects of pressure and type of gas on particle-particleinteraction and the consequences for gas-solidfluidization behaviourCitation for published version (APA):Piepers, H. W., Cottaar, E. J. E., Verkooijen, A. H. M., & Rietema, K. (1984). Effects of pressure and type of gason particle-particle interaction and the consequences for gas-solid fluidization behaviour. Powder Technology,37(1), 55-70. DOI: 10.1016/0032-5910(84)80006-6

DOI:10.1016/0032-5910(84)80006-6

Document status and date:Published: 01/01/1984

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Powder Technology. 37 (1984) 55 - 70 55

EFFECTS OF PRESSURE AND TYPE OF GAS ON PARTICLE-PARTICLE INTERACTION AND THE CONSEQUENCES FOR GAS-SOLID FLUIDIZATION BEHAVIOUR

H.W_Piepers, E.J.E.Cottaar, A.H.?A_Verkooijen and K.Rietema Department of Chemical Engineering Eindhoven Universit-y of Technology, The Netherlands

Abstract

The fluidization behaviour of cracking catalyst has been studied up to pressures of 15 bar with different fluidization gases (Ar, W2, HP)_ A number of parameters of both the homogeneous and heterogeneous flui dized bed has been examined experimentally. Tine experimental results reveal that the minimum fluidization velocity (U f) is independent of the pressure_ an3 the maximum bed expansion (H bp)

The bubble point velocity (Ubp) at this velocity increase with

increasing pressure. This also holds for the dense phase voidage (cd) and the dense phase gas velocity (Ud) in the bubbling bed. The bubble size decreases drastically with increasing pressure_ However, the above-mentioned parameters are also strongly dependent on the t:/pe of fluidization gas used, The cohesion constant of the powder was measured, using a tilting bed technique. The results reveal that the cohesion constant increases with increasing pressure_ Analysis of the results of adsorption measurements of the different gases to the solid reveals for the ad- sorption as well as for the cohesion and for the beu expansion the same pressure dependence_ It is believed that the gas adsorption influences the cohesion between the particles and hence the elasticity modulus introduced by Rietema and Mutsers [1,21. The increasing elasticity modulus with increasing pressure also explains the increasing bed expansion with pressure.

Introduction

The interest in fluidized bed behaviour at high pressures has increased considerably in the last years, because knowledge of the fluidization behaviour at high pressures is important to predict the bed performance as a chemical reactor C31_ tiany investigators 14-91 who studied the effect of pressure or gas density on fluidized bed behaviour reported that high pressure results in an increase of bed expansion and minimum bubble point velocity U for group A powders of Geldart's classification tlO1. From the resu% of their experiments on 48 gas-solid systems Abrahamsen and Geldart 1111

0032-5910/84/53-00 0 Elsevier Sequoia/Printed in The Netherlands

56

carrel ated the minimum bubble point velocity, U $

, and the maximum non-bubbl ing bed expansion ratio, lib /H,f- Actor ?ng to this carrel ation

ubP ity and ?he.viscosity of the fluidizing gas increases wJt~5th_‘Od5@_

according to pg - .u

i n6:g%esOw11~ _

The maximum bed expansion ratio, Hbp/Hmp gas density and gas viscosity according to

% e eff& of pressure on bubbling has also been studied extensively [5,8,9,12-141. Evidence is available that in beds of fine powders in- creasing pressure leads to the occurrenc- * of smaller bubbles _ A qua1 i- tati ve explanation [ 123 has been given in terms of increased bubble splitting by Taylor instabilities with increased dense phase expansion. Another explanation was advanced in terms of bubble stability based on particle pick-up from the lower surfaces of the bubbles [8,9]. However. none of the investigators explain why for fine powders the non-bubb7 i ng bed expansi on increases w. i th increasing pressure _ There is also no agreement on the mechanism that is determining the decreasing bubble size with increasing pressure. In both phenomena, however, the properties of the dense phase play an important part. Rietema and Mutsers 11,2] assert that interparticle cohesion forces play an important part in the non-bubbling bed expan- sion of fine powders. According to these authors the interparticle cohesion forces give rise to a powder structure in the bed with a certain mechani cal strength , even in the expanded state. They developed a theory in which they assigned a certain elasticity, E, to the powder structure , resulting from the interparticle cohesion forces. From this theory they derived a cri terium rel sting the elasticity modulus E of the bed to the voidage of the bed, &bp, at the bubble point velocity

3.4 2 2 pP dP g =

p2Ebp (1)

In a “human centrifuge” Rietema and Nutsers 1151 studied the fluidi- zati on behavi our at accelerations whi ch exceeded gravi tati onal accel e- rati on _ The theory they developed predicts exactly the maximum non- bubbling bed expansion in the centrifugal field. The aim of the present study is to investigate the influence of pres- sure and type of gas on non-bubbling and bubbling bed behaviour and to explain this behaviour in terms of interparticle forces, It is be1 i eved that adsorption of gases to the sol id influences the cohesion between the particles and hence the elasticity modulus, E.

ExDeri mental

Fl ui d bed experiments

A diagrammatic flowsheet of the fluidization apparatus is shown in figure 1. The fluidized bed consisted of a glass tube 82 mm ID, 100 mm OD and 145 cm 1 ong, which incorporated a porous brass gas dis- tributor, The top and bottom sections were made of brass and the whole system was held tight by means of four compression bars and O-ring seals . The fluidizing gas, initially fed from -a pressurized cylinder, was circulated in a closed-loop system by means of a compressor. Before and after the compressor a pressure vessel was installed to dampen flow rate fluctuations _ The gas fiow rate in the system was measured

57

FigzJre 2 Schematics of the aper4me-ntaZ szt-xp.

with a set of calibrated rotameters. Pressure tappings connected to a water manometer were used to measure the pressure drop over the bed and the static pressure was given by a manometer connected to the top section of the bed. A movable capacitive probe, mounted in the bed, was used to measure the bubble frequency. The height of the bed was measured by reference to a tape on the out- side of the glass tube. The same batch of cracking catalyst (d =59_4 urn, p =887 kg/m3) was used in all experiments_ The powder fell inPGeldart's gpoup A 1101_ Hith an average packed bed height of about 60 cm this powder was fluidized with N2, Ar and H2' at pressures u, n to 15 bar, The physical properties of the gases are given in table 1. At each flow rate, the pressure drop and the bed height were recorded-

gas density viscosity

I (Wm3) (Ns/m2)

Hz 0.084 87.6 x 1O-7

hi2 1.165 173 x lo-7

Ar 1.662 220 x 10-7

The mi nimum f 1 ui di zati on vel oci ty, Umf was determined by the inter- secting of the two linear portions 01 the velocity-Fressure drop curve. From a graph of the bed height,H,against the superficial velocity the bubble point Velocity, Ub , mined_ The bed voidage w s %

andthe maximilm bed height,Hb,,were deter- calculated from: .

58

E =I- W/ppxHxA (2)

For the heterogeneously f 1 ui di zed bed, the height of the dense phase, Hd, the dense phase gas velocity, Ud’ and the bubble hold-up, d, were calculated from so-call ed collapse experiments, a method suggested by Rietema 1163. Tine bed was first fluidized for several minutes at a superficial gas velocity higher than Ub - At a given moment, t=O, gas supply to the bed is suddenly shut o’Ff and the level of the bed

the

fa7ls ouicklv to a certain value HI, because of the Quick release of gas bubbles jn is much slower

the bed. Below this&value H1, the setiling of (figure 2) -

the bed

Figure 2 CoZZapse experiment. TotaZ bed hei_aht as fvnction of time.

The co1 lapse experiments were carried out at pressures up to 15 bar for both PI2 and At-, The time and the bed height were continuously recorded on a videofilm. A result of the analysis of a videofilm of such a collapse experiment is Shown in figure 2. The rate at which the top of the bed comes down below HI, -dH/dt, is equal to the super- ficial gas velocity through the dense phase,Ud. before shutting off the gas. The dense phase expansion,Hd , is obtained by extrapolation towards t=O, as shown in fioure 2. Values of the dense E , were cal &l ated from equation (2). Tde bubble hold-up,& is defined as the fraction bubble and as such may be calculated from

phaje voi dage,

vol ume/bed vol ume,

6 = (Hm-Hd)/Hm (3)

With Ar and N as fluiditing gas bubble frequencies were measured at a height of 6g cm above the gas distributor at pressures up to 15 bar and superficial gas velocities up to 5 cm/s.

Ti 1 ti ng f 1 u-i di zed bed experiments

In classical equipment for the determination of the cohesive and fri c- tional properties of powders relatively 1 arge compressive stresses are used. Consequently the cohesion of the powder, the shear strength, T

59

at zero compressive stress 0, is determined by extrapolation to a=O. The tilting fluidized bed is partictilarly suited to measure the shear strength of powders at low compressive stress _ The ti 1 ting bed _ con- structed as an ordinary fluid bed (diameter 10 cm, height 20 cm), was mainly fabricated of thick perspex in order to observe the bed materi al directly. To facilitate pressurized operation the bed was placed in a closed-loop system. The bed could be tilted very slowly and without shocks to an angle CY with the horizontal by means of a smoothly running electric motor combined with a speed reducing gear. Mhen in a fluidized bed a powder layer is formed by collapse from the heterogeneous?y fl uidized state and consequently the bed is ti 1 ted over an angle a the following forces act upon a 1 ayer with height H (figure 3)

a = pp(l-s)g H cos a

T = pp(l-&)g H sin a (4)

It can be seen that the shear strength, T, is largest at the bottom and failure will occur here first. When at the same time gas is passed through the bed a force due to the pressuredrop.AP, also acts on the 1 ayer (fi gure 4). Consequently:

o = pp(l-e)g H cos a - AP

T = pp(l-e)g H sin a (5)

From this it will be clear that the shear strength T at which failure occurs can be measured with decreasing values of the normal stress by increasing the pressure drop over the powder layer. In this way the cohesion can be determined more accurately.

AP

Figure 3 Force baihnce for a Figm=e 4 Force baZance _f.or a powder Zayer in a pmder layer in a tilting tiZtingbe8 (nogas ,cZow). bed <with gas ftml.

In order to measure the influence of pressure on the cohesion properly a number of precautions must be taken to eliminate other influencing parameters. The experiments were done with cracking catalyst in an atmosphere of Ar at pressures up to 11 bar for gas velocities 4Jmf. Care was taken to carry out the experiments with a powder bed of constant porosity and identical consolidation history. To measure small normal stresses and to avoid wall effects the bed heSght was 0.02 m.

60

Analysis of results

Minimum fluidization velocity (Umf>

For Ar and N the measured values of the minimum fl uidi zation velocity, U mfwithin t e experimental scatter did not show any systematic depen- i; dence on pressure. This is in agreement with the findings of other investigators 114-81 for fine powders. ds .reases a little with pressure. This

With H2 as fluidizing gas U is due to a change in the @d

voi dage, cmf. For H i ncreas i ng 2

the bed voidage E f decreases a little (4%) with pressur . On the other hana with Ar and N2 the bed voidage

Emf increases with increasing pressure. This is in agreement with the results of Sobreiro et al. 161 who reported that for a fine powder in a N2 atmosphere cmf seemed to be consistently lower at ambient pressure than at high pressure. However, for all pressures em was found to *increase with increasing gas viscosity. The results Q figure 5) could very we1 1 be represented by the 1 aminar part of the equation of Ergun 1173:

From this equation it follows that (Umf( l-cmf))jEif should be independent of pressure.

37

I

2- WI

k - * Y- .-. A_

xE l- * o .-.-.-_ .

E W

T - 2 CF

00 0 3 6 9 12 15

P (bar)

Figure 5 TJze variatia of U mf

with pressure (aAr, A N2, P Hz'.

Bubble point Velocity (Ub,) and bed voidage !gb,)

The bubble point ve7ocity, U be’

showed the characteristic behavi our found in fluidization of gro p A powders, pressure [4,6.71_

increasing with increasing

In figure 6 the results of the measurements are given. From this figure it is seen that the type of fluidizing gas has a very strong effect on

2.0

.56j I _c~I-c-l-L=-I i

! I_. __.-1-w

P (bar) J P (bar) _ ~-

61

the variation of U with pressure. With Ar the variation with pressure is strongest, whilgPfor H2 only a very weak dependence on the pressure was found. The same picture we get for the bed voidage at the bubble ooint velo- city,cbp. The dependence of

k with pressure is shown in figure 7.

The results reveal that I+, i Ereases steadi ly with pressure. Argon gives the highest voidage,Pwhile with H2 only a very weak increase of

cbP with pressure was found.

2.5 .68,

.A*

-54 - - I 1 -L-----T---~-- i 0 3 6 9 12 15 0 3 6 9 12 15

Figure 6 Fi_nuPe 7

The variation of U l p with PN2, I H&

The variation of E witir pressra2. pressure f e Ar, t 0 Ar, A iJ2, Ht?$f

in figure 8 the bed height at’ the bubble point velocity, H - _ as function of the gas density. In this figure the val uesb~~~k?~~~~ with the correlation of Abrahamsen and Geldart 1111 are also given. It can be seen that the dependence on the gas density given by this correlation is too low.

Bubb‘ling bed

Voi dage of the dense phase (IQ) and dense phase gas velocity (Ud)

With Ar and H 3

a series of collapse experiments were done for super- ficial gas ve oci ties between 2 and 5 cm/s and at pressures up to 15 bar. From these experiments Ed given in table 2,

and Ud were determined. The results are It can be seen that for the same gas velocity E and

Ud increase with increasing pressure and Ed and U are always big er

f! “9

-ti for Ar than for N2_ At low pressures the values o with increasing superficial gas velocity. However, a

and Ud decrease increasing pres-

sure the values are almost constant for all gas velocities_ This is in agreement with the results of other authors t8,181_

62

According to the correlation for the dense phase height,Hd,proposed by Abrahamsen and Gel dart 1183 Hd should increase with gas density according to p 0-O16_ This means that Hd should increase with 4.5% with an increak!e in pressure from 1 bar to 15 bar. However, the results reveal that the actual increase is 28% for Ar and 21% for N2. The dependence on the pressure or gas density given by this correlation is again far too low.

Bubble phase properties (9, f, Ubo, ub) Db)

The simplest model for a bubbling fluidited bed is the two-phase theory, which assumes EdTEmf and that all the gas in excess of that required for minimum fluldlzation passes through the bed as bubbles. The validity of this assumption has been questioned by several authors and Rietema Cl61 among others suggested that for fine powders during bubbling the dense phase has a higher voidage than that at Umf. Under very minimal assumptions Ri etema and 01 trogge El91 have derived that regardless of solids flow pattern around the bubble or occurrence of a bubble cloud the total gas flow balance can be written as

u, = (1+26)ud + 6$ (7)

From the collapse experiments the bubble hold-up,rS,is determined as function of the superficial gas velocity,Uo,and the pressure for Ar and N2_ With the values of d and the corresponding values of t&e dense phase gas ve‘locity U

-$’ the superficial_bubbie velocity Ub = 6Ub and

the mean rise veloc? y of the bubbles Ub could be calcula?ed.

63

With the capacitive probe the bubble frequency,f,was measured at a height of 65 cm above the gas distributor, for the same conditions under which also the collapse experiments were performed_ With the values of the measured bubble frequency,f,and the superficial bubble velocity,U amean bubble diameter can be estimated_ If we assume thatb?ie mean diameter of the bubbles is nb and the number of bubbles rising per square meter per second equals N it follows:

'bo = N x */6(Db3)

Hence for the mean bubble diameter Db we set:

' x 'bo Db= f (8)

The results of the measurements and calculations are given in table 2. The results show -as expected- an increasing bubble hold-up with in- creasing gas velocity_ The bubble hold-up did not show significant differences for Ar and N2 and there is an overall increase in d with increasing pressure. The superficial bubble velocity,Ubo,shows an increase with increasing gas velocity and Ubo decreases with pressure- This is mainly due to the fat t that Ud increases with pressure_ The lower values of Ubo for Ar than for N2 are also due to the higher dense phase gas velocity,U .for Ar. The same holds for the mean bubble rise velocity,$, though tde effect here is not so pronounced_ At higher pressures lTb becomes more or less constant for gas velocities of 3 cm/s and higher. Contrary to the bubble hold-up, 6, the bubble frequency,f,strongly increases with pressure and always a higher fre- quency was measured for Ar than for N2_ According to equation (8) a decreasing Ub and an increasing frequency with pressure yields a decreasing bubble 2iameter,nb,with pressure. It can be seen from table 2 that the bubble diameter for Ar is smaller than for N2 and for both gases_ there is an decrease in nb with pressure_ For the highest gas velocity the decrease in bubble diameter is about a factor 3 to 4 in the measured pressure range, The effect of pressure on bubbling behaviour is in agreement with the results of other authors C5,8,9,12-I41 who reported smaller bubbles at higher pressures_

Discussion

The elasticity modulus E

The results (figures 6, 7 anti 8) indicate that the homogeneous bed expansion,Hb increase is

,and so Ebp !?trongly

and Ubp increase with pressure and that the affected by the kind of gas used,

This means that the mechanical structure of the powder bed can with- stand higher disruptive forces at increasing pressure before the powder bed becomes unstable_ This can only be the result of higher interpar- title forces at higher pressures_ From the theory of Rietema and Mutsers El,21 it follows that no bubbling occurs as 7ong as:

64

-

-

_-I

-

-

A

65

With the measured values of E bp

the elasticity modulus E was calcu- lated for the different pressu es and fluidization gases b&.ed, accord- ing to the above equation. The results of these calculation are aiven in figure 9. From this figure it can be seen that E pressure and that Eb is hiahest for li and lowest P

increases tith gr Ar However,

this yields a distor?ed picture becausg at the same pressure the voidage reap is very different (figure I) for the three gases used.

P (bar)

r'igznre 9 The variation of Ebp L;ith pressmw. ( *_&P, A :J2, I 921.

According to Mutsers and Rietema 121 E decreases exponentially with the bed voidage. The same holds also for the cohesion constant and the tensile strength as is known from powder mechanics t20,211. To compare the elasticity modulus at different pressures the values of Ebo (figure 9) must be extrapolated to the same porosity value. - If we assume that at ambient pressure the effect of the gas adsorption can be neglected we get fo r the dependence of E on the porosity E: E = E exp117.9(& --E)]. According to thisOequation the elasticity modulus for a bed voidage eu=0_40_The results of the

E o was calculated calcu 4 ations are shown

in figure 10. It can be seen that both pressure and fluidizing gas have a strong influence on the elasticity modulus E40. With Ar as fluidizing gas the values of E40 are higher and also the increase with pressure is stronger than for N2. on pressure was found.

With Hz only a very weak dependence

The results are consistent with the dependence of U 6 and 7) on pressure and type of fluidization gas. ~Ph$eE,h&$~$~~ modulus E means that the powder structure can withstand higher disrup- tive forces, which results in a higher ~~~ and higher bubble point velocities Ubp.

66

0 3 6 9 12 15

Cohesion constant as function of the pressure

Because the elasticity modulus, E, is the resu7 t of the mechanica strength of the powder structure and therefore of the interparticle cohesion forces there must be a relation between the elasticity modutus, E, and the cohesion constant, C. The inf7uence of pressure on the cohesion and the coefficient of fric- tion was measured in the tilting bed in an atmosphere of Ar for pres- sures up to 11 bar. Classical Coul umb theory assumes the yield locus to be 1 inear and the fo?lowing relation describes it mathematically:

--I = C + tan 9 x o

In the experiments with cracking 2 atalyst a Coulomb behaviour was found for normal stresses down to 4 N/m _ From a p7ot of -r against CT the cohesion constant C and the friction coefficient tan Cp could be deter- mined. The resu7ts are given in figure 11. It can be seen that the coefficient of friction tan @ decreases with increasing pressure. However, the cohesion constant increases with pressure_ The increasing cohesion constant C must result in an increasing elas- ticity modulus,E, which is in agreement with the results given in figure 10.

Gas adsorption to the sol id

It is not exactly known which mechanism causes the cohesion to increase and so the elasticity modu7us with increasing pressure. However, by increasing the pressure not only the gas density is changed but the amount of adsorbed gas to the solid can a7so be influenced. With a differenti al pressure technique the adsorption of Ar, l-12 and N2 to the solid was measured at pressures up to 15 bar.

67

The results of these measurements are given in figure 12 and it can be seen that with H2 on7y a rel ative7y sma7 I amount of gas is adsorbed which increases a lttt7e with pressure. However, the adsorption of Ar and N2 is considerably greater and the influence of pressure is stronger.

10

8

6

4

2

0

-26

P (bar) -

0 3 6 9 12

Figure 22 The pressure for Ar.

varia-t‘ion of the cohesion constrozt C ant! tag + wifh

6 9 12 15 P (bar)

Comparing fi gures 10, 11 and 12 it can be seen that both the elasticity modulus E 30’. the cohesion constant C and the gas adsorption show the same tren wl th pressure, namety.increasing with increasing pressure.

The foregoing can be summarized as follows: due to the adsorption of gas the cohesion between the particles increases. It should be expected, therefore, that the elasticity modulus EqD, which is related to the cohesion force, also increases with increasing gas adsorption at higher pressures and this explains the higher bed expansion,Hbp,at higher pressures.

Bubbl ing bed properties

The effect of pressure on bubbling in fine powders has been studied by Guedes de Carvalho et al. 19 I and they found that up to 6 bar the total bed expansion was constant and thereafter the expansion increases significantly. They explain this by postulating that there is a higher bubble hold-up, 6, caused by smaller bubbles which rise slowly. The maximum bubble size observed in the bed did not exceed 2 cm even at the largest velocities. They interpreted this as an indication that larger bubbles are not stable in the bed. Subzwari et al. 1127 also found that the tctal bed expansion was con- stant below 6 bar. Their data showed, however, that the bubble ho1 d-up, the bubble frequency and the bubble size decrease with increasing pressure. They discount the theory of Suedes de Carvalho 1’91 and suggest that the increased total bed expansion must be due to a higher average dense phase voidage, cd, with increasing pressure. They explain the smaller bubbles in terms of increased bubble splitting by Taylor i nstabi 1 i ties connected with the increased dense phase expansi on. However, our results indicate tha t the total bed expansion, H,, is not constant below 6 bar, but that the total bed expansion incredses with pressure from 1 to 15 bar. Contrary to the results of Subzwari et al. Cl21 the bubble hold-up does not decrease but steadily increases with pressure . The increasing total bed expansion is not only the result of an in- creasing bubble hold-up 191 but is also due to a higher dense phase voi dage, cd, with increasing pressure. In contrast with the findings of Guedes de Carvalho C93 the bubble size did exceed 2 cm. However, for both Ar and N2 the bubble size decreases with increasing pressure. Al though the decrease in bubble size with pressure becomes less at higher pressures, the results do not indicate a stable bubble size.

Conclusions

The “qua1 i ty” of fluidization of abed of fine cracking catalyst im- proves with pressure but is also dependent on the type of fluidization gas used. This can be explained by an increasing elasticity modulus E of the powder structure with increasing pressure. The higher elasticity modu’i us E is the result of an increase of the cohesion between the particles. ?his increase in cohesion is probably due to an increased gas adsorption to the sol id at higher pressures.

References

1.

2. 3.

Rietema K. and Mutsers S.M.P.: Proc.Int.Symp.on Fluidization and its Applications, Toulouse (1973) 28. Mutsers S.M.P. and Rietema K.: Powder Technology 18 (1977) 239. Verkooijen A.H.M., Rietema K. and Thoenes D.: Pror 4th Eng.Found. Conf. on Fluidization, Kashikojima, Japan (1983) to be published.

69

4.

5.

6.

:: 9.

10.

:::

13.

14_

15.

1’;:

:9”:

20. 21.

Godard K. and Richardson J-F.: Inst.Chem.Eng_Symp_Ser. 30 (1968) 126. Guedes de Carvalho J.R.F., King D.F. and Harrison 0.: Proc.Conf. on Fluidization, Cambri dge ( 19 78) 59 _ Sobreiro L.E.L. and Monteiro J.L.F.: Powder Technology 33 ( 1982)95. King D.F. and Harrison D.: Trans.Inst.Chem.Eng. 60 (198v 26. Guedes de Carvalho J.R.F.: Chem.Eng.Science 36 (-81) 413, Guedes de Carvalho J-2-F. and Harrison D_: IiZt_Fuel Symp_Ser_ 1, (1975) Bl-1, Geldart 0. : Powder Technology 7 (1973) 285, Abrahamsen A.R. and Geldart D-7 Powder Technology 26 (1980) 35 _ Subzwari M-P,, Clift R. and Pyle D.L.: Proc _ Conf _ oii-Fl ui di zati on, Cambri dge (1978) 50. Varadi T. and Grace J.R.: Proc.Conf.on Fltiidization, Cambridge, (1978) 55. King D.F. and Harrison D.: Proc. 3rd Eng.Found.Conf.on Fluidization Henniker (1980) 101. Mutsers S.M.P. and Rietema K-s Powder Technology 18 (1977) 249_ Rietema K.: Proc. Int_Symp. on Fl uidizati on, Eindhzen (1967) 154. Ergun S. : Chem_Eng.Prog. 48 (1952) 89, Abrahamsen A.R. and Geldax D. : Powder Technology 26 (1980) 47. Rietema K. and Oltrogge R.D.: Symp. on Fundamentaland Applied Fluidization, Tampa, Florida (1968) 31E. Williams J.C. and Birks A.H.: Powder Technology 1 (1967) 199 _ Fairley R. and Valentin F.H.H.: Powder Technology 1 (1967) 344, -

Notations

A C

‘b

Edp

Ebp $I

:

Hbp

!d m imf

P

G U0 d ;bp Urnf Dbo Wb OL tan 9 E

Ed

area of bed cohesion constant mean bubble diameter sauter mean diameter of particles el as ti city modul us elasticity modulus at the bubble point elasticity modulus at a bed voidage of D-40 bubble frequency acceleration of gravity height of bed bed height at the bubble point height of the dense phase maximum height of the bubbling bed bed height at minimum fluidization bubble frequency absolute pressure time coordinate superficial gas velocity superficial gas velocity in the dense phase bubbl e point vel oci ty minimum fl ui di zation vel oci ty superfi ci al bubbl e vel oci ty mean bubble rise velocity mass of powder in the bed tilting angle coefficient of friction bed voi dage voidage of the dense phase

Cm21 [N/m21 Em3 [mj I N/m2 1 1 N/m2 I IN/m21 [l/s1 fm2/s 1

Iml [ml Cl/m2_sl [N/m23 IS1 Cm/s 1 Wsl tm/sl

f:::;

:Z’ Erzdl

51; E-l

70

‘bp voidage at the bubble point

Emf voidage at minimum fluidization bubble hold-up

T shear stress [N/m21 (5 normal stress C N/m*

Eg gas density 4 [kg/m 1

VP particle density viscosity of gas

C kg/m31 INS/m27