The size and frequency of bubbles in two- and three-dimensional gas-fluidised beds

The Sii

and Frequency of Rnbbks

in Two- and Thr~imensional

Gas-Flnidised

Beds

D. GELDART Posrgmduam School of SIu&s

in Poxder

Technolog_b, Unircrsrr_t of Bradford. Bradford (Gr. Bnmin)

(Receiwd

March

SUMMARY The

frequency,

concentration

and size

of

bubbles

bursting at the surfaces of TV * and three-dimensional gas-jluidised beds har;e been measured_ Bubble diameters in the 30 bed (30.8 cm diant.) arefound to be Iarger than those in the 20 bed (68 x 1-27 cm) under corresponding conditions. A theoretical model is dereloped which shows that this is due to the occurrence of out-of--fine coalescence and to departure from the strict two-phase theory_ Equations are dericed which enable the sizes of bubbles in a 30 bed to be cakdated from data gathered in a 20 bed.

l- INTRODI_XXION

A considerable amount of iundamental research into gas fluidisation is carried out using twodimensional (2D) beds. No practical bed can be truly 2D and the term is generally understood to describe equipment of rectangular cross-section in which the thickness is ‘l-2 cm and the width 20-100 cm. There are obvious advantages to be gained by using this type of bed when studying the growth, velocity and coalescence of bubbles_ Given suitable lighting bubbles can readily be seen and photographed; only small quantities of air and powder are required; modifications to the gas distributor are made easi!y and cheaply, and the capital cost of the bed and ancillary equipment is low compared with a cylindrical bed of diameter equaltothewidtbofthe2Dbed_ However, there are sufficient differences between 2D and 3D beds to raise doubts as to the validity of applying data gathered from 2D beds to 3D situations. In a 2D bed bubbles are restrained by the walls in one dimension, becoming slugs when viewed from the side of the bed It is not surprising, therefore, that when 2D and 3D bubbles of the same frontal diameter are compared those in the 2D bed are

26, 1970)

found to have lower rise velocities*_ Bed expansion is greater in a 2D bed than in a 3D bed under comparable conditions of gas flow rate, particle properties and bed height2, and it might be expected that solids flow patterns would be somewhat diEerent when powder is completely restricted in one dimension Because of the diffkulty and expense involved in measuring bubble sizes directly in 3D beds with X-ray tine film3-& or other means4-s-67 it is tempting IO assume that the 2D bed really does represent what mould be observed by cutting a vertical section through a bed of circular or square cross-section_ In the absence of any other information, such an assumption seems reasonable and has been made’ u hen predicting conversions in fluidised bed chemical reactors_ Recently, however, some evidence has come to light*-’ which suggests that bubble sizes .Ire not the same at least when compared at the -m-face of 2D and 3D be?-= snder otherwise identical conditions. In this paper a theoretical model is proposed which demonstrates the relationship between the numbers and diameters of bubbles obsemeb at the surfaces of 2D and 3D beds. Data obtained by making measurements in a 2D bed can be transformed to give the sizes of bubbles bursting at the xurface of the corresponding 30 bed Evidence is presented in support of the view that the visible bubble flow rate is less than U- U,.

2-

DKITRIBUTlOS

OF GAS BE-ITVEEN DENSE AhC

BUBBLE

PHASES

Consider the flow of gas between planes AA’ and BB’ in Fi_e l_ A mass balance gives : U-A=

U,(~-+,)-A-+C~-A-U_ i-m-

Ljd-Q,-

A

(1)

42

D.

GELDART

(&b)3D

n3D-(Vb)3D (VA),,

=

where n3D = number of bubbles/second unit area in a 3D bed. Eliminating ~~ from eqns (4), (5) and 6

Fig_ I_ Bubbhng

IlCiiid

bed.

cutting

U-kzD-

U, = n,,-(A&,

(7)

U-kso-

U=n3D-(P&D

(8)

where ie- total gas flow =

(flow through dense phase) t (visible bubble flow rate) + (superimposed gas flow through bubble)

where U,, = superficial gas velocity

in the dense

Phase, occupied by &b = fraction of cross-section bubbles, U, = average absolute nse velocity of bubbles at level BB’, m = dimensionless measure of gas flow across horizontal cross-section of bubble. Although in some gas-fluidised systems dense phase velocities may be greater than Uog even when gas bubbles are injected into the bed, there is evidence to show that U, z U,-, for denser. larger particles*“_ Equation (1) then becomes :

U = U,[l+ Now

(m-

l).zb] +cb- U,

(2)

writing k=

l+(m-l)Eb

U-kUo=q,-UA

(3) (4)

Now consider n bubbles per second passing a horizontaI Iine of Iength L in a 2D bed, each bubble having an area (A&o and 2 velocity (U&, The total bubble area cutting the line per second is n-(.4&. The average velocity of this area is (U,),, so that at any time a length of the line equal will be occupied by bubbles. to u - (&J,,/(lJ&.n The fraction of the Iine so OCCUpied (&b) is ?I - (Ab)& Writing nzD for njL L-W.-&, kD(&b)ZD

=

@b)LD (UA)2D

By a similar argument it can be shown that

(5)

kzo = 1+

(mlD-

l&b)ZD

(32)

k3D

+3D-

l)(&b)3D

(3b)

=

1 +

The terms on the LHS of eqns (7) and (8) represent the visible bubble flow rate. In the strict two-phase theory, supported by the theoretical analysis of some workers”*‘*, m=l so that k=l and U-U, represents the visible bubbIe flow rate. Other theoretical ar~alyses*~ put nz=,=2; nr~,=3_ Experimentally deduced values of mzD have been reported which are higher than these theoretical values; up to5in~ocases’“-‘Jandinonecaseashi~as40’5. There does appear to be growing experimental evidence in support of the idea that k > 1.

3.

BUBBLE

FREQUENCY

TaEAsummw

In this paper it is not appropriate to discuss in full detail the meaning and relevance of bubble frequency measurements made with capacitance probes and from tine film, as it will form the subject of 2 separate publication. However, some general comments are essential for the development of a theory of 2D-3D transformation. Suppose that it is desired to find the total-number of bubbles/second passing BB’ (Fig. 1). This might be done in two ways: directly, by counting the total number passing the line over a period of several seconds; or indirectIy by measuring the “point” bubble frequency at sample points XYZ If the second technique is used it is necessa ry to consider the length over which the bubbles are beiig detected Levenspiel16 has shown theoretically that in a 3D bed a probe at P wiII record all bubbles (diameter db) whose centres pass within 2 distance db/2 of it This is demonstrated by considering bubbles in Fig Z(2). Bubbles 1 and 3 will be detected ; bubbles 2 and 4 will not Only bubbles whose centres approach nearer than f&J,, will be recorded and the detection area is therefore $r(&)&_ Even though Povder

Techd,

4

(1970/71)41-55

-BUBBLES

IN GAS-FLUIDISED

43

BEDS

measurements (3D bed)

parts of bubbles 1 and 3 lie outside the area and parts of bubbles 2 and 4 lie inside, if a sufficiently long stipling time is taken in a randomly bubbling bed, the shaded area outside the detection area should equal the shaded areas inside. It is then argued that the total number of bubbles, total area available n=f3D-

detection area

(9)

This argument cannot be faulted theoretically, but in practice some modifications are necessary_ First, the bubbling may not be completely random. If the sampling point happens to be in a region which is either deficient in bubbles or has an excess of them (as in a bubble track), the frequency recorded will not be applicable to the bed as a whole. Secondly, even though there may be random bubbling over the greater part of the bed crosssection, there may be a deficiency of bubbles at the walls It is rare to observe segments of bubbles, such as bubbles 5 and 6 (Fig_ 2(a)), at the wall ofa fluidised bed, although the author has observed it when fluidising very large particles (> 1 mm). For most fluid&d systems, however, particles move downwards at the walls in a relatively bubble-free redon. The maximum correction which may have to be made ought to be based on the idea that no bubble centre approaches the wall nearer than i(d&,, as with bubble 2 (Fig- 2(a)). This reduces the total area available from L? to CL- (db)J’ for a square bed and to 7r/4 [O- (d&D]2 for a circular bed of diameter D. Equation (9) becomes

Dividing both sides by the total cross-sectional area of the bed: n3D

=f3D

- ;

- &

D- (4)3D =

-

D

>

or

(10)

(11)

where f ;,,

is the effective point frequency_ and

By a similar argument the detection length in a 2Dbed is (dJZD (see Fig. 2(b)), and n

1

'D=f2D-@-J-&-

F- &)2Dl

w

L

or

113)

4_ RELATIONSHIPBETWEEX 3D BEDS.

BUBBLE THEORY

SIZE

IS

2D

AhD

Eliminating n3D between eqns (8) and (1 l), and nzD between eqns. (7) and (13) :

U-k2D-

Dividing

u,

=

f ;D - &

- &)2D

(15)

eqn. (14) by eqn. (15) and rearranging:

POKdm TechmL, 4 (1970/-n) 41-55

D. GELDART

where f,=fraction of sphere of diameter (16 occupied by wake_ Substituting and rearran_tig eqn. (16):

(17)

Assume now that the strict two-phase theory applies, random bubbling dcurs (even at the walls), (&,)~r, < L: (d,& e D: then f&=&,: f;r,=j-3r,: k,,=k,,= 1. No direct experimental data is available in the literature on the relationship between (LL and (_L)x,. It seems reasonable to assume that, for a given powder, (L&u= (LL,Equation (17) becomes: hg

This equation shows that in an ideal system, if then (d&,/(d&,= 1, but. unfortunately there appears to be no way of knowing a priori whether there is an explicit relationship between f& and fir, in corresponding beds. It is possible, however, to deduce a relationship. Consider two modes of bubble growth : (a) by in-line coalescence only, (b) by out-of-line coalescence only. In the first mode the trailing bubble of a pair (diameter dT) accelerates and catches up with the leading bubble (diameter dL) without any displacement laterally_ In the second case the trailing bubble, in catching up the leading bubble, has to move laterally a distance equ.11 to or greater than $(dT+ dL)_ If mode (a) only occurs, it might be expected that the probability of coalescence would he the same in both 2D and 3D beds, given the same distributor- If mode (b) only occurs then Fig 3 is relevant. This shows an artificially regular array of bubbles Imagine that at levels 1 and 2 the upper and lower rows of bubbles have the same diameter in

f3,J_f2,,= 3x/8,

b$

,”

and 3D beds, (i-e (d&or = (d&or : C4hz growth a bubble in the upper row must coalescence with a bubble from below or above it. Some of these bubbles will coalesce and some will rise for a dis-

b 2D2_ For further overall bubble

3. Out-of-line coalescence model.

tance without coalescence_ However, because a bubble in the upper row can coalesce with 2 bubbles (4 half-bubbles) in the 3D case and only 1 bubble (2 half-bubbles) in the 2D case, it appears probable that relatively more coalescences will occur in the 3D bed than in the 2D bed, resulting in the 3D bubbles higher in the bed (say at level 4) being larger than those in the 2D bed at level 4 (at the same gas flow rate). Thus even in an ideal&d system it appears possible for (d&,/(d&, > 1. If this does occur, then eqn_ (18) shows that j&,/f2-, c 3x/8. This is as far as it is possible to go without recourse to experimental data which must inevitably be gathered from non-ideal systems. Experimental results will be discussed in detail later, but it is relevant at this point to anticipate the discussion and refer to Fig. 4, which shows that f&/f 2D-z 3x/8 for most bed depths and flow rates of interest_ It witi be recalled that these effective bubble frequencies are based on averages of 6 measured values corrected for wall effects If it is assumed that point frequencies are affected only by bed level and mode of coalescence (see this data, Fig 5, and the results of other workersr:X then the Pow&r

TechnoL.

4 (1970/71)41-55

BUBBLES

IN GA!+FLUIDISED

45

BEDS

bubbles. Equation (17) includes the ratio (U-k,, ir,), and it cannot be assumed that W/W-kmin a real system km= ksD= 1 or indeed that k,,= k ZD The measurements made in this study allow this question to be resolvti Although (f&, can be determined from tine film, it is not possible to other than by X-ray techniques. determine (j&n In the absence of any information to the contrary, it seems reasonable to assume that. for a given powder, (LAD = (AL.

5. RELATlOlSHlP IN 2D

Fa,e 4. Point frequency (3D)/Pomt

frequency (2D) rs U-

ci,

BJXWEEN A&C

3D

BUBBLE

COM-XXTRATIOS

BEDS-?Z3&

naD

In the previous section a relationship was established between (4J3D and &JzD, but it was not found possible to express the relation solely in terms of data gathered from the 2D bed It is desirable now to examinc theoretically the relationship between nso and n, and to attempt fo express n3n in terms of 2D data. It has been shown (eqn. 11) that n3D

=

f

;D

- $

- (d,f:,

b&)$D

NOW

1

(19)

=(db):D-(db):D

and from eqn. (13)

Substituting in eqn (11)

D S d, so that f
finding that f&,/f&,-z 3x/8 suggests that the occurrence of (&),,/(dJ,, > 1 may be due, at least in part, to the presence of some degree of out-of-line coalescence_ This may not be the only reason for the larger 3D

Also,

k3,=k,,=

l_

(from eqn. (20)):

are known, (n3D)rdal can be found and =d_fzD used in eqn. (8) (kxD= 1). Expressing (V,),, as 2 - [7~(4,)~~] j6, then

Ifn3D

PowdeT Technor., 4 (1970/71) 41-55

D. GELDART

46

The use of eqn (21) in a real (non-ideal) system would result in values of (r~~u)_~~,which would be too high relative to measured results (compare eqns. (21) and (20)). However, in a non-ideal system FC,D> 1, and althOU~ (n3D)idul might be too high, the ratio used to determine (d&u in eqn. (22), might be sufficiently close to ;;I P), (%Lai. -3o- U&n,, to give predicted diameters close to those measured_ This question will also be examined later in relation to the experimental results.

Procedure 3D bed. Bed depths (measured at U,) from 10 to 60 cm in increments of 10 cm were used_ At each bed depth 11 flow rates were studied. These corresponded to values of u- u0 of O-5-3 crn/sec in increments of 05 cm/see and 4-8 cm/set in increments of 1 cm/set. At each flow rate a 5 second burst of tine fihn running at 64 frames per set was taken with the camera located vertically above the bed. 2D bed- Cine films taken at 64 frames per set by Kelsey” were available. Ten flow rates at each of 5 bed depths (5,10,20,40 and 80 cm) had been filmed up to a maximum value of U - Ue of 9 cm/set.

Equipment The bed used for the 3D work was 30.8 cm diam. and constructed of perspex_ The distributor consisted of perforated metal having holes 32 mm diam. on a triangular pitch with 5.1 mm centres Two sheets of filter paper under the plate prevented particles falling into the wind box and provided a pressure drop across the distributor of at least 50% of the pressure drop across the bed at the working gas flow rates. The 2D bed was 68 cm x 1.27 cm in cross-section and also constructed of perspex. A distributor identical to that in the 3D bed was used. Variable area flowmeters were used to measure -air flow rates at a standardised pressure of 3.45 bar gauge (50 lbJin’)_ Pressure drops were measured using water manometers

Analyses of film Unless otherwise stated ah measurements were made at the bed surface. (a) 20 bed. Film was projected half full size onto a screen Bubble diameters had been measured previously and have been published”. The maximum width of at least 50 bubbles were measured at each flow rate and 4 bed depths and the mean size determined from

Materials The powder used was a washed river sand having a wide size range and mean sieve size of 128 jnn. A full sieve analysis is shown in Table l_ The mininium fluidisation velocity C&,, as determined by pressure drop-velocity plots, was found to be 12 cm//se in the 3D bed and virtnally independent of bed height U,, was found to be height dependent in the 2D bed and varied from 128 cm/xc for a S-cmdeep bed to 1.74 cm/set at a bed depth of 80 cm.

TABLEl:

(db)m_

= (q)*

Total numbers of bubbles bursting at the bed surface in a given time were also available. having been counted visually. Several random checks ofthesetwo parameters were made and found to be in good agrement v&h the published results. Although frequencies determined by a capacitance probe were also available, it was felt desirable to determine point frequencies visually direct from the film at more than the 3 positions used with the probe. Accordingly, 5 dually spaced points in a horizontal plane (at approximately 14-cm intervals and 5 cm from each end of the bed) were sampled_ A bubble which enclosed a point was counted_ (b) 30 bed. Measurements of bubble numbers, freouencies and diameters were more difficult to make in the cylindrical bed. At high flow rates

S~ZEAXN_YSSOF~A~~~D

BSsiece

+72

-72+s5

-8SflOO

-lootl20

-lID+lW

--1sO+170

-170-k200

-200

WC%

6

18

21.4

15.4

15

II-4

5.4

7.4

Pmrder Tec.fmd_. 4 (1970111) 41-55

BUBBLES

IN GAS-FJXJiDISED

(U- Ue >6 cm/xc), so much sand was flung up when more than 2 bubbles burst simultaneously that relatively few eruptions could Ix found suitable for measurement. Bubble numbers could, however, be counted even at high flow rates At lower flow rates the eruption sizes of 40-50 bubbles were measured at each flow rate. The maximum size of the eruption was measured just before it burst by matching circular discs of card (differing by 05 cm diam.) with the image. Side lighting in conjunction with a black powder of low density which remained at the bed surface, provided good contrast at all but the highest flow rates. A bubble’s diameter was estimated as 5 eruption diameter*_ The mean diameter of a distribution was calculated as I

47

BEDS

The experimental data are presented in graphical form in Figs. 5-9, and using these graphs Tables 2 and 3 were prepared Table 2 presents the smoothed data and derived results for the 3D bed, and Table 3 does the same for the 2D bed.

The numbers of bubbles were counted by considering each quadrant of the bed in turn. Point frequencies were measured by averaging the results from 6 positions on the bed surface (see Fig 2(c)). Since it was the bubble causing the eruption, not the disturbance itself which had to be counted, care was taken not to count cases in which the eruption enclosed the sample point but the bubble (5 eruption size) did not_

I OOOll

, 1

2

3

7

4 5 ” -lJ&n/sec)

1

6

7

6

9

FI_~ 7. No. of bubbles cm? per see breaking sotface of 3D bed ITS v--c&.

14

001 1

hg.

2

3

6. No. of bubbles;cm

4

* 5

. 6

. 7

per set breaking surfaa m. u-u,

1

I

8

9

l-

of 2D bed

= This is based on X-ray measurements in a 3D bed” photographic measurements in ZD hed~~~*~4

and Fig 8 Bubble diameter rs_ U-

Polrder

Te~l_I

f&,-3D

bed_

(197Of71)41-ss

D. GELDART

48

for the 3D bed and an almost straight line for the 2D bed. The ratio (d&n/(dJ20 appears to be independent of U- ET,, (Fig 1 l), but decreases slightly with increasing bed depth. The mean value Liesaromd 1.5. It cannot be assumed that this ratio would a!so be applicable to other powders or gas distributors. Cooke et aL8 employed a coarser powder (- 10 mesh coal) having a minimum fluidisation velocity about 5 times that of the powder in this study, and treed drilled plate distributors Their data shows that (&,)sn/(d&o z 3 in beds of similar depths to those used in this work but at a higher value of U - U, (19 cm/set)

diameter Figures 8 and 9 demonstrate the linear relationship between bubble diameter and U- U,. A cross plot of bubble diameter OS_initial bed height for constant U - U,, (Fig. lob) also gives a straight line Bubble

H&m) F 10 a20 040

Fig 11.Dmmeterof3Dbubble/Diameterof~Dbubblecs

Bubble

U-

UO.

frequency

As other workersX7 have found, point bubble frequency is relatively insensitive to increases in U- U, (Fig 5) but decreases steadily with in-creasing bed height (Fig 12) For bed depths greater than 20 cm and flow rates greater than about 4 cm/set, f&, is slightly smaller than fin At bed heights 320 cm fzD declines exponentially as e-&o, where a=0_0116 CIII-~-

1

I

lo20304050601080 t-b (cm)

Fig

IQ Variation of bubble diameter zad concentration initialbadheight; U--u,,=6cm,kc

with

Fis

I2

Point bubble

frequency / fJ-7_$,=6an/s~

OS. initial bed

Powder Techml.,

hi&t

4 !1970/71)

Ho;

41-5s

BUBBLES

IN GAS-FLUIDISED

Bubble concentrarion This term is used to describe n2n and nSD the numbeqs of bubbles bursting at the surface per cm per set and per cm2 per set respectively. A comparison of Figs 6 and 7 shows that nzD is much less sensitive to flow rate than n3,, for U - Z&,> 3 cm/set. n3,, does not appear to exhibit a maximum as does nZD, but visual observation at U - V,,C 05 cmjsec revealed that nsD fell sharply, indicating that the maximum occurs at about U - U,, = 1 cm/set This is in agreement with earlier published results” concerning the number of bubble tracks The reason for the smaller dependence of n2, on U - U,, may bc explained in terms of the idealised model discussed in Section 5. For U- U,, >4 em/set, let n,, = (n2& - e-b2D~(U--Uo) and The slope

n30_ b,,

(,+&_e-b~~

(u--u01

ln(nd -Wh), = (U-UO),--(U-W,

Now substitute for n3D from eqn. (21X remembering that j&, is independent of U - U, :

-

_ -

b 3D

=

XL,

experimental results indicate, for U - U, ~4 cm/see, b,, z 4bzD

The

39

BEDS

The variation of n2, and n3D with initial bed height also merits some comment Figure 13 is constructed by cross plotting the data from Figs 6 and 7 for U- U, values of 4, 6 and 8 cm[sec It the numbers of does not n ecessarily represent bubbles present at these levels in a very deep bed. There is some evidence that the bubble concentration at a given level in a deep 2D bed is greater than that at the surface of a bed of that depth The same might be true of 3D beds, but it is clearly impossible to confirm this using photo-graphic techniques. However, it does seem reasonable to compare surface results in both beds. The first point to notice is that ilow rate has a much greater effect on bubble concentration in the 3D bed. The 2D curves are much cIoser together, and this agrees with the general findings of Grace”_ Secondiy. the slopes of the 33

c~~rves are greater

zd

this can also model.

be

explained by considering the idealised Rewriting eqn. (13) as n ZD= fZD - S_rD

(9

where xzD= (C&J;&, for bed heights greater than about 20 cm and Thy,

fzD = foeed

(24j

xZD = xOe -cHo

(3)

the slope of the rz2n LS i-I, curxe (P-J

*=zseentobec+-n The slope of the njD 2x H,%rve

P3D =

In (n3Dh

can

(j?& is

(n3&

HI-H2

Now the idealised model in which (d&,= &JBD and in-line coalescence only occurs, eqn (21) applies. ln(lSf,,-x&), P3D

=

Substituting

-ln(l_Sj2D-x&J2 H,-H,

for fzD a;ld xzD from eqns fizD =

(24) and (25)

2-i-a

This ratio holds even if c and a are not constant, ie- if the nzD CT HO plot is not a straight line The tangent to the ZD Lxrve at any point divided by the tangent to the 3D cxxve at the same value of HO and U-U, would still be given by (2cta)/(c+a)If the dsta for U-U,=6 cm/set at HO=50 cm PcmrdbTechmL

4(1970/71)41-55

50

D. GELDART

are taken, a=0.0116, c=OXKI65 and /?3D//3ZD= 1.3. The measured figure was found to be 1.55, once again indicating the departure from ideahty- The maximum posstble vialue of &,//?.,n for in-line coalescence conditions is 2 and this would be approached if n< c. However, if out-of-line coalescence predominates &J&n could be larger than 2 Grace” has shown that a high value of the product fi-db increases the average bubble velocity (relative to the dense phase) at any level where the mean bubble diameter is d, since U, = U&-e-’

5 fi “b)

where U, is the rise velocity of an isolated bubble of diameter db- The maximum value attainable is evidently XJ, for very high value of j?-db_ It is of interest to note that &(d& >&,(d&,, resulting in 3 D bubble velocities even higher than those calculated merely on the basis of (d&,_ Visible bubble jlow rate In column 8 of Tables 2 and 3 the visible bubble flow rate has been calculated for the 3D and 2D beds respectively, and in column 10 this is expressed as a fraction of U - U,,, which would be the visible bubble flow rate if the strict two-phase theory applied. It can be seen that the measured bubble flow rate at the surface is SO-70% of U- ZJ, for the 3D bed and 4&60x for the 3D bed. Kelsey’s” results using a variety of distributors, including drilled hoies, show that U -kK2,,- U,, is always smaller nearer the distributor than the value at the surface This observation has also been made by other workers1g*20 using only porous plate distributors Pyle and Harrison’o suggest the “missing” gas may be in the dense phase, giving values of U, up to 6Uo. This explanation must be discounted, however, for if the missing gas were to be diverted into the slower moving interstitial phase, an appreciable expansion of the bed would occur_ In fact, measured bed expansions are lower than would be expected if the visible bubble flow rate were as high as U - U,, and are consistent with the U- kUo values meas~red’~. It would appear, therefore, that gas may be short-circuiting from bubble to bubble, perhaps just prior to the coalescence process However, it is puzzhng to find (U-kUe)/(U-Uo)
Inter-relation between n, f and d,, Equations (11) and (13) are theoretical relationships, expressing nsn and nzD in terms of effective bubble frequency and bubble diameter. In this study each of the parameters in these equations has been measured independently and thus should provide a means for testing the validity of the equations_ In column 11, Table 2, n3u is calculated from cqn. (11) and the results should be compared with the measured values in column 4. Inspection of the columns reveals that the agreement is good for U - U. P- 3 cm/set for bed heights up to 40 cm_ For the two deeper beds, however, the predicted values are too low when U - U. >4 cm/set. This suggests either that the measurement of fsu or the correction factor [(D-d&D]’ was too low. The latter seems the more likely explanation, since it must be remembered that this factor was based on the idea that no bubbles had centres closer to the wall than that at the higher flow rates (c;,),,/z. It I-s posstble in the deep beds (when bubble diameters are >9 cm), a proportion of the bubbles did occur close to the wall. As was pointed out earlier, there was a considerable amount of sand thrown about under these conditions and it is not possible to verify whether the measured frequency has been overcorrected. Columns 11 and 4 should also be compared in Table 3 where nzD is listed. The agreement here is less good, the predicted values of n2n being in every case higher than measured values. There is no obvious explanation for this and further in\-estigation is required_ Relation benveen f, d,, a$ U- U, By eliminating nsn between eqns. (8) and (11) it is easily shown that

@oLn = If k,,=

1.

2(U-kk,,-

U,)

(26)

x-v

J~D

faD=fsD this becomes

&)m =

quf

Uo)

3D

-

Kunii and Levenspiel” first derived a similar equation to (26a) by adopting a slightly different approach, and in their equation the numerical wnstant is 1.5. This apparent discrepancy is due to different deftitions of(d&,, since the other workers used the “‘equivalent volume diameter”, whereas in this work db is the frontal diameter_ GeMart1s showed in an earlier paper that at low flow rates Ponder

Tec%wL, 4 (1970/7X) 41-55

BUBBLES TABLE

RESUiTS

DATA

USIUG

7

1

2

3

HO

u-u,

(dh

I;D

(cm)

(cmfs)

(cm)

30

40

50

60

5

6

BEDS

FROM FIGS.

51 5.6,8,-3D

BED

8

9

10

n3D f (d&

@$

u--x,, u-u,

(s-9

(m’s)

(-)

1.46

11 %

/;&(d&

(-)

(cm-‘s-1)

L7 247 23 218 211 207 20

0.73 070 067 0.63 062 061 0.58

O-42 033 0.26 021 0 17 0 136 0.115

0.63 061 060 061 061 OJ9 OZ.4

022 0.149 (1.103 a077 0058 0045 O-035

0.50 050 049 0.50 O-48 048 048

0 OS9 0062 00-15 0034 O-027 OCll 0016 0_06S O-042 0_03 0.022 0016 0.102 0008

O

24 285 3.3 37 4 I5 46 50

027 023 0 19 0 16 0 134 0.112 0095

225

OS45

26 28 295 305 3.15 3.2

0.825 080 0.775 0.75 0 725 0.705

1.9 ,I4 224 228 228 228 225

208 268 3-15 375 4-25 465

2 3 4 5 6 7 8

3.0 36 4.2 48 54 60 66

0 12 0 10 0 083 007 O-058 0048 004

I 95 1.95 1.95 1.95 195 195 1.95

0815 O-78 0.735 0715 068 0.65 0615

1.59 151 1.43 1.39 1.32 I.27 12

127 1.83 241 301 3.6 41 43

z92 234 205 1.87 175 1.67

2 3 4 5 6 7 8

3.4 42 49 5.7 6-4 7.2 80

0064 0 052 0042 Q034 0 028 0023 0_019

IO 1.15 I.2 13 I.4 1.45 1.50

081 0 745 0.7 1 0.66 063 0.59 0.55

081 086 085 086 0.88 0.86 083

1.0 1Jl 1.94 747 288 337 3.82

17

2 3 4 5 6 7 8

41 5.0 5.9 685 7.7 8.7 9.6

0043 0034 O-027 0022 0017 00135 0011

I.20 1, 125 1.30 1.30 135 1.40

0.75 (L7 Q66 061 Cr.565 (L515 O-413

O-9 0.84 as3 080 074 07 0.6

1.15 1.67 118 277 305 3.50 384

246

1.78 1.67 1.67 1.68

0.X 055 055 0.55 OJl 0.50 O-18

2 3 4 5 6 7 S

4.7 5.8 68 7-9 90 10.0 110

0_029 O-023 O-O!8 0014 6.~11 0_0085 ooce5

07’ 0.66 0.61 055 050 O-46 041

1.19 l-68 z

1.0

o-72 066 061 O-55 050 O-46 041

265 3.04 3.34 3.42

2335 193 1.70 158 150 I.43 137

0.59 0.56 0.55 053 051 048 O-43

a041 O-025 O-017 O-01 1

2 3 4 5 6 7 8

53 65 7.8 9.0 103 116 IL8

O-023 OeO18 0.014 0011 OSlO85 O-CM366 O-0052

0.68 O-62 055 050 0.44 039 0.34

0.6 1 OS6 050 0.45 0.40 0_35 0.31

134 1.94 2-59 3.14 3.63 395 43

24 195 1.75 1.62 155 150 1.44

0_67 0.65 065 063 061 056 054

O-028 0017 0_0105 on07 OCW8 QOO33 0 0024

10

20

4

2: DERIVED

IN GAS-FLUIDISED

O-9

1.61

1.61 I.47 148 15 1.49 I.50

‘0 1.84

0006 0_00+3

4

Pomier Tlxhml..

4 (197Ol71)

41-55

1

2

3

4

5

HO

U-30

(d&x,

nzo

IZD

(4

(cm/s)

(cm)

(cm-‘s-1)

(s-z)

10

2 3 4

5 6 7 8

1.2 1.5 1.8 21 24 27 3.0

076 o-9 09 076 067 061 0.57

I.2 1.6 1.95 225 2.45 255 26

2 3 4 5 6 7 8

1.7 '1 24 29 34 3.8 43

O-44 04S 043 036 0335 032 032

09 13 165 1.95 2.2

2 3 5 5 6 7 S

26 35 4 4s 5-4 60 68

0.27 031 027 0.23 020 0.19 0.175

07 1.1 1.4 1.6 1.7 18 I.8

0113 0.101 0090 0083

05 08 1.0 1.1

20

40

80

2 4 6 s

r-

4 6 8 10

6 L-d, (-?

8

ID

n 2D - g

(s-1)

(cm/s)

- (d&,

9

10

+$

u ;?;

(-)

11 %

(-)

o

o

fld(db),, (cm-‘s-l)

O-99

12

032

1

1.58 191 218 235 245 25

0.65 I.20 1.75 1.93 228 262 3.03

072

0.99 O-98

080 088 095 1.0 098 O-98

0.4 OA4 0.39 0.38 038 038

1.05 1.06 1.04 O-98 09 084

088 l-26 1.58 1.85 209 216 216

0.76 12.5 143 1.78 2.27 272 3.34

O-76 O-91 1.0 1.13 125 1.25 1.2

038 0.42 0.36 037 038 0.39 042

052 06 06 064 0.61 0.57 051

091 090

066 1.03 1.31 1.49 157 164 1.61

112 224 25 3.18 3.64 402 4.8

0_92 128 1.4 1.53 1.53 1.55 iS2

0.56 a75 0.62 0.64 061 0.58 0.6

0.25 o-295 033 031 029 027 023

094 O-91 OS8 085

047 0.73 088 094

1.06 215 345 4-9

I.0 I.2 133 I-37

053 0.54 0.57 0.61

0118 0121 011 0094

097

0.96 096 O-95

098 O-97

O-96 0.95 O-95 094

z

7

O-94 O-95

094 094 O-93 0_92

(UU,-, -c 2 cm/set) when the bed is not evenly bubbling, the numerical constant (K) could have much hrgher values than 2 but approached 2 asymptotically with increasing U - U,,. An equation such as (26a) could offer a very convenient method for determining dt, without the necessity for tedious photographic measurements, which are in any case limited to the bed surface Equation (26) ought to be more accurate, but the _ _ determmatron of U-J+, - ZJ, requires a prior knowledge of IIso and db, whereas eqn. (26a) apparently requires only the determination of the average value of _&, and a kuowledge of U- U,. A similar equation can be derived for the 2D bed by eliminating n=n between eqns. (7) and (13), from which

cd_)

o ZD

which, for k,,=

=

‘0 _t"-'2D37L

1 and no wall 1.7(U-

(db)lD

=

&)

f;D

(27)

effects reduces to

Lie)

fzD

(27a)

In column 9 of Tables 2 and 3 [jiD - (db)3D]/(U - U,,) and [j& - (db)2D]/( U - U,,) have been calculated. Calling these numerical vahes KsD and K,, respectively and plotting them on Fig. 14 reveals that, as previously found by a less accurate method, K,, is high at low flow rates and gradually declines to a steady value. This value appears to be about 1.6 and is lower than the theoretical vahte of 2 However, the latter is based on an ideal system in which and kaD= l_ It seems that the departure f;D=fJD Pon~TechnoL.

4(197OJ71)41-55

BUBBLES

IN GAS-FLUIDISED

53

BEDS

Cal

from ideality reduces both U- U. and fsD by about the same factor, resulting in a value of K,, only a little below theory_ The same is true of Ku.,, which has a value of about 1.3 compared with the theoretical value of 1.7. Thus the use of eqns (26a) and (27a) would enable bubble diameters to be estimated from the excess gas flow rate U - U, and an average value of fin or fsD compiled from at least 5 values measured at different positions in the bed. If a capacitance probe were to be used at various Ievels in a deep bed then careful calibration against film records wou!d be required Prediction

x-.-. 10

20

30

”

so H,km)

so

’

60

‘-T&

70

of 30 results fkom 20 data

Equation (17) is obviously of little direct help in predicting 3D bubble sixes solely from 2D data, although it does reveal the factors involved in causing (d&,/(d&, > 1. It is evident from Tables 2 and 3 that a combination of these factors causes larger bubbles in the 3D bed_ It was suggested earlier that the use of eqns. (21) and (22.) might enable (d& to be calculated with a satisfactory degree of accuracy. In Fig. 15(a) it can be seen that, as expected, the predicted values Of n3D are too high When these values are used in eqn. (22), in which the visible bubble flow is written as u-I&,, not u-k-m -U,,, then the values of (d&n shown in Fig. 15(b) result. The predicted values of (db)3D show reasonable agreement with

measured values except in the deeper beds. In Fig 16 the data are plotted ns_ U- U. for 3 bed heights and it can be seen that the agreement between predicted and experimental values of (db)3Dis good

Fig 16 (d,,),, predicted from ?D data compared with m-red zalues; -predicted, neasured

at high flow rates and acceptable at low flow rates NaturaIly the validity of this transformation procedure would have to be verified for other fluidised systems PotentiaIIy, however, it offers a rapid and convenient way of estimating bubble diameters in POlFdiT

TedmoL.

4 (X970/71)

41-55

54

D. GELDXRT

3D beds by making measurement of only the total ntimber of bubbles passing various levels/xc and the a\reragepoint frequency at that level. This can be done from films quite rapidly (10-H minutes are required to analyse a S-set burst of film from which one value of n,,+S values of fiD may be found)_ The time-consuming tedious and eye-straining measurement of SO bubble diameters could thus be avoided, unless a check of the non-ideality of the system were required, i.e. to determine whether the visible bubble flow rate is < U- U,,. The transformation procedure described above could not be used in its present form for gas/solid systems in which the interstitial gas rises faster than the bubbles. A technique for dealing with this is being developed and it is hoped to publish a theory- together with experimental data, in due course_

COSCLUSIOhS

(1) Equations for a 2D-3D transformation have been developed for an ideal&d system (in-line coalescence, visible bubble flow rate U- U,,). They show that at the same bed height and gas flow rate, bubble sizes ought to be the same in both 2D and 3D beds. (2) The diameters of bubl$es bursting at the surface of a 3D bed have been found experimentally to be larger than those at the surface of an otherwise identical 2D bed of the same depth, at the same flow rate. (3) It has been shown that this may be due to a combination of two factors: (i) the occurrence of out-of-line coal escence (which could occur even in a system in which the strict two-phase theory applied); (ii) the non-ideal&y of the system, Le. the visible bubble flow rate was somewhat greater in 3D beds < 30 cm deep than in 2D beds of the same depth_ In both systems the visible bubble flow rate was appreciably less than U - U,. (4) The equations developed for the ideal&xi model can be applied to a real system without appreciable error. The bubble concentration in the 3D bed can be calculated fkom (eqn. (21)) 7z-jD

=

Go 1s -

fip

where q. is the number of bubbles passing a level in the 2D bed/unit bed length/second and fzD is the average point fkquer~cy at that 1eveL Both these

parameters are readily measured from tine film. The diameter of bubbles at that level in the 3D bed may then be calculated from (eqn. (22))

(db)fD=

8+ (7u-u, -;)

The validity of this technique for other gas-solid systems needs to be established. (5) The equations (d&n and

1_6(U-

=

U,)

f3D

(d&D =

13(U-

U,)

f2D

fit the experimental results reasonably well and offer a way of estimating bubble diameters at levels other than at the surface from a knowledge of point bubble frequency and the excess gas flow rate. Any technique for measuring j-a;D or fzD must be representative of the whole bed at that level and-must be calibrated very carefully against a visual method.

LIST OF SYMBOU

k

slope horizontal cross-sectional area of fluidised bed vertical cross-sectional area of bubble slope slope Frontal diameter of bubble frontal diameter of trailing bubble of a coalescing pair frontal diameter of leading bubble of a coakscing pair diameter of cylindrical fluidised bed point bubble frequency point bubble frequency corrected for wall fraction of sphere of diameter db occupied by wake height of fluid&d bed ai minimum fiuidisation velocity l+(??I-l)Eb

K

constant in eqn. d, =

Id

width

zl

-%a b

C db

4 4 D

5* f,

Ho

of fluidised bed

cm-’ cm’

cm2 cm-l cm-’ cm cm

cm s-1

S

-1

cm -

WJ- = Uo) J

cxn

Powder Techml, 4 (197Oj71)41-55

BUBBLES IN GAS-FLUIDISED

m n n2D n3D

N u

UO u*

dimensionless measure of gas through-flow in bubble number of bubbles passing a level per 5ec bubble concentration in 2D bed bubble concentration in 3D bed number of bubbles superficial gas velocity in empty bed superficial gas velocity in dense phase minimum fluidisation velocity absolute rise velocity of bubbles in a swarm rise velocity of single bubble rise velocity of bubble relative to dense phase volume of bubble

55

5 P. K. BALWGARTR

AXD R. I_ PIGFORD. Dens@ tluctuations in tluidised beds, Am. INI. C&m_ figrs_ J_ 15(1960) 115 6 A. B. WH ITMUD AND A. D. Youxc. Fhndisation performance in lars scale equipment Proc_ inrcm S_tmp- on

FIuidisation. Eindhorcn. 1967. p ‘94 7 J. E. ELLIS, B. A. PARI-RlDGE A.D D. I. LLOYD.Oqdch>dro-

8

ems-’ 9

cm s-* CIIls-'

10

m-Is-’ clns-’

I1

cm s-’ KXi-2 cm-’ cm-l

WJ slope fraction of bed cross-section occupied by bubbles Subscripts 2D and 3D refer to two- and threedimensional beds respectively. REFERENCES I D. L. Pvtx A>D D. HARRISOK The rising velocny of bubbles in tv.o-d~ensmnal fluid&d beds C&m. fig. SC__ 22 (1967) 531. 7 D. GELDARTAND J. R. K-, The inftuenaz of the was distnburor on bed expannon. bubble six and bubble frequenq in fluidtscd beds, Fhr~disarmn, INL Chem. En_ns (London), 1968. P_ 114. 3 P_N. ROU’E*D B. A PARTRIDGE. An X-t-a> study of bubbles in fluid&d beds+ Trans Insr Ckm. Engrs, ;13 (1965) T157. 4 R-Ton, R. MAYO, H KOHIWA,Y NAGAI. K NAMGAWA A>D S Yu, Behaviour of bubbles in the gas-solid fluid&d Lxd, A-rzgukuA-oguklr. 4 (1966) 142

BEDS

12

9enation of butanes in a _Q fluid&d bed. Rui&uz~zmz_ Inst. Chem Engrs (London). 1968. p_ 43. *.= D. F. W~t_t_t~w.s. M J. Coorcs&W. HARRIS, J. HIKmctics of ox>gcn consumprion m fluiducd bed carboniscr. Ruidirorion. Inst. chum. En_=. (London), 1968. p_ 21. I- D~vm AXD J. E R~MARDSO~.Gas interchanaehctuccn bubbles and the continuous phase in a flmdtscd bed. Tranx INI. C&m. Engrr.. 44 (1966) T-293_ J. R GRACE AX?) D. HARRLVJE.The behaviour of ffccip bubbhng fluid&i beds. Chem. Eng Ser.. 2 (1969) 497 A\D P_ N Ron% Analytic of _&xs fiou in a B. A. PARTRIDGE bubbhng tluidised bed v\hcncloud forma.ion s-curs. Trrmr. zitsr. Chcm Eizgrs_,44 (1966) 129. R D. TooAXD H. P. Jonxsro~;~. Gaseous fluidiarion of sohd particles, Chem. fig_

Progr.. 48 (1951) 110

x AND D H~RRIS~X. On rhe 13 M_J.Locxm-r_J.ED~vmso t-o-phase theoq of fluidtsation, C&m. Eng. Sri_ 22 (1967) 1059_ 14 M. R. JL~DD,Ph. D. dzssermrron, Unw. of Cape Town. 1965 IS J. R KEIZY. %I_SC zhaii. Univ. of Bradford. 1970 16 0. -PIEI_ personal communication May 1969_ 17 D. Kmm_ K YOsHIDl\AXD I. Ht~urcr The b&cxi;iour of freei> bubbhq fluidised beds. Proc. Imcm. S>mp_ on FIuldkarion, Eindho~ m. 1967. p 243. 18 D. GELDART,The expansion of bubbling flu’dtscd beds. Poradrr TeAmal_, I (1968) 355. 19 J. R GRACE. Ph.D. rhcszs. Unit of CambndSc 1968. 10 D_ I_ hu _&SDD. HXZRISOS. An eqm-imental rnxesti~t~on of the tuo-phase theory of fluidtsatton Chnn Eng SCL -72 (1967) 1199. 2 t D. Ku-a ASD 0. LEVESSPIEI_,BubblrnSbed r?odcl. I&. Eng. C&m. Fxmtiemak. 7 (1968) 446 ” J. S. M. Bormuu, J. S. G~DRGEA~J H. B-iw. Bubbie chains in _wtkidiscd beds. C&m. Eng. Progr. Swnp. Scr_.

6-7 (1966) 7_ 23 J. Btmx, Projecr reporr, Univ. of Bradford. 1969. 23 D. GELDARTA\D R CRA?~ELD, The fhxidisaiionof beds of large pat-ticks. to be published

Powder TechnoL. 4 (1970,W)

41-55