Heat transfer in a fluidized bed. Part II. Interpretation of the heat transfer coefficient on the basis of solids movement

Powder

Technology.

175

30 (1961)

0 EIsevier Sequoia S.A..

175 - 184 Lausanne - Printed in The Netherlands

Heat Transfer in a Fluidized Bed. Part II. Interpretation of the Heat Transfer Coefficient on the Basis of Solids Movement Q. E_ J. J. AM_HOELEN*

and S. STEMERDING

ChemicolEngineeringLaboratory,

(Received

September

30. 1960;

Uniccrsity

ofGroningen,

z’ijfnbor2pll

16. 97di

_4G Grorzingen

[The

AYcthcrlands)

in revised rorm March 9, 1961)

SUMhIARY In Part I the influence of the thermal driving force on the coefficierzt for heat exchange with a fluidtied bed was described_ The extent of this was clearly related to the kind of powder fluidized, i-e. powders which e_xhibited dense phase expansion were strongly influenced as opposed to those which did not exhibit dense phase expansion. The latter yielded heat transfer coefficients which could be reasonably predicted by a theoretical model based on a combination of solids movement and unsteady-state heat conduction, using well-known models from the literature_ The slight influence of the driving force on the experimentally determined heat transfer coefficients could also be anticipated when accounting for the temperature-dependent thermal properties of gas and solids. For powders exhibiting dense phase expansion, prediction with the proposed model proved to be impossible_ Here, the solids behaviour seems to be governed by random movements which are difficult to relate to fluidizing parameters. Tentative calculations suggested that the bubble frequency may be a useful correlatingparameter.

1. INTRODUCTION

At present it is generally recognized that heat transfer to a fluidized bed cannot satisfactorily be explained on the basis of a film model, Le. a stationary resistance to heat transfer, but more appropriately on the basis of au unsteady-state heat conduction model. *Prsent address: N.V. Ned. Gasunie, P-0. Groningen. The Netherlands-

Box 19,

This heat conduction takes place in those parts of the bed which, following the solids movement, contact the heat transfer surface for some period. The solids movement, which is mainly induced by the rising bubbles, may have a more or less distinct pattern near the wall of a fluidized bed and the pattern will determine the heat transfer coefficient_

“_ SOLIDS

CIRCUL_4TION

The solids movement in a fluidized bed is induced by the rising bubbles. Thus observations in 2D-fluidized beds show that bubbles are not merely gas pockets but may have a wake in which solids are carried upwards. From this wake, an exchange with the surrounding dense phase takes place continuously. Obviously, the rate of eschange relative to the rising velocity determines whether the solids concerned are transported over any appreciable height. Evidently, the same holds for the balancing stream of descending solids. Two very different exchange situations are visualized in Fig. 1. In Fig. l(a), the solids exchange is supposed to be small relative to the upward motion, thus creating a large-scale circulation pattern, in contrast to Fig. l(b), where strong eschange results in a diffusion-like pattern. Experiments carried out by de Groot [l] confirm the occurrence of both, rather extreme, cases. Large-scale circulation patterns appear to be typical for large-diameter fluidized beds, whereas the diffusion-type pattern is usually found in relatively small-diameter beds. It must be pointed out, however, that these findings apply to powders which fluidize homogeneously (so-called _. powders). Whether these results are pertinent to the behaviour of B powders is unknown.

1'56

-- =b l%f -1 T i --I-----

From photographic measurements Baeyens and Geldart concluded that the upward velocity due to drift action uda equals 0.38~~ on the average. Combining this result with eqn. (2) and solving for pP yields

-1 _-- I L 1

1-_-

u,

f,_(l

=

1

1-t

+

-fw

o-=fd/fw)

_ 6 _ u

(3)

b

--6(1

+fd)

------_I----

UP 1 w - - ---------

t

DISCUSSION

UP1 +t --I-

CP J --_-

* _‘_

UP 1

+t

Either expression for up contains the term hub, which is usually assumed to be equal to the excess gas flow over the minimum fluidization rati, thus:

----

7 - --

su b=KO-Umf

(b)

(a) Fig. l_ (a) Overall circulation tion with strxg exchange.

pattern and (b) circula-

On theoretical grounds it is to be expected that the rising bubbles will also exert a drift action on the surrounding solids, which means that due to the bubble passage solids are first pmhed aside and subsequently Ixansported upwards (Bayens and Geldart [ 23, Rowe (in 131, p_ 156))_ Here, too, a downward stream must balance the net upward flow of solids_ Kunii and Levenspiel [ 43 derived an expression for the solids velocity which accounts only for the wake action whereas that derived by Baeyens and Geldart [2] includes both wake and drift action as will be discussed below. Kunii and Levenspiel’s expression for the downward solids velocity is

f, up = l-fw-6(1

-6-ub + 2f,.)

-(I)

where 6 is the bubble fraction, ub the (true) bubble velocity and f, the wake fraction. The wake fraction as used here is given by the ratio of wake volume to the volume of the bubble plus wake. The slightly modified expression of Baeyens and Geldart reads

(4)

Experimental investigations, however, revealed that the visible bubble flow rate per unit area QB/AD may be far less than u. - K,.,.,~ (Grace and Harrison [ 51, Grace and Clift [ 61, Geldart [ 73 )_ A relevant correction factor as proposed by Grace and Harrison [ 51 reads = (Ko

(QB/ADLH~A

Up

-&+S--

fd (2)

-

(

~fh,r

(5)

fu-

=

-

1-ff,--6(1+2fw) %llr uo

-

up

=

f..-(1 1

-

+

-fw

(

(uo -&If)

(6)

(uo

(7)

hIIf

(Kunii and Levenspiel),

and

o-=fd/f,.) --6(1

+fd)

-

umf

l-4.46 UO

-

4nr

>

-

%lf)

and Geldart).

3. UNSTEADYSTATE

- fw 1 -fd

u,r)

From experimenti data Grace and Harrison concluded that n is about equal to 4-4. Up to now, a ready explanation for this rather high value is not available; bubble coalescence is supposed to be a probable cause. Assuming that n = 4-4 is a representative value, the expression for the solids velocity reads

(Baeyens

=6-

-

HEAT CONDUCTION

3.1 Mathematical model As indicated in the Introduction, transfer in fluidized beds is supposed

heat to take

177

place by means of a temporary solids-to-wall contact, during which heat is transferred by conduction_ For this one-dimensional heat flow the governing equation reads

(Pc,).ffg =;

(

hff g

1

(8)

The subscript ‘eff’ refers to the effective value of the pertinent physical properties of the powder mass, i.e. (PC,),~~ is the effective volumetric specific heat of the dense phase, of and Lif the effective thermal conductivity the dense phase. The increase of heff due to percolation of gas appears to be insignificant as long as the particle diameter does not exceed appros500 pm ([ 31, p_ 487). The use of heff implicitly assumes that the dense phase structure does not change during heat transfer. That this assumption may be invalid was extensively discussed in Part I. The effective volumetric specific heat depends on the specific heat of both gas and solids in a ratio that is given by the porosity_ Due to the low density of the gas relative to the density of the solids, the contribution of the gas is almost negligible_ 3.2 Initial and boundary conditions In solving the equation for the unsteadystate heat conduction, various boundary conditions may be applied. The initial condition appears not to be contioversialr at t = 0 a sudden temperature difference IT!. - T,l is applied. It must be noted, however, that this may be correct in a theoretical model, but hardly realized in practice; see the discussion in Part I. The boundary conditions most frequently applied are t>

0

x=0

T=

T,

x--

T=T,,

In fact, these conditions were used by Mickley and Fairbanks [S] in their theoretical model. In Section 4 we shall deal with this model in more detail. A few authors objected to the assumption of a semi-infinite medium and proposed a layer of finite thickness as van Heerden [ 91 did. However, there is inadequate experimental evidence to sustain this alternative assumption and to justify abandoning the postulate of a semi-in&rite medium. The boundary condition x = 0, t > 0, T = T, has been the subject of discussion, too.

The main objection here is that the ensuing expression for the heat transfer coefficient would yield an infinite value at contact times approaching zero_ Although very short contact times are hard to realize in practice, it nevertheless appeared that the heat transfer coefficients remain finite at diminishing contact times, as was shown by Baskakov [lo] _ Some departure from the model is to be espected because the dense phase for a shallow penetration depth cannot be treated as a one-phase system. To account for this, a correction term, known as contact resistance, Rk, may be included. The expression used by Baskakov [lo] is &z-e-

d,

L-

x efl

2x eff \\-

xi-

the effective thermal a- represents conductivity of the dense phase in the vicinity of the wall. Although this quantity is difficult to define exactly, the thermal conductivity may be taken as an average of that for a porosity of the undisturbed packing and that for a porosity E = 1.

Here, Lff

4. THE

HEAT

TO MICKLEY

TRANSFER AND

MODEL

ACCORDING

FAIRBANKS

In the model of Mickley and Fairbanks [ 81, ‘packets’ of dense phase are supposed to be brought to the heat transfer surface under the action of the rising bubbles. During contact with the surface, unsteady heat conduction occurs. After some time, the packets are replaced by fresh ones. Applying the equation for unsteady heat conduction (eqn. 8) with

t=o

x>o

T=T,,

t>o

x=0

T = T,-

x=m

T = T,,

yields for the momentary coefficient of heat transfer to one particular packet after a contact time t:

=t.

local

=

IF?

This result has to be combined with the distribution of contact times of the packets in order to obtain the average heat transfer coefficient. For two types of distribution -

Ii8

i

Tb i Yg$,

Tb

I i

separated from the surface by a E~titious layer with a contact resistance as discussed in Section 2. The temperature distribution at a given moment is outlined in Fig. 3. According to Baskakov [lo], the following initial and boundary conditions can be formulated: t=o

(b)

(a)

Fig_ 2_ (a) Higbie type or packet movement Danckwerk type of packet movement.

x>O

T=T,

X=0

T=

T’ = T,v +Axcrr

-R,

and (b)

the continuous replacement of Higbie type and the ‘random‘ or Danckwerts type distribution - the model was elaborated by Mickley and Fairbanks_ The accompanying solids behaviour is depicted in Figs. 2(a, b). For the Higbie type of movement. it can easily be shown that the average heat transfer coefficient is given by

with the contact

resistance

6, Rk = h eff

d

s vc

P 2h

efr

w

The expression for the instantaneous heat transfer coefficient is rather complicated, but can be simplified to

&Y= 1

Rk + 0_5R,

with

R,

=

V

7ir

@PC, )efi

(12)

provided RJR7 < 2_ The usefulness of eqn. (12) was demonstrated with esperimental results by Baskakov [ 101.

in which 7 represents the time of contact between solids and surface. For a continuous flow of solids with velocity uP along a wall with height L, the time of contact T equals L/u,. The height (or surface) average heat transfer coefficient then also follows from eqn. (lo)_ The Danckwerts type of solids movement leads to the time and surface average valuer c = J&c,

L&C

(11)

where < represents the probability of a packet a’i any place at the surface to be replaced by another lump per unit of time. In general the surface average heat transfer coefficient may be formulated as a = J(XPC,

LrrS

Thus (I can be regarded as the combined effect of a physical property and a dynamic term S. The latter is often hard to predict, but is clearly related to the intensity of the solids movements

5. THE TACT

HEAT

TRANSFER

MODEL

WITH

CON-

RESISTANCE

In this model the dense phase, for which the above physical picture holds, is now

Fig. 3. Temperature tact resistance_

distribution

6. COMBINATION

OF

SOLIDS

HEAT

in the case of con-

TRANSFER

AND

MOVEMENT

In the espression (10) for the heat transfer coefficient, 7 represents the time of contact between solids and the heat transfer area and, as has been pointed out, the contact time T may be replaced by L/u, for a uniform downflow of solids along the heat transfer surface. However, the solids velocity as derived either by Kunii and Levenspiel or by Baeyens and Geldart appears to be an average over the height of the fluidized bed. As shown by

1'79

Baeyens and Geldart, the local velocity may be height-dependent_ In our investigation the heat transfer area (of height L) is located at an average elevation h in the fluidized bed which has a total height E-I_Since the dimensions of the equipment used are almost identical to those of Baeyens and Geldart, it seems reasonable to assume a similar relationship to hold. According to these authors, then, the ratio ~~~~~~ equals 1.35 at hlH = 0.55, where up,., is the velocity prevailing at an elevation h in the fluidized bed and np is the mean velocity _ Summarizing, we have the following equations: Heat

transfer

for

solids flow without ---Y

x

and with contact

the

case

contact L

of a Higbie

type

resistance: (13)

resistance:

In Figs. 4(a - c) the measured and calculated heat transfer coefficients have been plotted for glass beads of 200 pm, for quartz sand of 135 pm and for catalyst powder oE 145 pm; similar date pertaining to fluidized bed temperatures of 40 “C and 180 “C - at driving force zero - have also been included_ The data for the smell particles were not plotted because of the large difference between the measured and calculated values. The coefficients were calculated on the basis of a solids downflow as predicted by the model of Baeyens and Geldart [ 21 and include a contact resisticeConversely, the same model can be used to calculate which uniform solids velocities correspond to the observed heat transfer coefficients_ The resulting velocities are presented as a function of the escess gas rate in Fig_ 5. It is seen that these quantities are essentially proportional to each other_

i. THE

where I?,=

DIFFERENTIAL

TERIPERXTURE-DEPENDENT

-

d,/2.

0-7&r

whereas the espression for the solids velocity up,., equals up from eqns. (6) (Kunii and Levenspiel) or (7) (Baeyens and Geldart) after multiplication by 1.3 5. Heat transfer coefficients predicted by means of the combined equations are shown in Table 1, together with esperimentally determined heat transfer coefficients_ These coefficients pertain to a fluidized bed temperature of 105 “C and a driving force zero, and have been taken from the esperiments reported in Pert I. As can be seen from the table, there is reasonable agreement between predicted and measured heat transfer coefficients for the larger particles_ For the smaller particles, however, the predicted heat transfer coefficients are far too low, suggesting that the estimated velocities are higher in practice, or, more probably. solids movements are characterized by ‘random’ movements instead of steady downflow. This, in fact, was observed in the 2-D fluidized bed experiments, where the fine particles were frequently disturbed by the rising bubbles. Of course, this results in a decrease in the effective contact time and an increase in the average heat transfer coefficient.

EQUATION

WITH

THERMAL

PROPERTIES

Having elucidated the most workable models for fluidized bed heat transfer, we may revert to a subject which has been touched upon in Part I, viz. the fact that the heat transfer coefficients may depend on the thermal driving force since the thermal properties of the solid material depend on temperature_ We note that in the differential eqn. (S), (p~,)~~~ and hen may be temperaturedependent. If so, an analytical solution of eqn. (8) may prove to be impossible_ One of the few esceptions occurs if the effective thermal diffusivity ~~~~ appears to be temperature-independent (see Carslaw and Jaeger [ 123 )_ For this case it is useful to introduce into the basic eqn_ (S) the following quantity I o=

T

1

J LrrdT

x effo

0

Equation

a0 G =

(8) then transforms

into

a20 =eff

s

After transformation of the initial and boundary conditions by means of the quantity

0,083 0,ll 0,12

014

96d

0.040

21.1

206

0,020 0,03G

0.060

20/l

4,22

0,03a

10,o

1O.G

0,030

0,083

3.02

31,G

0,048 0,072

7 ,G

17,G

nnrl ‘ll~corcllcnlly’

colculnlctl

9G,4

68.3

26,G

20,7

10,l

3.9

21,G

10.2

3,9

74

1,3

I,8

14,o

267

209

139

87

60

37

81

68

3h

190

147

9G

202

lG1 200

43 A,1

200

227

182

131

118 165

107

1BG

126

212

210

IGl

22h

179

11A

2DG

282

210

80

67

36

78

55

31

183

138

91

221

175

14B

lG9

974

12,9

39,G

194

74

135

20,3

7.7

12,9

7.1

2,9

234

13,6

a,9

13,G

90G

ia4

173

7,l

7,0

247

“PI1 (lo-3

9,o

121

wllh rcsintoncc

2AG

“C)

13G

126

a’ (W/m2 ,wdho;t

mlddle gas vcloclty.

The jo values cquol lhc cxprcssion

“ml

1 - 4,46 u0 -llmf

1GG

MG

270

170

119

83

61

1lG

79

19

255

189

121

339

GBR

290

241

wllh lhc lormuln

280

209

162

108

78

49

109

7G

I0

227

173

113

271

21G

110

22R

186

174 19G

wlth rcfilslnncc

d (W/m” “C) wllllollt contnct

in cqns, (G) and (7) nntl hnvc been cnlculnlctl

[ 111.

m/n)

q

036 0,875 0927

0,17 0829 q 0,7H q

q

GAG q 0,73 q

q

= OS92

q

q

q

100

1.02

q

q

0,3rl

0,9G

q

n

3,12

X 10m2

m/e

m* “C/W

- 516

m d86

q

8.06

X IO’”

m/s

q

140

El130

[ 51 for the

I(W%&i?&i q 2OR TiK- I,20 x lOwa m2 “C/W

u,f

I(Tn,ATmi fin = 9.3 X 10N4 m2 T/W

Ti~~6.1x10’4m2”C/W ii,! q 2,18X lo+ m/s

I(‘fn,AT&~

urn1q 1.0 x 10m3 m/n

I(‘&,,ATm m ,520 I?K q 1.4 X low4 m2 “C/W

ulnf - 1,2G

m2 “C/W X 1V2 m/s

3 X 10m4 /(T,,,ATh&?&, lijf q 7,l x 10-O

II,.,,~

lip

-F

X IO+ m/s

I(TII,A7’tiw~

II,“~ - 11,13

datn

ol Crncc and Harrison

$0’0,93

[,I m0~34

I”

90

Id q 0,94

Iw q 040

iil

fd

Iw q O/IO

/,I jo

I,

/II JO

I,,

jo

/,I

f,u

Adrllllonol

(7\, = 105 “C; A7’n 0 “C)

6 cnlculnlcd with the nolldn vcloclly nccorrling to Dnoycns nnrl Gclrlnrt

codriclcnls

142

m/s)

tnlnsbr

(W/l? T)

I&,

hcut

1,7

3,7

“Ill1 (lo-”

6 colcul~~lcrl wlth 1110 sollda vcloclty nccordlng to Kunll nntl Lcvc~~~plcl

dolonnlncd

Tho numcricol values of the woke lrocllon I,, hnvc llccn tnkcn from Rowe nnd Parlridgc The vnlucu or /d hnvc been lnkcn from Dncycns nnd Gcldnrt [ 21,

146pm

Cat. powder

Cnt, powder 70 Mm

Cot. powder 40 pm

136 pm

nun11

O,OG3

31.2

Quark

0,037

0,028

13,7

20.b

bcoda

200 pm

Glnss

0,oxl

0.09

0,022 0,02G

1.98

6.20

llcnrh

110 - Il,f 6 ( low3 m/.3) (n)

UT c~pcrlmcnlnlly

7G /Am

Glnae

Malarlnl

CompnriEon

TABLE 1

181

zz b-1eo /

wNln4-c

/’

300-

g$

I

s_

_-

a

-_-;-I_+_ __ _-_ _ __-_* /- __-_ --_-*

200-

-

IOC-

oLI 0

10

20

__L-._.__.__ 10

(a)

_._....._.__-._50 ‘0 10.'m/s U.-U,",

or Wh-n"C 1 3oot

___-

-_-

10

0

(b)

20

30

50

‘0

UrUrnl

lo-‘mll

Ez

*rm=-c 300-

I

(cl

200

-

100

-

On

25

50

75

Fig. 4_ Heat ficients;---

transfer coefficient as a function of gas velocity calculated heat transfer coefficients_

125

100 &b-U.,

(a through

Id'IW5

c). L

Measured

heat

transfer

coef-

=,rr =

x @=rrP&C,

x err P&C,

0

0

Evaluating 0 b and 0, using the expression and subsequent substitution in eqn. LfZ yields after some re arrangements

for (15)

V

h err0PZfCp

at = f(Ti,.4T) in which f(Tb,4T)

= 1 + bTb

0

irt

+ CT;

+ ($b

+ cT,,)AT

+

+ $c(nT)2

(18)

where 4T = T, - T,,_ Thus the surface-average heat transfer coefficient for a Higbie type of solids flow along the wall is

lhefTOP%cIl

n = 2f(Tb,4T)

Fig. 5. Solids velocities as calculated from the heat transfer coefficients as a function of excess- gas rate.

0, the above differential solved_ With the equality

equation

can be

/ -

1

0

(19)

Ii7

whereas the Danckwerts renewal yields c = f(Tb,aTIJherroPd*rCp

type

of contact

(20)

OS

If in addition a contact resistance according to Baskakov has to be accounted for, Z can be satisfactorily approsimated by

-1 1

1

(21)

2f(T,,4T)

it follows

that

or

-1

1 ot=

J&

(15)

$:::b

New, the dependence espressed as a polynomial:

x erf

=

x rrrO(l + bT +

CT’

The subscript ‘0’ refers to ence temperature_ It was assumed that the was constant. i-e_ that the ture does not vary with Hence we put

respectively.

of hrrr on T is + ___)

(16)

an arbitrary referdense phase density dense phase structhe temperature.

(pc,)err = Pa*&,

c,

0

(1 + bT+cT’

It then holds that

Correction measured

+ ___)

(17)

for heat

the influence of 4T transfer coefficients

on

the

mentioned in Part I, the driving force appeared to exert only a slight influence on the experimentally determined heat transfer coefficients in the case of B powders (see Part I, Fig. 4). It was therefore supposed that this influence could be caused by the temperature dependence of the physical properties_ To support this assumption, the thermal diffusity ucrI was determined experimentally over a temperature range from 30” to 150 “C, using the transient temperature response of a cylinder loosely packed with the pertinent solids material; the AT increments over the range mentioned amounted to 10 “C at the maximum. Within the experimentaI accuracy no significant influence of T on a,rr -.

4T

in which p zr represents the density of the dense phase_ From the assumption that a,rr is independent of T it follows that cp must depend on Tin an analogous way to h,ri, thus

=, =

(22)

I

193 TABLE

TABLE

2

Experimentally

determined

Temperature range 30”

- 160

thermal

E

=cr f_i (10

“C

Quartz sand 135 pm Cl= beads 75 urn Glass beads 200 pm Catilyst powder 70 pm

diffusities

0.45 0.35 0.40 0.4-l

l-i5

1.35 1.45 l-85

Efrective

= CnT

f(Tb,AT

e 0.03 2 0.03 ? 0.03 + O-03

Quartz

sand

135 pm 75 pm Glass beads 200 firn Catalyst powder 10 pm Catalyst powder 70 urn Catalyst powder 115 pm

Glass beads

= 0)

W-‘&T)-

*It may be noted that there exists no Iundamcntal reason to assume that this result will apply generally.

Numerical

of the constants cp o

Quartz sand 135 pm Glass beads 75 pm Glass beads 200 m Catalyst fraction 70 _

700 723 600 69s

in eqn.

(16)

(J/kg ‘Cl

6

(lo-3

2.8 353 3.25 2.95

oc-1)

0.43 0.40 0.40 0.59 0.56 0.4-l

&fro (W/m’

“C)

0_17i 0.13’7 0.153 0.046 0.019 0.06’7

6. CONCLUSIONS

In the case of 200 pm glass beads, quartz sand of 135 pm and the catalyst powder of 145 pm, the experimentally determined heat transfer coefficients correspond reasonably well with the heat transfer coeffkients based on a combination of the Higbie model with a contact resistance and a solids circulation expression. Due to inaccuracies in the measurements as well as to uncertainties in the parameters of the solids movement model, it is difficult to decide whether the Kunii and Levenspiel model of the Baeyens and Geldart model best applies. On the whole, the Higbie type heat transfer model with contact resistance combined with the model of Baeyens and Geldart seems to yield the closest fit. These conclusions also apply, although with less reliability, to the 75 pm glass beads. \Vhen comparing the two classes of cases, uti. those that can be interpreted on the basis of the uninterrupted solids circulation and those that can not, it appears that the former class pertains to powders that can be classified as B powders and the latter to powders that are A powders, according to the classifkation diagram of Baeyens and Geldart [ 13) ; see Part I. Although these conclusions do not

3 values

conductivities &f

(the contact resistance was neglected)_ After application of this correction to the results obtained with glass beads of 200 and 75 pm and quartz sand of 135 pm and with the 145 pm catalyst fraction, it appears that no or only a minor residual dependence on AT is left. See dotted lines, Part I, Figs_ 4(b), 4(a), 4(c) and 4(g) respectively_ With the 40 pm and 70 pm catalyst fractions, however, the residual dependence is still quite considerable; for this the mechanisms may apply that have been discussed in Part 1.

TABLE

thermal

m* /se=)

could be detected*. The results are shown in Table 2. The temperature dependences of the specific heats of the solids used were drawn from literature data and are given in Table 3_ The coefficients b and c refer to general eqns. (16) and (17). The Aeff values (corrected for E variations) are shown in Table 2. Using the above results, the measured heat transfer coefficients can now be corrected for the influence of AT in the following way : 5 (after correction

4

c (lo-6 -4.5 -6.71 -6-35 -5.75

“C’)

lS4

necessarily prove to be true for other - Le. larger - fluidized bed systems, there are several indications in the literature that may sustain the concept of different solids behavour dependent on particle properties. We may therefore refer to the well-known heat transfer correlations of Vreedenberg [ 141, in which he esplicitly distinguishes between correlations for fine, light particles (A powders) and coarse, heavy particles (B powders). With regard to solids behaviour, measurements carried out by Whitehead [ 151 point to circulation patterns when fluidizing coarse material (B powders)_ The good fit of heat transfer coefficients in a fluidized bed of coarse particles measured by Schmalfeld (in [IS] ) with a theoretical model elaborated by Werther 1161 in which he combines solids circulation and unsteady heat conduction further endorses this attempt at predicting heat transfer coefficients.

S

; A P 7

Subscripts of the gas g of the solids s eff effective value of the physical of the wall b” of the bed mf at minimum fluidization

1 2

ACKNOWLEDGEMENTS

4

The authors are indebted to E. Panman, P. J. _A. L&I.de Pont and H. H. van Heiningen for their esperimental and theoretical contributions The stimulating discussions with Dr -4. A. H. Drinkenburg in ‘ihe course of the investigation are greatly appreciated_

5 6 ‘I 8 9 10

2 fd

.!A CP T

R, AT

OF SYMBOLS

solids velocity, m/s wake fraction, (-) drift fraction, (-) superficial gas velocity, m/set specific heat, J/kg “C temperature, “C contact resistance, m’ “C/W temperature difference, “C

“C

property

REFERENCES

3

LIST

probability, [l/set] heat transfer coefficient, W/m’ bubble fraction, (-) thermal conductivity, W/m “C density, kg/m3 c-ontact time, set

11 12

13 14 15 16

J_ H_ de Groat. Proc. Znt_ Symp. on Fluidization. Eindhouen, 1967, p_ 348. J. Baeyens and D_ Geldart. Z+oc_ Znl_ Symp. on Fluidization, Toulouse. 1973. p_ 182. J. F_ Davidson and D_ Harrison, Ffuidization. Academic Press, London, 197 1 _ D. Kunii and 0. Levenspiel. Znd. Eng. Chem. Fundam.. 7 (1968) 446_ J_ R_ Grace and D_ Harrison, Clrem_ Eng_ Sci., 24 (1969) 49i_ J_ R_ Grace and R_ Clift. Chem_ Eng. Sci_. -39 (19’74) 32’7. D. Geldart, Powder Technol.. I (1967/68) 355. H. S. Mickley and D. F_ Fairbanks, AZChE J.. I (1955) 3i4. C. Mn Heerden, A_ P_ P_ Nobel and D. W. van Krevelen, Znd. Eng. Chem.. 45 (1953) 1237. A. P_ Baskakov et al_.Powder TechnoL. 8 (1973) 2’73. P_ N. Rowe and B. A. Partridge, Tmns_ Inst. Chem. Eng.. 13 (1965) T157. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids. Clarendon Press, Oxford, 2nd edn., 1959_ J. Baeyens and D_ Geldart, Proc. Znt_ Symp. on Fluidizntion. Toulouse. 1973, p_ 263. H_ A. Vreedenberg, Chem_ Eng. Sci_. II (1959) 274_ A. B. Whitehead, G. Gartside and D. C. Dent, Powder Techaol.. I4 (1976) 61_ J. Werther, Powder TechnoL. 15 (1976) 155_