ELSEVIER Powder Technology 80 (1994) 99-109

POWOER TECHNOLOGY

Quantitative estimation of bubble size in PFBC

P. Cai”, M. Schiavetti”, G. De Michele”, G.C. Grazzinia, M. Micciob>* “ENEL Xpsi. (Italian Electricity Board), R&D Depar?ment, CRT, Vii A. Pisano 120, 56122 Piss, Italy

bUniversitri di Parma, Department of Inahtrial Engineeting, Vllle delle Science, 43100 Parma, Italy

Received 2 July 1993; in revised form 17 March 1994

Abstract

This paper presents a study on the quantitative estimation of bubble size in pressurized fluidized bed combustors (PFBC). A generalized correlation of bubble size which fits the experimental data quite well has been developed for PFBC on the basis of a comprehensive analysis of previous work. The correlation takes into account the different flow regimes at different pressures and gas velocities, as well as the special variation of bubble size within the lower pressure range of the bubbling regime. Using this correlation, an ideal operating area of the PFBC can also be defined from the viewpoint of hydrodynamics and mass transfer. In addition, the correlation can be applied to atmospheric operations to overcome problems encountered by the previous correlations under high gas velocities (e.g. turbulent fluidization regime).

Kqworak Fluidized bed; Combustors; Bubble size

1. Introduction

Bubble size has been well recognized as one of the most important parameters of conventional gas-solid fluidized beds. Most rate phenomena in fluidized beds have been found to depend on the bubble size, e.g. bubble rising velocity, gas rate of interchange between phases, heat transfer, fine particle elutriation, etc. How- ever, little reliable information is presently available on bubble size under high pressure, which is crucial for the design and accurate modeling of PFBC. This study, therefore, was aimed at providing a satisfactory solution for bubble size estimation under high pressure.

Though many experiments have been done on bubble size under pressure, as seen by the large volume of publications in this area, no generalized conclusions have been drawn. Some attempts at summarizing the effect of pressure on bubble size were concluded with ‘no consistent results could be derived from the reported data’ [l] due to the following factors:

(1) work carried out by different investigators em- ployed different experimental setups and operated under different conditions:

*Author for correspondence.

(2) some results from different researchers under similar experimental conditions were controversial;

(3) the mechanism involved in some phenomena remains unclear, e.g. the bubble size has a slight initial growth with increasing pressure [2].

There are also problems in the correlation of bubble size under atmospheric pressure. For instance, whereas it has been observed that bubble size decreases with increasing gas velocity under relatively high gas velocity, almost all the currently available bubble size correlations predict a monotonic increase in bubble size with gas velocity.

It seems that the problems in correlating both low pressure and high pressure bubble size data have mainly resulted from the simple mixing of data. It has been found that bubble size shows different behavior under different flow regimes [3-51. In addition differences in measurement technique, form of the data, etc., can lead to differing results for the same problem. Thus, previous attempts which correlated data belonging to different regimes and different experimental arrange- ments failed to obtain generally applicable bubble size correlations which could cover the whole operating range of fluidized bed combustors. This work, therefore, starts by determining the general diagram for bubble size variation with pressure and gas velocity, D, =f(P, u),

0032-5910/94/$07.00 0 1994 Elsevier Science S.A. All rights reserved SSDI 0379-6779(94)02834-B

Tab

le

1 Pr

evio

us

rese

arch

on

bu

bble

si

ze

unde

r el

evat

ed

pres

sure

Aut

hor(

s)

D

T P

4

4 +

J P

owde

r G

as

U-U

, H

,,$

H

(m”;

”

Mea

suri

ng

Flow

R

esul

ts

Con

clus

ions

(m)

(“C

) (b

ar)

(pm

) (k

g/m

? gr

oup

(m/s

) (m

) m

etho

d re

gim

e an

d co

mm

ents

Kno

wlt

on

[311

0.

292

Can

ada

and

Mac

-Lau

ghlin

[291

0.30

5 x

0.30

5

Gue

des

de

Car

valh

o

et

al.

[301

Subz

war

i

et

al.

1331

Kin

g an

d H

arri

son

WI

0.1

2-D

0.46

x

0.01

5

0.1

Row

e an

d

Mac

-Gill

ivra

y

[81

0.19

Kaw

abat

a 2-

D

et

al.

0.3

x

u31

0.01

Bar

reto

et

al.

[281

Chi

test

er

et

al.

WI

0.1

2-D

0.10

2x

0.01

9

Row

e et

al

.

[21

Sch

wei

nzer

and

Mol

crus

[321

0.17

5 x

0.12

5

0.4

ambi

ent

1,

10,

20.3

0,

40,5

4,

68

ambi

ent

l-89

ambi

ent

l-28

ambi

ent

1-7

ambi

ent

l-10

ambi

ent

1, 4

ambi

ent

I-8

ambi

ent

1-12

ambi

ent

8, 3

2

64

ambi

ent

l-80

ambi

ent

1.54

0

777

1470

co

al

287

1291

co

al

267

1170

lig

nite

229

1121

lig

nite

244

1180

ch

ar

320

1571

co

ke

290

3912

si

deri

te

650

2480

1

2600

29

20

1

64

2900

211

2900

74

2690

59

950

100

2860

64

2900

10

1 29

00

475

2900

58

3186

1 1 sand

FCC

1 Sic

300

2630

430

2580

600

2590

98

1260

71

1260

53

1260

68

880

195

1247

203

1116

sand

sand

sand

zeol

ite

zeol

ite

zeol

ite

FCC

coal

char

450

1417

A

N3

262

3186

SE

56.5

24

95

-1

157.

4 26

30

sand

1065

26

30

sand

B

A/B

A/B

A

A

A/B

B

B/D

D

A

B

A

A

A/B

A

A/B

B

A

B

B

B

A

A

A

A

A/B

A/B

B

B

A

B

D

NZ

u<

0.

44

dire

ct

bubb

ling

no

quan

titat

ive

whe

n P

>

10 b

ar,

bed

and

resu

lts

for

Db

tend

s to

be

sm

ooth

er,

turb

ulen

t i.e

. sm

alle

r bu

bble

s

air

R-1

2 bu

bblin

g,

fluc

tuat

ion

slug

ging

am

plitu

de

=f(

P,

(I,

and

part

icle

pr

oper

ties

)

turb

ulen

t

pres

sure

fluc

tuat

ion

anal

ysis

slug

ging

is

le

ss

pron

ounc

ed

at

high

pres

sure

an

d he

at

tran

sfer

co

effi

cien

t

also

in

crea

ses

64

+n

part

icle

an

d C

O1

syst

em

show

sm

alle

r D

b

at

high

er

pres

sure

(P>

5 ba

r)

bubb

le

size

de

crea

ses

with

in

crea

sing

pres

sure

for

syst

em

of

64

/Lttt

, D

,, de

crea

ses

with

incr

easi

ng

P,

othe

rs

not

affe

cted

bubb

le

size

de

crea

ses

with

in

crea

sing

P

an

d de

crea

sing

U

the

rela

tion

betw

een

Db

and

H

beco

mes

w

eake

r,

whe

n P

in

crea

ses,

an

d

bubb

le

is

also

fl

atte

r

Db

decr

ease

s w

ith

incr

easi

ng

P

at

the

sam

e U

co2

U<

O.2

6

N

0.6-

1.4

air

U=

O.O

32

air

0.02

7

0.31

0.

16

air

N

CO

2

0.14

air

U=

O.O

27-

0.08

8

0.3

0.2

air

0.05

N2

u=o.

o5

NZ

O

-O.2

13

1.37

2 0,

0.

762

1.07

7

1.37

2

1.67

6

W

0.02

8 0.

039

0.25

X

-ray

bu

bblin

g D

,,=f(

P)

d <

0.8

R11

5

0.4

0.2

capa

cita

nce

0.6

prob

e

dire

ct

no

quan

titat

ive

visu

al

resu

lt fo

r

obse

rvat

ion

bubb

le

size

tine

bubb

ling

max

. bu

bble

si

ze

phot

ogra

phy

(onl

y 59

W

)

X-r

ay

bubb

ling

slug

le

ngth

and

slug

ging

X-r

ay

bubb

ling

bubb

le

volu

me

=

f(P

, rr

)

bubb

ling

Db

=f(

H,

P)

tine

phot

ogra

phy

X-r

ay

bubb

ling

Db

=fV

’, pa

rtic

le

prop

erti

es)

phot

ogra

phy

bubb

ling

no

quan

titat

ive

and

resu

lt fo

r D

b

turb

ulen

t

no e

vide

nce

for

max

imum

st

able

D

b up

to

8

bar;

high

P

tr

ansf

ers

bed

to

turb

ulen

t re

gim

e

Db

decr

ease

s w

ith

incr

easi

ng

P

afte

r

a sm

all

incr

ease

up

to

10 b

ar

incr

easi

ng

P

caus

es

smal

ler

bubb

le

size

, w

hich

is

mor

e ev

iden

t

for

coar

se

part

icle

s

bubb

ling

ratio

n of

av

erag

e

and

bubb

le

size

to

turb

ulen

t th

at

at

2 ba

r

Sella

kum

ar

and

Zak

kay

VI

Cai

et

al

.

(31

Alm

sted

t

and

Zak

kay

P71

Olo

wso

n an

d

Alm

sted

t

[ill

Alm

sted

t

[71

Wei

mer

an

d

Qua

rder

er

A

HO

fiilla

lU??

and

Yat

es

PO1

Cha

n et

al

.

(141

tape

red

780

8-17

be

d 85

0

0.9

x 1.

2

to

0.9X

2

with

tube

bund

le

0.28

4 am

bien

t l-

8

0.39

4 27

with

tube

bank

2.4

0.2X

0.3

ambi

ent

l-16

with

75

0-90

0 10

,16,

cool

ing

20

tube

s

0.13

am

bien

t l-

83

0.17

8 x

ambi

ent

I-80

0.12

7

0.38

1 am

bien

t l-

32

700

2400

476

834

1057

84

4

280

706

165

711

98

2580

53

1667

65

1172

566

1330

486

3000

486

2200

962

2200

700

2600

-1

-1

-1

-1

sand

FCC

FCC

-1

0.75

0.82

0.82

0.8

900

2600

0.

95

66

850

108

850

171

850

450

1417

0.

66

695

1488

0.

66

184

3186

0.

68

100

2565

sa

nd

200

2565

sa

nd

400

2565

sa

nd

400

1507

co

ke

400

1251

ch

ar

B/D

B

B/D

A/B

A

A/B

A

A

B

B

B

D

B/D

D

A

A/B

A/B

B

B

B

A

B

B

B

B

c u=

o.94

1.

14

capa

cita

nce

air

1.23

2.

13

prob

e

air

cJ<

1.2

0.

3 0.

2 pr

essu

re

0.6

0.4

fluc

tuat

ion

0.9

0.6

anal

ysis

air

cJ=

o.9

Hr=

1.

23

0.14

ca

paci

tanc

e

0.97

pr

obe

air

0.14

.6

0.87

0.

05

capa

cita

nce

prob

e

air

u=o.

95

Hf =

2.5

- 0.

58,

3.0

1.34

2.10

capa

cita

nce

prob

e

gash

(M

.06

1.33

pr

essu

re

bubb

ling

D,,

=f(

dp,

U-

U,,

P);

fl

ucto

atio

n D

b fr

om

U,

and

anal

ysis

D

-H

equa

tion

N2

NZ

0.02

8 0.

5 0.

2S

0.03

9 0.

40

X-r

ay

0.03

2.

03

0.45

7

0.05

0.

762

0.06

7 1.

067

0.08

5 1.

334

0.10

4 1.

555

pres

sure

bu

bblin

g lb

=fl

p,

u -

&I,

H

) pr

obe

and

turb

ulen

t

turb

ulen

t m

ean

pier

ced

leng

th

of

bubb

le

=f(

T,

P, H

)

bubb

ling

fluc

tuat

ion

and

ampl

itude

=

turb

ulen

t fi

P,

U,

part

icle

prop

erti

es)

bubb

ling

I,,

(in

the

cent

er

and

of

the

bed)

=

turb

ulen

t f(

P,

U-

U,J

1, =

m

P, H

)

bubb

ling

Db

=f(

P)

and

turb

ulen

t

mea

n pi

erce

d le

ngth

of

bubb

le

decr

ease

s w

ith

incr

easi

ng

pres

sure

and

incr

easi

ng

gas

velo

city

high

er

pres

sure

ca

uses

smal

ler

bubb

le

size

an

d

low

er

tran

sitio

n ve

loci

ty

to

turb

ulen

t re

gim

e,

whi

ch

is

true

fo

r al

l

the

part

icle

s

lb d

ecre

ases

w

ith

incr

easi

ng

P a

nd

decr

easi

ng

U -

(Iti

1, i

ncre

ases

w

ith

incr

easi

ng

P,

lb a

H”,

II

= 0

.09-

0.49

, n

decr

ease

s w

ith

incr

easi

ng

T

Db

decr

ease

s w

ith

incr

easi

ng

P a

nd

decr

easi

ng

u-u,

Db

decr

ease

s ov

er

mos

t

the

rang

e of

P a

nd

show

s

mor

e no

n-un

ifor

m

dist

ribu

tion

in

radi

al

dire

ctio

n

incr

easi

ng

pres

sure

caus

es

the

decr

ease

of

bubb

le

size

‘Or

Ho.

bM

ixed

ga

s,

M,=

17

, vi

scos

ity

at

ambi

ent

cond

ition

= 1.

56~

IO-’

P

a s.

‘Aft

er

re-c

orre

latin

g.

dDen

sity

=28

kg

/m’

at

4 ba

r.

‘Bur

ning

co

al.

102 P. Cai et al. / Powder Technology 80 (1994) 99-109

by carefully collecting, analyzing and summarizing the previous results on bubble size under different pressures, and rearranging these results according to the type of flow regime. Upon establishing the general diagram, a bubble size correlation for operating ranges of practical interest for PFBCs is obtained.

2. General variation of bubble size with pressure

In this part, efforts are made to answer the following questions:

(1) What is the trend in the bubble size variation when changing the operating pressure and gas velocity?

(2) What is the relation between bubble size and flow regime?

(3) What, if any, is the valid range of the previous correlations on bubble size in PFBC?

Table 1 summarizes the major findings of the effect of pressure on bubble size and includes different flow regimes (bubbling, turbulent and slugging regimes) and particle systems (groups A, B and D of Geldart’s classification). Most work did not indicate the operating flow regime, though many of them covered more than one regime. Hence, the flow regimes are identified in Table 1 according to the correlations of Cai et al. [3]. Since the characteristics of bubbles are closely related to the operating conditions and measuring position, such information is also listed in Table 1. It is worth mentioning that, although in Table 1 only work done in the past 15 years (the period of rapid development of PFBC) is included, several much earlier papers related to this area are also of importance. For example, Lanneau [6] measured a fluid-dynamic parameter de- fined as ‘bed non-uniformity’ and reported its variation under different pressures and gas velocities; hence, Lanneau’s results can now be regarded as an anticipation of pressure effects on bubble size under different flow regimes.

As shown in Table 1, most researchers reported a monotonic decrease in bubble size with increasing pres- sure under the conditions of constant U or constant (V-U,&, whereas contrasting observations were re- ported by Almstedt [7] and Sellakumar and Zakkay [l], etc. These seemingly controversial results, however, appear to be quite clear and consistent upon relation to the detailed analysis of the flow regime associated with the data:

(1) At constant pressure, with increasing gas velocity, the bubble size increases under the bubbling regime [8,9] and decreases under the turbulent regime [1,3].

(2) At constant U or (V-V,,,& the bubble size de- creases with increasing pressure in both the bubbling and turbulent regimes except when gas velocity is very low. In the latter case, increasing pressure exhibits a dual effect on bubble size, i.e. there is a small initial

increase in bubble size in the lower pressure range (less or much less than 10 bar) and then a decrease with a further increase in pressure. This phenomenon was reported by some investigators [2,3] and was also revealed in the data of others [lO,ll]. This special region, where bubble size increases with increasing pressure, diminishes or even disappears with increasing

(U-Klr) [31. A qualitative description of the above conclusions

is presented by a three-dimensional diagram of D,=f(P, U) (or &=f(P, U-U,,)) in Fig. 1 for a given type of particle and a given vertical position H in the bed. In this figure, UC denotes the onset velocity of the turbulent fluidization regime, and PC is the particular operating pressure which distinguishes the different trends in bubble size with increasing pressure.

Fig. 2 depicts a two-dimensional plot of some possible experimental pathways abstracted from the surface of Fig. 1 under constant operating pressure [Fig. 2(a)] or constant gas velocity [Fig. 2(b)]. It is seen that starting from different positions of the surface in Fig. 1 may result in very different patterns of variation in bubble size with pressure and gas velocity. This clearly explains why some observations were ‘inconsistent’ with most findings in Table 1.

A typical plot of the reference results is shown in Fig. 3. When comparing the plot of the experimental data [Fig. 3(a)] on the surface of Fig. 1 with the original figure [Fig. 3(b)] of Hoffmann and Yates [lo], it turns out that Fig. 1 can well represent the experimental results. All the other results in Table 1 can also be illustrated by an area or lines on the three-dimensional surface of Fig. 1 without any contradiction.

Fig. 1 also provides information on the validity of the currently available bubble size correlations. For example, the equation of Darton et al. [12] is seen to

Turbulent Fluidization

Bubbling ’ Fluidization

Regime

Fig. 1. Three-dimensional diagram of D,=f(P, U) for a given particle system at a tied value of H. The suitability of Darton et al.‘s equation [12] is shown as an example: (1) original applicable area, (2) possible extrapolation area.

P. Cai et al. I Powder Technology 80 (1994) 99-109 103

PI < P2 < PJ < P, (4

Gas Velocity

(b) 1 -- U = U, (A-B Line)

2 -- U = U, (C-D Line)

3 -- u = us (E-P Line)

PI PP P3

Pressure P4

Fig. 2. Experimental paths of the possible tests under constant

pressures. Bubble size as a function of (a) gas velocity and (b) pressure.

be suitable only for bubbling fluidization operations below or near ambient pressure.

3. Data collection, analysis and conversion

There are two primary considerations when using data for the purpose of correlating bubble size for PFBCs. First, among those data listed in Table 1, Group B particles and the lower part of Group D particles, which are of practical interest for PFBCs, were selected. Moreover, as the influence of an immersed tube is beyond the scope of the present work, only the quan- titative data from free beds (without immersed internals) were chosen. Therefore, the data employed for cor- relation include the works of Kawabata et al. [13], Rowe et al. [2], Weimer and Quarderer [9], HoEmann

H=O.Z5m

a, 0 ? 2.0 0 m U-U,f=0.039m/s

l U-U,f=0.028m/s

1.0 0 1 2 3 4 5 6

P, MPa

Fig. 3. HoEmann and Yates’ [lo] data (a) on the surface ofD,=f(U, P)

and (b) replotted (original data).

and Yates [lo], Chan et al. [14], Olowson and Almstedt [ll], and Olowson [15].

As seen in Table 1, these selected data exhibit a variety of background. (1) The data have various meanings of ‘average’, such as:

(i) the time-averaged pierced length of bubbles (Zb) at a local point (Olowson and Almstedt, 1990);

(ii) the time-averaged bubble diameter at a local point;

(iii) the time- and radial directionally-averaged bubble diameter of a bed section at a certain measuring height

%I,, PJOI. (2) The particles employed are widely spread over the Group B classification and a small part of Group D. (3) The data were obtained at different bed heights. (4) Measurement of bubble size was performed by various methods, such as:

(i) X-ray observation;

104 P. Cai et al. I Powder Technology 80 (1994) 99-109

(ii) local probe detection; (iii) indirect determination from bubble rising velocity.

(5) The experimental facilities differed in size and shape:

(i) two-dimensional bed [13]; (ii) relatively small sized three-dimensional bed [2]; (iii) relatively large sized three-dimensional bed [14]. Apparently, these differences must be taken into

account in order to successfully correlate the data. The following procedures are proposed to eliminate these differences.

(1) Define the bubble size D, as the diameter of a sphere having the same volume as the bubble, i.e. the equivalent bubble diameter, thus following the common practice in FBC design and modeling. It should be noted that bubbles under turbulent fluidization usually do not have a regular spherical or near-spherical shape. The equivalent bubble diameter is the most suitable parameter to describe the bubble size under this regime, because it has been widely employed in the design and modeling of FBCs.

(2) Establish a standard for bubble size at ambient pressure, by which all the data can be converted to a comparable scale so as to utilize as many data as possible and to eliminate the divergence among the data from different investigators.

(3) Clarify the relationship between particle prop- erties and bubble size under the same excess gas velocity to simplify the treatment.

(4) Clarify the relationship between bubble size and bed height at different pressures, which is very important in making full use of the data.

For engineering applications, it is convenient to cor- relate the ‘time- and radial directionally-averaged’ bub- ble size D,, to the distance H above the distributor in the bed. Then, the ‘time and spatially-averaged’ (equivalent) bubble size in the whole bed, D,, can be obtained by integrating Dbh over the bed height from H=O to H=H,

The general procedure for converting the selected data into a suitable form for determining the correlation is as follows. The first step is to convert the time- averaged pierced length of bubbles (lb) into time- averaged bubble diameter (Dbh) by the following equa- tion:

D,, = cz, (2)

where C is a constant. Eq. (2) has been proved by several investigators using different methods (both the- oretical and experimental), e.g. Chan et al. [14] obtained C= 1.43 by means of bubble rising velocity calibration. Thus, the following equation is valid:

When the standard for bubble size under ambient

pressure, D,,, St, is established, the data can be converted to comparable values for bubble size by Eq. (4);

D - DM,, st h(P-q

bh(P-P) - I (4) b(P-1)

where the pierced bubble length (1,) is the radial averaged result at a certain vertical position H. The Olowson and Almstedt [ll] data on pierced bubble length, which are local values, are calibrated according to additional information kindly supplied by them [15]. But the data of Chan et al. [14] are employed directly because such information is not available.

To eliminate any possible divergence introduced by the individual measurements of D,, carried out by the various investigators, the following data conversion was also performed.

D -Dbh, St D

bh(P-p bh(P-P) -

( 9 D bh(P- 1) measured

(5)

Since divergence also exists among the results of bubble size under ambient pressure, the standard values

of Dbh, it should come from a suitable correlation for FBC systems under low gas velocities (i.e. within the valid operating range of the correlation), rather than the experimental results of any individual investigator. After comparing those correlations available in the literature, the equation of Darton et al. [12], one of the most commonly used equations in FBC, was chosen as the standard for Dbh,s,:

D bh, sf = 0.54( U - U,,,f)o.4Ho.sg- ‘.’ (6)

Eq. (6) was also particularly recommended by Baron et al. [16], who found this equation to be more accurate than several others. The integrated form of Eq. (6) along the bed height according to Eq. (1) was reported by La Nauze et al. [17]:

D, = 0.3( U - lJ,,,,)“.4H,o.8g - ‘.* (7)

The effect of gas distributors on the data and on the original equation of Darton et al. [12] is not taken into account in this work. Considering that in FBCs the bed height is usually much larger than the length of the distributor-controlled zone, the influence of different gas distributors was considered negligible for the purpose of the present work. For the same reason, data from shallow beds were not employed. In this way, the data from Table 1 can be fully utilized in spite of the difference in distributors.

P. Cai et al. I Powder Technology 80 (1994) 99-109 105

5. Mathematical modeling of D,,=f(P, U, H)

Generally, each particle system has its specific surface &=f(P, v) at a certain position H. Therefore, it is very difficult to obtain a general convenient D,, =f(P, u> expression for different particle systems. On the other hand, most bubble size correlations at ambient pressure assume that different particle systems have the same bubble size as long as they are operated under the same excess gas velocity [12,18-201. Such treatment makes the term (U- U,.,J the only parameter which needs to be taken into account for particle properties. In other words, all the particle properties seem to be reflected in the value of (U- Umr) term.

For elevated pressures, if such a treatment of particle properties is also reasonable, the estimation of bubble size will be greatly simplified. In fact, some experimental results in the literature support this idea, e.g. Chan et al. [14] found that several PFBC particle systems show similar bubble sizes under the same (U- Umf) (see Fig. 34 of their paper). Another feature to be noted is that the variation in particle density in various PFBC systems is negligibly small. Therefore, for typical PFBC particle systems, the bubble size can be assumed to be &,, =f(P, U- I&), without further considering the par- ticle properties.

To follow the above approach, the value of U,, i.e. the transition velocity between bubbling and turbulent fluidization regimes, is further required. The line (U-UC) in Fig. 1 can be expressed as:

a&l

a(u-- Vn~) UC & =o (8)

It can be seen from Fig. 1 that the simplified treatment of D,,,,=f(P, U-U,,,,) could only be valid under the condition that (UC-- Umf) is constant for all particle systems below a certain pressure. With this condition, the line of (U- U,,,J = (UC- U,.,J does not change with particle properties. The calculation of (UC - Umf) was carried out for typical PFBC particle systems and the results are listed in Table 2. The equations of Cai et al. [3] and Thonglimp [21] were used to calculate UC

and Urn, respectively. It is seen from Table 2 that all the particle systems

of Group B have a similar value for (UC - U,,,,) at the same pressure, implying the possibility that a unique surface of D,, =f(P, U- U,.,,J can express the bubble size of any Group B particle system. The averaged values of (UC - Umf) in Table 2 also provide data for Eq. (8) during the surface fitting.

Once the expression for D,, for a certain distance from the distributor H has been established, D,, as a function of the bed height needs to be defined. Under ambient pressure, many investigators concluded that Dbh aH” (n = 0.8 according to Darton’s equation [12]

Table 2

Values of (U, - U,,,r) of typical Group B particles (T= ambient)

PP d, U, - U,t (m/s)

(kg/m? (mm) P=l bar P=3 bar P=S bar P=7 bar

2500 2500

2500 2500

2500 3000

3000

2000

Mean

0.2 0.73 0.3 0.76

0.4 0.76

0.5 0.74

0.6 0.71 0.2 0.76

0.3 0.78 0.4 0.78

0.5 0.75 0.6 0.71

0.2 0.69

0.3 0.71

0.4 0.73

0.5 0.72

0.6 0.70

0.74

0.39 0.28 0.23

0.39 0.28 0.22

0.37 0.26 0.20

0.34 0.24 0.18

0.32 0.21 0.16

0.40 0.29 0.23

0.40 0.28 0.22

0.37 0.26 0.20

0.34 0.23 0.18

0.31 0.21 0.16

0.37 0.27 0.22

0.37 0.27 0.22

0.36 0.26 0.20

0.34 0.24 0.19

0.32 0.22 0.17

0.36 0.25 0.20

1.30

1.20

1.10

1.00

0.90

0.80

d 0.70

0.80

0.50 0

0.40

0.30

0.20 I

10 20 30 40 50 60 70 P, bar

Fig. 4. The relationship between bubble size and bed height at

different pressures (data from beds without internals). (0) Hoffmann

and Yates [lo], (0) Kawabata et al. [13], (A) Chan et al. [14], (A)

Darton et al. [12], (Cl) Rowe [18].

and n = 0.75 according to Rowe’s equation [ 181). Under elevated pressures, it has been found that, in the case of the free bed (without internals) with a sufficiently large bed diameter and with Group B particles, an increase in pressure does not significantly change the relationship between D,, and H from that of the ambient systems, as shown in Fig. 4. But in the presence of immersed tubes, a much weaker relationship between D,, and H was found. From Fig. 4, an estimation of IZ = 0.8 can be made, which is the same as at ambient operation [12]. Hence, D,, can be expressed as

Dbh = HOTf(P, U - U,J (9

106 P. Cai et al. ! Powder Technology 80 (1994) 99-109

The solution to Eq. (9) at a position H should be visualized as the surface shown in Fig. 1. The three- dimensional mathematical model of the surface in Fig. 1 can be described by a modified Gaussian function:

xexp[-cP2-d(U-U,32-JP(U-Umr)] (10)

where k, a, b, c, d and f are positive coefficients. By adjusting these coefficients, the converted experimental data of D,, can be satisfactorily fitted using Eq. (10). The result of fitting Eq. (10) is the final form of the bubble size correlation developed in this work.

5. Results and discussion

The following bubble correlation for PFBC systems is obtained:

D,, = 0.38H”+‘( U - Urn,)”

xexp[-cP-d(U-U,,,t)2--fP(U-Umf)] (11)

where a = 0.06; b = 0.42; c = 1.4 X 10m4; d = 0.25; f= 0.1. Fig. 5 compares the result of Eq. (11) with exper-

imental data. Fairly good agreement between them can be observed. The average errors of the fitting are &- 11.8% and &- 0.01 m for relative and absolute values, respectively. Considering the essential divergence among the data used for the correlating, the errors are quite small.

The equivalent bubble size of the whole bed was calculated by substituting Eq. (11) into Eq. (1) and integrating over the bed height H,:

0.24

0.20

0.16

E -0.12

-S

SO.08 a

O.Og.“,‘,“,,“‘,,,‘,“,,“,” 00 0.04 0.08 0.12 0.16 0.20 0.24

Dbh,exp l m

Fig. 5. Comparison between predicted bubble sizes and experimental

data. (0) OIowson and Almstedt [ll], (0) Kawabata et al. [13], (A) Hoffmann and Yates [lo], Rowe et al. [2], (0) Weimer and Quarderer [9], (B) Chan et al. [14].

D, = 0.2Hp.8P(U- U,,,,)”

xexp[-cP-d(U-Umt)2-P(U-UmJ] (12)

for a = 0.06; b = 0.42; c = 1.4 X 10e4; d =0.25; f= 0.1. Fig. 6 gives an example showing the good agreement

between the bubble size values measured by experiment and those calculated using Eq. (11). Comparing Eqs. (11) and (12), t i can be concluded that a change in H or Hf values affects only the absolute value of bubble size, rather than the variation trend.

Figs. 7 and 8 present the Dbh =f(P, U- Umf) surface of Eq. (11) and D, =f(P, U- U,,,J surface of Eq. (12) with H = 0.7 m and Hf = 1 .O m, respectively. The variation in bubble size with pressure and gas velocity shown in the figures is in agreement with all the experimental findings listed in Table 1.

0.06 3

:::: I- O 10 20 30 40 50 60 70

P, bar Fig. 6. Comparison between predicted bubble sizes and experimental

data from Hoffmann and Yates [lo]. (-) This work; (0)

U-Umf=0.028 m/s, N=0.25 m; (0) U-lJ,t=0.039 m/s, H=0.40

m [data after conversion according to Eq. (S)].

Bubbling Fluidization

Regime

Turbulent Fluidization

/ Regime

Fig. 7. A typical surface generated using Eq. (11) (H= 0.7 m).

P. Cui et al. I Powder Technology 80 (1994) 99409 107

Bubbling Fluidization

Turbulent Fluidization

/ Reg’me

zF--- ’ rr-\’ _.$+‘ Area of No

Obvious Bubbles

Fig. 8. The variation of equivalent bubble size within a wide pressure

range (Hf = 1 m).

(1) When the excess velocity is very low, the bubble size is only slightly affected by operating pressure, in agreement with the observation made by King and Harrison [22] in the system of 0.475 mm particles under 1 to 10 bar pressure.

(2) In the bubbling regime, for a constant and rel- atively low value of the excess gas velocity, the bubble size as a function of pressure goes through a maximum. After an initial small growth within the range P<P,, the bubble size continuously decreases with increasing pressure (P > PC). In the turbulent regime, PC disappears and the bubble size decreases monotonically. In general, the higher the excess velocity, the greater the rate of decrease in bubble size with increasing pressure. These features are identical with the findings of Cai et al.

131. (3) The transition velocity (UC) from bubbling to

turbulent regime decreases with increasing pressure, which agrees fairly well with experimental results [3].

(4) In the turbulent regime, the bubble size decreases with increasing pressure and gas velocity and approaches zero when both pressure and gas velocity are high enough, as can be clearly seen from Fig. 8. This is supported by the experiments of Chitester et al. [23]. They reported that no clear void (bubble) could be seen at 64 bar. In fact, in this case, bubbles should have a size of the same order of magnitude as the particle size.

(5) From the viewpoint of hydrodynamics, PFBCs should ideally be operated within the turbulent regime, when the bubbles exhibit smaller sizes and a more marked tendency to break up or reorganize; hence, a more efficient heat and mass transfer is expected in the combustor. Using Eq. (12), the ideal operating area for PFBCs can be roughly determined, as shown in Fig. 9.

Bubbling Fluidization

Turbulent Fluidization

Fig. 9. Ideal operating area of PFJ3Cs from the viewpoint of hy-

drodynamics (using the surface of H,=l m as an example).

=0.16 e

n 0.12

0.08

0.04

O%b ’ ‘0.13 ’ ‘oh ‘0.b ’ ‘1.12 ‘1.15

U-Umf, m/s

Fig. 10. Comparison between (1) Darton et al’s [12] equation and

(2) this work at ambient pressure.

An additional result of this work is the bubble size correlation for ambient pressure operations. The cor- relation was obtained by setting P= 1 bar in Eqs. (11) and (12) and neglecting the term cP2 at small values of p:

Xexp[-0.25(U-U,,)*-O.l(U-U&l

De= 0.2lZ~I;~(iJ- Umf)o.42

(13)

Xexp[ -0.25(U- U,J’-O.l(LT-U&] (14)

Fig. 10 shows the comparison between Eq. (13) and Darton’s equation [Eq. (6)] by plotting D,, vs. U- U,,,,. In the bubbling fluidization regime, the two lines ob- tained by the two equations are very close to each other up to (U- U,,,J~0.45 m/s, indicating that the

108 P. Cai et al. / Powder Technology 80 (1994) W-109

predictions of the two equations are similar at low gas velocities. When (U- V,,,3 > 0.45 m/s, with increasing gas velocity, Darton’s equation predicts a monotonic increase in bubble size, whereas the bubble size obtained by Eq. (13) reaches a maximum at the flow regime transition velocity U, and then decreases. The variation in &, with gas velocity given by Eq. (13) is close to that observed in practical fluidized bed systems [3], while Darton’s equation cannot be employed for the estimation of bubble size in the turbulent fluidization regime. The incorrect behavior of Darton’s equation in this regime is due to the fact that it was derived from data under low gas velocity and verified within the bubbling regime only.

Almost all the other bubble size correlations for ambient pressure (e.g. [18-201) have the same problem as Darton’s equation. None of them could be applied to the turbulent regime. Thus, Eqs. (13) and (14) are recommended for a better prediction of bubble size at ambient pressure since they can cover both the bubbling and turbulent fluidization regimes.

Eqs. (ll)-(14) are expected to be applicable under the following conditions:

(1) Group B particles - use of the present corre- lations is likely to extend to the lower Group D region for the interest of FBC, but this should be experimentally confirmed.

(2) Gas velocity - both bubbling and turbulent fluidization regimes.

(3) Pressure - 1 to 70 bar. (4) Bed - having no strong effect of immersed

internals and for a bed height up to H,= 1.56 m (i.e. the highest value of H,,, used for the correlation). It is likely, however, that an extrapolation beyond that value is possible, as the dependence of D, on Hf is well established.

(5) Temperature - from room temperature to the typical operating temperatures of PFBCs. Because all the data employed were converted to the same standard defined by Darton’s equation, the temperatures should be within the same range as Darton’s equation. The latter was considered suitable for FBC operating tem- peratures, as pointed out by La Nauze [24] and De Marco et al. [25]. In addition, the trend of variation in bubble size with increasing temperature as predicted by Darton’s equation and Eqs. (ll)-(14) is in agreement with experimental observations (e.g. [3,26]).

6. Conclusions

(1) A generalized bubble size correlation has been developed which is fairly consistent with experimental findings in both bubbling and turbulent fluidization regimes. It is applicable to PFBC systems with or without

loose packed internals and can be used under pressures as high as 70 bar.

(2) A generalized bubble size correlation for ambient pressure has also been obtained which is recommended for the estimation of bubble size, especially under turbulent fluidization regimes where conventional cor- relations are not applicable.

(3) The ideal operating area of PFBCs has been defined from the viewpoint of hydrodynamics and mass transfer, and is within the turbulent fluidization regime.

(4) Investigation of the effect of particle properties on bubble size revealed that all the systems of Group B particles of interest to FBCs show similar bubble sizes at the same excess gas velocity.

(5) Investigation of flow regime transition found that all the Group B particles exhibit r’ nilar values for excess velocity (U, - U,,,r) during flow regime transition.

Acknowledgements

The authors express their appreciation to Professor P.U. Foscolo of L’Aquila University (Italy) for valuable discussions. Dr P.A. Olowson and Professor A.E. Alms- tedt of Chalmers University of Technology (Sweden) are also acknowledged for kindly supplying data.

List of symbols

D

Db

Dbh

D bh, st

D,

4

L?

H

H* H mea HIId lb

p”-

PC

T u

UC

bed diameter (m) equivalent bubble size (i.e. diameter of a sphere having the same volume as the bubble

(m) equivalent bubble size at a certain height in the bed (m) standard bubble size at ambient pressure (at a certain height in the bed) (m) equivalent bubble diameter in the whole bed

(m) particle diameter (average) (m) gravitational acceleration (m/s’) vertical distance from the distributor (m) expanded height of the fluidized bed (m) measuring height (m) bed height at minimum fluidization (m) pierced length of bubbles (m) molecular mass (kg/kmol) operating pressure (bar) pressure corresponding to the peak point of bubble size in the curve of constant (excess) gas velocity (see Fig. 1) (bar) operating temperature (“C) superficial gas velocity (m/s) transition velocity between bubbling and tur- bulent fluidization regimes, (m/s)

P. Cai et al. I Powder Technology 80 (1994) 99-109 109

u mf superficial gas velocity at minimum fluidi- zation (m/s)

Greek letters

PP particle density (kg/m3)

9 shape factor of particles

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WI

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