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2 Fluidization fundamentalsN. FUEYO and C. DOPAZO

2.1 Introduction

2 .1 .1 Fluidizatio n

Fluidization is an operation by which a bed of solid particles acquire sfluid-like properties by passing a gas or liquid through it .

Fluidization can be best pictured by considering a vessel partly filled witha column of solids (Figure 2 .1). Through a perforated bottom, a gas is injecte dinto the vessel . The sequence of events, as the gas flow rate increases, i stypically as follows :

• For low flow rates, the gas simply flows through the particle interstitia lspace. The particles remain stationary, the bed behaves like a porou smedium and is called a fixed bed (Figure 2 .1a) .

• As the gas velocity increases, there is a point at which the gas-particl edrag compensates for the bed weight . Then, interparticle distance sincrease, the bed expands, and the particles appear to be just suspende din the gas . This is called incipient or uniform fluidization (Figure 2.1b) .

• The gas volume flow rate at that point at which fluidization start sdivided by the bed area is called the minimum fluidization velocity :

4

4. 4

*4 * l a 44• á

+ w••*i i

;•: i1 *'11

•,• ` ••4 *

.0 ,

(a)

(b)

(c)

(d)

(e)

(f )

Figure 2.1 Fluidization states : (a) fixed bed, (b) incipient, (c) bubbling, (d) slugging, (e) transport ,(f) liquid—solid .

umf AGmf

v

FLUIDIZATION FUNDAMENTALS

39

The bed voidage (i .e . the gas volume per unit of the bed volume) at thi sstage is called the voidage at minimum fluidization conditions, Em f .

• For higher mass flow rates, large instabilities will generally develop ,causing the appearance of up-moving bubbles . The fluidization is thensaid to be aggregative, heterogeneous or bubbling (Figure 2.1c) . Aminimum bubbling velocity (U mb) and the corresponding bed voidag e(smb) are defined similarly to the minimum fluidization conditions .

• A further increase in the gas mass flow rate may result, especially i nlong, narrow columns, in the formation of slugs . Slugging occurs whenseveral bubbles coalesce, and the resulting cavity occupies the whole cross-section (Figure 2.1d) .

• At larger Reynolds numbers or, equally, mass flow rates a turbulen tregime appears characterized by short-lived irregular and intermitten tslug-like voids and particle clusters moving through the bed . Simulta-neously, pressure fluctuations decrease in amplitude and increase infrequency .

• Finally, for sufficiently high velocities, a significant fraction of solid swill be carried out of the vessel. This regime is termed fast fluidization ,or lean-phase fluidization (Figure 2.1e). Should the whole bulk solidsbe continuously renewed, the pneumatic transport domain is reached .

The general fluidizing behaviour as set out in preceding paragraphs hassome exceptions .

• The fluidization of liquid–particle systems is generally homogeneous, i .e .there is no formation of bubbles as the liquid velocity increases (Figure 2 .1f).

• Fine, cohesive powders are difficult to fluidize and exhibit channelling ,i .e . the appearance of low-resistance passages through which th efluidizing gas flows .

• Also, certain types of fine particles admit homogeneous fluidization fo ra range of gas velocities before the onset of bubbling .

2 .1 .2 Chapter layout

This chapter is set out as follows :• After this introductory section, the Huid dynamics of particles ar e

discussed (section 2 .2). Relevant topics include the geometrica lcharacterization of particles ; particle drag and terminal velocity ; andGeldart's classification of particles according to their fluidization behaviour .

• Attention is then turned in section 2 .3 to bubble dynamics, encompassingthe issues of jet penetration,bubble shape and rising velocity, flow fiel din and around the bubble, and bubble size and stability .

40

PRESSURIZED FLUIDIZED BED COMBUSTIO N

• Once the dynamics of particles and bubbles have been discussed, th edynamics of the bed are considered (section 2 .4): bed pressure drop, be dvoidage, minimum fluidizing velocity, minimum bubbling velocity ,two-phase theory and freeboard phenomena (entrainment, transpor tdisengaging height and elutriation) .

• The issues of dimensional analysis, similarity and scaling are dealt wit hin section 2 .5 .

• Then, the effects of high operating pressures are summarized in thesection 2.6. The influence on minimum fluidizing velocity, bed voidage ,bubbling characteristics and freeboard phenomena is considered .

• An elementary introduction to heat transfer in fluidized beds is presentedin section 2 .7. Thermal similarity and scaling rules are discussed .

• Finally, some speculative remarks are briefly outlined in the closin gsection 2 .8 .

2.2 Particle dynamic s

Particle characteristics can be expected to play an important role in the Hui ddynamics of fluidized beds . In practical applications, such as pressurize dfluidized bed combustion (PFBC), particles are present in a range of size sand have irregular shapes . Shapes and size distributions are the characterizin gproperties of the ensemble of particles, and are dealt with next in this chapter .

Once the geometrical characterization of particles has been achieved ,attention will be turned to the key (and related) concepts of particle dra gand terminal velocity . Finally, the fluidizing behaviour of different particletypes is discussed .

2.2 .1 Particle geometric characterizatio n

2.2 .1 .1 Particle shape . The characterization of the particle shape byreference to the spherical shape is achieved through the definition of asphericity factor O s :

Asphere

(2 .2 )*s=

particl e

where Aparticle is the particle surface area, and Asphere is the surface area of asphere having the same volume as the particle . Values of Os are therefore inthe interval 0 < * s 1 . The sphericity of pulverized coal is around 0 .696(Shirai, 1954) .

Heywood (1962) has defined a volumetric shape factor k as follows :

Vk = D3

(2 .3 )A

i :

e


4 1

where D A is the diameter of the sphere with the same projected area as th eparticle .

2.2.1 .2 Particle size distribution (PSD) . In PFBC, the solid phase is amixture of particles of different sizes . The classification of particles into group sof similar sizes is often effected through a system of sieves with decreasin gsieve aperture (Figure 2 .2) .

If y i is the mass fraction of solids retained by the ith sieve, and D i andD i_ 1 are, respectively, the diameters of the ith and the previous sieve, the na mean diameter can be calculated a s

Dp =Y id.t

1

with

Di _ 1 +Di

di =

2

It can be readily shown that the mean diameter defined through eqn (2 .4 )is such that the surface-area-to-volume ratio of a particle with size Dp isequal to that for the whole ensemble .

This mean diameter is called the Sauter mean diameter (SMD) and isparticularly relevant in interphase processes (such as drag forces or hea texchange) in which the interface area plays a major role .

For non-spherical particles, it can be shown that, using the sieve metho d

_ _ *IIA+ái w%r*! _

----------- -

-- ----- -

4D

D 1

D 2

D 3

Y1

Y 2

Y3

Y 4

Figure 2.2 Sieve system .

42


leading to eqn (2 .4), the mean diameter would be * S DP (provided that allparticles have the same sphericity O s ) .

Modern apparatus such as those based on laser diffraction and phase-Doppler principies can save time and significantly reduce errors in PS Ddeterminations .

In PFBC one has to deal with a mixture of particles of different sizes ,densities, shapes, composition, etc . The use of joint particle size—densitydistributions may have some advantages from a dynamical standpoint .

2.2 .2 Particle drag

The particle drag is the mechanism by which momentum is exchange dbetween the particle and the surrounding fluid . Particle drag determines theparticle terminal velocity, a paramount parameter in fluidized bed designthat is discussed in the next section .

The drag exerted by stagnant gas on a moving, spherical, isolated particl eis given by

D1 = 2 p9 AP CDIAu2

(2 .6 )

where p9 is the gas density, A P is the projected area of the sphere, Au is theparticle/fluid relative or slip velocity and CD1 is a drag coefficient .

CD1 is a function of the particle relative Reynolds number:

— pg4uDPRep -

a function of Rep . Many

I Í ¡

STOKES

1

10

10 2

103

10 5

10 6

10 7

Re

Figure 2 .3 Variation of CD1 with Re .

Figure 2.3 depicts the variation of CD1 as

100

1 0

C D1

1

1


43

correlations have been proposed to approximate, in a piecewise fashion, th eCD —Re relationship. A typical one is (Kunii and Levenspiel, 1969)

Cm = Re

for Re P < 0 . 4P

CD1 =Re0/2

for 0.4 < ReP < 500

(2 .8 )P

C D1 = 0.43

for 500 < ReP < 2 x 10 5

The classical book by Clift et al. (1978) provides a lavish compilation ofpublished correlations, and their range of deviation from experimental data .

A significant departure from the spherical particle behaviour can b eexpected if the particles have irregular shapes, as it is the case in man yfluidized bed applications . The matter is further discussed in the next subsection .

2.2 .3 Particle terminal velocity

The terminal (or settling) velocity u t is the (constant) velocity reached by afree-falling particle in a stagnant medium.

Its importance as a fluidization parameter stems from the fact that, fo rgas velocities larger than the terminal one, particles will be transported ou tof the bed. It therefore provides an upper bound to the range of operatin ggas velocities, the lower bound being, of course, the minimum fluidizingvelocity . The particle terminal velocity is also thought to play a major rolein bubble stability (discussed in section 2 .3 .6 below) .

The particle terminal velocity is obtained by balancing the gravitational ,buoyancy and drag forces. For an isolated, spherical particle, the balance i s

-6Pg(p — p 9 ) = zAP CDlP Y Au2

(2 .9 )

Introducing the CD1 values given by eqn (2.8) and noting that Au = uts i fug = 0 (stagnant medium), the following values are obtained for u t :

g(p— P 9 )Dpu =`S

1811

4 g(PP — P 9 )2g 2 1" 3u _ D[ts

P 225

11P 9

u = D [ 3.1g (P P — Pg)1 11 2is

P

P q9

for Re P < 0.4

for 0.4 < ReP < 500

for 500 < ReP < 2 x 10 5

(2.10)

A characteristic relaxation time, tR =(18/2 P), is customarily defined as

44


the time required for a spherical particle to reach u ts starting from rest underan acceleration g(pp — p 9)/pp , for Rep < 0.4 .

For arbitrarily shaped particles, the corresponding correlations can b eobtained by using, in eqn (2 .9), the appropriate values for CD .

A simpler alternative is the multiplication of the terminal velocity for th espherical particle by a sphericity-dependent correction factor :

ut =rluts

Pettyjohn and Christiansen (1948) suggest tha t

= 0.8431n ((/),/0 .065) for Re < 0 . 2

rl = 5.31 — 4.880,

for Re > 1000

Clift et al. (1978) indicate that terminal velocity is known to correlat epoorly with sphericity, and suggest the use of Heywood's volumetric shap efactor k (defined in section 2.2.1) for the intermediate range of Reynoldsnumbers Re < 750. The correction factors are

0.104 + 1 .538k for NDÁ = 1

0 .127 + 1 .526k — O .lk2 for NDÁ = 100 .5

0.1975 + 1 .575k — 0.45k2 for NDÁ = 10

(2 .13)

0.166 + 1 .496k — 0.3k2

for NDÁ = 10 1 . 5

0 .0665 + 1 .907k — 1 .05k2 for NDÁ = 10 2

For the Newton regime (the nearly horizontal part of the curve in Figure2.3, i .e . 750 Re - 3.5 x 105), the terminal velocity for irregularly shapedparticles does not depend strongly on Re, but it does on Mv . Clift et al.(1978) recommend the correlation by Barker (1951):

u t = 0.49(Mv + 1)1/36 [ gMvDs 11 2

(1 .08 — os)*

Finally, Kunii and Levenspiel (1969) have plotted values o f

N (=C RE2 =4gDp9(pp—p)

= 4 MvG )D

D

t

3u2

3

a

versus values of Ret(= Dpu t /v) as a function of ¢ s . The plot thus provides thevalue of Ret (i.e . ut ) given the physical properties of the gas and the particle (i .e.ND and 4's) .

(2 .11)

(2 .12)

0.1<Mv< 7.6

(2 .14)


45

2 .2 .4 Particle fluidization characteristics

It has already been indicated in section 2 .1 .1 that the fluidization behaviou rdepends on the characteristics of the particles . Geldart (1973) identifiedparticle size and gas particle density difference as the key characteristic sinfluencing gas fluidization behaviour . His classification (depicted in Figur e2.4a) includes four types of particles, which are described below .

• Group C includes small and/or light particles. Beds of group C particle sare prone to channelling (i .e . the appearance of low-resistance channelsthrough which the gas flows), and are therefore very difficult to fluidize .

• Group A particles are larger than group C ones . They do allow stablefluidization and, unlike particles in other groups, they exhibit homogeneou s(i.e . bubble-free) fluidization for a range of gas velocities . Typically, theminimum bubbling velocity is two or three times the minimu mfluidization velocity . Once bubbles appear, they generally rise faste rthan the interstitial gas .

• Group B particles are normally larger and heavier than group A ones .Bubbling starts at the minimum fluidizing velocity, and most bubblesrise faster than the gas .

• Group D particles are larger and/or heavier than those in other groups .Group D particles fluidize heterogeneously, and require, by reason o ftheir weight, much higher fluidization velocities than particles in group sA and B; and all but the largest bubbles flow slower than the gas .

It is important to note that Geldart's work was conducted at atmospheri cconditions (temperature/pressure) . The effect of pressure on fluidizatio n

400 0

300 0

2000

Pp — Pg

1000

kg/m 3 s

100

1000

Dp µm

Figure 2.4 (a) Geldart's (1973) particle classification according to fluidization behaviour .

PRESSURIZED FLUIDIZED BED COMBUSTION

2

r

46

1/ 3

D* =ArI/s

_ D [Pg (Pp P5 ) gp

P

1- 2

Figure 2.4 (b) Grace's (1986) flow-regimes map .

characteristics is dealt with later in this chapter . It should also be born inmind that, in practical applications, particles are not mono-sized but hav ea range of sizes . More recent work has in fact shown the importance of fine sin the fluidization characteristics (Abrahamsen and Geldart, 1980) .

A number of alternative classifications have been published in the literature ,among which Grace's (1986) deserves to be singled out . Grace uses twodimensionless parameters (a dimensionless diameter Dp = Ar 113 and a

2 .

Tp ia l

a tPfd id imte

a


47

dimensionless velocity u* = u [p2/(p i, — p9)gµ] il3) to represent the map offlow regimes encountered in gas particle fluidization . Grace's map (representedin Figure 2.4b) draws on a wider database than Geldart's (including gase sother than air and temperatures and pressures other than atmospheric), an dproposes new boundaries between groups A—B and B—D on the basis of th eadditional data .

2.3 Bubble dynamic s

The fluidizing behaviour of gas particle systems has been set out in thepreceding section . It was noted here that, for most particles of practica linterest (and in particular for FBC), bubbles form in the bed for sufficientl yhigh gas flow rates .

Bubbles play, in fact, a key role in several important aspects of the fluidize dbed performance, notably :

(1) Mixing —The upward motion of bubbles in a fluidized system greatlyenhances mixing, and hence promotes the uniformity of bed propertie s(e.g. heat and mass transfer).

(2) Bed expansion — The bed height is a function of the bubble-phas evolume within the bed .

(3) Through flow — It will be shown below that fast-moving bubbles carrywith them a cloud of gas and particles that circulate through the bubbl ebut are not exchanged with its surroundings . This through-flowhampers mixing, and may cause the elutriation of unburned particles .

(4) Elutriation — The phenomenon of elutriation is compounded by th ebursting of bubbles at the bed surface, which throws particles into th efreeboard zone .

2.3 .1 Jet penetration and bubble formatio n

The physics underlying bubble formation is not well understood . Someplausible speculations are advanced in section 2 .8 . However, literatureabounds in phenomenological descriptions of the process .

The fluidizing gas is introduced into the bed through the distributor, locate dat the bottom of the bed . In a comprehensive study on the subject of je tpenetration and bubble formation, Massimilla (1985) has identified fiv edifferent flow patterns leading to the formation of gas bubbles from the ga sdischarge at the distributor . However, he suggests that qualitative difference smay in part be attributed to diffiiculties associated to the photographi ctechniques employed .

Rowe et al . (1979) have suggested the use of only two modes of gas discharge :

48


a stable jet and a succession of bubbles . Massimilla (1985) indicates that theflow pattern evolves from the chain-of-bubbles type to the permanent-je ttype as the particle size increases .

Hirsan et al. (1980) have defined three different jet penetration length s(depicted in Figure 2 .5) :

• LB , the penetration of bubbles formed at the jet tip finto the bed beforelosing their momentum, such loss being evinced by the significan tdeviation of the bubble from the vertical direction.

• LMAX, the maximum length of the succession of cavities attached to the jet .• LMIN , the jet penetration length .

Several correlations have been published for the jet penetration length s(see Massimilla (1985) for a comprehensive listing) . Inspection of thesecorrelations reveals that jet penetration :

(1) decreases as particle density and size increases ;(2) increases with bed pressure .

The effect of orifice diameter Do on L/Do is, however, controversial; whilesome correlations show no influence, others show dependence ; and, furthermore ,this dependence does not always display the same trends. For PFBC, thecorrelation by Hirsan et al . (1980) has the merit of using the complete fluidizin gvelocity uef (i .e . the velocity at which the whole bed is fluidized), rather tha nthe minimum fluidizing velocity u,,, f , as independent variable . For widel ydistributed particle sizes (as is the case in FBC) uc f is more significant than

and it is also more sensitive to the effect of pressure (u, f is furtherdiscussed below in section 2 .6 .1). This correlation reads

*f*

-0.24

Do= 26.60

°90

0 .67

[-uPPV 9 P

L B

(2 .15)

LMAX

Figure 2.5 Jet penetration lengths .


49

DO%

P u 10.83[

-0 .54

=19.3 L 9Pp .\/gDn

!leí

2.3.2 Bubble shape

The shape of a bubble in a fluidized bed is either nearly spherical or a`spherical cap' . The term spherical cap refers to a spherical shape with a rea rindentation (Figure 2 .6) . The indentation is usually filled with particles tha tmove upwards with the bubble, forming the so-called wake . The work ofRowe and Partridge (1965) indicates that the angle 0 (Figure 2.6) increasesas the particle diameter does; and an increase in operating pressure has th eopposite effect (Chiba et al., 1985) . Cranfield and Geldart (1974) show thatbubbles with group D particles are nearly spherical .

2.3.3 Bubble rising velocit y

The analogy between gas–liquid and gas–solid systems is often employe dto illustrate the behaviour of bubbles in a fluidized bed (Davidson et al . ,1977) . Kunii and Levenspiel (1969) point out that the shape of the bubblesis the same in both cases ; that smaller bubbles rise more slowly than largerones in both cases, and that the rise velocity in gas–solid fluidized beds i sin close agreement with that in gas–liquid systems .

The theoretical work of Davies and Taylor (1950) showed that the velocityof an isolated bubble rising in a liquid in the absence of wall effects is given b y

(2 .17 )

Measured values of rise velocity of bubbles in fluidized beds show that th eequation by Davies and Taylor (1950) for liquid–gas systems is applicabl eto emulsion-bubbles systems, with the coefficient ranging from 0.57 to 0.85(Kunii and Levenspiel, 1969) . Thus, for a (widely used) value of 0.711, the

Figure 2 .6 Bubble spherical-cap shape .

50


rise velocity is

ubo, = 0.711(gDb ) 1/2

(2 .18)

In the presence of walls, the isolated-bubble velocity u b , needs to becorrected . Wallis (1969) suggests :

U b = 1.13ubw e -Db/Dv

(2 .19 )

Equation (2 .19) is applicable to 0 .125 D b/Dv 0.6; outside this range,the bubble is small enough for wall effects to be negligible, or large enoughto be considered a slug .

Bubble interactions also change the bubble size . The following equation,originally devised by Davidson and Harrison (1963) from continuityconsiderations, has proven to give an approximate bubble rise velocity whe nbubble interactions are considered :

ub=ubw+u —u mf

(2.20)

2.3 .4 Flow-field in and around the bubble

A number of models have been proponed for the velocity and pressure field swithin and in the vicinity of a bubble in a fluidized bed; see Cheremisinoff(1986) for a summary . One of the earliest, and perhaps more widely used ,models is that by Davidson and Harrison (1963) . Davidson's model rests onthe following assumptions :

(a) The dense phase is treated as a continuum that flows around the bubble .(b) The gas and solid velocity are linked through Darcy's law for porous media :

(u 9 — up) = — kVp

(2 .21)

(c) Voidage is constant in the particulate phase .(d) The fluidizing gas is incompressible.(e) The bubble is circular (2D) or spherical (3D) in shape .(f) The bubble boundary is an isobaric surface .(g) The (unperturbed) pressure gradient away from the bubble is the sam e

as the pressure gradient under minimum fluidizing conditions (se esection 2 .4 .1 below) .

With (b), (e) and (d), pressure is the solution of a Laplace equation, wit hboundary conditions given by assumptions (e), (f) and (g) . For the coordinatesystem shown in Figure 2 .7 (which moves with the particle), the solution i s

3U mf

p

k8 f(r— R2*cos 0

m

(2.22)

FLUIDIZATION FUNDAMENTAL S

Figure 2.7 Bubble and co-ordinate systems .

Assumption (a) yields, for the particle velocity components ,

R 3ur ,P - - ub (1 — 3) COS 6

3

ua .p = u b (1 + R3) sin B

The gas stream function can be obtained by combining eqns (2 .21), (2 .22 )and (2 .23) :

R3((

3 2

1) { 1 — sin 2 0 (2.24)= :f (a —mf 11 2

a=Ub Emf

(2 .25)umf

2-a+2 1/ 3

R

R (2 .26)* a — 1

The flow pattern is different for a > 1 and a < 1, as evinced by eqn (2 .24)(see Figure 2.8) :

• For a < 1, eqn (2 .25) yields Ub < umf /E mf (i .e . the bubble moves slowe rthan the interstitial gas) . A plot of eqn (2 .24) shows that the fluidizinggas enters the bubble through the bottom, and leaves through the top .There is a toroidal region of gas that circulates around bubble equator ,moving up with it . The size of this torus increases as Ub approaches um f /E mf .

(2 .23 )

52

PRESSURIZED FLUIDIZED BED COMBUSTION

F

P

st 1a

a

Figure 2 .8 Gas circulation patterns within the bubble. Left : slow bubble ; right : fast bubble.

• For a > 1, the bubble rises faster than the interstitial gas. A bubbl ecloud appears (of radius R, in eqn (2.24)), which fully surrounds th ebubble and is impervious to the gas outside it . R, is infinity forUb = u,,,f /Emf, and decreases as the bubble velocity increases . The gas inthe bubble circulates (as in the slow bubble case), from bottom to top,and returns through the cloud .

Murray (1965) relaxes Davidson's hypothesis on the flow of the particulatephase by including a momentum equation of the solids . Compared wit hDavisdon's, Murray's model predicts smaller, non-concentric clouds, whic his believed to be closer to reality ; but Davidson's provides a better predictio nof the pressure field around the particle (Jackson, 1971) .

The main qualitative difference between Davidson's or Murray's model sand reality concerns probably the shape of the lower part of the bubble ,where the pressure difference between the bubble and the emulsion draw sgas into the bubble . The ensuing instability results in the kidney-shapedindentation described in section 2.3 .2 . Solids are carried with the gas intothis indentation, forming a wake that travels upwards with the bubble. Theentrainment and shedding of solids by the wake is thought to play a majo rrole in solids mixing in a bubbling fluidized bed (see Rowe and Partridg e(1962)) .

The particle wake is usually taken as roughly completing the boundin gsphere. Clift et al. (1978) have provided a correlation for the ratio o fwake-to-bubble volume in gas–liquid systems as a function of the bubbl eReynolds number, resulting in

YW= 0.037Rebt .4 ,

3 Reb 110

(2 .27)b

w(c

al

D

w 1

vaanD ¿

2 . ;

B L

m ¿

sa l

insres


53

2.3 .5 Bubble siz e

Bubbles are supposedly the driving force of mixing in fluidized beds ; andhence bubble size (and the closely related bubble velocity) are cardinalparameters in the characterization of the state of the bed.

Bubble size changes in the bed as a consequence of the coalescence an dsplitting processes . There are a number of correlations in the literature fo rthe axial evolution of bubble size, mainly for group B and D solids (Hori oand Nonaka, 1987) .

Mori and Wen (1975) suggest tha t

dDb = 0. 3dZ D (Dbm — D b )

(2 .28 )v

and hence

Dbm — Db 0

where Dbm is the maximum bubble size, given by Mori and Wen (1975) a s(cgs units) :

Dbm = 0.652[Av(u — umf )] o . 4

and Dbo is the initial diameter (cgs units) :

— umf )A v1 0 .4

(2 .30)

(2.31)D b o

0 .347=

[

(un o T

Darton et al. (1977) have suggested :

D b = 0.54(u — um f)2/5(Z + Z0 ) 4/5g -1/5 (2.32)

where Zo is found by making Z = 0 and replacing D b with Dh o .More recently, Horio and Nonaka have proposed a correlation that i s

valid for group A particles as well as group B and D . It allows for coalescenceand splitting, and converges to the correlations by Mori and Wen (1975) an dDarton et al. (1977). See Horio and Nonaka (1987) for details .

2.3 .6 Bubble stability (maximum bubble size )

Bubbles in a fluidized bed grow primarily by coalescence ; and, beyond amaximum size Dbm ,they become unstable and split into smaller ones .

There are two leading theories for the splitting mechanism, which, for th esake of the present description, can be termed `top to bottom' and `bottom to top' .

The `top to bottom' theory suggests that, as the bubble grows larger ,instabilities of the Taylor kind develop at the leading boundary of the bubble,resulting in a curtain of particles `raining down' through the bubble (Figure 2 .9a) .

The `bottom to top' theory originates from the gas velocity field predicte d

Dbm — D b = e-0.3Z/Dv

(2 .29 )

54


b • •

• b•

m •* , *os

••

• •

o 1,llb

a rv

(a)

(b )

Figure 2 .9 Bubble split-up; (a) top to bottom, and (b) bottom to top .

by Davidson's model . The gas velocity in the bubble wake tends to dra wwake particles into the bubble . This circulation velocity u, is roughly th esame as Ub, the bubble velocity . Hence, if the particle terminal velocity u t is

smaller than Ub, particles will be drawn into the bubble (Figure 2 .9b) .Therefore, the bubble stability criterium according to the `bottom to top '

theory is as follows :

ub <u,

ub = 0.711(gDb)o .5 = u t maximum size D b of stable bubble

—* unstable bubble

—> stable bubble

ub > ut (2 .33 )

Unfortunately, the calculation of the maximum stable size using eqn (2 .33 )is affected by the uncertainties in the knowledge of u t (wide-ranging particl esizes, shape factors, clustering, particle–particle interactions) .

2.4 Bed dynamics

Once the Huid dynamic behaviour of particles and bubbles has bee nestablished in preceding sections, we now turn our attention to the overal lbed behaviour. Topics to be discussed include : pressure drop, bed voidage ,minimum fluidizing velocity, minimum bubbling velocity, two-phase theor yand freeboard phenomena (entrainment, transport disengaging height an d

elutriation) .

2.4.1 Pressure drop

Figure 2.10 represents the variation of pressure drop with fluidizing velocit y

in an ideal fluidized bed . While the bed remains fixed, the pressure drop


55

Ap

APm f

► -

)

Y

Umb

Umf

U

U

(a)

(b )

Figure 2 .10 Pressure drop versus fluidizing velocity ; (a) ideal and (b) real .

increases linearly with velocity, as is the case for a Darcy-type flow. For agiven velocity (u„, f in Figure 2 .10a), the drag on the particles is large enoug hto hold up the bed, and fluidization begins . As velocity increases, interparticl edistances increase and hence so does the bed height; but pressure drop remain sconstant .

In real beds, the departure from this ideal behaviour takes place in tw oregions (Figure 2.10b):

• The pressure—velocity graph shows a `hump' where the sioping andhorizontal sections meet. This corresponds to the èxtra' force neededto ùnpack' or ùnlock' the particles from their packed state . Sometimes ,and particularly for widely distributed sizes, the transition is a smoot hcurve (dashed line in Figure 2.10b) .

• As velocity increases in the fluidized state, the appearance of bubble scauses fluctuations in pressure (shown in Figure 2 .10b). Looking a tpressure as energy per unit volume, the kinetic energy equation applie dto the fluid in the bed indicates that the pressure drop, Ap, compensate sthe viscous dissipation and transfers energy from the fluid to the movin gparticles through the work of viscous and pressure surface forces . Shouldthe solid distribution along the bed not be uniform, the fluid migh taccelerate on its upward motion and the pressure drop would als ocontribute to the fluid kinetic energy increment . The viscous and pressur eforce work will increase particle agitation and interparticle distances ,being responsible for bed expansion .

The pressure drop is traditionally given by Ergun's equation (Ergun ,1952) :

AP= 150 (1

E

E)+ 1 .751

— eP9

D

z(2.34 )3

(`Nsp )2

3*s

p

where L and e are the bed height and voidage for a superficial velocity

56


u, and P = p — pggZ. The first term contains the viscous effects whilethe second is ascribed to fluid inertia . The ratio of the second term tothe first is proportional to the Reynolds number based on u and D p , asshould be expected . The characteristic dynamic pressure pgu2 can beused to make Ergun's equation dimensionless :

AP _150(1—a)2

1

L +1 .751—EL

pgu 2

e3

ReosD, *SDp

E3

Dp

where Re,IsDp = (pgu&Dp/u) .Near minimum fluidization conditions a more suitable dimensionless

form may be

iAP = 150 (1

E) Reo,D, +1 .75s ReosD,

(2.36)

(1 — E)(p p — pg)gL

E a ArosDp

E Ar4sD p

where Ar,sD, = [W)SDp)3(pp — pg)gpg]/µ2 is the Archimedes numbe rbased on (p sDp .

At the onset of fluidization the left-hand side of eqn (2 .36) is unity,u = umf and e = Emf . Then

21 =

150 (1 —E3

E) Reasnp + 1 .75 E3 1

ReAeOsDP

(2.37)O s D P

¢ S D P

Allowance has been made in the aboye equations for the presence o fnon-spherical particles, through the form factor Os ; and Dp can be taken asthe mean diameter for particles with a size distribution .

2.4 .2 Bed voidage

The bed voidage E m f is the volume fraction of space occupied by the ga sunder minimum fluidization conditions. Em f can be determined experimentall yby a number of methods, for instance by measuring the bulk density andrelating it to the gas and particle densities :

Pb = Em fpg + (1 — Emf)pp

(2 .35)

or

Pp Pg

Pp(2.38)E

Pp — Pi, — Pbmf =

The bulk density can be calculated from the bed height (Lmf ), its cross-


57

sectional arca (A V ) and the mass of solids in the bed (M). Then:

E mf = 1 —AV

MfPn

(2.39)

Values of E mf for different materials have been tabulated by Leva (1959) .

2.4 .3 The minimum fluidizing velocity u m f

umf is probably the single most important parameter in determining th eperformance of a fluidized bed. The experimental calculation of Umf, whenpossible, is indeed very simple: it suffices to reproduce, experimentally, th eu ~ Ap curve of Figure 2.10. The precise value of umf is difficult to determin efrom the raw graph due to the rounded transition of the curve between th efixed and the fluidized states ; but a simple prolongation of the straigh tsections, with the crossing point representing the minimum fluidizationcondition, is frequently the device used to overcome this difficulty . Whendirect measurement of umf is not feasible, a number of correlations have bee ndeveloped from other physical and geometric quantities . They are dealt withbelow .

2 .4 .3 .1 Umf from the Ergun equation, with known

and Emf . When q5 s and

Emf are known, eqn (2 .37) yields a quadratic equation for umf (or, alternatively ,Remf) :

1 .73Remf + 150 (1

2 3mf) Re m

f — Ar = 0

(2 .40 )Os Emf

Os Emf

It is important to note that the coefficients in the aboye equation are ver ysensitive to (even small) changes in s mf . Such changes may be brought abou tby bed expansion near the minimum-fluidization velocity ; or indeed bychanges in bed temperature, on which s mf depends (Botterill, 1989) .

2.4 .3 .2 umf from the Ergun equation, with correlations for Emf and * S . Thepresence of *s and E mf in eqn (2 .40) for u mf is cumbersome as the uncertaintie sin their determination are carried over to U mf .

Wen and Yu (1966) have proposed constant values for the coefficients o feqn (2.40) . Thus :

3 — 14;1

2 3mf ' 1 1O s E mf

Os Emf

The approximation was made in the following ranges :

0 .38 *Emf *0.94; 0.14 `Ys

1 .0; 8 .1 x 10 -4 *D p /DV ,0.25

58


The resulting u mf equation i s

24.5Re,2nf + 1650Re mf - Ar = 0

(2 .42)

which yields an average deviation of ± 34% in Remf with respect toexperimental values .

2 .4 .3 .3 umf from correlations . There is a wealth of published correlations fo rRe mf, both for spherical and for irregular particles . These correlations should,of course, be used with due care to respect the range of conditions for whichthey were obtained.

Thus, Baeyens and Geldart (1973) propose the following correlation fo rDp > 100 µm :

21.7Re2 + 1833Re 1 - 07 - Ar = 0

(2 .43)

For Dp < 100µm, Abrahamsen and Geldart (1980) suggest the followin gcorrelation by Baeyens (1973) (SI units) :

9 .4 x 10-4 [(Pp - Pg)g]° .934D P . 8

Pumf =

µ0.87 0.06 69

An extensive listing of published correlations can be found in Couder c(1985); he recommends, for non-spherical particles, the one by Thonglimp (1981) :

Re mf = [31 .62 + 0.425Ar]°• 5 - 31 .6

(2 .45)

(2 .44)

f

1

2.4 .4 Minimum bubbling velocity

The fluidization characteristics of several types of particles have bee nestablished in section 2.2.4. It may be recalled that group A particles (smallparticles and/or small gas-solid density differences) exhibit homogeneou sfluidization for a range of fluidization velocities before bubbling appears .Also, high pressures (such as those found in PFBC systems) may cause th eappearance of homogeneous fluidization conditions in systems that fluidiz eheterogeneously at atmospheric pressure, due to the increase in the gas densitywith pressure .

With bubbles playing such a key role in bed performance, it is understandabl yinteresting to be able to predict the gas velocity at which the transition fro mparticulate to bubbling fluidization takes place .

Romero and Johanson (1962) have suggested that the transition is marke dby the value of the product of four non-dimensional groups which characteriz ethe quality of fluidization . Thus


59

n

:d

Frmf Re,„f Mu f > 100 yields bubbling fluidizationv

FrmíRemf Mu Lmf < 100 yields particulate fluidizatio nv

Geldart (1973) suggested a correlation linking the minimum bubblingvelocity umb and Dp :

umb = KDp (2 .47 )

with K being a constant which takes a value of 100 when the fluidizing ga sis air at ambient conditions and cgs units are used .

Broadhurst and Becker (1975) have, in turn, suggeste d

A

o . sRemb =

r

(2.48)9.8 x 10 4 Ar-o.sz (pp/p 9 )o .2a + 35 . 4

Finally, Abrahamsen and Geldart (1980) have found that fines exert a nimportant influence in the fluidization conditions, and propose a correlatio nfor Umb that includes F, the fraction of fines (Dp < 45 pm):

0 .0 6

umb = 2.07 D0 .347 e0.716F .

(2 .49)p

2.4 .5 Two-phase theory

The attention is now turned to the split of the gas flow between the emulsio n(i .e . the mixture of particles and interstitial gas) and the bubble phase .

Grace and Clift (1974) have classified the net volume flow rate of gastraversing any reactor cross-section into four categories, two of them arisingfrom bubble flow and the other two from emulsion flow . These are usted below .

• Bubble flow:(1) the upward convection of bubbles, also termed the visible bubble

flow (GB );(2) the flow of gas relative to the bubble (see Davidson's theory in

section 2 .3 .4), also called through flow (G T ) ;• Emulsion:

(1) the flow of gas, relative to the particles, through the interstices o fthe emulsion phase (G1 ) ;

(2) the net flux of interstices moving with the particles (Ge ) .

Therefore the total flux i s

G total = Avu = G B + GT + GI + G E

(2 .50)

Ge is zero of the voidage of the upward- and downward-moving emulsio nis the same, and is usually neglected .

(2 .46)

60


GB is usually evaluated by considering that all the gas flow in excess o fthat required for minimum fluidization passes through the bed as bubbles :

GB = Av(u — umf)

(2 .51 )

This hypothesis, first formulated by Toomey and Johnstone (1952), i sknown as the two-phase theory .

Inserting eqn (2 .51) in eqn (2 .50), neglecting G E and substituting G T and

G 1 by functions of the mean through-flow velocity Ü BT and mean superficia lgas velocity in the emulsion phase 17 , the following equation results :

u = (u - Umf) + uBT£B + ?E(1 - £ B)

(2 .52)

Experimental results indicate that eqn (2 .51), although widely used, tend sto overestimate the bubble flow . Almstedt and Ljunsgrom (1987) and Almsted t(1987) have found large deviations from the visible bubble flow rate predicte dby the two-phase theory. Design factors like the internals, tube bankconfiguration and bed geometry play a major role in visible bubble flow rate .At the same time, the quantitative impact of operating variables like pressur eand temperature has not been fully established . Clift and Grace (1985) sugges tthat this difference results in increased through-flow and interstitial flow .

To correct GB , an alternative equation to eqn (2 .51) is used, as follows :

GB = Y(u — umf) x Av

(2 .53 )

with Y generally taking values between 0 .6 and 0 .8 .

2.4.6 Entrainment, transport disengaging height and elutriatio n

Entrainment and elutriation are phenomena taking place in the vessel spac elocated aboye the dense phase and known as the freeboard (Figure 2 .11).Particles from the dense phase cross the (not always sharply defined) interfac eand enter the freeboard. This is known as entrainment, and is caused b ybubbles bursting at the interface and projecting particles into the freeboard .

As particles move upwards in the freeboard against the gravitational force ,they lose momentum, and eventually fali back onto the bed if their termina lvelocity is greater than the gas velocity in the freeboard . Thus, the fractionof solids in the freeboard decreases with height as increasingly smaller particle sreverse their velocity, until it finally becomes constant (this has been depicte din Figure 2 .11) . The height at which this happens is called the transport

disengaging height, (TDH); and this process of segregation of finer and coarse rparticles is called elutriation .

2 .4 .6.1 The splash zone mechanism . Although there is widespread agreemen tin the role that bubble eruption plays on entrainment, there is som econtroversy on the detailed mechanism; and it has been variously suggeste dthat particles are splashed from the bubble wake or from the bubble roof .


6 1

FRACTION OF SOLIDS

• %f

b

• ,

•

wi

P

Figure 2.11 Dense phase, freeboard and solids fraction .

In a theoretical and experimental study of the subject, Pemberton an dDavidson (1986a) conclude that both mechanisms are relevant .

Thus, for group B particles and u/umf < 10-15, roof particles are ejected .As a bubble crosses the surface, particles from the roof rain back to the be duntil the roof thickness is of the same order as the mean diameter of theparticles in the bed, and then the bubble bursts ejecting the roof into the freeboard .

For group A particles, and for group B particles with u/umf > 10-15,bubbles are much closer to each other ; and, as they reach the bed surface ,they usually coalesce. This results in wake particles being ejected into thefreeboard .

The rate of entrainment yielded by the second mechanism is much greate rthan that resulting from the first one ; and the transition between both explains ,according to Pemberton and Davison (1986a), the fast increase in entrainmen twith fluidization velocity which has been reported by many authors .

2 .4 .6 .2 The elutriation constant K. It is generaily accepted that the rate a twhich particles of a given size D . are removed from the bed is proportionalto the mass fraction of particles with that size in the bed. Thus

dy i M

dt

K i Av y t (2 .54)

with K. having dimensions of kg/m 2 s .Kunii and Levenspiel (1969) have proposed a model for the freeboard

based on the co-existence of three distinct phases: a homogeneous mixtur eof gas and completely dispersed solids, which moves upwards ; projectedagglomerates (or particle `parcels'), also moving upwards ; and particle parcel s

)

Y

i ted

í

62


moving downwards. With further hypotheses for the dispersion of upwar dmoving agglomerate into upward moving homogeneous phase and th etransformation of upward moving parcels into downward moving ones, Kuni iand Levenspiel are able to deduce an expression for the elutriation constant ,and for other freeboard phenomena . However, the lack of numerical value sfor some of the model constants (notably the interphase exchange rates )renders the model of little practical use . A more practical three-phase mode lhas been proposed more recently by Pemberton and Davidson (1986b) .

More commonly, correlations are used to determine the value of theelutriation constant K i . There is a wide variety of them in the literature (see ,for instance, Ling Wan Lin et al. (1980) or Geldart (1985)) .

By way of example, the correlation by Geldart et al . (1979) (probably oneof the simplest ones) will be cited here . It reads

Ki = 23.7 e - s .4u*iu

(2 .55)pgu

However, Geldart (1985) points out the scatter of experimental point saround the correlation is sometimes greater than + 100%, and that up tofive-fold under- or overpredictions are not uncommon when the correlation sare applied to data other than those from which they are generated .

2.4 .6 .3 Transport disengaging height . It is widely agreed that the entrainmen trate decreases exponentially with height in the freeboard . There is ampleexperimental evidence of this exponential decay; and it is theoretically evincedby the three-phase models of Kunii and Levenspiel (1969) and Pemberto nand Davidson (1986b) .

Large et al. (1976) propose the following expression for the entrainmen tflux at a distance z aboye the free surface :

Ei(z) = Ei(oo) + E i (0)e -° Z

(2.56)

where Ei(0) is the flux of solids of size Di ejected at z = 0 (the bed surface) ,and Ei(oo) is the elutriation flux, given by

Ei( oo ) = Kiyi

(2 .57)

The net flux can be obtained by summing up eqn (2 .56) for each sizecomponent i . Since, it is argued, a is a weak function of D i , the following

expression is obtained :

E(z) = E(oo) + E(0)e -°Z

(2 .58 )

The values of a and E(0) must be obtained experimentally, by measurin g

E(z) at several heights .Wen and Chen (1982) propose a slightly modified expression :

E(z) = E(oo) + [E(0) — E(co)]e- ° Z

(2 .59)

1c

with De, being the equivalent diameter of the bubble at the bed surface .

2.5 Dynamic similarit y

In order to study the fluidized bed hydrodynamics the following approache scan be followed .

• The partial differential conservation equations for a two-phase flo wsystem with appropriate boundary and initial conditions can be solved .

• Global relationships among variables can be obtained through the us eof integral equations applied to the bed control volume.

• Full size experiments can be conducted and measurements can be carrie dout for a limited range of operating conditions .

• A judicious combination of dimensional analysis establishing well-defined scaling rules and small-scale experiments can be used t oextrapolate laboratory results to large commercial units at high temperatur eand, in some cases, high pressure .

The two-phase-flow governing equations are still open to some mino rcontroversies . If the disperse phase is treated as a continuum, forms of th econservation equations have been presented, for example, by Anderson andJackson (1967) and Whitaker (1966) . Should the flow be turbulent, time-averaged versions of these equations have been established by Aliod andDopazo (1990) and Balzer and Simonin (1993) . Fueyo (1990) has develope da two-fluid metodology in which the two-phase, turbulent flow is represente din terms of alternating particle-rich and particle-lean parcels . Such a strategycan be easily adapted to model the exchange processes between bubbles an ddense phase in a fluidized bed .

It is pertinent to remark that average moment equations using, for example ,k-a or Reynolds-stress turbulence-models may well prove to be insufficien tfor a detailed simulation of fluidized bed dynamics . The correct predictionof bubble formation and evolution will most probably require the solutio n


63

By taking zTDH such that E(z) — E(co) < 0.O1E(oo) one readily obtain s

1

E(0) — E(oo )zTDH — aln

0.01E(co )

Wen and Chen point out that the value of a is in the range 3 .5—4.6/m,with a recommended value of 4/m . They also suggest the following correlationfor estimating E(0):

E¡ O)

3 .5 n0 . 5

E" = 3 .07 x 10-9 P9R29 (u — u,,,f ) 2 .5 in kg/m 5 s

(2 .61)Av -D e°

(2 .60)

64


of time-dependent two-phase laminar/transitional/turbulent flows . Largeeddy simulation (LES) will be probably the best option in a near future .

Integral equations provide global answers, and have been exploited to th elimit .

Measurements in commercial units are hampered by their cost, the limite daccess to different points in the bed, by the parameters that can be monitored ,by the reduced range of operating conditions and by difficulties in repeat-ability .

On the other hand, dynamic-similarity theory and small-scale mode lexperiments help at a reduced cost to identify the parameters controlling th ebed dynamics and the basic flow mechanisms, to determine strategies fo rpart-load operation, to compare dynamic characteristics of different be ddesigns, to explore a wide range of geometric and operating variables an dto assess the influence of design modifications. Even tube and wall erosio nin commercial units could, in principie, be projected from laborator ytests .

It is essential for a rational use of the dynamic similarity principies t odefine a complete set of dimensional variables or its dimensionless counterpart .This is presented next in this chapter .

2.5 .1 Basic dimensionless parameters

A few dimensionless groups have been introduced in the previous section sof this chapter . Several authors have presented complete sets of the dimensionles sparameters that control the bed hydrodynamics . Writing the dimensionles sequations corresponding either to two continua (fluid and solids) or to singl eparticles, Glicksman et al . (1994) obtain the following governingparameters :

u2 p 9 uDp p 9 uDv Pp QS os , PSD, bed geometric ratios, turbulencegDv u

u P 9 Pp u

dimensionless parameters, etc .

(2 .62)

The Froude number, Fr = u 2 /gDv, is based on the bed characteristi cdimension, Dv . The Reynolds number, Re = p9uDp/u based on Dp , or ratherReo„ = p 9 (Du)Dp/u, determines the fluid drag regime for the particle motion .Re DV = p 9 uDv/u indicates the type of overall flow regime in the bed ; bubbleformation and evolution should probably be determined by ReDV or, evenbetter, by ReDo = pguoDo/u, where uo is the gas exit velocity at the orifice sof the distributor of characteristic diameter D 0 . The ratio pp/p 9 has beenshown by Geldart to be an essential parameter in the solids fluidizatio nbehaviour . Q S is the average solids feed rate per unit area from outside th ebed through the bottom. PSD stands for particle size distribution .


65

Glicksman et al . (1994) also conclude that the motion of particles near awall is controlled by the same parameters . They also develop a simplifiedset of dimensionless parameters, namely

u

u Pp D v Q S2

-

PSDgDv umf pg L pp u

In what Glicksman et al. (1994) cala the viscous limit, the ratio pp/pp i sexcluded from the previous list .

2.5 .2 Scaling law s

In order to maintain a dynamic similarity between a small-scale model (m)and a commercial unit (c), the dimensionless parameters listed aboye mus tbe identical for the two beds . Using the full set yields

uDp

m —

)e 'C9Dv/ m — gDv/ (pg)(2 .64)

(Pg UDv)

m

(PUDv)(P)

P—

((Qs)Pg cu m

After some algebra, eqn (2.64) leads to

D pm Dym _ (V )ZJ3 =

(u)2

Ppm _ Pg m

Ppc

Pg c

If it is further assumed that the Huid behaves as a perfect gas and it sdynamic viscosity varíes with temperature according to the law (y/µo) (T/To ) "where Po and To are reference values of µkT, and n 0.67, then

ay' d.

Vm Pc M c Tm\' +n(2 .66 )

Mm and M e are the mean molecular masses of the fluidization gases usedin the model and the commercial unit, respectively .

The previous relations explicitly yield the influence of pressure, temperatur eand type of gas upon the scaling rules .

(2 .63)

(2.65)

Vc P. Mm

66


2.6 Pressure effect s

With the leading theme of this work being PFBC, it seems pertinent t oallocate a section in this fundamentals chapter to the effect of operating pressure .

Because of the (relatively) late appearance of PFBC, and because of th edifficulties and cost involved in conducting experiments at high pressure, th ebody of literature in pressurized systems is scanty by comparison with th epublished results for atmospheric systems . Botterill (1989) points out that ,because of experimental constraints, the size of the research rig is ofte nrestricted; wall effects may therefore play a significant role ; and extrapolationto larger, pilot- or industrial-scale systems (where wall effects are likely t obe less important) is not straightforward .

A more detailed discussion of pressure effects follows in this section ; but ,by way of summary, it may be asserted that :

• the main effect of pressure on fluidization behaviour is exerted throug hthe increase of the gas density; high temperature changes the fluidizationcharacteristics through changes in density and, most importantly ,viscosity; and that

• Elevated pressures cause, at least for group A particles, a smootherfluidization, which is brought about by smaller bubbles .

2.6.1 Effect on minimum fluidization velocity U mf

The minimum fluidizing velocity was obtained in section 2.4 .3 by equatin gthe pressure drop given by Ergun's equation and the pressure drop requiredto support the bed weight . u,,,f was then given by eqn (2.40) . Reworking thisequation slightly, one gets :

1 .75

ose.

(Dpµ f Pyl2

+ 15012E3,mf (D

µ f p9/ 9Dp9(

p

P — p9) 2.672

f

s f

A

B

The (theoretical) effect of pressure and temperature on u,„f can beascertained by examining eqn (2 .67) .

Thus, if f (the group in parentheses in eqn (2 .67)) is moderate (Le. theparticles are small), then only term B is important on the left-hand side o fthe equation ; and hence f is proportional to ,u' . u,„f is therefore expecte dto decrease with temperature, since µ increases with T; and, because thereis no dependence (or only a weak one) on density, pressure should have n osignificant effect .

For higher Re,,,f (larger particles) then term A is dominant, and u,,, f i sproportional to p9

1/2 . Therefore increasing pressure should decrease u,,,f ;and temperature would be expected to have the inverse effect . (


67

1

The experimental evidence, in general, confirms this theoretically predicte dbehaviour .

Chitester et al . (1984) studied the fluidization characteristics of severa lsolids (coal, char and Ballotini) at pressures up to 6.485 MPa and at ambien ttemperature, and have suggested the following modified form of the Ergunequation :

Re,,,f = ✓28 .72 + 0.0494Ar — 28 .7

(2 .68)

Knowlton (1977) points out that u mf has little meaning in beds with widesize distribution, since the fluidization of smaller particles occurs at lowervelocities, resulting in partial fluidization and segregation of finer materia lto the bed surface. Knowlton suggests, for such systems, the use of a completefluidizing velocity (u cf ), or the velocity at which the whole bed is fluidized .He indicates that u,f can be calculated by computing the minimum fluidizingvelocity um fi for each size-interval i, and then taking U, f as the weightedaverage of all the um fi using the mass fraction of particles in the size interva las weights :

Ucf

Yi umfi

(2 .69 )

For the calculation of um fj , he uses the Wen and Yu (1966) correlatio n(eqn 2.42). He found good agreement between the calculated u c f andexperimental results for siderite and lignite particles with pressures rangin gfrom 103 kPa to 8.27 MPa at ambient temperature .

2 .6 .2 Effect on bed voidag e

Knowlton (1977) has studied the effect of pressure on bed expansion at ucffor widely sized solids with average sizes of approximately 250µm. Heconcludes that there is no clear correlation between pressure increases an dbed expansion .

Sobreiro and Monteiro (1982) have investigated the behaviour of (mainly)group B powders at pressures up to 3 .5 MPa. Their work suggests that emfis independent of pressure, while £mb increases with pressure for particles clos eto the A—B boundary . King and Harrison (1982), who worked with grou pA and group B particles, report similar trends ; and so do Jacob and Weime r(1987, 1988) for group A powders . Weimer and Quarderer (1984) studied th eeffect of temperature as well as pressure on the fluidization of a group Apowder . Their results corroborate the aboye trends in respect of pressure;and additionally conclude that an increase in gas temperature (or µ for thatmatter) also increases e .

In a recent work with larger particles (1 .51 mm SMD), Bouratona et al .(1993) suggest that e is determined by a single non-dimensional parameter

68


Y, given by

y=^gu

(2 .70)9p

3

1

The use of Y as a scaling parameter successfully regroups the expansiondata from high pressure experiments, but fails to regroup the atmospheric ones .

2.6 .3 Effect on bubbling characteristic s

As indicated in the introduction to this section, the main effect of high pressur efluidization is on the size of bubbles . It has been generally established thatincreased pressure results in smaller, more frequent bubbles ; and that thes eeffects are more pronounced for smaller group A particles than for large rgroup B ones.

Chan et al. (1987) have studied the effect of pressures up to 3 .2 MPa inbubble size, frequency and velocity for group A and group B powders . Theirfindings confirm the general trends outlined in the preceding paragraph .From their data, they suggest the following correlation (imperial units) fo rthe bubble diameter :

D b

1 .O96L0.64D*.064(u — umf )0 .6 5

1 .43

^ 0 .088 ^0 ..04 5P

Db

O .319Lo.81D°.37 p0 .51 (u — umf )0 .5 9

1 .43

^0.48

pg > 0.61b/ft 3

(2 .72)

A two-part correlation is needed because D b decreases with pp at lowpressures (low pg), but increases with pp at high pressure .

With respect to minimum bubbling velocity, there is general agreemen tthat, for group A powders, an increase in pressure widens the range ofvelocities umf —> U m b in which the bed admits homogeneous fluidization .

Jacob and Weimer (1987) indicate that, for their group A particles and fo rpressures up to 12.4 MPa, the ratio umb/umf is well correlated by the expressio nsuggested by Abrahamsen and Geldart (1980) :

umb

2300p9.126 p0 .52 3e0 .716 F

f —0 .8 g0 .934(p — ^ )0 .934u

D 0 . 8m n

g

2.6 .4 Effect on entrainment and elutriatio n

An increase in pressure results in smaller fluidizing velocities umf , and hencein an increase in bubble flow u-umf. Since entrainment is mainly caused b ybubbles erupting at the bed surface, it is therefore expected that it shouldincrease with pressure . Additionally, the particle terminal velocity decrease swith increasing gas density, thus bolstering the entrainment/elutriation processes .

p g < 0.61b/ft 3

(2 .71)

(2.73)

2 .

h(inthoI

ve

char

com D

be

se cco 'boim


69

Chan and Knowlton (1984) have studied the effect of pressure (up t o3.1 MPa) on TDH for a group B solid, and found a linear increase in TD Hwith pressure . They also report that all the (until then) published correlation sfail to predict TDH accurately enough .

2.6.5 Some dynamic scaling considerations

Equation (2 .65) has shown that the bed dimensions, particle size, operatin gsuperficial velocity and the volumetric flux of solids scale with powers of th ekinetic viscosity ratios. It is a common practice to conduct exhaustive testin gin laboratory models, aiming at rational projections of these data to full siz eunits under real operating conditions . Glicksman et al. (1994) show that i tis feasible to scale down the hydrodynamics of atmospheric FBC to laborator ymodels with dimensions one fourth or one fifth of those of the commercia lunit, operating with air at ambient temperature . On the other hand, a similarattempt for pressurized fluidized bed combustors is not an easy task . Withthe scaling rules previously discussed, a laboratory model of a typical PFBCat 1 .2 MPa and 860°C, operating with air at ambient temperature andatmospheric pressure, turns out to be approximately the same size as thecommercial combustor. Using fluids with smaller kinematic viscosities doe sallow a significant reduction in the model dimensions. One-half to one-thir dsize reductions can be readily obtained by changing the model fluid .

2.7 Heat-transfer concepts

Fluidized beds display rather uniform temperature profiles and very efficien theat transfer characteristics with quite high thermal conductivity . This is duein bubbling beds to the intense mixing induced by the bubbles, or rather bythe underlying flow"pattern, which leads to heat transfer coefficients of th eorder of 200–500 W/m2 (Grace, 1986) . These peculiarities make fluidized bed svery attractive for combusting conditions .

Combusting beds transfer heat to walls . In order to estimate heat exchangecharacteristics, quantitative information on bed/surface heat transfer coefficientsare required, apart from the radiative properties .

Heat transfer is strongly linked to the bed hydrodynamics . Energy-conservation partial differential equations could be added to the mass an dmomentum equations ; and this system, supplemented with appropriat eboundary conditions, could, in principle, be solved numerically .

It is important to remark that hydrodynamic similarity, as explained i nsection 2 .5, does not imply thermal similarity . Thermal dimensionless group scould be readily obtained from the dimensionless energy equation . Maintainingboth dynamic and thermal similarity is in general difficult to achieve, if no timpossible .

It is apparent that even a superficial review of the heat transfer literature

70


in fluidized beds is well beyond the scope of this chapter . Only a fewphenomenological ideas and the basic thermal dimensionless groups ar epresented in this section ; and some thermal scaling rules are discussed .

2.7 .1 Phenomenology

The transfer of heat by convection-enhanced conduction (radiation can als obe included) from a bed to a wall can be expressed as

q = hAT

(2 .74 )

where q is the heat flux in W/m2 , h is the overall average heat transfe rcoefficient in W/m2K and AT is a characteristic temperature differencebetween the bed and the wall .

A dimensionless Nusselt number is defined a s

Nu = hDp

(2 .75 )k g

where kg is the gas thermal conductivity. Nu can be interpreted as the rati obetween the actual heat flux and the conductive heat transfer .

The bed/wall heat transfer phenomenology can be described as a sequenc eof intermittent events of either a dense phase or an almost particle-free ga scoming close to the wall and exchanging energy with it . The dense phas emay be a highly loaded emulsion, cluster or packet of particles, while th egas may be a bubble with varying small amounts of particles . The overal lheat transfer coefficient can be written as

h = yh p + (1 — y)h g

(2 .76 )

where hp and h g stand for the average heat transfer coefficient when eitherthe particulate or the gas phase, respectively, is in contact with the wall . yis the intermittency factor or the average fraction of the wall area occupiedby the particulate phase . For dense beds the first term in eqn (2 .76) maydominate, while for small solids concentrations or in the freeboard regionthe second term may become important .

2.7 .2 Thermal dimensionless groups

The use of the II-theorem leads to a new set of thermal dimensionles sparameters in addition to those specifying the bed hydrodynamics . Thefollowing list is commonly obtained :

ha Dp v g Cpp kp T T aT 3 Dp TW

DI LH, y, —,

,

,

,

, e W , eB, —'D ,

heat transfer dimensionles sk g

ag Cpg k g Z R k g

TC

Dv vsurface geometry, turbulent heat transfer dimensionless parameters (2 .77)

C ,

f

n

B

ti .

h,

)

r


71

h Q stands for either h p or h9 , ag = kg/ g Cpg is the gas thermal diffusivity, v g /agis the gas-phase Prandtl number, Cpp and Cpg are the specific heats atconstant pressure for the solids and the gas, respectively, kp and kg are thethermal conductivities of solids and gas, respectively . iT and zH are thecharacteristic thermal and hydrodynamic times, whose ratio can be expressed a s

2

2

iT _ppCppuDp _ D p /ap

'LH

kPLH

LH/u

where L H is the characteristic heat exchange length along the flow, and apis the particle thermal diffusivity . D p2 /ap is the time required by a particle tomodify its temperature by a prescribed amount over a distance of the orde rof Dp due solely to thermal diffusive effects . LH /u is the residence time ove rthe heat transfer surface . 6 is the Stefan-Boltzmann constant, and the grou p

(2.78 )

;)6T 3 Dp

6T4 AT

k g

k g AT/Dp T(2 .79)

o

e1 s

le11

6 )

Y

ry)n

;s she

can be interpreted as a ratio between radiation and conduction heat fluxestimes the relative temperature increment. Here T stands for a bedcharacteristic-temperature, and ATis of the order of the bed temperatur e(TB ) minus the wall temperature (Tw) . T* will in general be equal to TB orbetween Tw and TB . ew and EB are the emissivities of the wall and the bed,respectivey . DI is a characteristic length of the internals .

For bed temperatures well below 1000°C the radiation heat-transfercomponent is estimated to contribute between 5 and 10% to the overall hea tflux. In that case UT3Dp/kg, ew and eB would drop from the aboye list .

Should 'zT be much greater than 'rR , then the partirles will not modify theirtemperature along the heat transfer length and will therefore remai napproximately at TB . It is pertinent to remark that Glicksman et al . (1994)use an ill-defined hybrid which inappropriately combines gas and particlethermodynamic properties .

It is also to be recalled that a basic hydrodynamic parameter, the Reynoldsnumber, is a key group to estimate the Nusselt number .

2.7 .3 Dense-phase heat transfer

Baskakov (1964), using an emulsion renewal model for asymptotically smal ltimes, provides the following expression for the Nusselt number :

:ss

(hpDp) - (

hwDp)

- 1

(heDp) -

I

k g

kg

k g(2.80)

77)

hw and he are the wall and dense-phase average heat transfer coefficients,

72


respectively . The emulsion Nusselt number can be estimated (Glicksman etal., 1994) as

h e Dp J1

t

e e (T,y'2

*R

(kkk gy / 2kg

where t' is the dimensionless time using 'r R as reference, and ke and E Q arethe emulsion conductivity and voidage, respectively. Should the radiativ eheat transfer be important and the particle diameter be large compared tothe infrared radiation wavelength (Glicksman and Decker, 1982), k e in eqn(2 .81) is to be replaced with ke + kr , where

kr : 9Dp aT3

(2.82)

is an effective radiative conductivity .Several correlations for the wall Nusselt number exist . In general, they

reduce to the general functional form :

(2 .83)

2.7 .4 Gas-phase heat transfe r

From the analysis of extensive data sets, Linst and Glicksman (1993 )approximate the gas-phase Nusselt number by the expression

h9Dp=

(Ph41)vf

, Prg , void geometry

(2.84)k9

The heat-transfer surface-length, LH , can be used instead of Dv in eqn (2.84) .

2.7.5 Heat transfer scaling rule s

It is convenient to perform heat transfer measurements in small-scal elaboratory models . Projection of those measurements to large-scale commercia lunits requires, generally speaking, that all the dimensionless groups definedaboye, namely the various Nusselt numbers and, consequently, the ratios o nwhich they depend, be identical for model and industrial unit . Dynamicsimilarity is indeed a previous step in order to possibly guarantee therma lsimilarity . Commonly, it is impossible to match both thermal and hydrodynamicdimensionless groups. Rational partial similarity criteria should then b eapplied, discerning the essential controlling parameters .

One of the most attractive heat transfer scaling applications consists inusing cold bed measurements to project a hot bed behaviour . Ensuringdynamic similarity between the two beds and assuming that radiation i sunimportant, the following dimensionless parameters should still be matched :

(2 .81)


73

Pr, kp , keTTT , Eg

ek kg g R

(2.85)

1

1)

) .

leal

) nica li c3 e

inngis

For bubbling dense beds or for PFBC, the first term in eqn (2 .76) will dominateover the second one . Moreover, assuming that the emulsion Nusselt numbe rin eqn (2 .80) is the dominant contribution and considering the model heat-transfer implied by eqn (2 .81) with the radiative correction, ke + k„ alreadydiscussed, thermal similarity would require (Glicksman et al., 1994) :

5✓ 1 — C e (t\1'2 r(ke + kr)kp

]

1I 2

`t-

R/ L

kg

5 HOT BE DV

✓ 1 — Ee

(T

1/2

r(k kp)J1

1/ 2

—

t

rR/ L k 9

*COLD BED

After simplification and given the dynamic similarity between the two beds ,eqn (2.86) reduces to

[Pr9

D QT 3+ p = Pr

(2.87)Cpg k g 9 kg

]HOT BED

g Cpg k g COLD BED

At the other end, if the wall Nusselt number in eqn (2 .80) is the dominantcontribution, thermal similarity can be guaranteed when the group hwDp/kg ,

given by eqn (2 .82), is identical for the cold model and for the hot unit .

2.8 Some closing remarks

In closing this introductory chapter on the fundamentals of the hydrodynamic sof fluidized beds, a few comments seem pertinent .

• The process of bubble formation seems to be rather poorly understood .As a speculation, it is likely that bubbles and concentrated vorticityregions in the flow are closely related . In the transition regime of asingle-phase mixing layer or an axisymmetric jet from laminar t oturbulent, spanwise or azimuthal coherent vortices appear that ma ymerge and grow through vortex pairing; these may interact withstreamwise or axial vorticity leading to complicated flow patterns . Fortwo-phase flows, these concentrated vorticity regions would tend tocentrifugate the heavy/large particles leaving an almost particle-fre eempty core (Lazaro and Lasheras, 1989) . Even for the gravitationalsettling of particles in homogeneous turbulence, Maxey (1987) finds tha tparticles move towards regions of high strain rate and/or low vorticity .Along this speculative fine of reasoning, one might be tempted toestablish a complete analogy between bubbles and coherent vortica lstructures and bubble coacence/growth and vortex pairing . A word

(2.86)

D A

D b

D b m

Db o

Di

DoDp

DS

E i (zEi ( 0

E i (c

F

Fr

LMIN

MMMvND

LMA :

gGashkkkKiLLB

L mf

NDA

Nu

P

74


of caution is in order not to reach premature conclusions . However,this topic deserves some close scrutiny .

• Bubble formation has also been approached as a stability problem b ysome authors (Hernandez, 1990 ; Batchelor, 1993).

• Dimensionless diagrams establishing clear-cut distinctions in the dynami cbehaviour of different particles as a function of their density, size, shape ,velocity, flow nature, etc. are very much needed . Existing diagrams suc has those of Geldart (1973) and Grace (1986), although useful, are notcompletely satisfactory . One strong recommendation would be to us edimensionless groups with physical meanings wherever possible .

• The numerical simulation of the fluidized bed hydrodynamics is anothe rfield in which much work is required. It has already been pointed outthat capturing significant flow features may require a time-dependen tLES methodology . This might be at present very time-consuming andhence inadequate for industrial applications . However, research intothis field may significantly benefit from the LES treatment o fwell-established Lagrangean or Eulerian conservation equations, integratedfor simple geometries .

• The rigorous inclusion in the mathematical formulation of interparticlecollisions, Van der Waals and electrostatic forces, combustion/flo winteractions, etc., are goals that should be achieved in the near future .

2.9 Nomenclature

The main symbols used in this chapter are usted below . For each symbol,the information usted includes the symbol meaning, its SI units and, i fappropriate, the equation in which the symbol is defined or first used .

2.9 .1 Latin

a Constant 1/m (eqn 2.56 )A Area m2Ap Particle area m2

A r

CD

CD1

Archimedes

p P9(P — P9)Dp 9number, =

=

(eqn 2 .8)

Ga MvP

Drag coefficient for non-spherica lor non-isolated particle s

Drag coefficient for a spherical, isolate dparticle

Cp Specific heat J/kgK (eqn 2.77)D 1 Particle drag for an isolated particle kgm/s 2 (eqn 2.6)

75

mmm (eqn 2.30)m (eqn 2 .31 )m (eqn 2 .5)mm (eqn 2.4)

mkg/m 2 (eqn 2.56)

v

1,if

FLUIDIZATION FUNDAMENTAL S

Diameter of a sphere with the sam eprojected area as the particle

D b

Bubble diameter

Dbm

Maximum bubble diamete rDbo

Initial bubble diameterD .

Diameter of the ith size group of particlesDo

Distributor orifice diamete rDp

Mean diamete rD s

Diameter of a sphere having the sam esurface as the particle

E;(z)

Entrainment flux for particles of size D .E i(0)

Entrainment flux of particles at the bedsurface

E(co) Entrainment flux of particles aboye the TDH(i.e. elutriation rate )

F

Fraction of fines (Dp < 45µm)

u2Fr

Froude number, = - -Dpg

g

Gravitational accelerationGa s

Galileo number based on D s, = Ds p g/µ 2h

Heat transfer coefficien tk

Darcy constantk

Volumetric shape factork

ConductivityK .

Elutriation constan tL

Bed heightL B

Bubble penetration lengthL,,,f

Bed height at minimum fluidizationconditions

Length of the succession of cavities at thedistributor orific e

LM,N

Jet penetration lengthM

Mass of solids in the be dM

Molecular weightMv

Density ratio, Mv = ( pp — p 9 )/p 9ND

Dimensionless diameter group ,3 2

ND = CD Re, = 3 Mv gDZp9 = 3Ar

N D A

NuND based on DANusselt number (eqn 2.75 )

PP

PressureP=p—p9Lg

PaPa (eqn 2.34)

D A

m/s 2

W/m 2 s (eqn 2.74 )

(eqn 2 .3 )W/mKkg/m 2s (eqn 2.54 )mm

(eqn 2 .15)

mLMAX

m

(eqn 2.16)mkgkg/kmo l

76


PL

Pierced length of bubble m (eqns 2 .7 1and 2.72)

vPr

Prandtl number, _ — (eqn 2.77)ag

R b

Bubble radiu sR,

Bubble-cloud radiusmm (eqn 2.24)

Rep

Particle Reynolds number (eqn 2.7 )Re s

Reynolds number based on D s and thesuperficial gas velocity r,

Re t

Reynolds number based on u t

TDH

Transport disengaging height m PAu

Slip velocity, = (up — u g m/s su

Velocity m/s tUb

Bubble rising velocity m/s (eqn 2.19) o .

ubc

Rising velocity of an isolated bubble m/s (eqn 2.17 )u,,,f

Minimum bubbling velocity m/s (eqn 2 .1) Uu t

Particle terminal velocity m/s (eqns 2 .1 1and 2.14)

uts

Terminal velocity of a spherical, isolated m/s (eqn 2.10)particle

uo

Gas velocity at the distributor orificeVp

Particle volumem/sm3

R

A lAl

Vw

Bubble wake volume m3 Al .

Yi

Mass fraction of particles of size D . in thebed

kg/kgAl:A n

z

Height aboye the bed surface m Ba(

Z

Height aboye distributor m (eqn 2.28) Bae

2.9 .2

Greek

Bai

Ba r

a

Thermal diffusivity m2/sBa<Bat

Bed voidage Bot

e

Emissivity1

Sphericity correction for u t

I7

PBo L(eqn 2 .11 )

P

Gas dynamic viscosity kg/m es A

v

Gas kinematic viscosity m2/s Br o

P

Density kg/m 3 ChaCh a

a

Stefan-Boltzmann constant, = 5 .669 x 10 -8 W/m 2K4 Che

i

Characteristic time s

(eqn 2.78) M

O s

Sphericity (eqn 2 .2)Chi t

InN.

Chit.1

Clift2.9 .3

Subscripts/superscriptsCl

Clift.i

Isolated particle

)


77

BubbleBed

• Commercial uni t

• Complete fluidization

• EmulsionGasRelative to particles of diameter D .ModelMinimum bubblin gMinimum fluidizatio nParticleSphereTermina l

• OrificeVesselWall

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n t

ig,

on;o ,

*o .Ed .

• i nset.

R.

194.

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