Epstein N., Grace J.R. Spouted and Spout-Fluid Beds: Fundamentals and Applications

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Conical spouted beds 87

Table 5.4. Correlations for calculating bed expansion in conical spouted beds.

Authors Correlation

Gorshtein and Mukhlenov

ε = 2.17 (Re/Ar)

0.33

)

0.5

(tan(γ/2))

−0.6

(5.21)

Markowski and Kaminski

(ε − ε

)/(1 − ε) = 39.7(F

)

1.23

)

3.91

)

1.44

(5.22)

Olazar et al.

(ε − ε

)/(1 − ε) = 215 (F

)

1.74

)

3.91

1.95

(5.23)

1.0

0.8

0.6

0.4

key

γ (°)

–2

–1

Figure 5.3. Bed expansion for two contactors of different angles. Glass spheres, H

= 0.06 m,

= 0.05 m, d

= 4 mm, ρ

= 2420 kg/m

have been proposed in the literature, as listed in Table 5.4. When the minimum dilute

spouting velocity is known from Eq. (5.20), the correlations shown in Table 5.4 can be

used to estimate the bed voidage at minimum dilute spouting.

5.4 Gas ﬂow modeling

The models for the prediction of gas ﬂow distribution and tracer dispersion in conical

spouted beds are based on the deﬁnition of streamtubes according to their geometry

and on the knowledge of gas velocity through the spout. Only the dispersion problem is

discussed here.

5.4.1 Spouted bed

The geometric model proposed for tracer dispersion in the spouting gas is established

in direct relationship with contactor and inlet geometry,

Figure 5.4. The conservation

equations have been developed from the geometric model by introducing the following

assumptions: (1) the gas travels through the spout zone in plug ﬂow; (2) in each volume

element of the spout zone, there is mass transfer through the interface toward the annular

zone; (3) the gas ﬂowrate is the same in each of the streamtubes of the annular zone;

88 Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

Spout

Figure 5.4. Streamtubes and differential volume elements of the gas ﬂow model for the spouting

regime.

(4) in each streamtube, the deviation from plug ﬂow is deﬁned by a dispersion coefﬁcient,

D, which is the same for all streamtubes; and (5) the concentration gradient obeys Fick’s

law.

The mass conservation equation for a tracer ﬂowing in unsteady state, for a differential

volume element in the annular zone, is

∂C

∂t

= D

∂

∂ρ

− u

∂C

∂ρ

∂C

∂ρ

(5.24)

with the following initial and boundary conditions:

for t ≤ 0, C

= 1 (negative step function) (5.25)

and for t > 0, at the inlet of the jth streamtube,

∂C

∂ρ

= (C

−C

. (5.26)

At the outlet of the jth streamtube,

∂C

∂ρ

= 0. (5.27)

In a differential volume element of the spout zone (Figure 5.4),

∂C

∂t

∂C

∂z





= 0 (5.28)

Conical spouted beds 89

with the following initial and boundary conditions:

for t ≤ 0, C

= 1 (negative step function) (5.29)

and

for t > 0, when z = 0, C

= C

(5.30)

and

when z = z

∂C

∂z

= 0. (5.31)

The model requires knowledge of gas velocity along the spout, which has been

measured for more than 900 systems.

Assuming a ﬂat proﬁle in the spout, axial

velocity is given by

= 1.6 × 10

−2



1 + 300H

+ 3.8



0.6

φρ

)

0.8



1 + K



msi



(5.32)

where, with all terms in SI units,

K = exp[−C

(z − C

)/z], (5.33)

= 6.2D

− 0.77 × 10

−2

φρ

)

0.33

+ (0.10 − 1.69 × 10

−5

), (5.34)

and

= 3.68 − 5.24 × 10

−4

. (5.35)

The numerically calculated values of the dispersion coefﬁcient (for glass spheres,

rice, and polystyrene) in conical spouted beds are compared,

in Figure 5.5, with the

values obtained in the literature for other gas–solid contacting methods. The values of

U for conical spouted beds in this ﬁgure are those at the upper level of the static bed

(of diameter D

). The diverse range of operating conditions studied explains the wide

spread of the results for conical spouted beds, with values of D between 2.6 × 10

−4

and

6.1 × 10

−2

/s.

5.4.2 Dilute spouted bed

The geometric model proposed for gas ﬂow is established in direct relationship with

contactor and inlet geometry, as in Figure 5.4,

but in this case there is no well-deﬁned

spout zone, given that voidage is approximately uniform throughout. The gas dispersion

coefﬁcients are found by ﬁtting the experimental tracer concentration data to the mass

balances for the streamtubes; they are of the same order of magnitude as those for ﬁxed

beds (Figure 5.5) and substantially lower than those for ﬂuidized beds.

90 Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

1.0

flu

idized beds

–1

–2

cyli

drical

spo

ted bed

ical

spo

ted beds

fixed beds

–3

–4

0.01 0.1 1.0 10

dilute

spouted beds

large particle

fluidization

U (m/s)

D(m

/s)

fast fl

idizatio

Figure 5.5. Dispersion coefﬁcients for spouting and dilute spouting in conical contactors.

5.5 Particle segregation

Conical spouted beds perform well for the stable treatment of beds formed by particles

of different size and density, with low segregation. In fact, the studies carried out indicate

that beds of glass beads of signiﬁcant size distribution

25,32,33

show less segregation for

spouting than for ﬂuidization.

Figure 5.6 shows weight fractions of the particles of greater diameter at different radial

and longitudinal positions in a bed for a system with pronounced segregation (50 wt%

each of 1- and 7-mm glass spheres).

The samples were extracted by a probe connected

to a suction pump. As observed in Figure 5.6, the weight fraction of larger particles,

, passes through a minimum that corresponds approximately to the interface between

the spout and annular zones. For the higher bed levels, the mass fraction of the larger

particles, X

, passes through a maximum, corresponding to an intermediate position in

the annular zone.

In addition to less segregation, the inversion of segregation is an important difference

from ﬂuidized beds. In conical spouted beds, the component of greater size is in greater

proportion at the top of the bed. This can be explained by the fact that the larger particles

(theoretically jetsam) falling back from the fountain into the annulus tend to follow

radial positions near the annulus–spout interface, and so describe shorter trajectories

than smaller ones. The segregation produced in the fountain is important in this situation,

in which the heavier components (by size or density) describe shorter trajectories than

Conical spouted beds 91

0.6

key

z (m)

0.225

0.200

0.175

0.125

0.100

0.075

0.050

0.100

0.175

0.200

= 0.225 m

0.125

0.075

z = 0.05 m

0.5

0.4

0 0.05 0.10

r (m)

Figure 5.6. Weight fraction of particles of greater diameter at different radial and longitudinal

positions. Binary mixture of glass spheres: 50% each of 1- and 7-mm particles. γ = 33

◦

, D

0.03 m, H

= 0.18 m, u

msi

= 1.05.

lighter ones. This has also been observed for segregation by both size and density by

Piccinini et al.

through the transparent wall of a conical-cylindrical spouted bed.

Bed segregation in binary mixtures has been characterized by a mixing index that is

analogous to that proposed by Rowe et al.

for ﬂuidized beds:

= (X

)

, (5.36)

where (

)

is the weight fraction of the particles of greater diameter in the upper half

of the bed volume and

is the weight fraction of the same particles in the whole bed.

The ternary mixing index has been calculated using a distribution function

= f (d

)

/ f (d

), (5.37)

where f (d

)

is the average value of the particle size distribution function in the upper

half of the bed volume and f (d

) in the whole bed, deﬁned by

f (d

)

= (d

svu

− d

)/(d

− d

) ( 5.38)

and

f (d

) = (d

− d

)/(d

− d

). (5.39)

Segregation attenuates as D

and u

increase, and is more pronounced as static

bed height increases. On changing the proportion of the solid components, segregation

reaches a peak for mixtures with the same weight fraction of each component and is

more pronounced as the ratio between the particle diameters deviates further from 1.

The stability of spouting regimes in conical spouted beds with inert particle mixtures

has been studied by Bacelos and Freire

using the statistical analysis of pressure ﬂuc-

tuations. These authors used glass spheres with a size range from 0.78 to 4.38 mm and

binary, ﬂat, or Gaussian distributions. They applied the same analysis as San Jos

eetal.

92 Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

Fountain

Pariphery

Annulus

ANNULAR ZONE SIGNAL

SPOUT ZONE SIGNAL

Spout

Fountain

Core

Figure 5.7. Signals from optical probe in annulus and spout. From: San J os

eetal.

to quantify seg regation. Their main conclusion is that their conical spouted beds produce

more particle segregation than those of San Jos

e et al. for particle size mixtures. Given

that cone angle is the main difference between the beds used by both research teams –

◦

in the study of Bacelos and Freire

and 36

◦

in the case of San Jos

eetal.

–the

smaller angle used by San Jos

e et al. must be responsible for the lower segregation. In

fact, particle circulation rate is much higher with the smaller angle, which enhances

particle mixing and reduces segregation.

5.6 Local properties

Knowledge of local properties in a conical spouted bed is essential for developing models

of gas and solid ﬂow. The properties of interest are (1) spout and fountain geometry;

(2) local bed voidage in the annulus, spout, and fountain; and (3) velocity and trajectory

of particles.

Experimental studies have been carried out at pilot plant scale using glass spheres

and plastics (d

= 1–6 mm) in conical contactors of different geometry (γ = 28–45

◦

=0.03–0.05 m, D

=0.062 m) and under different operating conditions (H

=0.05–

0.30 m and U/U

from 1 to 2).

5.6.1 Spout and fountain geometry

The interface between the spout and the annulus has been detected by the differences

in the signals from an optical probe

(Figure 5.7). When the tip of the probe is in the

annulus, the corresponding signal is formed by wide peaks. Likewise, the two zones of

Conical spouted beds 93

0.10

key Material

Glass beads

Extruded PS

HDPE

Expanded PS

0.08

0.06

0.04

0.02

0 0.04 0.08 0.12 0.16 0.20

z (m)

(m)

Figure 5.8. Evolution of spout diameter at minimum spouting with bed level for different

materials. γ = 33

◦

, D

= 0.03 m, H

= 0.16 m. Particle densities: glass beads 2420 kg/m

;

extruded PS 1030 kg/m

; HDPE 940 kg/m

; PP 890 kg/m

; expanded PS 65 kg/m

.FromSan

Jos

eetal.

the fountain, the core or particle ascending zone and the periphery or particle descending

zone, may be delimited based on the signals from the optical probe. When the probe is

in the core, the delay times indicate that particles are ascending. Qualitatively, the shape

of the spout zone is similar in all conical spouted beds and has a pronounced expansion

near the inlet of the contactor, followed by a neck, and then a further expansion toward

the fountain.

The geometry of the contactor and the operating conditions (including particle density)

greatly inﬂuence the geometry of the spout. Figure 5.8 shows that as the solid density is

increased, the spout diameter signiﬁcantly increases, the neck is more pronounced, and

its position reaches higher levels in the bed.

The average diameter of the spout is much higher than the inlet diameter. A correlation

has been developed for its estimation, based on statistical analysis

= 0.52G

0.16

0.41

−0.19

0.80

msi

)

0.80

, (5.40)

with γ in radians and other terms in SI units (kg, m, s).

Fountain height increases as air velocity is increased and particle size and shape factor

are decreased. A correlation proposed for its calculation is

= 1.01 × 10

−2

−0.14

)

1.14

)

0.83

×(H

)

−0.52

msi

)

4.80

−0.12

−1.45

, (5.41)

with γ in radians and ρ

in kg/m

5.6.2 Bed voidage

In conical spouted beds and on the basis of experimental observations, bed voidage

in the spout decreases with increasing height.

More recently, bed voidage has been

measured using an optical ﬁber probe in the t hree zones of conical spouted beds: spout,

94 Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

1.0

ε (0)

key material

Glass spheres

Extruded PS

HDPE

Expanded PS

0.8

0.6

0.4

0.2

0 0.04 0.08 0.12 0.16 0.20

z (m)

Figure 5.9. Longitudinal proﬁle of bed voidage at spout axis for different materials at minimum

spouting: γ = 33

◦

, D

= 0.03 m, H

= 0.18 m, glass spheres of d

= 3.5 mm. Particle densities

as in Figure 5.8.

annulus, and fountain.

Figure 5.9 shows the results for different materials under given

operating conditions. As obser ved, bed voidage has a maximum value at the central

point of the gas inlet, where it is almost unity for s olids with a density similar to glass.

This maximum voidage decreases as solid density is decreased.

Bed voidage in the fountain has a similar proﬁle for all the systems studied.

First, it

decreases radially from the axis to the core–periphery interface, then increases sharply in

the periphery, and, ﬁnally, decreases slightly near the external surface of the fountain. It

has also been observed that particle shape factor does not signiﬁcantly affect the voidage

in the two regions of the fountain, which shows that a greater fountain volume caused

by a decrease in sphericity leads to an increase in the amount of solids in the fountain.

5.6.3 Particle velocity and trajectory

According to studies on conventional spouted beds, the solids ﬂow rate in the lower

conical section is much greater than in the upper cylindrical section.

The few papers

dealing with particle velocities in conical spouted beds reveal the peculiar characteristics

of particle velocity in the spout.

39–41

Thus, the maximum velocities in the spout of

conical spouted beds are higher than those corresponding to cylindrical spouted beds

and are reached closer to the base. Furthermore, annulus particles near the spout–annulus

interface descend relatively quickly just before entering the spout.

41,42

The vertical component in the spout (Figure 5.10a) reaches a maximum at the axis,

which differs from what was observed by He at al.

for conventional spouted beds.

These authors observed that the maximum particle velocity in the spout is displaced

slightly from the axis in the upper section of the bed. This displacement of maximum

velocity was theoretically predicted by Krzywanski et al.,

who attributed it to the radial

movement of the particles and interparticle collisions in the spout.

Conical spouted beds 95

6.0

(m/s)

5.0

key z (m)

Extruded PS

4.0

0.02

0.03

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.18

3.0

2.0

1.0

0 0.01 0.02 0.03 0.04 0.05

r (m)

(a)

0.20

0.16

Extruded PS

key

z (m)

(b)

r (m)

(m/s)

0.12

0.08

0.04

0 0.02 0.04 0.06 0.08 0.10

0.02

0.05

0.09

0.13

0.17

0.18

Figure 5.10. Radial proﬁles of vertical component of particle velocity in (a) spout and (b) annulus.

γ =33

◦

, D

= 0.03 m, H

= 0.18 m, extruded polystyrene of d

= 3.5 mm. From San Jos

eetal.

An equation similar to that proposed by Epstein and Grace

is valid for predicting

the radial proﬁle at any level in the spout:

= ν

(0)



1 −







. (5.42)

The exponent m decreases from a value of 3 at the bottom of the contactor to a value

of 1 at the top of the bed. Epstein and Grace

reported that 1.3 ≤m ≤2 for conventional

spouted beds, but these values provide suitable proﬁles only at the intermediate section

of the bed. Although Olazar et al.

developed a procedure for accurate estimation of this

parameter, a linear decrease from the bottom to the surface provides acceptable results.

The vertical component of downward particle velocity in the annulus (Figure 5.10b)

reaches a maximum at an intermediate position between the spout and the wall. Com-

parison of these velocities with those for conventional spouted beds in both spout and

annulus shows that they are greater in conical spouted beds, especially for steeper

contactor angles.

96 Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

0 0.05

key material

Glass spheres

Extruded PS

LDPE

Expanded PS

0.10 0.15 0.20

z (m)

Figure 5.11. Effect of solid density and shape on solids cross-ﬂow from annulus to spout. γ = 33

◦

= 0.03 m, H

= 0.18 m, d

= 3.5 mm. Particle densities: LDPE 923 kg/m

, other particle

densities as in Figure 5.8. From San Jos

eetal.

When spout shape, local voidage, and vertical components are known, the horizontal

components (and, therefore, velocity vectors and particle trajectories) can be determined

by solving the mass conservation equations for differential volume elements in the spout

and annulus with the corresponding boundary conditions

∂

∂z

[

(

1 − ε

)

]

∂

∂r

[

(

1 − ε

)

]

= 0. (5.43)

This procedure also allows calculation of the solids crossing from the annulus into the

spout, although this may also be determined by an optical probe to count the particles

ascending in the spout.

Figure 5.11 shows the crossﬂow proﬁle as a function of bed

height. The most important solids crossﬂow is seen to occur near the contactor bottom,

and the other zone of preferential solids crossﬂow is at an intermediate longitudinal

position in the spout, located approximately at the spout neck. Solids crossﬂow at the

neck is very low for the lighter solid (expanded polystyrene) where this neck is barely

discernible.

The cycle times in conical spouted beds are signiﬁcantly shorter than those in cylin-

drical ones, especially for steep contactor angles. This is because of the higher particle

velocities and, hence, higher solid ﬂow rates. For example, for a conical bed of H

0.25 m, the average cycle time ranges from 2.5 to 3.5 s.

5.7 Numerical simulation

Compared with other multiphase s ystems, numerical simulations of spouted beds, espe-

cially conical ones, have received less attention, and simulated results are limited and

disputed. Lu et al.

48,49

simulated conical spouted beds operated by San Jos

eetal.

taking the kinetic theory approach for granular ﬂow. They investigated the effect of the