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

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Empirical and analytical hydrodynamics 47

Dividing Eq. (3.81) by Eq. (3.82) results in

−dP/dz

(

−dP/dz

)

= sin



π z



. (3.83)

Expressing −dP/dz in terms of U

by means of Eq. (3.45),

(

−dP/dz

)

in terms of

by means of the same equation with U

= U

, and expressing the result in terms

of y(= U

) and β, deﬁned by Eq. (3.67), transforms Eq. (3.83) to

(

β − 2

)

y + 3y

2β − 1

= sin

π z

. (3.84)

If Darcy’s law prevails, for example, for small particles (d

< 1 mm), β →∞and

therefore Eq. (3.84) reduces to

y = U

= sin(π z/2H

), (3.85)

whereas for the opposite extreme of inviscid ﬂow, β = 2, so Eq. (3.84) becomes

y = U

= [sin(π z/2H

)]

1/2

. (3.86)

For coarse particles, Eq. (3.86) gives much better agreement with experimental data

than Eq. (3.85), which generally underpredicts for d

> 1 mm, b ut not quite as good

agreement as Eq. (3.84), especially

if U

in the expressions for y and β is replaced

by the measured value of U

aHm

Equations (3.65) and (3.85) intersect at z/H

= 0.343, below which values of U

from Eq. (3.85) exceed those from Eq. (3.65), and vice versa above z/H

= 0.343.

3.5 Pressure drops, proﬁles, and gradients

In this section, equations are derived for longitudinal pressure proﬁles and pressure

gradients, as well as total pressure drops, for spouted beds of height H as a function of

quantities (e.g., H

) that depend only on column geometry, ﬂuid properties, and particle

properties.

3.5.1 Empirically based modeling

Continuing with the Lefroy-Davidson

approach, introduced in the previous section,

and assuming that the pressure gradient at the top of a bed of maximum spoutable depth

is given by

(−dP/dz)

= (−dP/dz)

= (ρ

− ρ)(1 − ε

)g, (3.87)

substitution of this equation into Eq. (3.83) gives

−dP/dz

(−dP/dz)

−dP/dz

(ρ

− ρ)(1 − ε

= sin



π z



. (3.88)

48 Norman Epstein

It is assumed here and elsewhere, after Grbav

cetal.

and others,

31,34

that the pressure

gradient at level z for a given column geometry and ﬂuid and particle properties, is

independent of bed height H , at least when U ≈ U

If we now substitute Eq. (3.87) into Eq. (3.82), we arrive at

(−P

)

= (ρ

− ρ)(1 − ε

)g(2H

/π ). (3.89)

Eq. (3.89) is generalized to be applicable for all values of H and not just for H

writing

−P

= B(ρ

− ρ)(1 − ε

)g(2H /π). (3.90)

It follows from Eqs. (3.89) and (3.90) that

B =

−P

(−P

)

. (3.91)

The numerator in this equation is evaluated by differentiating Eq. (3.79) by z,so

−dP/dz = ( π/2H)(− P

)sin(π z/2H ), (3.92)

which at z = H yields

−P

/H = (2/π )(−dP/dz)

. (3.93)

The denominator in Eq. (3.91) is evaluated by putting z = H in Eq. (3.81), so

(−P

)

= (2/π)(−dP/dz)

/sin(π H /2H

). (3.94)

Combining Eqs. (3.91), (3.93), and (3.94) gives

B = sin( π H /2H

). (3.95)

If the entire bed of height H were at minimum ﬂuidization,

−P

= (dP/dz)

· H = (ρ

− ρ)(1 − ε

)gH. (3.96)

Combining Eqs. (3.90), (3.95), and (3.96), we obtain

(−P

)/(−P

) = (2/π)sin(π H/2H

). (3.97)

For a bed at its maximum spoutable height, H

, Eq. (3.97) simpliﬁes to

(−P

)

/(−P

)

= 2/π, (3.98)

that is, the pressure drop across a bed of height H

is 63.7 percent of that required for

ﬂuidization.

Pressure gradients determined by Eq. (3.88) and total pressure drops from Eq. (3.97)

tend to underpredict the corresponding experimental measurements to varying extents,

depending, at least in part, on the magnitude of radial pressure gradients near the ﬂuid

inlet nozzle, neglected in the derivation.

Substitution of (−P

) by Eq. (3.97) into Eq. (3.79) results in

(P − P

)/(−P

) = (2/π)sin(π H/2H

) cos(π z/2H ). (3.99)

Empirical and analytical hydrodynamics 49

At the inlet, z = 0, and therefore P − P

=−P

as given by Eq. (3.97), whereas at

the outlet, z = H and therefore P = P

.AtH = H

, Eq. (3.99) reduces to Eq. (3.80)

with (−P

)

given by Eq. (3.98).

3.5.2 Mechanistic modeling

Turning now to the f orce balance model of Figure 3.2, we start by dividing Eq. (3.45) by

the same equation applied to minimum ﬂuidization conditions, expressing the result in

terms of y(= U

) and β, and substituting for (−dP/dz)

according to Eq. (3.87):

−dP/dz

(−dP/dz)

−dP/dz

(ρ

− ρ)(1 − ε

+ K

2(β − 2)y + 3y

2β − 1

(3.100)

When Darcy’s law prevails, β =∞and the right-hand side (RHS) of Eq. (3.100) reduces

to y, whereas for pure inviscid ﬂow, β = 2 and the RHS equals y

. For a given value of β

between 2 and ∞, Eq. (3.100) can be solved for −dP/dz at speciﬁed values of z/H

using Eq. (3.75) to generate Y for a given value of z/H

, and hence the cor responding

value of y from Eq. (3.68). Less rigorously, but with little loss of accuracy, answers

can be generated with ease by substituting for y(= U

) in Eq. (3.100) according to

Eq. (3.65), so

−dP/dz

(−dP/dz)

2β(3x − 3x

+ x

) − 12x +39x

− 58x

+ 45x

− 18x

+ 3x

2β − 1

(3.101)

where x = z/H

. Eq. (3.101) correctly yields limiting RHS values of 1 − (1 − x

[1 − (1 − x

)]

, 0, and 1 for β =∞, β = 2, x = 0(allβ), and x = 1(allβ), respectively.

Eq. (3.101) gives

(−dP/dz) values over the range x = 0 to 1 that typically average

only 3 percent above those obtained by the use of Eq. (3.75) with Eq. (3.100). The method

of arriving at Eq. (3.101) has been labeled “hybrid” because it combines Eq. (3.100),

which is perfectly rigorous, with Eq. (3.65), which nominally applies only to Darcy ﬂow

but, as indicated above, is an excellent approximation even for non-Darcy ﬂow. Hybrid

methods are similarly applied, as will be shown, to pressure proﬁles and pressure drops.

To arrive at an equation for the longitudinal pressure proﬁle, Eq. (3.100) is written in

integral form with integration limits from z to H and P to P



−dP =(−dP/dz)



2(β − 2)y + 3y

2β − 1

dz. (3.102)

Substituting for (−dP/dz)

by means of Eq. (3.96), Eq. (3.102) can be transformed to

P − P

−P

h(2β −1)



[2(β − 2)y + 3y

]dx, (3.103)

50 Norman Epstein

where h = H /H

. Eq. (3.103) is again most rigorously solved numerically by determin-

ing y at various values of x from Eq. (3.75) in combination with Eq. (3.68). Adopting the

simpler hybrid approach, the term in square brackets within the integral of Eq. (3.103)

is equivalent to the numerator on the RHS of Eq. (3.101). The integration then gives

P − P

−P

h(2β −1)



2β{1.5(h

− x

) − (h

− x

) + 0.25(h

− x

)}

−6(h

− x

) + 13(h

− x

) − 14.5(h

− x

) + 9(h

− x

)

−3(h

− x

) +

− x

)



. (3.104)

At the Darcy’s law limit, β =∞, this equation reduces to

(P − P

)/(−P

) = [1.5(h

− x

) − (h

− x

) + 0.25(h

− x

)]/ h. (3.105)

The total pressure drop, −P

= P

− P

, is given by substituting x = 0in

Eq. (3.104) for the hybrid solution and Eq. (3.105) for the Darcy solution. The hybrid

solution is thus

−P

h(2β −1)

[2β(1.5h

− h

+ 0.25h

) − 6h

+ 13h

−14.5h

+ 9h

− 3h

+ 0.429h

], (3.107)

whereas the Darcy (β =∞) result is simply

(−P

)/(−P

) = 1.5h − h

+ 0.25h

. (3.108)

For the opposite extreme of inviscid ﬂow (β = 2), assuming H = H

– that is, h = 1,

Eq. (3.107) reduces to

(−P

)

/(−P

)

= 0.643, (3.109)

which is in excellent agreement with Eq. (3.98). However, the maximum value of

the spouted-to-ﬂuidized-bed pressure drop ratio often exceeds 0.64 experimentally,

11,31

probably because of radial pressure gradients and other disturbances near the ﬂuid inlet,

and is actually better represented by putting h = 1 in Eq. (3.108), so

(−P

)

/(−P

)

= 0.75. (3.110)

Similarly, the Darcy’s law-based Eq. (3.108) for pressure drop at any value of h and

Eq. (3.105) for pressure proﬁle often fortuitously give better predictions than the

corresponding, more rigorous, Forchheimer-based equations containing β deﬁned by

Eq. (3.67).

Empirical and analytical hydrodynamics 51

Determination of −P

experimentally for comparison with theory is best done either

by extrapolation of pressure measurements at different heights to z = 0 or by pressure

drop measurements across the whole bed, including the inlet during spouting, and for

the empty column, using the appropriate procedure to eliminate the effect of the inlet.

Unlike annulus voidages and ﬂuid velocities, which increase somewhat with increasing

superﬁcial spouting velocity, total pressure drops remain essentially constant.

3.6 Spout diameter

By a force balance on the spout–annulus interface in the cylindrical part of the col-

umn in which axial variation of the spout diameter is relatively small, including both

hydrodynamic forces and solid stresses based on hopper ﬂow of solids, Bridgwater and

Mathur

derived

32 fρ Q

(−dσ

/dz)(D − D

= 1. (3.111)

They then made the following simplifying assumptions:

(1) The volumetric gas ﬂow through the spout, Q

, above the cone but well below the

bed surface, is typically about one-half the total volumetric ﬂow through the bed,

that is, Q

 0.5π D

G/4ρ.

(2) Flow through the spout is equivalent to ﬂow of a dilute air-solids suspension through

a rough pipe with an equivalent sand roughness of d

/2D

and a Fanning friction

factor f ≈ 0.08.

(3) From Eq. (3.42), −dσ

/dz = (ρ

− ρ)(1 − ε

)g − (−dP/dz). Assuming again at

about the mid-height of the spouted bed that (−dP/dz) is approximately one-half of

(ρ

− ρ)(1 − ε

)g, and that ρ  ρ

, then (−dσ

/dz)  ρ

(1 − ε

)g/2 = ρ

g/2,

where ρ

= bulk density in the annulus.

(4) The analysis is limited to air spouting at ambient conditions, so ρ = 1.2 kg/m

(5) D

 D, so that D − D

 D.

With these assumptions, Eq. (3.111) reduces to

= (constant)G

1/2

3/4

(

)

1/4

. (3.112)

In parallel with this derivation, McNab

developed an empirical equation for the

average spout diameter

by statistically correlating literature data on

available at

that time, all for air spouting at ambient conditions in semicylindrical columns, using the

three dependent variables G, D, and ρ

of Eq. (3.112). The resulting McNab equation,

in SI units, is

= 2.0G

0.49

0.68

/ρ

0.41

. (3.113)

Eq. (3.113) has served well as a predictor, within about 10 percent of

observed

on the transparent ﬂat face of semicylindrical columns for spouting with air at room

52 Norman Epstein

temperature, provided that d

>1 mm. Even for full-column shallow beds (including

those with ﬂat bases, in which dead solids at the base form an angle of repose), in

which the spout conﬁguration is dominated by gas deceleration and solids acceleration

in the cone, and the spout diameter is measured by optical ﬁber probing, Eq. (3.113)

has required

38,39

only minor empirical modiﬁcations to account mainly for D

/D and

θ. However, for beds of normal depth in which the dominant contribution to

comes

from the par t of the spout well above the cone, distortion of the spout at the ﬂat face of a

half-column was found to diminish D

relative to that in the corresponding full column,

, via Eq. (3.113), was underpredicted

forspoutingof1.44-mm glass beads by as

much as 35.5 percent. For particles with d

≤ 1.0 mm, even with D

measured visually

at the ﬂat face of a half-column, Eq. (3.113) underestimated

by about 25 percent,

although it predicted the variation of

with G

(

= Uρ

)

correctly.

For spouting with air at elevated temperatures, or with gases other than air, assumption

(4) above is dropped and ρ reinserted as a variable in simplifying Eq. (3.111). We then

obtain, instead of Eq. (3.112),

= (constant)G

1/2

3/4

(

)

1/4

. (3.114)

Considering now each variable in Eq. (3.114), as well as gas viscosity μ, which rises

with temperature and could affect f and the shear stress at the spout wall, we can write

= f (D, G,ρ

g,ρ,μ). (3.115)

By dimensional analysis,

/D = f (DG/μ , ρρ

/μ

). (3.116)

In the study mentioned previously

for sand spouting with air at temperatures from

◦

C to 580

◦

C, bolstered by an earlier study

of spouting similar particles with air

up to 420

◦

C and with both helium and methane at 20

◦

C, the mean spout diameter was

determined based on the longitudinally averaged spout cross-sectional area from visual

and photographic

measurements of D

at the transparent ﬂat face of a half-column

⎡

⎣



{

(

)

}

⎤

⎦

1/2

. (3.117)

A total of 205 runs from the two combined studies

12,24

yielded, as a best ﬁt to a simple

power l aw relationship among the three dimensionless groups in Eq. (3.116),

/D = 5.59(DG/μ )

0.435

(ρρ

/μ

)

−0.282

(3.118)

with an average absolute deviation of approximately 5 percent. By rewriting Eq. (3.118)

= 5.59G

0.435

0.589

0.129

(

)

0.282

, (3.119)

Empirical and analytical hydrodynamics 53

we can see that the effect of each of the process variables is qualitatively similar to

that displayed by the simpliﬁed theory incorporated in Eq. (3.114), with an additional

small effect of viscosity. Eq. (3.119), unlike Eq. (3.113), is dimensionally consistent and

accounts for variations in gas density and viscosity.

To estimate the ﬁnal ﬂuid ﬂow split between the spout and the annulus, knowledge of

, the spout diameter at the top of the bed where the annulus ends and the fountain

begins, is desirable. Zanoelo and Rocha

have given detailed plots relating measured

spout radius, D

/2, to z/H for ﬁve conical-cylindrical spouted beds from three studies

in the literature.

24,40,43

They recorded values of

determined from these experimental

data, which were mainly at U /U

= 1.1. From these plots it is possible to obtain the

experimental values of D

. From the ﬁve values thus obtained, D

= 1.09 ± 0.04,

and from the corresponding recorded gas inlet diameters, the mean value of D

was

1.80 ± 0.3. This value is consistent with the argument of Lefroy and Davidson

that to

satisfy the conditions of both spout wall stability and gas-jet inlet momentum for spout

penetration, D

≈ 2D

3.7 Flow split between spout and annulus

At the common operating velocity of U = 1.1U

, we can determine U

from Eq. (3.77)

applied to z = H :

αU

= 1 −



1 −



, (3.120)

where U

is obtained from Eq. (3.32) and H

from Eq. (3.33). For air spouting at

ambient conditions, U

is given by Eq. (3.6) and

by either Eq. (3.113) or Eq. (3.119).

Taking D

as 1.09

, we can determine the fraction of total airﬂow through the annulus

as U

− D

)/UD

. The superﬁcial gas velocity, U

, leaving the spout can be

determined from the gas ﬂow balance,

= U



− D



. (3.121)

Let us apply this procedure to the ﬂuid–solid combination used illustratively in

Section 3.1 for a half-column of diameter D = 0.5 m, gas inlet diameter D

= 0.10 m,

and bed height H = 0.9H

. From Eqs. (3.33) and (3.34), respectively, the latter with b

taken as unity, U

= 1.934 m/s and H

= 2.70 m, so H = 2.43 m. As α(= U

)

at U/U

= 1.1 has been reported

for several diverse column dimensions, parti-

cle properties, and bed heights as approximately 0.9, this value is adopted here.

Then from Eq. (3.120), U

= 1.739 m/s and from Eq. (3.6), f or 5-mm spheres of

density 2500 kg/m

spouted with ambient air (ρ = 1.21 kg/m

), U

= 1.817 m/s.

With G = Uρ = (1.1 × 1.817)(1.21) kg/m

·s and ρ

= ρ

(1 − ε

) = ρ

(1 − ε

) =

2500(1 − 0.415) kg/m

, Eqs. (3.113) and (3.119) yield

= 0.0970 m and 0.0850 m,

respectively. Based on the average of those two values, D

= 1.09(0.0910) =

0.0992 m. The airﬂow fraction through the annulus is then 1.739(0.5

− 0.0992

54 Norman Epstein

(1.1 × 1.817)(0.5

) = 0.836, and from Eq. ( 3.121), U

= 6.6 m/s. Hence, the superﬁcial

exit spout velocity is about 3.3 times the superﬁcial spouting velocity of 2.0m/s.In

this example, U

/U = 0.87, which compares reasonably

with the recorded value of

/U = 0.89 for ambient air spouting at U/U

= 1.1 in a half-column of comparable

dimensions (D = 0.61 m, D

= 0.102 m) for particles of comparable size (3.2 mm ×

6.4 mm prolate spheroids) though less dense (ρ

= 1380 kg/m

), and a somewhat smaller

bed height (0.183 m). For a full column, D

would be larger, and thus both the annulus

ﬂow fraction and U

would be somewhat smaller than calculated above.

Recall that Eq. (3.120) and its antecedents have been derived on the assumption

that ε

= constant = ε

throughout the annulus, which, strictly speaking, applies only

at U = U

. The justiﬁcation for applying it at U = 1.1U

is that the void fraction

increase in most of the annulus

33,34

is then less than 0.02 and that the corresponding

small decrease in resistance to ﬂuid ﬂow is counteracted by a voidage decrease near

the spout wall, especially in the conical region of the column,

and by the downward

motion of the annular solids, which is absent at U

but entrains ﬂuid downward at

1.1U

. The use of the adjustable parameter α (= U

aHm

theoretically) provides

further accommodation of Eq. (3.120) to experimental data. As H is decreased, the

value of α apparently decreases.

At U > 1.1U

, however, the dependence

33,34

of ε

on U, H, and z renders Eq. (3.120) increasingly inapplicable, even with adjustments i n

α. Prediction of the changes in annular voidage and its distribution, a prerequisite for

improvements on equations such as Eq. (3.120), is perhaps best pursued by means of

computational ﬂuid dynamics (see Chapter 4).

References

1. H. Littman, M. H. Morgan III, D. V. Vukovi

c, F. K. Zdanski, and Z. B. Grbav

c. Prediction of

the maximum spoutable height and the average s pout to inlet tube diameter ratio in spouted

beds of spherical particles. Can. J. Chem. Eng., 57 (1979), 684–687.

2. J. F. Richardson and W. N. Zaki. Sedimentation and ﬂuidization. Trans. Instn. Chem. Engrs.

Part A, 32 (1954), 35–53.

3. A. R. Khan and J. F. Richardson. Fluid-particle interactions and ﬂow characteristics of ﬂuidized

beds and settling suspensions of spherical particles. Chem.Eng.Commun., 78 (1989), 111–

130.

4. N. Epstein. Liquid-solids ﬂuidization. In Handbook of Fluidization and Fluid-Particle Sys-

tems, ed. W.-C. Yang (New York: Marcel Dekker, 2003), pp. 705–764.

5. H. A. Becker. An investigation of laws governing the spouting of coarse particles. Chem. Eng.

Sci., 13 (1961), 245–262.

6. J. Eastwood, E. J. P. Matzen, M. J. Young, and N. Epstein. Random loose porosity of packed

beds. Brit. Chem. Eng., 14 (1969), 1542–1545.

7. K. B. Mathur and P. E. Gishler. A technique for contacting gases with coarse solid particles,

AIChE J., 1 (1955), 157–164.

8. P. P. Chandnani and N. Epstein. Spoutability and spout destabilization of ﬁne particles with a

gas. In Fluidization V, ed. K. Ostergaard and A. Sorensen (New York: Engineering Foundation,

1986), pp. 233–240.

Empirical and analytical hydrodynamics 55

9. B. Ghosh. A study of the spouted bed, part I – a theoretical analysis. Indian Chem. Engineer,

7 (1965), 16–19.

10. J. R. Grace and C. J. Lim. Permanent jet formation in beds of particulate solids. Can. J. Chem.

Eng., 65 (1987), 160–162.

11. K. B. Mathur and N. Epstein. Spouted Beds (New York: Academic Press, 1974).

12. Y. Li, C. J. Lim, and N. Epstein. Aerodynamic aspects of spouted beds at temperatures up to

580

◦

C. J. Serbian Chem. Soc., 61 (1996), 253–266.

13. E. R. Altwicker and R. K. N. V. Konduri. Hydrodynamic aspects of spouted beds at elevated

temperatures. Combustion Sci. and Tech., 87 (1992), 173–197.

14. R. Clift, J. R. Grace, and W. E. Weber. Bubbles, Drops, and Particles (New York: Academic

Press, 1978), p. 114.

15. B. Thorley, J. B. Saunby, K. B. Mathur, and G. L. Osberg. An analysis of air and solid ﬂow

in a spouted wheat bed. Can. J. Chem. Eng., 37 (1959), 184–192.

16. W. Du, X. Bi, and N. Epstein. Exploring a non-dimensional varying exponent equation relating

minimum spouting velocity to maximum spoutable bed depth. Can. J. Chem. Eng., 87 (2009),

157–162.

17. A. G. Fane and R. A. Mitchell. Minimum spouting velocity of scaled-up beds. Can. J. Chem.

Eng., 62 (1984), 437–439.

18. Y. L. He, C. J. Lim, and J. R. Grace. Spouted bed and spout-ﬂuid bed behaviour in a column

of diameter 0.91 m. Can. J. Chem. Eng., 70 (1992), 848–857.

19. G. A. Lefroy and J. F. Davidson. The mechanics of spouted beds. Trans. Instn Chem. Engrs.,

47 (1969), T120–T128.

20. S. Ergun. Fluid ﬂow through packed columns. Chem. Eng. Progr., 48:2 (1952), 89–94.

21. C. Y. Wen and Y. H. Yu. A generalized method of predicting the minimum ﬂuidization

velocity. AIChE J., 12 (1966), 610–612.

22. G. S. McNab and J. Bridgwater. Spouted beds – estimation of spouting pressure drop and the

particle size for deepest bed. In Proceedings of the European Congress on Particle Technology

(Nuremberg, Germany, 1977), 17 pages.

23. H. Littman, M. H. Morgan III, D. V. Vukovi

c, F. K. Zdanski, and Z. B. Grbav

c. A theory

for predicting the maximum spoutable bed height in a s pouted bed. Can. J. Chem Eng., 55

(1977), 497–501.

24. S. W. M. Wu, C. J. Lim, and N. Epstein. Hydrodynamics of spouted beds at elevated temper-

atures. Chem. Eng. Commun., 62 (1987), 251–268.

25. M. A. Malek and B. C.-Y. Lu. Pressure drop and spoutable bed height in spouted beds. Ind.

Eng. Chem. Process Des. Dev., 4 (1965), 123–128.

26. M. H. Morgan III, H. Littman, and B. Sastri. Jet penetration and pressure drops in water

spouted beds. Can. J. Chem. Eng., 66 (1988), 735–739.

27. Z. B. Grbav

c, D. V. Vukovi

c, F. K. Zdanski, and H. Littman. Fluid ﬂow pattern, minimum

spouting velocity and pressure drop in spouted beds. Can. J. Chem. Eng., 54

(1976), 33–42.

28. T. Mamuro and H. Hattori. Flow pattern of ﬂuid in spouted beds. J. Chem. Eng. Japan, 1

(1968), 1–5.

29. R. L. Brown and J. C. Richards. Principles of Powder Mechanics (Oxford, UK: Pergamon

Press, 1970), p. 70.

30. A. E. Scheidegger. The Physics of Flow through Porous Media,3rded.(Toronto:Univ.of

Toronto Press, 1974), p. 155.

31. N. Epstein, C. J. Lim, and K. B. Mathur. Data and models for ﬂow distribution and pressure

drop in spouted beds. Can J. Chem. Eng., 56 (1978), 436–447.

56 Norman Epstein

32. N. Epstein and S. Levine. Non-Darcy ﬂow and pressure distribution in a spouted bed. In

Fluidization, ed. J. F. Davidson and D. L. Keairns (Cambridge, UK: Cambridge University

Press, 1978), pp. 98–103.

33. Y. L. He, C. J. Lim, J. R. Grace, and J. X. Zhu. Measurements of voidage proﬁles in spouted

beds. Can. J. Chem. Eng., 72 (1994), 229–234.

34. Y. L. He, C. J. Lim, and J. R. Grace. Pressure gadients, voidage and gas ﬂow in the annulus

of spouted beds. Can. J. Chem. Eng., 78 (2000), 161–167.

35. A. Yokogawa, E . Ogina, and N. Yoshii. Trans. The distribution of static pressure and stress in

the spouted bed. Japan Soc. Mech. Engrs., 38 (1972), 1081–1086.

36. J. Bridgwater and K. B. Mathur. Prediction of spout diameter in a spouted bed – a theoretical

model. Powder Technol., 6 (1972), 183–187.

37. G. S. McNab. Prediction of spout diameter. Brit. Chem. Eng. & Proc. Technol., 17 (1972),

532.

38. M. J. San Jos

e, M. Olazar, M. A. Izquierdo, S. Alvarez, and J. Bilbao. Spout geometry in

shallow spouted beds. Ind. Eng. Chem. R es ., 40 (2001), 420–426.

39. M. J. San Jos

e, S. Alvarez, A. O. de Salazar, M. Olazar, and J. Bilbao. Spout geometry in

spouted beds with solids of different density and different sphericity. Ind. Eng. Chem. Res.,

44 (2005), 8393–8400.

40. Y. L. He, C. J. Lim, S. Qin, and J. R. Grace. Spout diameters in full and half spouted beds.

Can. J. Chem. Eng., 76 (1998), 696–701.

41. N. Epstein and P. P. Chandnani. Gas spouting characteristics of ﬁne particles. Chem. Eng.

Sci., 42, (1987), 2977–2981.

42. E. F. Zanoelo and S. C. S. Rocha. Spout shape predictions in spouted beds. Can. J. Chem.

Eng., 80 (2002), 967–973.

43. C. J. Lim and K. B. Mathur. Residence time distribution of gas in spouted beds. Can. J. Chem.

Eng., 52 (1974), 150–155.