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

Подождите немного. Документ загружается.

Computational ﬂuid dynamic modeling of spouted beds 77

Further fundamental and experimental studies on the kinematic properties of the two

phases are needed to improve the accuracy of the CFD models.

Chapter-speciﬁc nomenclature

A experimental constant in Schaeffer model, N/m

column diameter, m

particle diameter, m

coefﬁcient of restitution of particle

drag

drag force, N

lift,q

lift force, N

n,ij

normal contact force, N

external body force, N

t,ij

tangential contact force, N

vm,q

virtual mass force, N

0,ss

radial distribution function

I turbulence intensity

moment of inertia, kg m/s

spring stiffness of normal force

spring stiffness of tangential force

s

energy diffusion coefﬁcient, J/m

rolling resistant torque, N

n experimental constant in Schaeffer model

solid pressure predicted by friction model, N/m

kinetic

solid pressure predicted by kinetic theory, N/m

solid pressure, N/m

interaction force between phases, N

v vertical particle velocity, m/s

phase q velocity, m/s



particle velocity ﬂuctuation, m/s

Greek letters

gas volume fraction

volume fraction of phase q

solids volume fraction

s,max

maximum solids volume fraction

β ﬂuid-particle interaction coefﬁcient, kg/m

Ergun

ﬂuid-particle interaction coefﬁcient in Ergun equation, kg/m

Wen–Yu

ﬂuid-particle interaction coefﬁcient in Wen–Yu equation, kg/m

dissipation of granular energy, kg/m

particle displacement in normal direction, m

particle displacement in tangential direction, m

78 Xiaojun Bao, Wei Du, and Jian Xu

coefﬁcients of dissipation in normal direction, kg/s

coefﬁcients of dissipation in tangential direction, kg/s



granular temperature, m

solid bulk viscosity, Pa s

solid viscosity predicted by friction model, Pa s

gas viscosity, Pa s

kinetic

solid viscosity predicted by kinetic theory, Pa s

solid shear viscosity, Pa s

turbulent viscosity, Pa s

gas density, kg/m

density of phase q, kg/m

particle density, kg/m

solid stress tensor, N/m

Reynolds stress tensor, N/m

φ internal friction angle, degrees

switching function in Gidaspow model

1,4

ω angular velocity vector

energy exchange between ﬂuid or solid phase, J/s

Subscripts

g gas

i, j individual particle

kinetic kinetic theory

max maximum

min minimum

n normal direction

q phase (gas or solid)

r radial direction

s solid phase

t tangential direction

z axial direction

r radial direction

References

1. D. Gidaspow. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions

(London: Academic Press, 1994).

2. R. S. Krzywanski, N. Epstein, and B. D. Bowen. Multi-dimensional model of a spouted bed.

Can. J. Chem. Eng., 70 (1992), 858–872.

3. Y. He, G. Zhao, J. Bouillard, and H. Lu. Numerical simulations of the effect of conical

dimension on the hydrodynamic behaviour in spouted beds. Can. J. Chem. Eng., 82 (2004),

20–29.

Computational ﬂuid dynamic modeling of spouted beds 79

4. H. Lu, Y. He, W. Liu, J. Ding, D. Gidaspow, and J. Bouillard. Computer simulations of gas-

solid ﬂow in spouted beds using kinetic-frictional stress model of granular ﬂow. Chem. Eng.

Sci., 59 (2004), 865–878.

5. H. Lu, Y. Song, Y. Li, Y. He, and J. Bouillard. Numerical simulations of hydrodynamic

behaviour in spouted beds. Chem. Eng. Res. Des., 79 (2001), 593–599.

6. S. Y. Wang, Y. R. He, H. L. Lu, J. X. Zheng, G. D. Liu, and Y. L. Ding. Numerical simulations

of ﬂow behaviour of agglomerates of nano-size particles in bubbling and spouted beds with

an agglomerate-based approach. Food and Bioprod. Proc., 85 (2007), 231–240.

7. P. C. Johnson, P. Nott, and R. Jackson. Frictional-collisional equations of motion for particulate

ﬂows and their application to chutes. J. Fluid Mech., 210 (1990), 501–535.

8. M. Syamlal, W. Rogers, and T. J. O’Brien. MFIX Documentation. US Department of Energy,

Federal Energy Technology Center, 1993.

9. W. Du, X. Bao, J. Xu, and W. Wei. Computational ﬂuid dynamics (CFD) modeling of spouted

bed: Assessment of drag coefﬁcient correlations. Chem. Eng. Sci., 61 (2006), 1401–1420.

10. W. Du, X. Bao, J. Xu, and W. Wei. Computational ﬂuid dynamics (CFD) modeling of spouted

bed: Inﬂuence of frictional stress, maximum packing limit and coefﬁcient of restitution of

particles. Chem. Eng. Sci., 61 (2006), 4558–4570.

11. W. Du, W. Wei, J. Xu, Y. Fan, and X. Bao. Computational ﬂuid dynamics (CFD) modeling of

ﬁne particle spouting. Int. J. Chem. React. Eng., 4 (2006), A21.

12. Z. G. Wang, H. T. Bi, and C. J. Lim. Numerical simulations of hydrodynamic behaviors in

conical spouted beds. China Particuology, 4 (2006), 194–203.

13. P. A. Shirvanian, J. M. Calo, and G. Hradil. Numerical simulation of ﬂuid-particle hydrody-

namics in a rectangular spouted vessel. Int. J. Multiph. Flow, 32 (2006), 739–753.

14. Z. H. Wu and A. S. Mujumdar. CFD modeling of the gas-particle ﬂow behavior in spouted

beds. Powder Technol., 183 (2008), 260–272.

15. O. Gryczka, S. Heinrich, and J. Tomas. CFD-modelling of the ﬂuid dynamics in spouted

beds. In Micro-Macro-Interactions, ed, A. Bertram and J. Tomas (Berlin: Springer, 2008),

pp. 265–275.

16. L. Schiller and A. Naumann. A drag coefﬁcient correlation. Verein Deutscher Ingenieure, 77

(1935), 318–320.

17. C. Y. Wen and Y. H. Yu. Mechanics of ﬂuidization. Chem. Eng. Progr. Symp. Ser., 62 (1966),

100–111.

18. M. Syamlal and T. O’Brien. Computer simulation of bubbles in a ﬂuidized bed. AIChE Symp.

Ser., 85 (1989), 22–31.

19. D. Gidaspow, R. Bezburuah, and J. Ding. Hydrodynamics of circulating ﬂuidized beds:

Kinetic theory approach. In Fluidization VII, ed. O. E. Potter and D. J. Nicklin (New York:

Engineering Foundation, 1991), pp. 75–82.

20. D. L. Koch and R. J. Hill. Inertial effects in suspension and porous-media ﬂows. Ann. Rev.

Fluid Mech., 33 (2001), 619–647.

21. M. A. Van Der Hoef, R. Beetstra, and J. A. M. Kuipers. Lattice-Boltzmann simulations of

low-Reynolds-number ﬂow past mono- and bidisperse arrays of spheres. J. Fluid Mech.,

528

(2005), 233–254.

22. R. Beetstra, M. A. Van Der Hoef, and J. A. M. Kuipers. Drag force of intermediate Reynolds

number ﬂow past mono- and bidisperse arrays of spheres. AIChE J., 53 (2007), 489–501.

23. O. Gryczka, S. Heinrich, N. G. Deen, M. v. S. Annaland, J. A. M. Kuipers, and L. M

orl. CFD

modeling of a prismatic spouted bed with two adjustable gas inlets. Can. J. Chem. Eng., 87

(2009), 318–328.

80 Xiaojun Bao, Wei Du, and Jian Xu

24. R. B

ettega,R.G.Corr

ea, and J. T. Freire. Scale-up study of spouted beds using computational

ﬂuid dynamics. Can. J. Chem. Eng., 87 (2009), 193–203.

25. R. B

ettega,A.R.F.deAlmeida,R.G.Corr

ea, and J. T. Freire. CFD modelling of a semi-

cylindrical spouted bed: Numerical simulation and experimental veriﬁcation. Can. J. Chem.

Eng., 87 (2009), 177–184.

26. Y. L. He, C. J. Lim, and J. R. Grace. Scale-up studies of spouted beds. Chem.Eng.Sci., 52

(1997), 329–339.

27. K. G. Santos, V. V. Murata, and M. A. S. Barrozo. Three-dimensional computational ﬂuid

dynamics modelling of spouted bed. Can. J. Chem. Eng., 87 (2009), 211–219.

28. C. R. Duarte, M. Olazar, V. V. Murata, and M. A. S. Barrozo. Numerical simulation and

experimental study of ﬂuid-particle ﬂows in a spouted bed. Powder Technol., 188 (2009),

195–205.

29. J. Ding and D. Gidaspow. A bubbling ﬂuidization model using kinetic theory of granular ﬂow.

AIChE J., 36 (1990), 523–538.

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

31. C. S. Campbell. Granular material ﬂows – an overview. Powder Technol., 162 (2006), 208–

229.

32. S. Sundaresan. Some outstanding questions in handling of cohesionless particles. Powder

Technol., 115 (2001), 2–7.

33. C. K. K. Lun, S. B. Savage, D. J. Jeffrey, and N. Chepurniy. Kinetic theories for granular

ﬂow: Inelastic particles in Couette ﬂow and slightly inelastic particles in a general ﬂow ﬁeld.

J. Fluid Mech., 140 (1984), 223–256.

34. R. A. Bagnold. Experiments on a gravity-free dispersion of large solid spheres in a Newtonian

ﬂuid under shear. Proc. Royal Soc. London Ser. A, Math. and Phys. Sci., A225 (1954), 49–63.

35. J. T. Jenkins and S. B. Savage. A theory for rapid ﬂow of identical, smooth, nearly elastic

spherical particles. J. Fluid Mech., 130 (1983), 187–202.

36. D. G. Schaeffer. Instability in the evolution equations describing incompressible granular

ﬂow. J. Diff. Eqns, 66 (1987), 19–50.

37. J. H. Ferziger and M. Peric. Computational Methods for Fluid Dynamics, 3rd ed. (Berlin:

Springer, 1999).

38. C. A. J. Fletcher and K. Srinivas. Computational Techniques for Fluid Dynamics Vol. 1,

Fundamental and General Techniques, 2nd ed. (Berlin: Springer-Verlag, 1991).

39. C. J. Freitas. Perspective: Selected benchmarks from commercial CFD codes. ASME J. Fluids

Eng., 117 (1995), 208–218.

40. R. J. LeVeque. Finite Volume Methods for Hyperbolic Problems (Cambridge, UK: Cambridge

University Press, 2002).

41. S. V. Patankar. Numerical Heat Transfer and Fluid Flow (Washington, DC: Taylor and Francis,

1980).

42. J. R. Grace and F. Taghipour. Veriﬁcation and validation of CFD models and dynamic similarity

for ﬂuidized beds. Powder Technol., 139 (2004), 99–110.

43. Y. L. He, C. J. Lim, J. R. Grace, J. X. Zhu, and S. Z. Qin. Measurements of voidage proﬁles

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

44. Y. L. He, S. Z. Qin, C. J. Lim, and J. R. Grace. Particle velocity proﬁles and solid ﬂow patterns

in spouted beds. Can. J. Chem. Eng., 72 (1994), 561–568.

45. M. J. San Jos

e, M. Olazar, S. Alvarez, M. A. Izquierdo, and J. Bilbao. Solid cross-ﬂow

into the s pout and particle trajectories in conical spouted beds. Chem. Eng. Sci., 53 (1998),

3561–3570.

Computational ﬂuid dynamic modeling of spouted beds 81

46. P. A. Cundall and O. D. L. Strack. A discrete numerical model for granular assemblies.

Geotechnique, 29 (1979), 47–65.

47. J. Link, C. Zeilstra, N. Deen, and H. Kuipers. Validation of a discrete particle model in a

2D spout-ﬂuid bed using non-intrusive optical measuring techniques. Can. J. Chem. Eng., 82

(2004), 30–36.

48. J. M. Link, L. A. Cuypers, N. G. Deen, and J. A. M. Kuipers. Flow regimes in a spout-ﬂuid

bed: A combined experimental and simulation study. Chem. Eng. Sci., 60 (2005), 3425–

3442.

49. S. Takeuchi, S. Wang, and M. Rhodes. Discrete element simulation of a ﬂat-bottomed spouted

bed in the 3-D cylindrical coordinate system. Chem.Eng.Sci., 59 (2004), 3495–3504.

50. T. Kawaguchi, M. Sakamoto, T. Tanaka, and Y. Tsuji. Quasi-three-dimensional numerical

simulation of spouted beds in cylinder. Powder Technol., 109 (2000), 3–12.

51. T. Kawaguchi, T. Tanaka, and Y. Tsuji. Numerical simulation of two-dimensional ﬂuidized

beds using the discrete element method. Powder Technol., 96 (1998), 129–138.

52. S. Takeuchi, X. S. Wang, and M. J. Rhodes. Discrete element study of particle circulation in

a 3-D spouted bed. Chem.Eng.Sci., 60 (2005), 1267–1276.

53. S. Takeuchi, S. Wang, and M. Rhodes. Discrete element method simulation of three-

dimensional conical-base spouted beds. Powder Technol., 184 (2008), 141–150.

54. W. Zhong, Y. Xiong, Z. Yuan, and M. Zhang. DEM simulation of gas-solid ﬂow behaviors in

spout-ﬂuid bed. Chem. Eng. Sci., 61 (2006), 1571–1584.

55. X.-L. Zhao, S.-Q. Li, G.-Q. Liu, Q. Song, and Q. Yao. Flow patterns of s olids in a two-

dimensional spouted bed with draft plates: PIV measurement and DEM simulations. Powder

Technol., 183 (2008), 79–87.

56. X.-L. Zhao, S.-Q. Li, G.-Q. Liu, Q. Yao, and J. S. Marshall. DEM simulation of the particle

dynamics in two-dimensional spouted beds. Powder Technol., 184 (2008), 205–213.

57. T. Swasdisevi, W. Tanthapanichakoon, T. Charinpanitkul, T. Kawaguchi, T. Tanaka, and Y.

Tsuji. Investigation of ﬂuid and coarse-particle dynamics in a two-dimensional spouted bed.

Chem. Eng. Technol., 27 (2004), 971–981.

58. M. I. Kalwar, G. S. Raghavan, and A. S. Mujumdar. Circulation of particles in two-dimensional

spouted beds with draft plates. Powder Technol., 77 (1993), 233–242.

59. S. Limtrakul, A. Boonsrirat, and T. Vatanatham. DEM modeling and simulation of a catalytic

gas-solid ﬂuidized bed reactor: A spouted bed as a case s tudy. Chem. Eng. Sci., 59 (2004),

5225–5231.

60. G. Rovero, N. Epstein, J. R. Grace, N. Piccinini, and C. M. H. Brereton. Gas phase solid-

catalysed chemical reaction in spouted beds. Chem. Eng. Sci., 38 (1983), 557–566.

61. N. V. Brilliantov and T. Poeschel. Rolling friction of a viscous sphere on a hard plane.

Europhysics Letters, 42 (1998), 511–516.

62. T. F. Miller and F. W. Schmidt. Use of a pressure-weighted interpolation method for the

solution of the incompressible Navier-Stokes equations on a nonstaggered grid system. Num.

Heat Transf., Part B: Fund

., 14 (1988), 213–233.

63. A. A. Amsden and F. H. Harlow. A simpliﬁed MAC technique for incompressible ﬂuid ﬂow

calculations. J. Comp. Phys., 6 (1970), 322–325.

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

65. T. Tsuji, M. Hirose, T. Shibata, O. Uemaki, and H. Itoh. Particle ﬂow in annular region of a

ﬂat-bottomed spouted bed. Trans. Soc. Chem. Engrs, Japan, 23 (1997), 604–605.

66. D. Roy, F. Larachi, R. Legros, and J. Chaouki. A study of solid behavior in spouted beds using

3-D particle tracking. Can. J. Chem. Eng., 72, (1994), 945–952.

5 Conical spouted beds

Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

5.1 Introduction

In spite of the versatility of spouted beds of conventional geometry (cylindrical with

conical base), there are situations in which the gas–solid contact is not fully satisfactory.

In these situations, the process conditioning f actors are the physical characteristics of the

solid and the residence time of the gas. Thus, conical spouted beds have been used for

drying suspensions, solutions, and pasty materials.

1–4

Chemical reaction applications,

such as catalytic polymerization,

coal gasiﬁcation,

and waste pyrolysis,

7–9

have also

been under research and development.

In fast reactions, such as ultrapyrolysis,

selectivity is the factor that conditions

the design, so the optimum residence time of the gaseous phase can be as short as

a few milliseconds. The spouting in conical beds attains t hese gas residence times,

which can be controlled within a narrow range. Nevertheless, until the 1990s, the lit-

erature on the principles and hydrodynamics of the ﬂow regimes in conical spouted

beds was very scarce, and its diffusion was very limited, partially owing to the

scant dissemination in the past of the research on such beds carried out in Eastern

Europe.

5.2 Conditions for stable operation and design geometric factors

Operating in conical contactors is sensitive to the geometry of the contactor and to

particle diameter, so it is necessary to delimit the operating conditions that strictly

correspond to the spouted bed regime and allow for the gas–solid contact to take place

stably. The ranges of the geometric factors of the contactor (deﬁned in Figure 5.1) and

of the contactor–particle system for stable spouting are as follows

Inlet diameter/cone bottom diameter ratio, D

. This ratio should be between 1/2

and 5/6. The lower limit is imposed by the pressure drop and by the formation of dead

zones at the bottom (a serious problem for operating with solids circulation). Exceeding

the upper limit gives rise to poor deﬁnition of the spout and an increase in instability

because of rotational movements.

Spouted and Spout-Fluid Beds: Fundamentals and Applications, eds. Norman Epstein and John R. Grace.

Published by Cambridge University Press.



Cambridge University Press 2011.

Conical spouted beds 83

γ/2

Figure 5.1. Geometry of the conical spouted bed reactor.

Cone angle, γ . The lower limit is 28

◦

, as the bed is unavoidably unstable for lower

angles. From a practical point of view, angles >60

◦

are not recommended because the

solid circulation rate is then very low, especially for deep beds.

Inlet diameter/particle diameter ratio, D

. Whereas for spouting in cylindrical

vessels,

bed instability occurs at D

≥ 30, stable operation with conical vessels is

obtained for D

between 2 and 60.

Maximum spoutable bed height. There is no maximum spoutable bed height in conical

spouted beds, at least not in the same sense as is found for d

> 1 mm in cylindrical

contactors.

Nevertheless, for large particles (glass spheres with diameters > 5 mm)

there is a maximum height because of the instability of the spouting regime within the

previously described range of the contactor geometric factors.

The cause of instability

is clearly slugging, which affects the whole section of the bed and whose for mation does

not bear a direct relation to any individual contactor geometric factor or to particle diam-

eter, but rather is a consequence of all the factors combined. In general, the maximum

spoutable bed height increases as the particle diameter decreases, as D

decreases,

and as the cone included angle increases.

Column diameter. The upper terminal diameter of the contactor, D

, is a parameter

that can be speciﬁed at will to treat any volume of solid, as long as the key geometric

factors for stability are maintained in the design and the remarks on trajectories of solid

particles and bed pressure drop are taken into account. An equation for calculating D

has

been deduced

from the relationships between the geometric factors of the contactors,

the correlation of minimum spouting velocity, and the correlation for bed expansion:

= [D

+ 2H

tan(γ/2)]

(1 + ψ) − D

ψ, (5.1)

where

ψ = 2.58 × 10

(u/u

msi

)

3.48

−0.52



+ 2H

tan(γ/2)



−0.53

[tan(γ/2)]

−1.84

1.95

(5.2)

with γ in radians.

84 Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

Table 5.1. Hydrodynamic correlations for minimum spouting velocity in conical spouted beds.

Authors Correlation

Nikolaev and Golubev

= 0.051 Ar

0.59

)

0.1

)

0.25

(5.3)

Gorshtein and Mukhlenov

msi

= 0.174 Ar

0.5

)

0.85

[tan(γ/2)]

−1.25

(5.4)

Tsvik et al.

msi

= 0.4Ar

0.52

)

1.24

[tan(γ/2)]

0.42

(5.5)

Goltsiker

msi

= 73 Ar

0.14

)

0.9

(ρ

/ρ)

0.47

(5.6)

Wan-Fyong et al.

msi

= 1.24 (Re)

)

0.82

[tan(γ/2)]

0.92

(5.7)

Markowski and Kaminski

msi

= 0.028 Ar

0.57

)

1.27

)

0.48

(5.8)

Kmiec

msi

[1.75 + 150(1 − ε

)/Re

msi

] =

31.31 Ar (H

)

1.757

)

0.029

[tan(γ/2)]

2.07

(ε

)

(5.9)

Choi and Meisen

= 0.147 (2gH

)

0.5

0.51

0.61

0.24

−1.36

[(ρ

− ρ)/ρ]

0.48

(5.10)

Olazar et al.

msi

= 0.126 Ar

0.5

]

1.68

[tan(γ/2)]

−0.57

(5.11)

Bi et al.

= D

)forD

> 1.66,

msi

= [0.3 − 0.27/(D

)

]



Ar(D

)[(D

)

+ (D

) + 1]/3 (5.12)

for D

< 1.66,

msi

= 0.202



Ar(D

)[(D

)

+ (D

) + 1]/3 (5.13)

5.3 Hydrodynamics

The hydrodynamics of conical spouted beds differ signiﬁcantly when compared to

conventional spouted beds (cylindrical with conical base). Consequently, the values of

the hydrodynamic parameters required for stable spouting – minimum spouting velocity,

pressure drop, and bed expansion – also differ.

11,14

5.3.1 Spouted bed

Minimum spouting velocity. Choi and Meisen

and Olazar et al.

found that the min-

imum spouting velocity in conical beds is approximately proportional to bed height,

whereas it is proportional to the square root of the height in conventional spouted beds.

Correlations developed to predict minimum spouting velocity in conical spouted beds

are listed in Table 5.1. The minimum spouting velocity in conical beds has also been

modeled based on equations developed from force and momentum balances in the s pout

region of the bed.

However, the applicability of these equations is limited owing to the

inclusion of parameters that are not easy to estimate.

The diameter of the cylindrical section of the bed, D

, should not be used in the

correlation for predicting the minimum velocity in conical beds because this velocity

will remain unchanged with variations in D

as long as the bed remains entirely in the

conical section (see Figure 5.1). Eqs. (5.3) and (5.8) through (5.10) include D

as a

parameter, which implies that they are valid only for the speciﬁc beds tested or for cases

in which the static bed upper surface diameter is D

. Other equations do not include the

cone angle, as the authors have not observed any angle inﬂuence in the range covered, or

Conical spouted beds 85

Table 5.2. Correlations for peak pressure drop.

Authors Correlation

Gelperin et al.

−P

g = 1 + 0.062 (D

)

2.54

((D

) − 1)(tan(γ/2))

−0.18

(5.14)

Gorshtein and Mukhlenov

−P

/P

= 1 + 6.65 (H

)

1.2

(tan(γ/2))

0.5

0.2

(5.15)

Olazar et al.

−P

/P

= 1 + 0.116 (H

)

0.50

(tan(γ/2))

−0.80

0.0125

(5.16)

Table 5.3. Correlations for spouting pressure drop.

Authors Correlation

Gorshtein and Mukhlenov

−P

(1 − ε

)g = 7.68 (tan(γ/2))

0.2

(Re

msi

)

−0.2

)

−0.33

(5.17)

Markowski and Kaminski

−P

/ρu

msi

= 0.19 (D

)

0.56

)

2.39

)

2..35

(5.18)

Olazar et al.

−P

(1 − ε

)g = 1.20 (tan(γ/2))

−0.11

(Re

msi

)

−0.06

)

0.08

(5.19)

only one angle has been studied, as in the case of Eqs. (5.8) and (5.10). All equations in

Table 5.1 are empirical except for Eqs. (5.12) and (5.13), which are based on the Ergun

equation and literature data. Nevertheless, these equations are deﬁcient in not including

bed height or cone angle because there are many combinations of these parameters

that provide the same D

, and u

msi

is not the same for any combination. In fact, these

equations underestimate u

msi

for small angles. Eq. (5.11) is based on widely varying

experimental conditions (D

= 0.02–0.062 m, D

= 0.063 m, H

= 0.02–0.55 m, D

0.36 m, γ = 28–45

◦

, d

= 0.5–25 mm, ρ

= 14–2420 kg/m

) and has recorded better

results over a wide range of experimental conditions.

This correlation is also valid for

mixtures of solids of different size,

as long as the Sauter mean diameter, d

, is adopted

as the characteristic mean diameter.

Peak pressure drop. As a rough approximation, this pressure drop is 1.5 to 2.5 times

higher than the spouting pressure drop. The few correlations for its prediction in conical

spouted beds are listed in Table 5.2. Eq. (5.14) does not include particle properties and

Eq. (5.16) is a modiﬁcation of Eq. (5.15) for a wide range of geometric factors and

particle properties.

Spouting pressure drop. Correlations for determining the spouting pressure drop are

listed in Table 5.3. Eq. (5.18) does not include cone angle because the authors used only

a37

◦

contactor. Eq. (5.19) provides better predictions than Eq. (5.17), mainly because it

has been obtained from a wider range of geometry, particle sizes, and particle densities.

A theoretical model has been proposed by Hadzismajlovic et a1.

for calculating the

pressure drop for conical beds. The solution of this model faces an important practical

difﬁculty, as it is necessary to choose an upper boundary that is suitable for integrating

the differential pressure gradient.

5.3.2 Dilute spouted bed

A peculiarity of conical spouted bed contactors is that the gas velocity at the inlet can be

increased substantially without losing the characteristic cyclic movement of the solids.

86 Martin Olazar, Maria J. San Jos

e, and Javier Bilbao

1200

800

Incipient

spouting

Incipient

dilute spouting

Dilute

spouting

Spouting

Transition

(Pa)

400

04812162024

(m/s)

Fixed bed

Figure 5.2. Pressure drop evolution with velocity and particle states in the contactor for different

ﬂow regimes. γ = 36

◦

, D

= 0.03 m, H

= 0.08 m, ρ

= 2420 kg/m

Figure 5.2 shows the evolution of bed voidage with gas velocity for a given system.

The bed passes through a transition state until it reaches a new regime with different

hydrodynamic characteristics, originally termed a jet-spouted bed,

but more accurately

called a dilute spouted bed.

Its general characteristics are (1) high gas velocity;

(2) bed voidage >0.75, depending on particle size and operating conditions; (3) cyclic

movement of the par ticles if the cone angle is sufﬁciently high; and (4) minor instability

problems that are sensitive to the contactor’s geometry.

The velocity required to attain dilute spouting is, according to Markowski and

Kaminsky,

more than 1.7 times that required for spouting. As shown in Figure 5.2, this

number is generally understated; it depends on particle properties and bed geometry.

The only correlation in the literature for predicting the minimum velocity for dilute

spouting is

dsi

= 6.9Ar

0.35

)

1.48

(tan(γ/2))

−0.53

. (5.20)

The range of variables encompassed by t his correlation is the same as for Eq. (5.11).

Bed expansion is of special interest for the design of conical spouted beds. Several

authors

14,28,29

monitored the voidage of conical spouted beds with the ratio of drag

forces to gravitational forces, F

. Typical expansion curves are shown in Figure 5.3

for two contactors of different cone angles. As observed, there are three zones, although

the change from one to the other is not clearly deﬁned. The ﬁrst one corresponds to bed

voidages between a loose-packed bed and the end of the spouting regime, the second from

the end of the spouting regime to approximately the beginning of the dilute spouting

regime, and the third to velocities higher than those for minimum dilute spouting.

Expansion depends greatly on initial bed height and particle size. In f act, a dilute

spouted bed is reached for a bed voidage of 0.76 when coarse particles (d

> 6 mm)

are spouted, whereas high voidages (>0.99) are required for particles smaller than

1 mm. A few correlations for the prediction of bed voidage variation with gas ﬂowrate