Yang Fluidization, Solids Handling, and Processing

Recirculating and Jetting Fluidized Beds 305

where

	Eq. (43)	P =		2RiWz			+	2Wz
			(Ro2 − Ri2 )(1 − ε mf )ρs					Ri (1 − ε i )ρs
	Eq. (44)	Q =			2Wz Wi
			π R	(R2	− R2 )H (1 − ε		mf	)(1 − ε	)ρ 2
			i	o	i			i	s

Equation (42) can be readily integrated, however, there are two limiting cases to consider.

Case I. Instantaneous Injection of Tracer Particles. If it is assumed that the tracer particles are injected instantaneously, Wt = Wto = a constant, Eq. (42) can be integrated with the boundary condition that XJ = 0 at t = t0 to give

Eq. (45)	X J	= 1 − exp[− P(t − to )]
	X o
	J

Case II. Uniform Injection of Tracer Particles. Since the large amount of tracer particles usually required more than 75 seconds to inject, the other limiting case would be to assume that the injection rate was uniform over the injection period, or

W = æWto ö×t ç ÷

Eq. (46) t ç t ÷

è w ø

Again, Eq. (42) can be integrated with the boundary condition that XJ = 0 at t = 0 to give

Eq. (47)	X J	=		1		{Pt − [1− exp(− Pt)]}
			t
	X o			w	P
	J

The equilibrium tracer concentration in the bed after complete mixing can be expressed as

		X Jo =					Wt
	Eq. (48)		π Hρ	s	[R2	(1 − ε	) + (R2	− R2 )(1 − ε	mf	)]
					i	i	o	i

306 Fluidization, Solids Handling, and Processing

The voidage inside the bubble street, ε i, can be calculated as follows

Eq. (49)	εi = ε mf	+ fB (1 - εmf )= ε mf +	nVB	(1 - ε mf )
		πRi2U A

where fB is the volumetric fraction of bubbles occupying the bubble street region at any instant; it can be evaluated from the following equation

	Eq. (50)	fB =	nVB
			π R 2U	A
			i

If the bubble frequency, bubble diameter, and bubble velocity are known, the solids mixing rate can be calculated.

The mechanistic model developed in the last section is applied to the data collected experimentally. Bubble diameter and bubble velocity calculations were based on the empirical equations obtained from frame-by- frame analysis of high-speed motion pictures taken under the respective operating conditions (Yang et al., 1984c). The equations used are:

For 0.254 m jet nozzle assembly

Eq. (51)	Bubble diameter	DB =12 .36 × G 0.155
Eq. (52)	Bubble velocity	U A = 0.711
				gDB
For 0.406 m jet nozzle assembly
Eq. (53)	Bubble diameter	DB = 0.0195× G0.620
Eq. (54)	Bubble velocity	U A = 0.35
			gD

In correlating the data, the solid exchange rate between the two regions, Wz, was assumed to be constant. The tracer concentration data were analyzed statistically and the solids circulation rates are reported in Table 2. The positive fluxes indicate that the net solids flow is from bubble

Recirculating and Jetting Fluidized Beds 307

street to annular region. The net exchange fluxes do not seem to depend on the jet velocity and the bed configuration. The solids circulation rate depends on the jet velocity, however, because higher jet velocities generate larger bubbles. The circumferential area surrounding the bubble street will then be larger and thus the solids circulation rate will be larger. The solids circulation rates derived on the basis of this model range from 47,500 to 73,400 kg/h, as shown in Table 2.

Comparison of the calculated and the experimentally observed tracer concentration profiles is good as shown in Figs. 38 through 42 for set points 3 employing the 0.254 m jet nozzle assembly.

The solids mixing study by injection of tracer particles indicated that the axial mixing of solids in the bubble street is apparently very fast. Radial mixing flux depends primarily on the bubble size, bubble velocity, and bubble frequency, which in turn depend on the size of the jet nozzle employed and the operating jet velocity.

Table 2. Statistical Analysis of Solids Mixing Data

Test No.	t0	Wz	2RiWz (H - J)
	s	kg/m2 s	kg/s
0.254 m Jet Nozzle Assembly
Set Pt. 1	8.3	1.0	13.2
Set Pt. 2	44.5	1.3	16.2
Set Pt. 3	10.2	1.3	17.7
Set Pt. 4	- 10.2	1.0	13.4
0.406 m Jet Nozzle Assembly
Set Pt. 1	- 1.2	1.0	14.7
Set Pt. 2	- 8.6	0.84	20.4

308 Fluidization, Solids Handling, and Processing

Solid Entrainment Rate into Gas and Gas-Solid Two-Phase Jets.

A mathematical model for solid entrainment into a permanent flamelike jet in a fluidized bed was proposed by Yang and Keairns (1982). The model was supplemented by particle velocity data obtained by following movies frame by frame in a motion analyzer. The experiments were performed at three nominal jet velocities (35, 48, and 63 m/s) and with solid loadings ranging from 0 to 2.75. The particle entrainment velocity into the jet was found to increase with increases in distance from the jet nozzle, to increase with increases in jet velocity, and to decrease with increases in solid loading in the gas-solid, two-phase jet.

It is well known that jets in a fluidized bed induce high solids mixing. In one extreme, jets can be permanent and flamelike, similar to jets in spouted beds. The solids circulation in this case is created by solids entrainment into the jet along the jet height. Studies of solids circulation in spouted beds have been reviewed by Mathur and Epstein (1974). Data on solids entrainment into a permanent flamelike jet immersed in a fluidized bed, however, are meager. In another extreme, jets can be a series of rapidly coalescing bubbles, called bubbling jets. Solids mixing in this case is induced essentially by the solids-carrying capacity of the bubble wake and by the bubble frequency.

Another kind of jet encountered in operating fluidized beds are those created by pneumatic transport of solid particles into the fluidized beds. Here we call them gas-solid, two-phase jets because the incoming jet streams have already entrained solid particles at different loadings. The momentum of these solid particles is not negligible, as already shown by Yang and Kearins (1980). With high-speed movies such particles can usually be seen to penetrate right through the roof of coalescing bubbles in the bubbling jet regime.

Regular and high-speed movies were taken of the tracer particle movement around the jets at different velocities and different solid loadings. The tracer particles used are red plastic pellets of similar size and density to the bed material. The movies were then analyzed frame by frame using a motion analyzer to record the particle trajectories and the particle velocities.

Typical particle trajectories observed in the movies are shown in Fig. 47 for a jet velocity of 62.5 m/s and a solid loading of 1.52. The time elapsed between dots shown in Fig. 47 was typically 5 movie frames, while the movie speed was 24 frames/s. The colored tracer particles were followed in the vicinity of the jet until they disappeared into the jet, as

310 Fluidization, Solids Handling, and Processing

The average particle velocity in each trajectory was determined by dividing the distance traveled by the particle with the total time elapsed. The particle velocity into the jet increased with the distance from the jet nozzle, decreased with increases in solid loading, and increased with increases in jet velocity. High-speed movies indicated that the entrained particles tended to bounce back to the jet boundary more readily under high solid loading conditions. This may explain why the entrainment rate decreases with increases in solid loading in a two-phase jet. A ready analogy is the relative difficulty in merging into a rush-hour traffic as compared to merging into a light traffic. The dependence of particle velocity on the distance from the jet nozzle is presented in Fig. 48 for a jet velocity of 62.5 m/s and a solid loading of 1.52. The regressional analysis of this dependence at different operating conditions is summarized in Table 3.

Figure 48. Experimental solid velocity into the jet.

Recirculating and Jetting Fluidized Beds 311

Table 3. Summary of Experimental Solids Velocity into the Jet at Different Operating Conditions

______________________________________________________________________________

Run No. Nominal	Solid Loading	Regressional Relationship	Correlation
Jet Vel.	Wt. Solid/Wt. Gas	w.r.t. Dist. from Jet Nozzle	Coefficient
(m/s)		(Vz = cm/s; Z = cm)

______________________________________________________________________________

GSF-1	1.21	0	Vz = 0.0788 Z + 4.707*	0.8158
GSF-3	1.21	0.40	Vz = 0.0688 Z + 5.265	0.4433
GSF-4	1.21	0.92	Vz = 0.0842 Z + 2.979	0.8959
GSF-5	1.21	1.52	Vz = 0.0765 Z + 2.692	0.9085
GSF-22	1.20	0	Vz = 0.1811 Z + 3.119	0.8705
GSF-23	1.20	0.51	Vz = 0.1212 Z + 3.597	0.8439
GSF-24	1.20	1.21	Vz = 0.1839 Z + 2.228	0.8953
GSF-25	1.20	1.99	Vz = 0.1115 Z + 2.000	0.9593
GSF-44	1.19	0	——	——
GSF-45	1.19	0.71	——	——
GSF-46	1.19	1.67	——	——
GSF-47	1.19	2.75	Vz = 0.0967 Z + 2.576	0.8750

______________________________________________________________________________

* Vz = solid particle velocity, cm/s; Z = distance from jet nozzle, cm

Model for Solids Entrainment into a Permanent Flamelike Jet. A simple model for solids entrainment into a permanent flamelike jet is described here. The jet is assumed to expand at an angle 2θ as shown in Fig. 49, where θ is commonly known as the jet half-angle. Although the existence of a jet half-angle for the jet in a fluidized bed is not universally accepted, employment of this concept considerably simplifies the development of the model. The concept may also be applicable to a bubbling jet (Anagbo, 1980). Material balance of solid particles in a differential element inside the jet gives

Eq. (55)	Wj + dWj -Wj = Vjz (2πr ×dz )(1 -ε z )ρs
or
Eq. (56)	dWj =Vjz ρs (1 -ε z )2πr ×dz

312 Fluidization, Solids Handling, and Processing

where Wj = solids circulation rate, Vjz = horizontal component of the particle velocity into the jet at z and ε z = voidage of emulsion phase at z.

Figure 49. Schematic of jet for solids entrainment model development.

Through geometric consideration, we find that

Eq. (57)	r = z × tanθ +	do
	2

where do is the diameter of the jet nozzle.

Recirculating and Jetting Fluidized Beds 313

Substituting Eq. (57) into Eq. (56), we have

					(1 - ε		)V	æ	d	ö
Eq. (58)	dW	j	= 2πρ	s		z		ç z × tanθ +		o	÷dz
								jz ç		÷
								è	2 ø

The overall entrainment rate into the jet or the rate of solid circulation induced by the jet is then

			= 2π ρ (1 - ε		)	Lj		æ	d	ö
Eq. (59)	W	j		z		ò	V	ç z × tanθ +		o	÷dz
			s					jz ç		÷
						0		è	2 ø

where Lj is the jet penetration depth and ε z is assumed to be independent of jet height. We found experimentally that the particle velocity into the jet, Vjz, is linearly dependent on the jet height, as expressed below:

Eq. (60)

Vjz = C1 z + C2

where C1 and C2 are two empirical constants. Some of those constants at different jetting conditions were reported in Table 3. Substituting Eq. (60) into Eq. (59) and integrating, we have

Eq. (61)

éC tanθ

1

æ C d

o

ö

C

d

o

ù

W

j

= 2πρ

(1 -ε

)

1

L3

+

ç

1

+ C

2

tanθ ÷L2

+

2

L

j ú

s

z ê

3

j

2

ç

2

÷

j

2

ë

è

ø

û

The correlation for the jet penetration depth, Lj , was developed earlier as shown in Eq. (23).

The solids entrainment rate into a jet in a fluidized bed can be calculated from Eqs. (61) and (23) if the empirical constants C1 and C2 and the jet half-angle θ are known. The jet half-angle θ can be taken to be 10° as suggested by Anagbo (1980), a value very close to 7.5° obtained from solid particle trajectories reported here. The real jet half-angle will be larger than 7.5° because of the truncation of the jet by the front plate of the semicircular bed.

The particle velocity can usually be approximated as shown in Eq. (27). When the jet velocity is low or the bed particles are relatively fine or of wide size distribution, the jet tends to be a bubbling jet. A separate model

314 Fluidization, Solids Handling, and Processing

for solids circulation is required. The present model can be used as a first approximation.

The model as formulated in this section cannot be used to predict a priori the solids entrainment rate into the jet because of the two empirical constants in Eq. (61). Lefroy and Davidson (1969) have developed a theoretical model based on a particle collision mechanism for entrainment of solid particles into a jet. The resulting equation for particle entrainment velocity is

		V	=	π 2e(1 + e)d p
	Eq. (62)	e
		V j		16r

assuming the dp /2r <<1.

The relationship between Ve and Vjz can be expressed as:

Eq. (63)	V		=	(1 - ε j		)	×	(ε j -εz )	×V
		jz
			(1		- εz	) (1 -ε z )			e

The first term, (1 - ε j) /(1 - ε z), corrects for the voidage difference between that in the jet and that in the emulsion phase. The second term, ( ε j - ε z) /(1 -ε z), takes into account the fact that only a fraction of the particles having the entrainment velocity Ve will be entrained, the remainder rebounding back to the jet wall due to collisions with the particles already in the jet. Substituting Eq. (63) into Eq. (62), we have

(1 - ε

j

) (ε

j

-ε

z

) π 2e(1 + e)d

p

Eq. (64)

V jz

@

×

×

×V j

16r

(1 -ε z ) (1 - εz )

Equation (64) predicts correctly the increase in solid entrainment into the jet with increases in jet velocity and the decrease with increases in solid loading in a two-phase jet. Since neither the voidage nor the particle velocity inside the jet were measured, direct verification of Eq. (64) was not performed.

The validity of extrapolating the data obtained in a semicircular model to a circular one is also of concern. Not much research was carried out in this area. Preliminary research results by Whiting and Geldart (1980) indicated that, for coarse, spoutable solids similar to the particles