Yang Fluidization, Solids Handling, and Processing

Fluidized Bed Scale-up

55

is sufficient for “macroscopic similarity” and sacrifices only cluster size similarity. This reduced law is identical to the bubbling bed scaling law developed by Horio (1986) and has been shown by Glicksman (1988) to be equivalent to the viscous limit scaling law presented in the previous section. Horio’s suggestion that the density ratio can be neglected because sacrificing cluster size does not necessarily alter the macroscopic behavior of a circulating fluidized bed is somewhat curious since his development is based on the CAFM which assumes clusters are the primary mechanism of particle transport.

From an operational standpoint, the paramount question for a designer is which scaling law must be followed. Since Horio’s reduced set is equivalent to the viscous limit, Eq. (66), it must be determined if the density ratio can be omitted in the scaling relationship, i.e., is the viscous limit an adequate set of scaling parameters or must the density ratio also be included as is done in the simplified scaling relationships, Eq. (53). As will be shown in a later section, density ratio is not a parameter which can be omitted when scaling circulating fluidized beds and the density ratio is generally required for scaling bubbling beds.

6.3Deterministic Chaos

Recent studies have indicated that fluidized beds may be deterministic chaotic systems (Daw et al.,1990; Daw and Harlow, 1991; Schouten and van den Bleek, 1991; van den Bleek and Schouten, 1993). Such systems are characterized by a limited ability to predict their evolution with time. If fluidized beds are deterministic chaotic systems, the scaling laws should reflect the restricted predictability associated with such systems.

Van den Bleek and Schouten (1993) have suggested that if two beds are properly scaled, the rate of information change in both systems will be the same. They suggest that two scaled beds will exhibit the same Kolmogorov entropy, or information generation rate, when measured on the same time scale. Hence, they adopt an additional method for verifying dynamic similarity requiring that the information group, Kt, remain constant (where t = dp /uo and K is the Kolmogorov entropy) along with the full set of scaling parameters we have described.

Further work is needed to determine in which regimes, if any, fluid bed behave as chaotic systems. Additional testing is needed to determine the sensitivity of important bed hydrodynamic characteristics to the Kolmogorov entropy, to quantitatively relate changes of entropy to

56 Fluidization, Solids Handling, and Processing

shifts in hydrodynamic behavior. It may be that the requirement of Kt equality may be an over-restrictive condition, that is, similarity of hydrodynamic characteristics such as solid mixing and gas-to-solid transfer may be achieved even when Kt varies considerably.

7.0DESIGN OF SCALE MODELS

7.1Full Set of Scaling Relationships

We will first consider the steps to design a model which is similar to another bed based on the full set of scaling parameters, Eqs. (37) or (39).

To construct a model which will give behavior similar to another bed, for example, a commercial bed, all of the dimensionless parameters listed in Eqs. (37) or (39) must have the same value for the two beds. The requirements of similar bed geometry is met by use of geometrically similar beds; the ratio of all linear bed dimensions to a reference dimension such as the bed diameter must be the same for the model and the commercial bed. This includes the dimensions of the bed internals. The dimensions of elements external to the bed such as the particle return loop do not have to be matched as long as the return loop is designed to provide the proper external solids flow rate and size distribution and solid or gas flow fluctuations in the return loop do not influence the riser behavior (Rhodes and Laussman, 1992).

Proper conditions must be chosen to design a scale model to match the dimensionless parameters of the commercial bed. To model a gas fluidized commercial bed, a scale model using air at standard conditions is most convenient, although several investigators have used other gases (Fitzgerald and Crane, 1980; Fitzgerald et al., 1984; Chang and Louge, 1992) or pressurized scale models (Almstedt and Zakkay, 1990; Di Felice et al., 1992 a,b). The gas chosen for the model, along with the gas pressure and temperature, determines the values of ρf and μ. The particle density for the model is chosen to match the density ratio, so that

	æ	ρ	f	ö		æ	ρ	f	ö
Eq. (67)	ç			÷	=	ç			÷
	ç	ρ		÷		ç	ρ		÷
	è		s	øm		è		s	øc

where the subscript m is for the model and c is for the commercial bed. For the remaining parameters, the form of Eq. (37) will be chosen for the

Fluidized Bed Scale-up

57

dimensionless parameters. Combining the Reynolds number based on bed diameter and the square root of the Froude number,

Eq. (68)

Rearranging,

Eq. (69)

ρf uo L

æ

3

ö

æ

3

ö

gL

=

ç

L2

g ÷

=

ç

L2

g ÷

μf

uo

ç

ν f

÷

ç

ν f

÷

è

øm

è

øc

					2
æ Lm ö			æ	(ν f )m ö			3
			= ç			÷
ç	÷			(
			ç		) ÷
è	Lc ø		è	ν f	c ø

All of the linear dimensions of the model are scaled to the corresponding dimensions of the commercial bed by the ratio of the kinematic viscosities of the gas raised to the two-thirds power. By taking the ratio of Reynolds number based on the particle diameter to Reynolds number based on the bed diameter

ρf uo L

μf

æ

ö

æ

ö

Eq. (70)

×

=

ç

L

÷

=

ç

L

÷

μ

ρ

u

d

f

f

o

p

ç d

÷

ç d

÷

è

p øm

è

p øc

The particle diameters in the model scale by the same factor as the bed diameter, by the ratio of the kinematic viscosities to the two-thirds power.

Equating the Froude number and rearranging,

	uom			1
Eq. (71)		=	æ	Lm ö2
			ç		÷
	uoc
			è	Lc ø

Thus, the velocity scales are the square root of the linear dimension scale. By satisfying both Eq. (69) and Eq. (71), the Reynolds number and the Froude numbers are kept identical between the model and the

commercial bed.

Combining Gs /ρs uo and the product of Reynolds and Froude number along with Eq. (69), it can be shown that

58 Fluidization, Solids Handling, and Processing

	æ Gs ö			æ		1
	ç	ρ ÷			ν f m	ö3
	è	s øm	=	ç		÷
					ν f c
Eq. (72)	æ Gs ö			ç		÷
	ç	ρ ÷		è		ø
	è	s ø
		c

so that the ratio of solids-flow to solids-density scales as the ratio of the cube root of the kinematic viscosity.

Once the model fluid and its pressure and temperature are chosen, which sets the gas density and viscosity, there is only one unique set of parameters for the model which gives similarity when using the full set of dimensionless parameters. The dependent variables, as nondimensionalized by Eq. (18), will be the same in the respective dimensionless time and spatial coordinates of the model as the commercial bed. The spatial variables are nondimensionalized by the bed diameter so that the dimensional and spatial coordinates of the model is proportional to the twothirds power of the kinematic viscosity, as given by Eq. (69)

		æ	2
	xm		ö3
Eq. (73)		= ç		ν fm	÷
	xc	ç	÷
		è		ν fc ø

Since the velocity scales with v1f 3 , the ratio of time scales can be expressed as

		æ	1
	tm		ö3
Eq. (74)		= ç	ν fm	÷
	tc	ç	÷
		è	ν f c ø

Similarly, it can be shown that the frequency scales as

					1
Eq. (75)	f	M		æ	ν fc ö3
			=	ç		÷
	fc			ç	÷
				èν fm ø

Fluidized Bed Scale-up

59

Table 1 gives the values of design and operating parameters of a scale model fluidized with air at ambient conditions which simulates the dynamics of an atmospheric fluidized bed combustor operating at 850oC. Fortunately, the linear dimensions of the model are much smaller, roughly one quarter those of the combustor. The particle density in the model must be much higher than the particle density in the combustor to maintain a constant value of the gas-to-solid density ratio. Note that the superficial velocity of the model differs from that of the combustor along with the spatial and temporal variables.

When modelling a pressurized hot bed (Table 2) the ambient temperature model fluidized with air has dimensions very close to those of the pressurized combustor. If another gas is used in the model, particularly a gas with a higher density, the model can be made much smaller than the pressurized combustor (see Table 3). Care must be taken to select a safe modelling gas and one which yields a solid density for the model which is available.

Table 1. Atmospheric Combustor Modelled by a Bed Fluidized with Air at Ambient Conditions

GIVEN:	Commercial Bed	Scale Model, full scaling laws
Temperature (°C)	850	25
Gas Viscosity (10-5kg/ms)	4.45	1.81
Density (kg/m3)	0.314	1.20
Derived from Scaling Laws:
Solid Density	ρ sc	3.82ρ sc
Bed Diameter, Length	D c	0.225D c
Particle Diameter	d pc	0.225d pc
Superficial Velocity	u oc	0.47u oc
Volumetric Solid Flux	(G s /r s ) c	0.47(G s /r s )c
Time	t c	0.47t c
Frequency	f c	2.13f c

60 Fluidization, Solids Handling, and Processing

Table 2. Pressurized Combustor Modelled by a Bed Fluidized with Air at Ambient Conditions

GIVEN:	Commercial Bed	Scale Model, full scaling laws
Temperature (°C)	850	20
Gas Viscosity (10-5kg/ms)	4.45	1.81
Density (kg/m3)	3.14	1.20
Pressure (bar)	10	1
Derived from Scaling Laws:
Solid Density	ρsc	0.382ρsc
Bed Diameter, length	D c	1.05D c
Particle Diameter	d pc	1.05 d pc
Superficial Velocity	u oc	1.01 u oc
Volumetric Solid Flux	(Gs/ρs)c	1.01(Gs/ρs)c
Time	t c	1.01t c
Frequency	f c	0.98f c

Table 3. Pressurized Combustor Modelled by a Bed Fluidized with Refrigerant Vapor 134a at Ambient Conditions

GIVEN:	Commercial Bed	Scale Model, full scaling laws
Temperature (°C)	850	20
Gas Viscosity (10-5kg/ms)	4.45	1.19
Density (kg/m 3)	3.14	4.34
Pressure (bar)	10	1
Derived from Scaling Laws:
Solid Density	ρsc	1.38ρsc
Bed Diameter, length	D c	0.334D c
Particle Diameter	d pc	0.334 d pc
Superficial Velocity	u oc	0.58 u oc
Volumetric Solid Flux	(Gs/ρs)c	0.58(Gs/ρs)c
Time	t c	0.58t c
Frequency	f c	1.7f c

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61

7.2Design of Scale Models Using the Simplified Set of Scaling Relationships

The simplified scaling relationships, Eq. (53), offer some flexibility in the model design since fewer parameters must be matched than with the full set of scaling relationships. When the fluidizing gas, the pressure and temperature of the scale model are chosen, the gas density and viscosity for the scale model are set. The model must still be geometrically similar to the commercial bed. There is still one free parameter. Generally this will be the linear scale of the model. For the simplified scaling relationships, the gas-to-solid density ratio must be maintained constant

æ ρ

Eq. (76) ç f

çè ρs

ö	æ	ρ	ö
÷	= ç		f	÷
		ρ
÷	ç		÷
øm	è		s øc

With ρf of the model set by the fluidized gas and its state, the solid density in the model follows from Eq. (76). Choosing the length coordinate of the model, Lm , which is now a free parameter, the superficial velocity in the model is determined so that the Froude number remains the same,

Eq. (77)

so that

Eq. (78)

æ	2	ö		æ	2	ö
ç	uo	÷	=	ç	uo	÷
ç		÷		ç		÷
è gL ø				è gL ø
		m				c

uom	=	æ	Lm ö1/2
uoc		ç		÷
		è	Lc ø

Note that in simplified case, the velocity scaling is not uniquely tied to just the gas properties as it is in the full scaling relationship. With uo and ρs set, the solids recycle rate can be determined by

	æ	Gs	ö	æ	Gs	ö
Eq. (79)	ç		÷	= ç		÷
	è	ρs uo øm		è	ρ s uo øc

62 Fluidization, Solids Handling, and Processing

	Eq. (80)	Gsm
		Gsc

=æç ρsm

è ρsc

ö æuom ö			æ ρfm
÷ ç		÷	= ç
ø è uoc ø			ç	ρ fc
			è

1

ö÷ æç Lm ö÷2 ÷ø è Lc ø

Finally the mean particle size for the model as well as the sphericity and particle size distribution must be determined. The particle size is determined by the need for equal values of uo /umf between the model and the commercial bed.

Eq. (81)

Eq. (82)

	æ	uo	ö	æ	uo	ö
	ç		÷	= ç		÷
	ç		÷	ç		÷
	è umf øm			è umf øc

(u

)

= (

)

æ

uom

ö

= (

)

æ

Lm

ö1/2

ç

mf m

umf

c

ç u

÷

umf

c

÷

è

oc ø

è

Lc ø

In general, umf is a function of the particle diameter and gas properties, as well as φ and εmf . Once the fluidizing gas and the length of scale of the model is chosen, the proper particle diameter is that which gives the value of umf needed in Eq. (82).

If both the model and commercial bed are in the region where the respective Reynolds numbers based on particle diameter and gas density are very low, then a single algebraic relationship can be developed. In that region

Eq. (83)

Eq. (84)

umf			~		ρs d2p
						μ
umf		m				æ dpm ö2					ρ
				=		ç				÷	sm
u											ρ
	mfc					ç d		÷
						è		pc ø			sc

μc				1
		uom	æ	Lm ö2
	=		= ç		÷
μm		uoc
			è	Lc ø

Eq. (85)

Eq. (86)

æ	ö2						ρ
ç		dpm	÷		=		sc
ç	÷						ρsm
è d pc ø
æ dpm ö2							ρf	c
ç				÷	=
ç	÷						ρfm
è		dpc ø

Fluidized Bed Scale-up

63

1

μm æç Lm ö÷2

μc è Lc ø

μm æ Lm ö1/2

μc çè Lc ÷ø

when both (Redp)m and (Redp)c < 20.

When the Reynolds number of the model and commercial bed are both very large

Eq. (87)	umf2 ~ dp	ρs
		ρ	f

Since the gas-to-solid density ratio of the model and the commercial beds must be the same to satisfy the simplified scaling relationships, Eq. (87) combined with Eqs. (81) and (78) becomes,

æ

ρ

ö

ç

s

÷

2

ç

ρ

f

÷

2

Lm

umfm

=

dpm è

øm

=

uom

=

Eq. (88)

u2

dpc

æ

ρ

ö

uoc2

Lc

mfc

ç

s

÷

ç

ρf

÷

è

ø

c

When both (Redp)m and (Redp)c > 1000

Eq. (89)

dpm

= Lm

dpc

Lc

Tables 4 and 5 show the values of the mean particle diameter for models of an atmospheric and pressurized commercial bed, respectively, for different selected linear scale ratios between the model and commercial bed.

By the use of the simplified scaling parameters, the linear scale factor can be changed as shown in Tables 4 and 5. Note that as the linear

64 Fluidization, Solids Handling, and Processing

scale factors are changed the particle diameters change much more slowly. The scale model of the 10-atmospheres bed has a mean particle diameter which is quite close to the mean particle diameter of the commercial bed. The model particles have a substantially lower density in this case.

It is not clear where cohesive forces will become important. The use of very dense particles (for the models of the one atmospheric bed) will cause a shift of the boundary of cohesive influence as given, for example, by Geldart’s classification. However, adequate experimental data is still lacking with such dense fine particles to definitely set the limits of cohesive influence.

Note that for completeness, the nondimensional particle size distribution, sphericity and the internal angle of friction (for slugging and spouting beds) should also be matched between the two beds.

Table 4. Scale Models of Atmospheric Commercial Hot Bed Using the Simplified Scaling Relationship

Commercial Beds		Particle Diameter of Model with
		Bed Linear Scale Factor
dp	umf	Lm/Lc = 1/4	Lm/Lc = 1/9
40 μm	7.45 × 10-4 m/s	10 μm	8 μm
60	1.68 × 10-3	15	12
100	4.66 × 10-3	24	20
200	1.86 × 10-2	49	40
400	7.42 × 10-2	98	80
1000	0.441	245	198

Commercial Bed: T = 800oC, P = 1 atm, ρs = 2500 kg/m3, gas:air Model Bed: ρs = 8960 kg/m3, gas:air at STP