Yang Fluidization, Solids Handling, and Processing

Heat Transfer in Fluidized Beds 185

The data of Fig. 20 also illustrates one of the most significant parametric effects, that of the solid concentration in the particle/gas suspension. Many of the parametric effects mentioned above (e.g., due to elevation, gas velocity, solid mass flux) can be attributed to their individual impact on solid concentration in the fast fluidized bed. Thus increasing gas velocity in order to increase particle entrainment to upper regions of the bed, or increasing solid mass flux to increase solid holdup within the bed, lead to higher solid concentrations and greater suspension densities—which correlates with higher heat transfer coefficients. This is illustrated by the data of Dou et al. in Fig. 20, which shows that for a given radial position (centerline or wall) the heat transfer coefficient correlates with cross-sectional-aver- aged suspension density. The increase of heat transfer coefficient with increasing average density is seen to be much more pronounced at the vessel wall than at the surface of a tube submerged at the bed centerline. It should be noted that these results include data obtained over a range of axial elevations, indicating that this latter parameter is secondary, influencing the heat transfer coefficient only through the effect of axial variations in suspension density.

The data of Fig. 20 also point out an interesting phenomenon—while the heat transfer coefficients at bed wall and bed centerline both correlate with suspension density, their correlations are quantitatively different. This strongly suggests that the cross-sectional solid concentration is an important, but not primary parameter. Dou et al. speculated that the difference may be attributed to variations in the local solid concentration across the diameter of the fast fluidized bed. They show that when the cross-sectional averaged density is modified by an empirical radial distribution to obtain local suspension densities, the heat transfer coefficient indeed than correlates as a single function with local suspension density. This is shown in Fig. 21 where the two sets of data for different radial positions now correlate as a single function with local mixture density. The conclusion is that the convective heat transfer coefficient for surfaces in a fast fluidized bed is determined primarily by the local two-phase mixture density (solid concentration) at the location of that surface, for any given type of particle. The early observed parametric effects of elevation, gas velocity, solid mass flux, and radial position are all secondary to this primary functional dependence.

The parametric effect of system pressure on the heat transfer coefficient was studied by Wirth (1995). They obtained experimental measurements of the heat transfer Nusselt number for fast fluidized beds

Heat Transfer in Fluidized Beds 187

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190 Fluidization, Solids Handling, and Processing

phase and the dense-particle phase plus thermal radiation, as expressed by Eq. (3). Many of the models combine individual phase convection contributions into a single convective coefficient which we can define as,

Eq. (16)

h c = fl hl + (1 - f l )hd

All correlations based on ambient temperature data where thermal radiation is negligible should be considered to represent only the convective heat transfer coefficient hc.

The simplest correlations are of the form shown by Eq. (15), in attempts to recognize the strong influence of solid concentration (i.e., suspension density) on the convective heat transfer coefficient. Some examples of this type of correlation, for heat transfer at vertical wall of fast fluidized beds are:

Wen and Miller (1961):

	Eq. (17)	hc d p
		kg

æ	C ps ö æ		ρ susp ö0.3
= ç		÷ ç		÷
ç	÷ ç		÷
èC pg ø è			ρs ø

æ	u	2	ö 0.21
ç		t	÷	Prg
ç g d p ÷
è			ø

where ut = terminal velocity of particle.

Kiang et al. (1976):

hc d p

Eq. (18)

k g

	æ	C ps	ö b	æ
	= ε a ç		÷	ç
		C pg
	s ç		÷	ç
	è		ø	è

ρ	susp	öc	æ	D	b	ρ	g	u	g	ö d
		÷	ç							÷	Prge
ρ s		÷	ç			μ g			÷
		ø	è						ø

where a, b, c, d, e are empirical constants.

Fraley et al. (1983):

Eq. (19)	(	)0.90	,	SI Units
	hc = 2.87 ρsusp
Divilio and Boyd (1993):
Eq. (20)	h c = 23.2 (ρsusp)0.55			, SI Units

Heat Transfer in Fluidized Beds 191

Werdmann and Werther (1993):

hc d p

æ

D

ρ

u

ö

0.757

(

)0.562

-4

ç

b

g

g ÷

Eq. (21)

= 7.46 ´10

ç

÷

ρsusp

kg

ρ g

è

ø

It should be noted that the suspension density (ρsusp) used in above correlations represent the cross-sectional-averaged value. Since Dou et al.

(1991) have shown that heat transfer coefficient is primarily dependent on local solid concentration in the vicinity of the surface (rather than cross- sectional-averaged concentration), the above correlations for convective coefficient at the wall of FFBs can be justified only if there is a generic similarity profile in solid concentration relating ε s at the wall to cross- sectional-averaged ε s. Further study of Eqs. (17–18) indicate that there are significant uncertainties with regard to the fundamental nature of appropriate heat transfer correlations. For example, such parameters as particle size (dp), bed diameter (Db), particle terminal velocity (ut), heat capacity of solid (Cps), and gas thermal properties (ρg , kg , μg , Cpg) are incorporated in some correlations and not in others. Obviously, each of these empirical correlations is useful only for the range of parameters covered by the correlation’s experimental data base.

In contrast to the purely empirical correlations, various authors have attempted to develop mechanistically based models for each of the contributions from particle convection, gas convection and thermal radiation reflected in Eq. (3). Of these, it is generally acknowledged that the dense phase convection (represented by hd) is the dominant contributor. The lean phase gas convection (represented by hl) is often thought to be small or negligible. The radiative contribution (represented by hr) is acknowledged to be significant for high temperature fluidized beds, at temperatures greater than 500°C. Various models proposed for each of these contributing mechanisms are briefly summarized below.

The lean/gas phase convection contribution has received the least attention in the literature. Many models in fact assume it to be negligible in comparison to dense phase convection and set hl to be zero. Compared to experimental data, such an approach appears to be approximately valid for fast fluidized beds where average solid concentration is above 8% by volume. Measurements obtained by Ebert, Glicksman and Lints (1993) indicate that the lean phase convection can contribute up to 20% of total

192 Fluidization, Solids Handling, and Processing

convective heat transfer when the average solid concentration becomes less than 3%, as is often found in upper regions of fast fluidized beds. A better approach is to approximate the lean phase convection by some equation for single phase gas convection, e.g., the Dittus-Boelter correlation,

	hl Db		æ	Db	ρ	g	u	g	ö	0.8	)0.33
		= 0.023	ç						÷	(
Eq. (22)	k g		ç		μ				÷	Prg
			è			g			ø

Some researchers have noted that this approach tends to underestimate the lean phase convection since solid particles dispersed in the up-flowing gas would cause enhancement of the lean phase convective heat transfer coefficient. Lints (1992) suggest that this enhancement can be partially taken into account by increasing the gas thermal conductivity by a factor of 1.1. It should also be noted that in accordance with Eq. (3), the lean phase heat transfer coefficient (hl) should only be applied to that fraction of the wall surface, or fraction of time at a given spot on the wall, which is not submerged in the dense/particle phase. This approach, therefore, requires an additional determination of the parameter fl , to be discussed below.

The contribution of dense phase convection is acknowledged to be most important and has received the greatest attention from researchers. Models for prediction of hd have followed two different, competing concepts. The first type of models considers the heat transfer surface to be contacted periodically by clusters or strands of closely packed particles. Similar to the packet model for dense bubbling beds, this approach leads to a surface renewal process whereby the heat transfer occurs primarily by transient conduction between the heat transfer surface and the particle clusters during their time of residence at the surface. Equation (4), from Mickley and Fairbanks (1955), can be used here to estimate the effective cluster heat transfer coefficient by treating clusters in FFBs as the equivalent of particle packets in bubbling beds. Experimental evidence supporting this concept is reported by Dou, Herb, Tuzla and Chen (1992). They obtained simultaneous measurements of the dynamic variations of solid concentration and instantaneous heat transfer coefficient at the wall of a fast fluidized bed, and found significant correlation between the two—indicating a direct dependence of the dense phase heat transfer on cluster contact. Application of this cluster type model requires information on solid concentration in the cluster (ε s ,cl ), average residence time of

Heat Transfer in Fluidized Beds 193

cluster on the heat transfer surface (τcl ), and the effective thermal conductivity of the cluster (kcl ). Figure 15 shows sample measurements of solid concentration in clusters as reported by Soong, Tuzla and Chen (1993) at various locations within the fast fluidized bed. Other measurements of ε s ,cl have been reported by Wu (1989), Louge et al., (1990), and Lints (1993). Lints report that the average solids concentration in the cluster can be correlated with the cross-sectional-averaged solid concentration across the bed,

Eq. (23)	ε s ,cl	= 1.23(	)0.54
		ε s,b

The average cluster residence time (τcl) could be correlated directly with operating parameters (e.g., ug , dp , Gs , H) if sufficient experimental data on hydrodynamics were available. Alternately one can assume that the clusters of finite length, Lcl , fall down over the heat transfer surface with a velocity Ucl . For any given position on the heat transfer surface, the cluster contact time would just be the ratio Lcl /Ucl . Experimental measurements of cluster velocity and length have been reported by Hartge et al., (1986), Horio et al., (1988), Lints (1992), and Soong et al. (1995). Figure 25 shows sample data reported by Soong et al. indicating that near FFB wall, cluster velocity are in the range of 1–5 m/s downward, and cluster length are in the range 0.5–4.5 cm. While there is general agreement about the range of values for these two parameters, there are inadequate data to develop general correlations. In the absence of such correlations, Lints and Glicksman (1993) suggest that an average downward velocity of 1 m/s be assumed for the clusters, and that the vertical length of the heat transfer surface be substituted for the cluster length. Clearly this will require additional research and analysis.

Finally, to utilize this cluster concept in modeling the heat transfer, one needs to determine the fraction of time that a spot on the heat transfer surface is covered by clusters, which can be assumed to be equal to the fractional of wall coverage by clusters at any given time. Sample measurements of fl have been obtained by Wu (1989), Rhodes et al., (1991), Louge et al., (1990), Rhodes et al., (1991), and Dou et al. (1993). Lints and Glicksman present a combined plot of some of these data, reproduced here in Fig. 26. They suggest an approximate correlation for the fractional wall coverage in terms of the cross-section-average solids concentration,

Eq. (24)	(1 - f	l	) =	3.5 (	)0.37 , at wall
				ε s, b

	(ww)
(s
/w)	Hl~N3l
AlI:)013A
CJ31Snl:)	'd31Snl~

oc -'

Processing and Handling, Solids Fluidization, 194

Figure 25. Sample data on cluster velocities and length. (From Soong, Tuzla and Chen, 1995.)