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Wall-to-bed heat transfer in gas—solid fluidized beds: Prediction of heat transfer regimes
223
Powder Technology, 69 (1992) 223-230
Wall-to-bed heat . transfer in gas-solid fluidized beds: prediction heat transfer regimes G. Flamant,
of
N. Fatah
Institut de Science et Ginnie des Mat&au
et Pro&d&
CNRS, BP No 5, 66120 Odeillo (France)
and Y. Flitris Universiv of Thessaly, Div. of Physics, Athens av. Pedio Areos, Voles 38334 (Greece) (Received
April 24, 1990; in revised form September
26, 1991)
Abstract When studying heat transfer between a fluidized bed and immersed surfaces, three modes of heat transfer can occur, VIZ. conduction (particle convection), gas convection and radiation. A classical definition of the dominant mechanisms is the following: (1) particle convection dominant: d, < 600 pm; (2) radiation significant: T,, > 900 K, (3) gas convection significant: dP> 600 pm. A new definition of the ‘heat transfer regimes’ is presented in a particle diameter vs. temperature diagram. The discussion on the transition between the various ‘regimes’ is based upon (i) the Saxena and Ganzha numbers related to the ratios classification scheme combined with Ergun’s equation, and (ii) dimensionless between conduction and radiation and between convection and radiation in porous media.
Introduction
Dynamics and heat transfer are interdependent in fluidized beds and, from this point of view, particle size and bed temperature are the more pertinent parameters for a classification of fluidization regimes. Nevertheless, classical schemes of powder classification are based only on particle diameter, whereas both particle diameter and temperature must be considered because of the marked variations in gas properties (p, and Pi) with temperature. The well-known Geldart classification [l] was proposed by considering the fluidization behavior of particles at ambient conditions. For example the transition between group B and D particles occurs at (p,-p,) > 4 000 kg mm3 and d,> 600 pm. The effect of temperature on the transition between regimes is not well understood and this approach is not appropriate for heat transfer considerations. In contrast, Decker and Glicksman [2] suggested a heat transfer-based classification. Their analysis is based on the comparison between the thermal time constant of the particles (T) and the mean residence time of the solid at the walls (&). For small particles, t,~- T, d, <400 ,um; heat transfer is dominated by particle convection (conduction through a layer of more than one particle depth). For large particles, t, s T, d, > 1500
0032-5910/92/$5.00
pm, the heat transfer is to a large extent due to gas convection, and the temperature decrease of particles adjacent to the wall is negligible as a first approximation. For intermediate particles both conduction and convection must be considered. This classification scheme does not account for the effect of temperature on the heat transfer mechanisms, and it cannot be used for high temperature design since the variation of the transition between the three groups is not clearly presented. The previous discussion deals with the variation of conductive (particle convection) and gas convective heat transfer. Concerning radiative heat transfer, combined with the two other modes of heat transfer, theoretical [3] and experimental results [4] are in good agreement for suggesting radiation to be a significant heat transfer mode for temperatures higher than 800 K for either small or large particles. Finally, when considering the classifications of Geldart and Decker, Glicksman, and the latter conclusion, one may propose a very simple heat transfer regime diagram as shown in Fig. 1. However, this diagram does not take into account how gas properties vary with temperature. This paper suggests another way for classifying dominant heat transfer modes as a function of temperature
0 1992 - Elsevier Sequoia. All rights reserved
224
The purpose of this paper is not to discuss the physical existence of the gas film. We assume that there exists a resistance to heat transfer (Rf) between the wall and the emulsion phase which may be regarded as an equivalent gas film of depth 8,. The transient conduction through this film is fast and the temperature profile is constant during the heat transfer. As a consequence, R, does not vary with time. The existence of the gas film does not change the conclusions on the heat transfer mode in the emulsion, which must be considered as a function of T and d,.
6
Particle convection and gas convection 0
1
3
2
4
Fig. 1. Heat transfer diagram based on ‘classical’ considerations. (1) Particle convection, (2) particle convection + gas convection, (3) convection, (4) particle convection + radiation, (5) particle convection + gas convection + radiation, (6) gas convection + radiation.
and particle diameter. We examine successively particle convection and gas convection, particle convection and radiation, particle convection + gas convection and radiation. The analysis is based on characteristic dimensionless numbers. The heat transfer component due to the bubble phase (being small in comparison with the exchange due to the emulsion) will be neglected as a first approximation.
Heat transfer mechanisms Assuming the additivity of the heat transfer components in the emulsion, i.e. the particle convection (h,), gas convection (h,) and radiation (h,), the total heat transfer coefficient may be written as: h,=h,+h,+h,
(I)
In eqn. (l), particle convection contribution h, may be expressed as: 1 -= h,
1 + _L = &+ h, h,
; c
(2)
The particle convection contribution is a result of two heat transfer mechanisms in series: steady state conduction through a gas film of depth 8, and transient conduction through the emulsion. As a consequence R, is given by: R,=d,lnA,
(3)
with 2
The Saxena and Ghanza [6] classification can be extended to include both hydrodynamics and heat transfer. For their group I, particle convection (conduction through emulsion) is the dominant mechanism of heat transfer, i.e. the fluid flow through the emulsion is laminar. This group is defined by: Re,, 1.6~ 106. According to this classification the bed heat transfer behavior changes drastically with temperature. For example, an air-fluidized bed of 1.4 mm diameter corundum particles is classified under group IIB at ambient conditions but under group I at 1173 K. As a result, gas convection may be neglected at high temperature but is significant at 300 K. Concerning particle convection (conduction through emulsion packet) and gas convection heat transfer, the analysis must be based on the value of Archimedes number at the transition between groups I and IIA. Mathur and Saxena, [6] recognized that the uniqueness of the limit Ar= 2.17X lo4 is not clearly established. Botterill et al. [S] suggested that the same transition occurs for Ar = 2.6 x 104, while Chen and Pei [9] proposed to take Ar=2X lo4 as the limit between fine and large particles. Previous propositions remain uncertain because they are based either on observations about the variations of the bed voidage at minimum fluidization with Reynolds and Archimedes numbers or on theoretical considerations. Recently Flamant et al. [lo] found the transition to be at Ar= 1.4 x lo4 on the basis of a comparison between experimental data measured with
225
corundum particles at high temperature (up to 1073 K) and theoretical results generated by the model of Flitris et al. [ll]. An illustration of the variation of these transitions in a temperature VS.particle diameter diagram is shown in Fig. 2 for corundum particles. According to this representation at ambient conditions, convection becomes significant when d,=500 and 600 pm for Ar = 1.4 x lo4 and 2.6 x lo4 respectively. At 1000 K this transition is observed when d,= 1.2 and 1.5 mm for the same values of Archimedes number. Another approach to the quantitative limits of particle convective and gas convective heat transfer mechanisms may be based on Ergun’s equation [12]: Arc
..,I
w
0.25 _
150(1- sIn3 Re + 1.75 Re2 #J55zIf mf $QCL Inf
(4)
The gas convective heat transfer is significant when the kinetic forces cannot be neglected. This phenomenon is observed when the second term of eqn. (3) becomes significant, i.e. when the minimum fluidization velocity increases with temperature. A U,,,, VS. T,, plot is shown in Fig. 3 in order to illustrate this comment. The transition between both domains is defined by: dU,,ldT=
0.75
0
(5)
This limit is plotted in Fig. 2 for two particle densities related to sand and corundum. A comparison between the curves in Figure 2 indicates very good agreement between both transitions obtained for Ar = 2.6 x lo4 and dU,,,,ldT=O. Equation (3) proves the sensitivity of the results with sphericity & and porosity .&. The variation of the
0.00 ~( 0.4
0,6
0,8
1.0
1.2
1.4
Ib’lo’K, Fig. 3. Variation of U,,,, with temperature on the basis of the Ergun equation for corundum, &,t=O.45, @,=0.8, 4=3950 kg mm3 (1) d,=2 mm, (2) d,=1.5 mm, (3) d,=l mm, (4) dr,= 500 pm. 3
Tb(lO'K) 1
l_ 3
Tb(lO'K)
I 0
I 1
I
I
I
2
1 3
dp(,o-‘m) Fig. 4. Influence of spheric&y and porosity on the particle convection-particle convection +gas convection transition (dU& dT=O), pp=2500 kg rnd3. A: particle convection, B: particle convection + gas convection. (1) OS= 0.5, (2) @,= 0.75, (3) QS= 1 (&=0.55); dotted hnes: (4) &=0.5, (5) &,,r=0.65 (+$I= 1).
0
1
2
3
Fig. 2. Limit between gas convection+particle convection and particle convection dominant heat transfer domains based on Archimede’s number and Ergun’s equation. A: particle convection, B: particle convection + gas convection. (1) Ar= 1.4x 104, (2) Ar=2.17~ 104; (3) Ar=2.6x104; dotted lines, dlJ,,,r/dT=O (&=0.5): (4) pr=3950 kg rnm3, (5) 4=2500 kg m-‘.
limit with & and &,r is illustrated for sand particles in Fig. 4. When the sphericity decreases, the surface area of the particle convection domain (dU,,,,ldT
the combined mechanism is characterized number NC: NC= AJWln2 uT3
by the Planck (6)
where A, is the effective thermal conductivity of the emulsion, K the extinction coefficient and n the refractive index. Radiation cannot be neglected when NC<5 and as a consequence, the boundary between both domains (particle convection and particle convection + radiation) may be defined by: NC=5 0
I
I
I
I
I
1
2
3
4
5
I 6
dpclo-‘m, Fig. 5. Limits between the three domains: (A) particle convection, (B) particle convectton +gas convection significant, (C) gas convection. (Bi: particle convection dominant, B,: gas convection dominant) (1) Ar=lAX 104, (2) Ar=1.3X10s, (3) Ar=1.6X 106; dotted lines: 4=3950 kg rnm3, dark lines: pr=2500 kg mm3.
heat transfer at 1000 K for particle diameters larger than 1 and 1.8 mm for &,,r=O.65 and 0.5 respectively. Thus the heat transfer is influenced by the variation of bed properties near the wall, in particular the emulsion voidage [13], and a definitive conclusion cannot be given. In any case the local porosity in the vicinity of the heat transfer surface &,, is greater than the meanvoidage at minimum fluidization conditions. Kubie [14] proposed an estimate of &, in the range 0< d/d,< 1 (where d is the distance from the wall). This analysis must be completed by the introduction of another limit: the transition between the domain where both particle conduction and gas convection are significant and the domain where gas convection is dominant. This limit was defined previously ::%n the basis of Archimedes number: Ar= 1.6X 106, shown in Fig. 5 for both corundum and sand particles. According to this representation, gas convection is dominant for 2.5
Particle convection
(7) Figure 6 illustrates this observation: the heat transfer coefficient becomes a nonlinear function of the temperature for NC< 5 and increases when the extinction coefficient (ZC)decreases. The limits NC= 5 (and NC= 0.5) are plotted in Fig. 7 with &,,I as parameter for air and corundum particles. The parameters A, and K have been estimated as follows [5, 161:
(8) 3 -,l-tnlr
K=
2
andn=l
4
(9)
Figure 7 indicates that radiation becomes significant at - 650 K for d, = 500 pm and &,,r= 0.5. For a given particle diameter the transition temperature decreases with increase in porosity. This estimation is in agreement with the two-flux model calculations proposed in [17] which indicate a radiative heat transfer coefficient ranging from 30 to 40 W mm2 K-’ for a bed temperature of 750 K (d,= 280 pm). On the other hand the ex-
1200
h,cwm-2K-'~
/
and radiation
This section is concerned with fixing a boundary for incipient radiation inside zone A previously defined in Figs. 2, 4 and 5. The combined conductive and radiative transient heat transfer inside a grey wall has been studied by Doomink and Hering [15] and the application of these general results to fluidized beds was proposed in [3]. The main results are the following: when radiation and conduction occur simultaneously inside an optically thick medium,
400
I_, 96
0.8
1.0
1.2
T~~Io’K?
Fig. 6. Influence of the combined conduction-radiation number N, on the heat transfer coefficient, after [3]; d,=300 pm. (1) lO
221
097
0
1
2
3
d,,c&m) Fig. 7. Particle convection-particle convection + radiation transition as a function of the porosity. A: particle convection, B: particle convectionfradiation N,=5, (1) &=0.5 (2) &,r= 0.65, (3) N,=OS, &=0.5.
perimental results obtained by Mathur and Saxena [18] show a significant radiative heat exchange for a 559 pm fluidized bed at 820 K: 28 W me2 K-l. More data are presented in a review of radiative and total heat transfer in gas fluidized beds proposed in [19]. According to [15], radiation is dominant for N< 0.005; our calculations show that radiant heat transfer is never dominant when combined with particle convection.
Particle convection + gas convection and radiation
091
OS1
093
0,5
0,7
099
Fig. 8. Effectwe emissivity of fluidized beds (.&=0.5) curves (1) [22]; (2) [23]; (3)[21]; (4) [24]; (5) [25]; (7’s= 1000 “C, (0) A1203 spheres, 2
Qr=
’
where cw and T,,, are the wall emissivity and the wall temperature respectively. Assuming a small difference between Tb and T,, and cw= 1, we have: Qr=4~,d’3(Tb-Tw)=h,(T,,-Tw)
Theoretical and experimental data dealing with gas convective and radiative heat transfer in fluidized beds are scarce. Chung and Welty [4] found that radiation is significant in a fluidized bed operating at 812 K: the radiative contribution reaches 8% at d,= 1 mm and 12% at d, =3 mm. On the other hand, a theoretical study [20] suggested a radiant contribution of about 33% in a large particle fluidized bed at 1 100 K. No general criterion was proposed in order to characterize the transition between the convection domain and the domain where both convection and radiation are significant. According to [2], in the domain where gas convection is dominant the temperature decrease of the particles adjacent to the wall is negligible. As a consequence the emulsion may be considered as a constant temperature medium whose emissivity is 4. The effective emissivity E, was defined by Brewster [21] on the basis of the two-flux radiative model. The results of this theoretical study are compared with other data in Fig. 8. The effective emissivity of a bed whose particle emissivity is 0.5 ranges between 0.7 and 0.8. Under the previous assumption, the radiative heat flux exchange between wall and bed may be expressed as follows (for T,< Tb):
t p
with F= (Tb + T,)/2. With this definition a convection/radiation number may be constructed as: N,, = (h, + h,)k
(11)
interaction
(12)
where the convective term accounts for particle and gas contribution. For large particle diameters, h, is roughly equal to hf (see discussion of heat transfer mechanisms) and we have:
(13) For calculations T = T,, is assumed. Radiation is significant for NPg= 10 and equal to convective heat transfer for NPe= 1. The main parameters of the convection/ radiation number NPgare the gas convective heat transfer coefficient h,, and the effective emissivity Q. Concerning h,, various correlations have been proposed in the literature. The model of Ganzha et al. [26] was shown to be adequate for powders in groups II and III [6], and it accounts for both convective contributions h, and h,:
228
Nu, = 8.95(1- t)“.667+ 0.12&O sP+‘43(l - &)o.133/p.8
3,2,
Tb(lO'K)
(14) with Nu,=Nu,+Nu, For Archimedes numbers in the range lo3
(15)
Recently Murachman [28] proposed a correlation on the basis of experimental results obtained at steady state near the distributor of gas-fluidized beds. Nu, = 0.03Re’ 4g
0
3
2
I
4
5
6
dp
(16)
Re<30
lc = 0.5, (2) .E= =0.75,
The comparison of correlations (14), (15) and (16) is illustrated in Fig. 9 for Nz= 1. N,* is a modified interaction number accounting only for gas convection, i.e. N,* = h,/h,. The extrapolation of the data for d, smaller than 1 mm is questionable (dashed zone). The main conclusion of this comparison is the good agreement between the correlations (14) and (15), whereas (16) overestimates the convective component for Reynolds numbers larger than 30. The transition between the domains: particle convection + gas convection dominant and convection and radiation significant (N, = 10) is a function of E,. Figure 10 illustrates this dependence for a bed porosity of 0.5. For particles 3 mm in diameter, radiation becomes non-negligible at 500 K and 630 K for E, = 1 and E, = 0.5 respectively. This result is coherent with the experimental data of [4], but radiation effects seem to be slightly overestimated when choosing the limit Npg = 10; the value Np,=5 is perhaps more appropriate. In addition, Fig. 10 indicates that radiative and convective contributions are equal at -1150Kfordp=3mmand E, = 0.7.
(3) l,=l.
L
/’
.’
.’
5E
-
--_-___F_,‘-__--
2 I
0
I 1
I
I
I
2
I 3
I
I
I
4
5
dpcro-f, Fig. 11. The proposed heat transfer diagram: (1) particle convection, (2) gas convection, (3) particle convection + radiation, (4) particle convection + gas convection, (5) particle convection + gas convection + radiatron.
Conclusion: a new wall-to-fluidized bed heat transfer diagram Taking into account the previous discussion on combined particle convection-gas convection, particle convection-radiation and particle convection +gas convection-radiation heat exchange, a new heat transfer diagram is proposed on the basis of the intersections between the three separate plots. Figure 11 shows the proposed diagram. Five domains are defined as a function of the bed temperature and the particle diameter. The thermal properties of air and sand have been used.
d+lo-‘mm, I, 0
I, 2
III 4
I 6
0
II 10
Fig. 9. Gas convection-gas convection + radiation transition (without particle convection) as a functton of heat transfer coefficient h, x =l, E== 1. (1) eqn. (14), (2) eqn. (15), (3) eqn. (16).
air: pp =353.45 T-l (kg mB3), Pi= 0.42 x lO-‘j TV3 (Ns m-‘) h,=5.66x10e5 sand: 4=2
T +1.1X1O-2
(W m-l K-l)
500 (kg mm3), A,= 1.75 (W m-r K-l)
229
Domain 1: Particle convection dominant; 2: Gas convection dominant; 3: Particle convection and radiation significant; 4: Particle convection and gas convection significant; 5: Particle convection + gas convection + radiation significant. The limits between the domains are determined for the following values of the parameters: (i) Particle convection/gas convection + particle convection (l/4 and 3/5): Ar= 1.4 x lo4 (ii) Gas convection/gas convection + particle convection (2/4): Ar= 1.6X lo6 (iii) Particle convection/particle convection + radiation (l/3): NC=5 (iv) Particle convection + gas convection/gas convection+radiation (4/5) N,,=lO (e,=l and eqn. (14)). Domain 5 may be divided in four subdomains 5a, 5b, 5c and 5d. Subdomain 5a is limited by the following non-dimensional numbers: Ar = 1.4 x 104, Ar = 1.3 x 105 and N,,,=lO. In this region the three modes of heat transfer must be taken into account because any of them may be considered as dominant. It corresponds roughly to group IIA of Saxena and Ganzha, particles 2 mm in diameter belong to this domain in the temperature range 600-l 100 K. Subdomain 5b is bounded by Ar= 1.4~ 104, Ar= 1.3X lo5 and ypg= 1. In this zone radiation is dominant in comparison with particle convection and gas convection. Particles larger than 1.5 mm at temperatures higher than 1200 K are concerned. Subdomain 5c is limited by&= 1.3 x 10s and N, = 1; in this zone particle convection may be neglected, but gas convection and radiation contributions are both large even though radiation provides the largest contribution. Subdomain 5d is bounded by Ar= 1.3 x loS, N,_ = 10 and Npg= 1. Gas convection is the dominant heat transfer mode in this zone. The large surface area of domain 5 may be considered as a surprising result. Nevertheless recent experimental data reported in [29] concerning gas and particle temperature distribution near a heat exchange wall immersed in a 3 mm corundum particle fluidized bed (subdomain 5a) confirm this estimation: a significant particle temperature decrease was measured, indicating that particle convection cannot be neglected. This diagram of heat transfer regimes shows clearly that gas convection may be neglected at high temperature even for large particles, for example above 1250 K for 1.5 mm particle diameter fluidized bed. Previous estimates indicate that radiation heat transfer is significant at rather low temperatures when combined with gas convection (at Tb> 600 K for d,= 1.5 mm). This observation points out the difficulties of
the estimates, convective heat transfer perhaps being underestimated. Finally this diagram must be considered as providing a first estimate of significant heat transfer mechanisms for design and modelling purposes.
List of symbols Archimedes number Ar = p,(p,, -p&f; g/p; back-scattering coefficient (Fig. 8) distance from the wall (m) particle diameter (m) heat transfer coefficient (W me2 K-l) extinction coefficient (m-l) refractive index (n = 1 for the calculations) combined convection-radiation number N,, = h, + h,/4e,aT3, combined conduction-radiation number NC= AJU4n2uT, Prandlt number, Pr = pgC,,/Ag thermal resistance Reynolds number Re = pg U,d,lp., temperature (K) velocity (m s-l)
Ar B d
? K
;lr, N, Pr R Re T
u
Greek letters emissivity E
effective emissivity of the emulsion thermal conductivity (W m-l K-l) dynamic viscosity (N s mm2) porosity density (kg rnm3) Stefan Boltzmann constant u= 5.67 x 10e8 -2 K-4 (Wm ) sphericity
? ; P
u @s
Subscripts
b C
f” g lot mf P ;
fluidized bed conductive emulsion film gas (convective) local minimum fluidization particle radiative total
conditions
References P. Geldart, Powder Technol., 7 (1973) 285. N. A. Decker and C. R. Glicksman, AIChE Symp. Ser. No 208, 77 (1981) 341. G. Fhmant and G. Amaud, ht. .J. Heat Mass Transfer, _ _ 17 (1984) 1725. T. M. Clung and J. R. Welty, AIChE I., 35 (1989) 1170. N. I. Gelperin, V. G. Einstein, in Fluidizarion, Academic Press, New York, 1971, pp. 471-540. S. C. Saxena and V. L. Ganzha, Powder Technol., 39 (1984) 199.
230 7 A. Mathur and S. C. Saxena, Powder Technol., 45 (1986) 287. 8 J. S. M. Botterill, Y. Toeman and K R. Yuregir, Powder TecZmoZ.,31 (1982) 101. 9 P. Chen and D. C. T. Pei, Znt J Heat Mass Transfer, 28 (1985) 675. 10 G. Flamant, Y. Flitris and D. Gauthier, Chem. Eng. Process., 27 (1990) 175. 11 Y. Flitris, G. Flamant and P. Hatzikonstantinou, Chem Eng. Commun., 72 (1988) 187. 12 S. Ergun, Chem. Eng. Prog., 48 (1952) 89. 13 C. L. Tien and M. L. Hunt, Chem. Eng. Process., 21 (1987) 53. 14 J. Kubie, Chem. Eng. Sci., 43 (1988) 1403. 15 0. G. Doornink aad R. G. Hering, I. Heat Transfer, ASME, 94 (1972) 473. 16 R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, McGraw-Hill, New York, 1972. 17 G. Flamant and A. Bergeron, Entropic, 145 (1988) 25. 18 A. Mathur and S. C. Saxena, AZChE J., 33 (1987) 1124.
19 S. C. Sazena, K. K. Krivastava and R. Vadivel, Exp. Thermal Fluid Sci., 2 (1989) 350. 20 R. L. Adams and J. R. Welty, AZChE J., 5 (1979) 395. 21 M. Q. Brewster, .Z. Heat Transfer, Trans. ASME, 108 (1986) 710. 22 G. K. Rubsov and N. I. Syromyatmkov, Zzv. Energetika, 6 (1963) 118. 23 V. A. Borodulya and V. I. Kovensky, Znt. J. Heat Mass Transfer, 26 (1983) 277. 24 G. Flamant, Ph.D. Thesis, I.N.P. Toulouse, No. 93, 1985. 25 J. S. M. Botterill and C. J. Sealey, Bit. Chem. Eng., 15 (1970) 1167. 26 V. L. Ganzha, S. N. Upadhyay and S. C. Saxena, Znt. J. Heat Mass Transfer, 25 (1982) 1531. 27 J. S. M. Botterill and A. 0. 0. Denloye, AZChE Symp. Ser. no 176, 74 (1978) 194. 28 B. Murachman, Thesis, I. N. P. Toulouse, April 4, 1990. 29 N. Fatah, G. Flamant, D. Hernandez and G. Olalde, in Fluidtkation: Recent Progr& en Genie des Procedes, Vol. 5, No. 11, Lavoisier, Paris, 1991, pp. 114-121.
Powder Technology, 69 (1992) 223-230
Wall-to-bed heat . transfer in gas-solid fluidized beds: prediction heat transfer regimes G. Flamant,
of
N. Fatah
Institut de Science et Ginnie des Mat&au
et Pro&d&
CNRS, BP No 5, 66120 Odeillo (France)
and Y. Flitris Universiv of Thessaly, Div. of Physics, Athens av. Pedio Areos, Voles 38334 (Greece) (Received
April 24, 1990; in revised form September
26, 1991)
Abstract When studying heat transfer between a fluidized bed and immersed surfaces, three modes of heat transfer can occur, VIZ. conduction (particle convection), gas convection and radiation. A classical definition of the dominant mechanisms is the following: (1) particle convection dominant: d, < 600 pm; (2) radiation significant: T,, > 900 K, (3) gas convection significant: dP> 600 pm. A new definition of the ‘heat transfer regimes’ is presented in a particle diameter vs. temperature diagram. The discussion on the transition between the various ‘regimes’ is based upon (i) the Saxena and Ganzha numbers related to the ratios classification scheme combined with Ergun’s equation, and (ii) dimensionless between conduction and radiation and between convection and radiation in porous media.
Introduction
Dynamics and heat transfer are interdependent in fluidized beds and, from this point of view, particle size and bed temperature are the more pertinent parameters for a classification of fluidization regimes. Nevertheless, classical schemes of powder classification are based only on particle diameter, whereas both particle diameter and temperature must be considered because of the marked variations in gas properties (p, and Pi) with temperature. The well-known Geldart classification [l] was proposed by considering the fluidization behavior of particles at ambient conditions. For example the transition between group B and D particles occurs at (p,-p,) > 4 000 kg mm3 and d,> 600 pm. The effect of temperature on the transition between regimes is not well understood and this approach is not appropriate for heat transfer considerations. In contrast, Decker and Glicksman [2] suggested a heat transfer-based classification. Their analysis is based on the comparison between the thermal time constant of the particles (T) and the mean residence time of the solid at the walls (&). For small particles, t,~- T, d, <400 ,um; heat transfer is dominated by particle convection (conduction through a layer of more than one particle depth). For large particles, t, s T, d, > 1500
0032-5910/92/$5.00
pm, the heat transfer is to a large extent due to gas convection, and the temperature decrease of particles adjacent to the wall is negligible as a first approximation. For intermediate particles both conduction and convection must be considered. This classification scheme does not account for the effect of temperature on the heat transfer mechanisms, and it cannot be used for high temperature design since the variation of the transition between the three groups is not clearly presented. The previous discussion deals with the variation of conductive (particle convection) and gas convective heat transfer. Concerning radiative heat transfer, combined with the two other modes of heat transfer, theoretical [3] and experimental results [4] are in good agreement for suggesting radiation to be a significant heat transfer mode for temperatures higher than 800 K for either small or large particles. Finally, when considering the classifications of Geldart and Decker, Glicksman, and the latter conclusion, one may propose a very simple heat transfer regime diagram as shown in Fig. 1. However, this diagram does not take into account how gas properties vary with temperature. This paper suggests another way for classifying dominant heat transfer modes as a function of temperature
0 1992 - Elsevier Sequoia. All rights reserved
224
The purpose of this paper is not to discuss the physical existence of the gas film. We assume that there exists a resistance to heat transfer (Rf) between the wall and the emulsion phase which may be regarded as an equivalent gas film of depth 8,. The transient conduction through this film is fast and the temperature profile is constant during the heat transfer. As a consequence, R, does not vary with time. The existence of the gas film does not change the conclusions on the heat transfer mode in the emulsion, which must be considered as a function of T and d,.
6
Particle convection and gas convection 0
1
3
2
4
Fig. 1. Heat transfer diagram based on ‘classical’ considerations. (1) Particle convection, (2) particle convection + gas convection, (3) convection, (4) particle convection + radiation, (5) particle convection + gas convection + radiation, (6) gas convection + radiation.
and particle diameter. We examine successively particle convection and gas convection, particle convection and radiation, particle convection + gas convection and radiation. The analysis is based on characteristic dimensionless numbers. The heat transfer component due to the bubble phase (being small in comparison with the exchange due to the emulsion) will be neglected as a first approximation.
Heat transfer mechanisms Assuming the additivity of the heat transfer components in the emulsion, i.e. the particle convection (h,), gas convection (h,) and radiation (h,), the total heat transfer coefficient may be written as: h,=h,+h,+h,
(I)
In eqn. (l), particle convection contribution h, may be expressed as: 1 -= h,
1 + _L = &+ h, h,
; c
(2)
The particle convection contribution is a result of two heat transfer mechanisms in series: steady state conduction through a gas film of depth 8, and transient conduction through the emulsion. As a consequence R, is given by: R,=d,lnA,
(3)
with 2
The Saxena and Ghanza [6] classification can be extended to include both hydrodynamics and heat transfer. For their group I, particle convection (conduction through emulsion) is the dominant mechanism of heat transfer, i.e. the fluid flow through the emulsion is laminar. This group is defined by: Re,,
225
corundum particles at high temperature (up to 1073 K) and theoretical results generated by the model of Flitris et al. [ll]. An illustration of the variation of these transitions in a temperature VS.particle diameter diagram is shown in Fig. 2 for corundum particles. According to this representation at ambient conditions, convection becomes significant when d,=500 and 600 pm for Ar = 1.4 x lo4 and 2.6 x lo4 respectively. At 1000 K this transition is observed when d,= 1.2 and 1.5 mm for the same values of Archimedes number. Another approach to the quantitative limits of particle convective and gas convective heat transfer mechanisms may be based on Ergun’s equation [12]: Arc
..,I
w
0.25 _
150(1- sIn3 Re + 1.75 Re2 #J55zIf mf $QCL Inf
(4)
The gas convective heat transfer is significant when the kinetic forces cannot be neglected. This phenomenon is observed when the second term of eqn. (3) becomes significant, i.e. when the minimum fluidization velocity increases with temperature. A U,,,, VS. T,, plot is shown in Fig. 3 in order to illustrate this comment. The transition between both domains is defined by: dU,,ldT=
0.75
0
(5)
This limit is plotted in Fig. 2 for two particle densities related to sand and corundum. A comparison between the curves in Figure 2 indicates very good agreement between both transitions obtained for Ar = 2.6 x lo4 and dU,,,,ldT=O. Equation (3) proves the sensitivity of the results with sphericity & and porosity .&. The variation of the
0.00 ~( 0.4
0,6
0,8
1.0
1.2
1.4
Ib’lo’K, Fig. 3. Variation of U,,,, with temperature on the basis of the Ergun equation for corundum, &,t=O.45, @,=0.8, 4=3950 kg mm3 (1) d,=2 mm, (2) d,=1.5 mm, (3) d,=l mm, (4) dr,= 500 pm. 3
Tb(lO'K) 1
l_ 3
Tb(lO'K)
I 0
I 1
I
I
I
2
1 3
dp(,o-‘m) Fig. 4. Influence of spheric&y and porosity on the particle convection-particle convection +gas convection transition (dU& dT=O), pp=2500 kg rnd3. A: particle convection, B: particle convection + gas convection. (1) OS= 0.5, (2) @,= 0.75, (3) QS= 1 (&=0.55); dotted hnes: (4) &=0.5, (5) &,,r=0.65 (+$I= 1).
0
1
2
3
Fig. 2. Limit between gas convection+particle convection and particle convection dominant heat transfer domains based on Archimede’s number and Ergun’s equation. A: particle convection, B: particle convection + gas convection. (1) Ar= 1.4x 104, (2) Ar=2.17~ 104; (3) Ar=2.6x104; dotted lines, dlJ,,,r/dT=O (&=0.5): (4) pr=3950 kg rnm3, (5) 4=2500 kg m-‘.
limit with & and &,r is illustrated for sand particles in Fig. 4. When the sphericity decreases, the surface area of the particle convection domain (dU,,,,ldT
the combined mechanism is characterized number NC: NC= AJWln2 uT3
by the Planck (6)
where A, is the effective thermal conductivity of the emulsion, K the extinction coefficient and n the refractive index. Radiation cannot be neglected when NC<5 and as a consequence, the boundary between both domains (particle convection and particle convection + radiation) may be defined by: NC=5 0
I
I
I
I
I
1
2
3
4
5
I 6
dpclo-‘m, Fig. 5. Limits between the three domains: (A) particle convection, (B) particle convectton +gas convection significant, (C) gas convection. (Bi: particle convection dominant, B,: gas convection dominant) (1) Ar=lAX 104, (2) Ar=1.3X10s, (3) Ar=1.6X 106; dotted lines: 4=3950 kg rnm3, dark lines: pr=2500 kg mm3.
heat transfer at 1000 K for particle diameters larger than 1 and 1.8 mm for &,,r=O.65 and 0.5 respectively. Thus the heat transfer is influenced by the variation of bed properties near the wall, in particular the emulsion voidage [13], and a definitive conclusion cannot be given. In any case the local porosity in the vicinity of the heat transfer surface &,, is greater than the meanvoidage at minimum fluidization conditions. Kubie [14] proposed an estimate of &, in the range 0< d/d,< 1 (where d is the distance from the wall). This analysis must be completed by the introduction of another limit: the transition between the domain where both particle conduction and gas convection are significant and the domain where gas convection is dominant. This limit was defined previously ::%n the basis of Archimedes number: Ar= 1.6X 106, shown in Fig. 5 for both corundum and sand particles. According to this representation, gas convection is dominant for 2.5
Particle convection
(7) Figure 6 illustrates this observation: the heat transfer coefficient becomes a nonlinear function of the temperature for NC< 5 and increases when the extinction coefficient (ZC)decreases. The limits NC= 5 (and NC= 0.5) are plotted in Fig. 7 with &,,I as parameter for air and corundum particles. The parameters A, and K have been estimated as follows [5, 161:
(8) 3 -,l-tnlr
K=
2
andn=l
4
(9)
Figure 7 indicates that radiation becomes significant at - 650 K for d, = 500 pm and &,,r= 0.5. For a given particle diameter the transition temperature decreases with increase in porosity. This estimation is in agreement with the two-flux model calculations proposed in [17] which indicate a radiative heat transfer coefficient ranging from 30 to 40 W mm2 K-’ for a bed temperature of 750 K (d,= 280 pm). On the other hand the ex-
1200
h,cwm-2K-'~
/
and radiation
This section is concerned with fixing a boundary for incipient radiation inside zone A previously defined in Figs. 2, 4 and 5. The combined conductive and radiative transient heat transfer inside a grey wall has been studied by Doomink and Hering [15] and the application of these general results to fluidized beds was proposed in [3]. The main results are the following: when radiation and conduction occur simultaneously inside an optically thick medium,
400
I_, 96
0.8
1.0
1.2
T~~Io’K?
Fig. 6. Influence of the combined conduction-radiation number N, on the heat transfer coefficient, after [3]; d,=300 pm. (1) lO
221
097
0
1
2
3
d,,c&m) Fig. 7. Particle convection-particle convection + radiation transition as a function of the porosity. A: particle convection, B: particle convectionfradiation N,=5, (1) &=0.5 (2) &,r= 0.65, (3) N,=OS, &=0.5.
perimental results obtained by Mathur and Saxena [18] show a significant radiative heat exchange for a 559 pm fluidized bed at 820 K: 28 W me2 K-l. More data are presented in a review of radiative and total heat transfer in gas fluidized beds proposed in [19]. According to [15], radiation is dominant for N< 0.005; our calculations show that radiant heat transfer is never dominant when combined with particle convection.
Particle convection + gas convection and radiation
091
OS1
093
0,5
0,7
099
Fig. 8. Effectwe emissivity of fluidized beds (.&=0.5) curves (1) [22]; (2) [23]; (3)[21]; (4) [24]; (5) [25]; (7’s= 1000 “C, (0) A1203 spheres, 2
Qr=
’
where cw and T,,, are the wall emissivity and the wall temperature respectively. Assuming a small difference between Tb and T,, and cw= 1, we have: Qr=4~,d’3(Tb-Tw)=h,(T,,-Tw)
Theoretical and experimental data dealing with gas convective and radiative heat transfer in fluidized beds are scarce. Chung and Welty [4] found that radiation is significant in a fluidized bed operating at 812 K: the radiative contribution reaches 8% at d,= 1 mm and 12% at d, =3 mm. On the other hand, a theoretical study [20] suggested a radiant contribution of about 33% in a large particle fluidized bed at 1 100 K. No general criterion was proposed in order to characterize the transition between the convection domain and the domain where both convection and radiation are significant. According to [2], in the domain where gas convection is dominant the temperature decrease of the particles adjacent to the wall is negligible. As a consequence the emulsion may be considered as a constant temperature medium whose emissivity is 4. The effective emissivity E, was defined by Brewster [21] on the basis of the two-flux radiative model. The results of this theoretical study are compared with other data in Fig. 8. The effective emissivity of a bed whose particle emissivity is 0.5 ranges between 0.7 and 0.8. Under the previous assumption, the radiative heat flux exchange between wall and bed may be expressed as follows (for T,< Tb):
t p
with F= (Tb + T,)/2. With this definition a convection/radiation number may be constructed as: N,, = (h, + h,)k
(11)
interaction
(12)
where the convective term accounts for particle and gas contribution. For large particle diameters, h, is roughly equal to hf (see discussion of heat transfer mechanisms) and we have:
(13) For calculations T = T,, is assumed. Radiation is significant for NPg= 10 and equal to convective heat transfer for NPe= 1. The main parameters of the convection/ radiation number NPgare the gas convective heat transfer coefficient h,, and the effective emissivity Q. Concerning h,, various correlations have been proposed in the literature. The model of Ganzha et al. [26] was shown to be adequate for powders in groups II and III [6], and it accounts for both convective contributions h, and h,:
228
Nu, = 8.95(1- t)“.667+ 0.12&O sP+‘43(l - &)o.133/p.8
3,2,
Tb(lO'K)
(14) with Nu,=Nu,+Nu, For Archimedes numbers in the range lo3
(15)
Recently Murachman [28] proposed a correlation on the basis of experimental results obtained at steady state near the distributor of gas-fluidized beds. Nu, = 0.03Re’ 4g
0
3
2
I
4
5
6
dp
(16)
Re<30
lc = 0.5, (2) .E= =0.75,
The comparison of correlations (14), (15) and (16) is illustrated in Fig. 9 for Nz= 1. N,* is a modified interaction number accounting only for gas convection, i.e. N,* = h,/h,. The extrapolation of the data for d, smaller than 1 mm is questionable (dashed zone). The main conclusion of this comparison is the good agreement between the correlations (14) and (15), whereas (16) overestimates the convective component for Reynolds numbers larger than 30. The transition between the domains: particle convection + gas convection dominant and convection and radiation significant (N, = 10) is a function of E,. Figure 10 illustrates this dependence for a bed porosity of 0.5. For particles 3 mm in diameter, radiation becomes non-negligible at 500 K and 630 K for E, = 1 and E, = 0.5 respectively. This result is coherent with the experimental data of [4], but radiation effects seem to be slightly overestimated when choosing the limit Npg = 10; the value Np,=5 is perhaps more appropriate. In addition, Fig. 10 indicates that radiative and convective contributions are equal at -1150Kfordp=3mmand E, = 0.7.
(3) l,=l.
L
/’
.’
.’
5E
-
--_-___F_,‘-__--
2 I
0
I 1
I
I
I
2
I 3
I
I
I
4
5
dpcro-f, Fig. 11. The proposed heat transfer diagram: (1) particle convection, (2) gas convection, (3) particle convection + radiation, (4) particle convection + gas convection, (5) particle convection + gas convection + radiatron.
Conclusion: a new wall-to-fluidized bed heat transfer diagram Taking into account the previous discussion on combined particle convection-gas convection, particle convection-radiation and particle convection +gas convection-radiation heat exchange, a new heat transfer diagram is proposed on the basis of the intersections between the three separate plots. Figure 11 shows the proposed diagram. Five domains are defined as a function of the bed temperature and the particle diameter. The thermal properties of air and sand have been used.
d+lo-‘mm, I, 0
I, 2
III 4
I 6
0
II 10
Fig. 9. Gas convection-gas convection + radiation transition (without particle convection) as a functton of heat transfer coefficient h, x =l, E== 1. (1) eqn. (14), (2) eqn. (15), (3) eqn. (16).
air: pp =353.45 T-l (kg mB3), Pi= 0.42 x lO-‘j TV3 (Ns m-‘) h,=5.66x10e5 sand: 4=2
T +1.1X1O-2
(W m-l K-l)
500 (kg mm3), A,= 1.75 (W m-r K-l)
229
Domain 1: Particle convection dominant; 2: Gas convection dominant; 3: Particle convection and radiation significant; 4: Particle convection and gas convection significant; 5: Particle convection + gas convection + radiation significant. The limits between the domains are determined for the following values of the parameters: (i) Particle convection/gas convection + particle convection (l/4 and 3/5): Ar= 1.4 x lo4 (ii) Gas convection/gas convection + particle convection (2/4): Ar= 1.6X lo6 (iii) Particle convection/particle convection + radiation (l/3): NC=5 (iv) Particle convection + gas convection/gas convection+radiation (4/5) N,,=lO (e,=l and eqn. (14)). Domain 5 may be divided in four subdomains 5a, 5b, 5c and 5d. Subdomain 5a is limited by the following non-dimensional numbers: Ar = 1.4 x 104, Ar = 1.3 x 105 and N,,,=lO. In this region the three modes of heat transfer must be taken into account because any of them may be considered as dominant. It corresponds roughly to group IIA of Saxena and Ganzha, particles 2 mm in diameter belong to this domain in the temperature range 600-l 100 K. Subdomain 5b is bounded by Ar= 1.4~ 104, Ar= 1.3X lo5 and ypg= 1. In this zone radiation is dominant in comparison with particle convection and gas convection. Particles larger than 1.5 mm at temperatures higher than 1200 K are concerned. Subdomain 5c is limited by&= 1.3 x 10s and N, = 1; in this zone particle convection may be neglected, but gas convection and radiation contributions are both large even though radiation provides the largest contribution. Subdomain 5d is bounded by Ar= 1.3 x loS, N,_ = 10 and Npg= 1. Gas convection is the dominant heat transfer mode in this zone. The large surface area of domain 5 may be considered as a surprising result. Nevertheless recent experimental data reported in [29] concerning gas and particle temperature distribution near a heat exchange wall immersed in a 3 mm corundum particle fluidized bed (subdomain 5a) confirm this estimation: a significant particle temperature decrease was measured, indicating that particle convection cannot be neglected. This diagram of heat transfer regimes shows clearly that gas convection may be neglected at high temperature even for large particles, for example above 1250 K for 1.5 mm particle diameter fluidized bed. Previous estimates indicate that radiation heat transfer is significant at rather low temperatures when combined with gas convection (at Tb> 600 K for d,= 1.5 mm). This observation points out the difficulties of
the estimates, convective heat transfer perhaps being underestimated. Finally this diagram must be considered as providing a first estimate of significant heat transfer mechanisms for design and modelling purposes.
List of symbols Archimedes number Ar = p,(p,, -p&f; g/p; back-scattering coefficient (Fig. 8) distance from the wall (m) particle diameter (m) heat transfer coefficient (W me2 K-l) extinction coefficient (m-l) refractive index (n = 1 for the calculations) combined convection-radiation number N,, = h, + h,/4e,aT3, combined conduction-radiation number NC= AJU4n2uT, Prandlt number, Pr = pgC,,/Ag thermal resistance Reynolds number Re = pg U,d,lp., temperature (K) velocity (m s-l)
Ar B d
? K
;lr, N, Pr R Re T
u
Greek letters emissivity E
effective emissivity of the emulsion thermal conductivity (W m-l K-l) dynamic viscosity (N s mm2) porosity density (kg rnm3) Stefan Boltzmann constant u= 5.67 x 10e8 -2 K-4 (Wm ) sphericity
? ; P
u @s
Subscripts
b C
f” g lot mf P ;
fluidized bed conductive emulsion film gas (convective) local minimum fluidization particle radiative total
conditions
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19 S. C. Sazena, K. K. Krivastava and R. Vadivel, Exp. Thermal Fluid Sci., 2 (1989) 350. 20 R. L. Adams and J. R. Welty, AZChE J., 5 (1979) 395. 21 M. Q. Brewster, .Z. Heat Transfer, Trans. ASME, 108 (1986) 710. 22 G. K. Rubsov and N. I. Syromyatmkov, Zzv. Energetika, 6 (1963) 118. 23 V. A. Borodulya and V. I. Kovensky, Znt. J. Heat Mass Transfer, 26 (1983) 277. 24 G. Flamant, Ph.D. Thesis, I.N.P. Toulouse, No. 93, 1985. 25 J. S. M. Botterill and C. J. Sealey, Bit. Chem. Eng., 15 (1970) 1167. 26 V. L. Ganzha, S. N. Upadhyay and S. C. Saxena, Znt. J. Heat Mass Transfer, 25 (1982) 1531. 27 J. S. M. Botterill and A. 0. 0. Denloye, AZChE Symp. Ser. no 176, 74 (1978) 194. 28 B. Murachman, Thesis, I. N. P. Toulouse, April 4, 1990. 29 N. Fatah, G. Flamant, D. Hernandez and G. Olalde, in Fluidtkation: Recent Progr& en Genie des Procedes, Vol. 5, No. 11, Lavoisier, Paris, 1991, pp. 114-121.