Ind. Eng. Chem. Res. 1992,31, 1147-1156 Draught Tube. J . Chem. Eng. Jpn. 1986,19,341-344. Myauchi, T.; Shiu, C. N. Flow of Fluid in Gas Bubble Columns. Kagaku Kogaku 1970,34,95&965. Narayanan, S.; Bhatia, V. K.; Guha, D. K. Suspension of Solids by Bubble Agitation. Can. J . Chem. Eng. 1969,47, 360-364. Pandit, A. B.; Joshi, J. B. Three Phase Sparged Reactors: Some Design Aspects. Rev. Chem. Eng. 1984,2,1-84. Pandit, A. B.; Joshi, J. B. Effect of Physical Properties on The Suspension of Solid Particles in Three Phase Sparged Reactors. Znt. J . Multiphase Flow 1987,13,415-427. Perry, R. H.; Chilton, C. H. Chemical Engineers Handbook; McGraw-Hill: Singapore, 1984,pp 5.63-5.64. Roy, N. K.;Guha, D. K.; Rao, M. N. Suspension of Solids in a
1147
Bubbling Liquid; Critical Gas Flow Rates for Complete Suspension. Chem. Eng. Sci. 1964,19,215-225. Shah, Y. T.Gas-Liquid-Solid Reactor Design; McGraw-Hill: New York, 1979; pp 32-36. Smith, D. N.; Ruether, J. A.; Shah, Y. T.; Badjugar, M. N. Modified Sedimentation Dispersion Model for Solids in a Three Phase Slurry Column. AIChE J. 1986,32,426-436. Ueyama, K.;Myauchi, T. Properties of Recirculating Turbulent Two Phase Flow in Gas Bubble Columns. AIChE J. 1979,25,25&266. Received for reuiew May 7, 1991 Revised manuscript receiued October 25, 1991 Accepted November 28,1991
Maximum Heat-Transfer Coefficient for an Immersed Body in a Bubbling Fluidized Bed Mayumi Tsukada and Masayuki Horio* Department of Chemical Engineering, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184, Japan
The heat-transfer coefficient between the bed and an immersed body is an important parameter for determining the surface temperature of a body, which affects the rate of combustion, reaction, drying, or heat treatment. In this paper the following simple correlation for the maximum heattransfer coefficient, hap,, was obtained within the accuracy of +70% and -50% by a comprehensive data set covering a wide range of operating conditions (d = 0.08-3 mm, d, = 4-60 mm, temperature = 293-1323 K and pressure = 0.1-8.1 MPa): Nu,,,, p R,,,d,/k, = (d,/d,)O.*, where d,, d,, and k, are particle diameter, body diameter, and effective thermal conductivity of a bed. In this correlation, k, was used instead of gas thermal conductivity. The correlation was successfully validated. Eleven previous correlations were tested and evaluated by the same data set.
Introduction So far, the following knowledge has been widely accephd with respect to the gas fluidized bed heat-transfer coefficient, h,, for “immersed bodies”, i.e. small and rather spherical bodies having sizes much larger than the size of bed particles but smaller than the bed dimensions, or for other heat-transfer surfaces. (1)The bed to surface heat-transfer coefficient increases with increasing fluidizing gas velocity. The rate of the increase jumps up a t the minimum fluidization velocity, umf.Above umfa renewal of the dense phase on the surface takes place. The renewal rate controls the heat-transfer rate. Below umf,there is no renewal of the dense phase and the heat-transfer rate is controlled mainly by the resistance between surface and interstitial gas. (2) If the gas velocity is increased continuously, h, reaches a maximum, ha,,,, and then starts decreasing gradually. This decrease is because of the increased occupation of the surface area by the bubble phase. (3)The above two features appear leas obvious for coarse particle beds, where the surface to interstitial gas heat transfer is more significant than it is in the case of fine particle beds. For the past two decades, fluidized bed to surface heat transfer has been investigated intensively with the focus being on fluidized bed combustor developments. Since the combustion efficiency of bubbling fluidized bed boilers is determined by the balance of the combustion rate and elutriation rate of char particles, the surface temperature of burning char is quite important. The surface temperature also affects NO, decomposition on the char surface. In this regard the heat transfer between the fluidized bed and immersed or floating bodies is quite significant. It is also related to fluidized bed drying and heat treatment. The heat-transfer characteritics of bed to immersed bodies would be a bit different from those of bed to heat
exchange surface because (1)immersed bodies are rather spherical in shape; (2)the diameters of immersed bodies to 0.1 in industrial processes range quite widely from m and often change with time; (3)in most cases immersed bodies move within the bed; and (4) surface temperatures of bodies relevant to industrial processes vary widely, depending on heat of surface reactions if they exist. With respect to the bed to immersed body heat-transfer coefficient,Ross et al. (1)determined a bed to burning char heat-transfer coefficient from combustion experiments. They also assumed fluidized particles as a continuum. On the same line, Prins et al. (2) tried to develop a RanzMarshall type correlation for ha. The correlation introduced a new difficulty since it required the knowledge of the bed effective viscosity. In our previous work (3),the bed to sphere heat-transfer coefficient was measured over wide experimental conditions (immersed body diameter d, = 6.35-19.05 mm, particle diameter d = 0.14-0.55mm, bed temperature Tb = 293-1123 K),an2 the following simple empirical correlation was obtained: Nu,,, E h,,,d,/k, = 10d,0.8d,+’.s (*20%) (1) In this correlation the effect of radiation was successfully included in the bed effective thermal conductivity k,. The objective of the present work is to obtain a more widely applicable but simpler correlation for the bed to immersed body maximum heat-transfer coefficient through comprehensive evaluation of previous data and correlations. Previous Works There are quite a few review on fluidized bed heat transfer (4-15), but not much information is available concerning bed to immersed body heat transfer. In this section previous works are reviewed, with a focus on this particular aspect.
0888-5885/92/2631-1147$03.00/00 1992 American Chemical Society
1148 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992
The common features between the heat transfer of the bed to heat exchange tube and that of the bed to immersed body allow us to apply to the latter some approaches previously developed for the former. When the particle size is sufficiently small, the surface renewal mechanism proposed by Mickley and Fairbanks (16)would be applicable to the heat transfer between bed and immersed body. They treated fluidized particles as a continuum and expressed the framework of the heattransfer mechanism by the following formula:
h, = ( 2 / ? r ' ~ 2 ) ( k , c ~ e / ~ c ) 1 ~ z ( 2 ) However, although the effective thermal conductivity of a bed can be predicted rather accurately, no reliable correlations are available either for bed effective viscosity or bed to body relative velocity. The latter parameter, a major parameter in the Mickley and Fairbanks model, is influenced by the local hydrodynamic conditions significantly. Since then, the packet concept has been modified and improved by many investigators. One recent expression is given by Kunii and Levenspiel (17). Another mechanistic approach to the theoretical estimation of bed to surface heat transfer is the particle contact model (18, 19). A refined version of it was developed by Martin (20,21)treating particle motion with an analogy to the molecular kinetic theory of gases but not of liquids. However, as stressed by Zabrodsky et al. (221, the mechanistic models mentioned above present considerable difficulty since their major parameters are very much dependent on fluidized bed hydrodynamics, which is quite complicated. Certainly, heat-transfer phenomena cannot be separated from fluidized bed hydrodynamics, but providing a practical correlation by avoiding too much complication with hydrodynamics is also needed. Zabrodsky (23)developed a purely empirical correlation for bed to surface maximum heat-transfer coefficients. Zabrodsky et al. (22) rewrote their original correlation into nondimensional form, where the particle Nusselt number, NU,-, was expressed as a function of the Archimedes number, Ar. As recognized from an early period, the Reynolds number correaponding to a certain flow regime c8n be expressed as a function of the Archimedes number. Accordingly, the Reynolds number corresponding to the maximum heattransfer coefficient has been also well correlated as a function of Ar (24). This would be the reason why successful correlations have been developed in the form of a Nu,, v8 Ar correlation, which may automatically include the reference velocity that controls the contact time. In the fluidized bed combustion research, attempts have been made to formulate the bed to immersed body heattransfer coefficient for predicting the char combustion temperature. Ross et al. (I)measured the temperature of burning coke particles and determined the immersed body Nusselt number, Nu,' = had,/ k:. They assumed the fob lowing expression for the total heat flux from coke surface: Q = (Nu,'k,'/d,)(T, - T b )
+ U C r , l ( T 2 - Tb4> ( 3 )
The modified effective thermal conductivity of the bed, k:, in eq 3, includes the gas film effect in the following manner: 1/kl
l / k , - d,/[k,(d, + 26J1
+ d,/[k,(d, + 26J1
(4)
where k , in the third term was estimated at the minimum fluidization condition with the Yagi and Kunii (25) cor-
TS
Tbs Tb Figure 1. Temperature profile near the heat-transfer surface.
relation for effective thermal conductivity of a fixed bed with gas flow: k , = k,O + O.lp,c,d,u,f (5) The thermal conductivity of a fned bed with stagnant gas, k,O, is calculated by the Kunii and Smith (26) correlation for an incipiently fluidized bed. Ross et al. (1) calculated k,' for three gas film thicknesses, i.e. 6, = 0, d,/10, and d,/6. Except for Russian works (27,28),the work of Ross et al. (1) would be the first attempted to determine the bed to body heat-transfer coefficient from combustion experiments. Their way of correlating the heat-transfer coefficient in Nu,' (i.e. with d, as a reference scale instead of d , and with the effective thermal conductivity of the bed instead of the gas thermal conductivity) was also a new approach in the fluidized bed heat-transfer study. The above mentioned approach was then succeeded by La Nauze and de Jung (29). They found that the Nusselt number cannot be correlated as simply as the Sherwood number, contrary to the expectation of Ross and Davidson (30). Recently a similar attempt was made by Prins et al. (2) to derive a correlation of the bed to immersed sphere heat-transfer coefficient in a bed of FCC particles. As mentioned already, they intended to develop a correlation having the same form as the Ranz-Marshall correlation. To avoid the previously mentioned difficulty accompanied with the continuum assumption, Donsi et al. (31) used gas viscosity instead of bed viscosity for the Reynolds number, Re,', to correlate the bed to body heat-transfer coefficients for beds of sand and glass beads. With regard to the correlations derived from char combustion works, there exists another difficulty. As far as we understand, the difficulty arises from the way they handled the radiation term. In eq 3 it can be read that light can be transmitted through a bed section for a certain distance greater than the particle diameter. Such treatment would be allowable for coarse particle beds, where the temperature boundary layer could be as thick as the particle diameter. However, for noncoarse particle systems, it is obvious that temperature boundary layers are not transparent. Expressing the convective term as the sum of particle convective and gas convective components, Baskakov et al. (32-34) experimentally determined effective emissivity, E:, as a function of Tb, T,, and bed properties. In their model e,:, is sort of a parameter compensating for the above mentioned difficulty. In eq 3 the radiation effect can be included in the first term via the effective thermal conductivity. The addition of a radiation term u4,,)(T,4 - Tb4) would be a double accounting of the radiation, if its effect is already included in the "convective" term through effective thermal conductivity. If the air gap between the surface and the bed cannot be neglected, as illustrated in Figure 1, a radiative term mr8(T84- Tb:) and T, - Tb,) should be added to a gas convective term hs,g,c(
Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 1149 Table I. Experimental Conditions of the Previous Works' symbol author &/mm particles
0
Tsukada and Horio (3)
58.8
0 Kharchenko and Makhorin
[3
silica sand
d,bm
~,,/(kg/m~)
gas
emf
Tb/K
4/mm
0.160, 0.460 2600;
0.45, 0.46
air
293-1123 6.35-19.05
alumina catalyst 0.140, 0.550 1580
0.54, 0.43
air
293-1123 6.35-19.05
220
silica sand
0.350
2600
0.43
flue gas'
573-1323 60.0
100
Cu, Ni, solder
0.136
8955, 8891, 0.42* 8875
air
293
(41)
A
Ziegler and Brazelton (42)
X
Pillai (43)
140
silica sand
0.243-0.755 2630
0.43
flue gas" 548-1283 6.0
0
Botterill et al. (44)
188
silica sand
0.380-2.32
2600;
0.42;
flue gas" 893
alumina
0.98
3430
0.42;
flue gas" 523-1223 9.50
-e
Shirai et al. (45)
280
silica sand
0.170
2660
0.45
air
43
Botterill et al. (46)
188
silica sand alumina
3.00 0.370
2400 3640
0.42* 0.42;
flue gasa 523-843 6.4-31.6 flue gaso 483-1223 6.4;
Prins et al. (37)
103
glass beads
0.131-1.011 2620-2850
0.42;
air
573
alumina
0.631
0.42;
N2
573-1173 4.0-18.0
FCC
0.103, 0.014 1830
W
4-
*e, A
-
-0- Prins et al. (2)
++,
4
Donsi et al. (31)
103
150
4 @
Mickley and Fairbanks (16)c
102
3430
Borodulya et al. (47)c*d
105
9.50 35.6
4.0-18.0
0.60) 0.7gb air
293
4.0-30.0
0.58; 0.76b He
293
4.0-30.0
silica sand
0.220-0.715 2500
0.42*
air
323
8.C-20.0
glass beads
0.073
2600
0.42;
air
323
8.0- 20.0
glass beads
0.083
2450
0.45
air
293
6.35
He
276
6.35
Freon 12 309
6.35
air
13.0
4 $ 4
293
12.7
silica sand
0.794, 1.225 2600;
0.42*
293
" Properties approximated by those for air. bDense phase voidage reported in their paper. eData from cylindrical heat-transfer surfaces. dBed pressure: 1.1-8.1 MPa. Pitch of tubes: 39.0 mm. OPressure is atmospheric, except for Borodulya et al. Values marked with an asterisk are assumed by the present authors. the particle convective term, which already includes the effective conduction. A more rigorous treatment of radiative heat transfer is of course possible if a computer simulation is conducted taking every particle layer of the bed into account. Borodulya and Kovensky (35) thus explained the above mentioned behavior of er,B) determined by Baskakov (32). Concerning the bed to immersed body heat transfer, the size of an immersed body would play an important role since the contact time or surface renewal time is roughly proportional to the body size. Some information is available in the literature. The value of h, or h,,, is proportional to ds4.06in Shirai et al. (36),to ds4.65in Tamarin et d. (28), to ds4.278with another complex term containing dJd, in Prins et d. (37),and to in Tsukada and Horio (3). In the case of the heat-transfer correlation for tubes, reported values of the exponent on tube diameter again range widely from -0.21 (Grewal and Saxena (IO)), -0.56 to -0.7 (Vreedenberg (38)),to -0.674 (Andeen and Glicksman (39)). However, the Grewal and Saxena (IO) correlation developed based on a thorough review of previous data has an exponent -0.21 which is similar to our previous result (3). Prins et al. (37) examined the effect of immersed body movement on the bed to body maximum heat-transfer coefficient. They reported that the body movement slightly reduced the heat-transfer coefficient. This is in
contradiction to the finding of Rios and Gibert (40),where moving bodies showed a higher heat-transfer coefficient. As reviewed above, more reliable information on the characteristics of the bed to immersed body heat transfer is still required. Data Set for Testing t h e Correlations for 4 I,mpI Previous experimental works on bed to spherical body maximum heat-transfer coefficients of interest to this research are summarized in Table I. The Mickley and Fairbanks data (16) and the Bolodulya et al. data (47) obtained for cylindrical bodies are also included in Table I. The reason why they are included is because the former expresses the effect of different gas species and the latter expresses that of pressures. It should be noted here that, except for our previous research, there are no works where all of the three conditions, i.e. bed particle size, immersed body diameter, and bed temperature, were changed systematically. The range of data used in our test is shown in Figure 2 in terms of the Nu,,, and Ar plot, from where it can be seen that the range of Archimedes number of the data set is sufficiently wide for a bench mark testing. Here it should be noted that though the classical Zabrodsky et al. (22) correlation seems to fit the data rather well, it cannot express the effect of immersed body size, which is our present concern. Furthermore, in the Zabrodsky et al. correlation, the effect of radiation is not rigorously treated.
1150 Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 Table 11. Correlations Tested by the Present Data Set author Modified Packet Model
correlation
Zabrodsky et al. (22) Denroy and Botterill (48) Baskakov (32)
Grewal and Saxena (IO)
Kharchenko and Makhorin (41)
NU,,, = 0 . 1 6 A ~ ~ . ' ~ P r '-/ ~c,~)'/~(c, (l pP/c, Nupwmax = O.365( Tb/273)0'82A~0'22
Pillai (43) Prins et al. (37)
N u , , = 4.175(d,/d,)-0.278ArmfT (fixed body) m
E
0.087(d,/d,)0~'28
fT e 0.844 Shirai et al. (36)
Bed to Sphere h, = 17.3(U O / u , ~ ) ~ ~ ~ ~ ~ ~ + ' ~ ~ D , O . ~ ~
Nu, = 5Ar0.207(d,/d,)0.65
Tamarin et al. (28) Prins et al. (2)
+ O.O756(Tb/273)
h, = (1- eb)(k,/d,)(2
+ 1.78Re,,,0.50Pr,0.33)
Res,, P pekds(u0 -
Pre
peCp/ke
log fie = 6.25(1 - c,)
- 3.88
h, = 0.30[kP(l- ~ , ) / d , ] ( R e , ' ) ~ ~ ~ A r ~ ~ ~
Donsi et al. (31)
Res' e
(UO- UrndPp(1
- Ce)ds/Pg
Present Correlation
Nu,,, = h,,,d,/k,
Correlations Tested In the present work the correlations tested were chosen from the following four groups: (1)a mechanistic model (modified packet model), (2) previous Nu,,, vs Ar type correlations, (3) previous correlations derived from experiments on immersed spheres, and (4) our present correlation (Nu,,, type). Correlations corresponding to categories 1 and 2 are taken only to see how well those correlations developed for surfaces different from "a body" can work. In Table I1 all correlations tested in this work are listed. The details of the derivation of the modified packet model and the new empirical correlation proposed in this paper are presented in the following. Modified Packet Model. For investigation of the difficulty and possibility of mechanistic approaches, a modified packet model is derived in the following. As already pointed out by many authors, the packet model overestimates h,,,,, except when bed particles are not
= (d,/d,)o~s
sufficiently small to neglect the gas gap between the heat-transfer surface and the bed. There have been two different approaches to treat the gas gap. In the first approach, the irregularly shaped gap has been approximated to a gas film (33,49-51), while, in the second approach a particulate layer of a porosity greater than that of the dense bed has been assumed (17, 52). In the present work the former approach of Baskakov et al. (33, 49) is adopted. They introduced a gas film resistance into the packet model, treating the resistance as a constant during the period of the packet contact. The overal heat-transfer coefficient, h,, for the period of no bubble contact was then derived as eq T1 in Table 11, where R, given by eq T2 and R, are the packet resistance at time T, and the gas film resistance, respectively. An advantage of such a mechanistic correlation is that the effect of d, is rigorously included in the expression for h, through the contact time 7,. When the heat-transfer
-
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1151
u
10'
i o 2 10' A r (-1
Figure 2. Data used for testing h,,, correlations on the Nu,,, vs Ar plane. Key: symbols, see Table I; -, Zabrodsky et al. correlation (22)).
surface is sufficiently large, the contact time of a packet does not depend on the size of the surface, but on the frequency of bubbling. On the other hand, when the heat-transfer surface is small, as in the case of most immersed bodies, r Ccan be expressed approximately by eq T3, where u, is the velocity of the dense phase relative to the immersed body. R, in eq T1 is, however, more difficult to predict. In the origlnal treatment of Baskakov et al. (491, the conductive heat transfer through the gas film was considered. When gas is not stagnant, as in the case of a fluidized bed, the gas convective component, h,,,, should be much larger than the gas conductive component and would hence control the overall heat-transfer rate, as discussed by Decker and Glicksman (53). In addition, the radiative heat transfer via the gas gap occurs in parallel with the gas convective transfer, especially in high-temperature operations. The radiation component can be expressed by eq T6. By the summation of the two heat-transfer components, R, can be expressed by eq T4. Since the gas convective term controls the overall heat-transfer coefficient for coarse particle beds, h , , can be estimated by the h, correlation developed for coarse particle beds. Equation T5 for hg,owhich may include a small gas conductive contribution, is the one fitted by Kunii and Levenspiel (17) to the Glicksman and Decker (54) data. The above model has four vaguely known parameters u,, tb, T,, and Tbs. For preliminary evaluation of mechanistic approaches, values of h, were calculated from eqs Tl-T6 for the set of data in Table I, assuming that u, 0.02 m/s, t b E 0.3, and T, e T b Tp Proposed Correlation. To derive a new correlation for hs,, the maximum Nusselt number, Nu,,, E h,,,d,/k,, was calculated for the present data set. The effective thermal conductivity of the bed, k,, was predicted from eq 5 and the Kunii and Smith (26) correlation for a fixed bed. The emulsion phase voidage t, was assumed to be emf. The minimum fluidization velocity, ud, in eq 5 was estimated by the Wen and Yu (55) correlation. The Kunii and Smith correlation and additional physical properties of particles and gases necessary to estimate k, can be found in the Appendix. As shown in Figure 3, Nu,,,, is almost independent of bed temperature. As shown in Figure 4, being proportional to dSo8,Nu,,, shows the same dependence as that of our previous correlation, eq 1. On the other hand, Figure 5 shows that Nu,,,, from the data set of Table I is rather proportional to dp-".8, while in eq 1developed for our own
10
10;02
io6 io8
Tb
(K)
Figure 3. Dependence of Nu,,, on bed temperature. Key: sym-
bols, see Table I.
4
* -e-
*%
*-
+
lo-' d, ( m r n l Figure 4. Dependence of Nu,,, on the immersed body diameter. Key: symbols, see Table I.
r*
0
E
v
*
10
d, Figure 5. Dependence of
(m)
NU,,,^^-".^ on d,. Key: symbols, see
Table I.
data it is to dp-"*5.Accordingly, Nu,,, can be simply expressed by Nu,,, = (d,/dp)o.8 (6) Figure 6 shows that all the data in Table I are correlated within the accuracy of +70% and -50% when the data of very fine FCC are excluded. The thickness of the thermal boundary layer around a body, 6,, can be approximately estimated from the Nusselt number by 6, k,/h, = d,/Nu, (7)
1152 Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 Table 111. Simplified Rating of Present and Previous Correlations for l ~ , , - ~ sensitivity to
modified packet model Zabrodsky et al. (22) Denroy and Botterill (48) Baskakov (32) Grewal and Saxena (IO) Kharchenko and Mackhorin (41) Pillai (43) Prins et al. (37) Shirai et al. (36) Prins et al. (2) present correlation
accuracy M H H H H H M H M L H
immersed body size A A A
B A A
bed particle size* B A A A B A A A A C A
bed temperature* B B B A A B B B A B A
bed pressure A A A A A C A B C B A
thermophysical properties gas particle A B B B B C B B A B B B B B B B
"Criteria for accuracy (with respect to 90% of the present data set (see part 1 of b): H, high (within f 50%); M, medium (within f80%); L, low (out of 180%).Criteria for sensitivity ratings: performance index I,, I, [a(ln h , , , , ~ ) / d x ] / [ d ( l n h s - , o b ) / ~ x ] ;x , parameter d,, dp, Tb, P,k , ck,k,, and cp;A, good correlation (Le., I, is always in the range of 0.5 I, 2); B, reasonable correlation (i.e., I, is always positive
or corre!ation performs well in practical but limited conditions); C, poor correlation (i.e., in some conditions I, becomes negative); -, the factor not considered (i.e., I, = 0). bAdditional comments: (1) data from fine FCC (Prins et al. ( 2 ) ,d, = 14 pm) were not used in the evaluation; (2) Pillai data were not used in the evaluation because their bed was too shallow.
10' v
X
E
z2 1 0 '
ddd, ( - 1 Figure 6. Test of the present correlation (eq 6) with the previous data. Key: symbols, see Table I.
From eq 7 lines can be drawn on the Nu,,, vs d,/d, plane from which the boundary layer thickness can be compared with particle size. As can be seen in Figure 6, the thermal boundary layer that develops around the body occupies a few particle layers when d,/d, is large (large body or fine particles). When d,/d, is small (small body or coarse particles), the temperature boundary layer thickness becomes as thin as a particle monolayer. However, more than half of the present data are in the range of &Idp 2 2. Accordingly, the present assumption of the bed as a continuum faces some difficulty, at least logically, in coarse particle systems of d , / d , < 20.
Assessment of Correlations The correlations listed in Table I1 are compared in Figure 7 with the set of observed data listed in Table I. Some details of the assumptions made for h- predictions can be found in the caption of Figure 7. For calculations of h,,, from correlations for h,, the following Todes correlation (24) was used to predict uopt., i.e. the superficial gas velocity for which the maximum heat transfer coefficient is obtained, since those correlations for h, in Table I1 do not show a maximum of h,. Re,,,
pguopt.dp/pg = Ar/(18
+ 5.22(Ar1I2))
(8)
For the case of FCC particle beds tested by Prim et al. (2), experimental data are available for uOpt.Since the esti-
mated values of uOpL for their fine FCC particles were too much lower than the experimental values, the experimental data were chosen for h , , calculation. The specific heata of particles necessary for the calculation of h,, are shown in Figure 12, in the Appendix. The data scattering seen in Figure 7 may be the result of both the limited accuracy of the experimental data and the limited applicable range of the correlations. The former limitation is inevitable, not only because of the experimental errors but also because of the difference in the methods for determining h , , among different authors. Results of the assessment are summarized qualitatively in Table I11 except for those of the Tamarin et al. (28)and Donsi and Ferrari (31)correlations, which show gross deviations from observed values. The assessment was made by checking how well each correlation can reproduce the effecta of changes in the immersed body size, the bed particle size, the bed temperature, the bed pressure, and thermophysical properties of gases and those of bed particles. Detailed discussions are made in the following subsections. Effect of Immersed Body Diameter. Since eqs T7T11, T13, and T14 in Table I1 do not contain the effect of the immersed body diameter, it is inevitable that the values estimated from them will scatter, as can be seen in Figure 7b-d,f,g. To show it quantitatively, hsmobs/ hs,m,,d, the ratio between observed and calculated vdues of h,,, is plotted against d, in Figure 8, whose data were chosen only from the works where d, was systematically changed. Since the Zabrodsky et al. (22) correlation does not contain the effect of d, but reproduces data rather well, values for the Zaas shown in Figure 2, h,-,,b/ h,,, brodsky et al. correlation are plotted against d, in Figure 8a. As can be seen again, observed h,, values roughly show a dependence on d, to the power of -0.2. The Prins et al. (37) correlation contains d, in a slightly complicated form. The data of hsa3aLob/hs-,d from the Prins et al. correlation (cf. Figure 8b) show little dependence on d,, but there remains a scatter of data for different particles and bed temperatures. A similar result can be seen in Figure 8c for the author's correlation (eq 6) which has a function of d, much simpler than the one of Prins et al. (37). Resulta from the modified packet model with a gas film resistance shown in Figure 8d look as if they can successfully reproduce the effect of d, on h,+,. However, it must be noted again here that parameters u,, q,, T,,and
Ind. Eng. Chem. Res., Vol. 31, No. 4, 1992 1153 n
x N
m
E
o\
-5 m n X
E
I
A
x N
\
3
v
m n
9
8 I :
n
Y N
E
\
e m n
9
i
m I :
*
U
N
N + O N -
0
0
0
3
3
0
O
3
A
x N 1'
IZ P
n
9 0 U
1I
, -u, , ,1, , , ,1 _,*
0,
N
0 3
d
-I
1154 Ind. Eng. Chem. Res., Vol. 31, No. 4,1992
a
-
1
I
10
10 0.5 2
5
10
20
50
d, imm)
b
z 10 \
7
a Y
m Y @
Y
10
t I
6 gQO
1
i
F 05 1 2 5 10 20 50 d, imm)
10-yo
C
2
lo3
temperature ( K )
Figure 9. Thermal conductivities of solids, gases, and fluidized bed dense phases (56,57).Key for k,: symbols, see Table I; *, estimated by Russell correlation (58);(a) epoE = 0.55; (b) epoR = 0.30; (c) quoted from ref 16.
2
5
10
20
50
d, imm)
2r-----l
Figure 8. Test of four correlations for the capability of reproducing the effect of the immersed body size on h,,,. Key: symbols, see Table I; (a) Zabrodsky et al. correlation (22);(b) Prins et al. correlation (37);(c) present correlation (eq 6); (d) modified packet model (IJ,= 0.02 m/s, cb = 0.3, cr,s = 0.4,T. = Tb).
Tbsof the packet model were assumed in the present calculation. Although the Grewal and Saxena (10) correlation was developed for the bed to tube maximum heat-transfer coefficient, it was also examined in the present assessment because, as mentioned earlier, the correlation has a rather wide background. The Grewal and Saxena correlation shows a dependence of h,,,, on dT similar to that of the present correlation. Effect of Bed Particle Diameter, Temperature, and Pressure. These factors seem to be expressed by the Archimedes number, as shown in Figure 2, where the simplest correlation of Zabrodsky et al. (22) is indicated
by a solid line. To estimate h,, for coarse particles more accurately, Denroy and Botterill(48) treated the effect of the particle convective and the gas convective component separately. In addition to these two heat-transfer terms, the Baskakov (32)correlation contains the third component, i.e. the radiative term. For beds of coarse particles and of high temperatures, the latter two correlations reproduce h , , a bit more accurately than the Zabrodsky et al. correlation, but the improvements are not very remarkable. Prins et al. (37) introduced the effect of the immersed body diameter into the above Nu,,, vs Ar correlation. However, the modification reduced the applicability of such a correlation in the high-temperature range. Accordingly, Prins et al. introduced an additional empirical factor, fT, to fit the observed data. Nevertheless, their correlation is still not applicable to high-pressure beds. The Grewal and Saxena (IO) correlation, which also includes the effect of the kize of the immersed surface, fails in the case of particles of sizes less than 150 pm. Effect of Thermophysical Properties of Gases and Particles. In Figure 7 the observed h,,, in the bed fluidized by helium cannot be reproduced well by most of the correlations. Quite a few correlations fail to reproduce the h,,, data from the bed of metal particles, as can be seen in Figure 7. Among the correlations tested, our correlation and the modified packet model with a gas film effect are capable of reproducing the effect of gas properties, though there still remains some scatter.
Conclusion The bed to immersed body maximum heat-transfer coefficient h,,, can be predicted by a simple correlation N u , , , = (d,/dJ0.* The correlation was tested with 11 previous correlations for h,,,, by the previous data. It was verified that the
Ind. Eng. Chem. Res., Vol. 31, No. 4,1992 1155 1
I
-
I
I
I
I l l 1
glass
-
CI
0.5
1
10 temperature
10 a (K)
Figure 10. Emissivity of particles (59).
10
A
x
t
0)
Y \
7 v
m
u
10
i
8.1MPa
\
I
tFreon12' I I 1
I
111
10 (K)
10
temperature
Figure 11. Specific heats of gases (56) (*, quoted from ref 16).
t-
,alumina
1
glass A
Ni cu
1 a A
solder
1Ofo'
lo3
ternperoture
(K)
Figure 12. Specific heats of solids (56,57) (* quoted from ref 12).
proposed correlation is applicable with the accuracy of +70% and -50% for fluidized beds of d, = 0.08-3 mm, with immersed bodies of d, = 4-60 mm in diameter, over the range of bed temperature Tb = 293-1323 K and pressure P = 0.1-8.1 MPa. No other correlations tested here were completely successful in accounting for all the heat-transfer factors (immersed body diameter, particle diameter, bed temperature, pressure, and thermophysical properties of gas and particle) tested in the above mentioned range. Nomenclature Ar = dp3pg(pp - pg)g/p,2, Archimedes number, dimensionless
c = specific heat, J/(kg.K) d = diameter, m D,= column diameter, m f T = factor defined in Table 11, dimensionless g = gravity acceleration, m/s2 h = heat-transfer coefficient, W/(m2.K) h, = bed to immersed surface heat-transfer coefficient, W/
(m2.K) I , = performance index in Table 111, dimensionless k = thermal conductivity, W/(mK) Nu, = h,d,/ ke, Nusselt number, dimensionless Nu, = h,d,/ k,, Nusselt number, dimensionless P = pressure, Pa Pr = c,p,/k,, Prandtl number, dimensionless q = heat flux, W/m2 R = thermal resistance, (m2.K)/W Reynolds number, dimensionless Re, = pguOdp/pg, Re,' = Reynolds number defined in Table 11, dimensionless T = temperature, K Tb = bed temperature, K u = gas velocity, m/s uo = superficial gas velocity, m/s u, = bed to immersed body relative velocity, m/s 6, = thickness of thermal boundary layer around a body, m 6, = effective thickness of gas film around a body, m t = bed voidage, dimensionless % = volume fraction of bubble phase in the bed, dimensionless t, = emissivity, dimensionless epore = particle porosity, dimensionless p = viscosity, P e s p = density, kg/m3 u = Stefan-Boltzmann constant, W/(m2.K4) 7 = time elapsed since start of contact, s rC= contact time, s Subscripts bs = fiit layer of particles attached to a heat-transfer surface e = effective value for dense phase c = convective, contact cal = calculated g = gas 1 = instantaneous max = maximum value mf = minimum fluidized obs = observed opt. = optimum p = bed particle, packet r = radiative s = sphere, surface of an immersed body T = tube 0 = reference Superscripts 0 = stagnant flow ' = parameter used by other investigators in a way different from that of the present work
1156 Ind. Eng. Chem. Res., Vol. 31, No. 4,1992
K E
kp/kg
Data for calculating k, are shown in Table I and Figures 9-11. The examples of ke values can be found in Figure 9. Specific heat of particles necessary to estimate h,,, in some correlations are shown in Figure 12. Literature Cited (1) . . Ross, I. B.: Patel, M. S.; Davidson, J. F. Trans. Inst. Chem. Eng.'1981,'59,83. (2) , , Prins. W.: et al. In Fluidization VI: Grace, J. R., et al., Eds.; Engineering Foundation: New York, 1989; p 677. (3) Tsukada, M.;Horio, M. A Simple Correlation for Fluidized Bed-to-Immersed Body Heat Transfer. Submitted for publication in Kagaku Kogaku Ronbunshu, 1991. (4) Vreedenberg, H. A. Chem. Eng. Sei. 1960,11,274. (5) Zabrodsky, S. S. Hydrodynamics and Heat Transfer in Fluidized Beds; MIT Press: Cambridge, MA, 1966. (6) Kunii, D.; Levenspiel, 0. Fluidization Engineering; John Wiley & Sons Inc.: New York, 1969; p 195. (7) Gelperin, N.I.; Ainstein, V. G. In Fluidization; Davidson, J. F., Harrison, D., Eds.;Academic Press: New York, 1971;p 471. (8) . . Botterill. J. S. M. Fluid-Bed Heat Transfer;Academic Press: New York, 1975. (9) Grewal, N. S.;Saxena, S. C. Znt. J. Heat Mass Transfer 1980, 23,1505. (10) Grewal, N. S.; Saxena, S. C. Znd. Eng. Chem. Process Des. Dev. 1981,20, 108. (11) Muchi, I.; Mori, S.; Horio, M. Fluidized Bed Reaction Engineering;Baifukan: 1984;p 177. (12) Xavier, A. M.; Davidson, J. F. In Fluidization, 2nd ed.; Davidson, J. F., et al., Eds.; Academic Press: New York, 1985; p 437. (13) Botterill, J. S.M. In Gas Fluidization Technology;Geldart, D., Ed.; John Wiley & Sons Ltd.: New York, 1986; p 219. (14) Horio, M. In Combustion and Thermal Engineering;Progrese of Chemical Engineering No. 21;Society of Chemical Engineering (Japan), Ed.; Maki Shoten: Tokyo, 1987;p 206. Kunii, D. Fluidization Engineering, 2nd ed.; (15) Levenspiel, 0.; Butterworth London, 1991. (16) Mickley, H. S.;Fairbanks, D. F. AZChE J. 1955,1,374. (17) Kunii, D.; Levenspiel, 0. Znd. Eng. Chem. Res. 1991,30, 136. (18) Botterill, J. S. M.; et al. Chem. Eng. Prog. Symp. Ser. 1966,62, 1. (19) Ziegler, E. N.; Koppel, L. B.; Brazelton, W. T. Znd. Eng. Chem. Fundam. 1964,3,324. (20) Martin, H. Znt. Chem. Eng. 1982,22,30. (21) Martin, H. In Heat and Mass Transfer in Fixed and Fluidized Beds; van Swaaij, W. P. M., Afgan, N. H., Eds.; Hemisphere Publishing Corp.: Bristol, PA, 1986;p 143. (22) Zabrodsky, S. S.;Antonishin, N. V.; Parnas, A. L. Can. J. Chem. Eng. 1976,54,52. (23) Zabrodsky, S.S. Tr. Inst. Energ., Akad. Nauk BSSR, 1958, No. 8,1. Cf. ref 22. (24) Todes, 0.M. In Applications of Fluidized Beds in Chemical Industry, Part ZI; Izd. Znanie: Leningrad, 1965;p 4. Cf. ref 7. (25) Yagi, S.; Kunii, D. AZChE J. 1957,3, 373. (26) Kunii, D.;Smith, J. M. AZChE J. 1960,6,71. (27) Tamarin, A. I.; et al. In Heat Transfer in Disperse System; Heat- and Mass Transfer, Pt. I, Vol. 6; A. V. Lykov Institute of Heat and Mass Transfer, Akad. Nauk BSSR Minsk, 1980; p 44. Cf. ref 28.
(28) Tamarin, A. I.; Galershtein, D. M.; Shuklina, V. M. J. Eng. Phys. 1982,42, 14. (29) La Nauze, R. D.; de Jung, K. In Proc. 7th Znt. Conf. Fluid. Bed Combust. 1983,1040. (30) Ross, I. B.;Davidson, J. F. Trans. Inst. Chem. Eng. 1982,60, 108. (31) Donsi, G.; Ferrari, G. Proc. 2nd World Congr. Particle Technol., Kyoto, Part 4 1990,254. (32) Baskakov, A. P., Ed. Processes of Heat and Mass Transfer in Fluidized Bed; Metallurgia: Moscow, 1978. Cf. ref 34. (33) Baskakov, A. P.; et al. Powder Technol. 1973,8,273. (34) Baskakov, A. P. In Fluidization, 2nd ed.; Davidson, J. F., et al., Eds.; Academic Press: New York, 1985; p 465. (35) Borodulya, V. A.;Kovensky, V. I. Znt. J. Heat Mass Transfer 1983,26,277. (36) Shirai, T.; et al. Kagaku Kogaku 1965,29,880. (37) Prins, W.; Draijer, W.; van Swaaij, W.P. M. In Heat and Mass Transfer in Fixed and Fluidized Beds; van Swaaij, W. P. M., Afgan, N. H., Us.; Hemisphere Publishing Corp.: Bristol, PA, 1986;p 317. (38) Vreedenberg, H. A. Chem. Eng. Sci. 1958,9,52. (39) Andeen, B. R.; Glicksman, L. R. ASMEIAZChE Heat Transfer Conf., St. Louis, MO 1976,76-HT-67. (40) Rios, G. M.; Gibert, H. In Fluidization VI; Kunii, D., Toei, R., Eds.; Engineering Foundation: New York, 1983;p 363. (41) Kharchenko, N. V.; Makhorin, K. E. Znt. Chem. Eng. 1964,4, 650. (42) Ziegler, E. N.;Brazelton, W. T. Znd. Eng. Chem. Fundam. 1964,3, 94. (43) Pillai, K. K.Lett. Heat Mass Transfer 1976,3, 131. (44) Botterill, J. S.M.; Teoman, Y.; Yuregir, K. R. AZChE Symp. Ser. 1981, 77 (208),330. (45) Shirai, T.; Hamada, T.; Udamura, T. Proc. 16th Fall Meet. SOC.Chem. Eng., Jpn. 1982,29. (46) . . Botterill, J. S.M.: Teoman,. Y.:. Yureair, - . K. R. Powder Technol. 1984,39,177. . (47) Borodulya, V. A.; et al. Znt. J. Heat Mass Transfer 1983,26, 1577. (48) Denroy, A. 0.0.; Botterill, J. S. M. Powder Technol. 1978,19, 197. (49) Baskakov, A. P. Int. Chem. Eng. 1964,4,320. (50) Koppel, L. B.; Patel, R. D.; Holmes, J. T. AZChE J. 1970,16, 456. (51) Kubie, J.; Broughton, J. Int. J. Heat Mass Transfer 1975,18, 289. (52) Gelperin, N. I.; Ainshtein, V. G.; Korotyanskaya, L. A. Znt. Chem. Eng. 1969,9,137. (53) Decker, N.; Glicksman, L. R. Znt. J. Heat Mass Transfer 1983, 26, 1307. (54) Glicksman, L. R.;Decker, N. Heat Transfer in Fluidized Beds of Large Particles. Report from Mechanical Engineering Department, MIT. Cambridge, MA, 1983. (55) Wen, C. Y.; Yu, Y. H. AZChE J. 1966,12,610. (56) Society of Chemical Engineering, Japan, Ed. Handbook of Chemical Engineering; Maruzen: Tokyo, 1988. (57) Society of Chemistry, Japan, Ed. Handbook of Chemistry; Maruzen: Tokyo, 1984. (58) Russell, H. W. J. Am. Ceram. SOC.1935,18,1.Cf. Perry, R. H., Chilton, C. H., Eds. Chemical Engineers' Handbook, 5th ed.; McGraw-Hill: New York, 1974;p 3-241. (59) Sala, A. Radiant Properties of Materials, Table of Black Body and Real Materials; Elsevier: London, 1986. Received for review May 30, 1991 Revised manuscript receiued November 22, 1991 Accepted December 10,1991
