Fluid-to-Particle Heat Transfer in Fluidized Beds D. Bhaltacharyya and D. C. T. Pei* Department of Chemical Engineering, Universify 01 Waterloo, Waterloo, Ontario, Canada
The fluid-to-particle heat transfer in fluidized beds was investigated for different particle sizes, shapes, densities, and thermal conductivities under unsteady-state conditions. A heating technique via microwave power was developed in achieving constant uniform solid temperature within the bed. The uniformity of bed temperature successfully eliminated the heat transfer through the particle-to-particle mode. Therefore, the transient measurements were carried out by merely cutting off the power without affecting the fluid flow. The parameter Ar/Re2, defined as the ratio of gravity force to inertia force, was found as the controlling parameter for heat transfer in fluidized systems. A correlation was presented expressing jh as a function of Ar/Resmz. The data were found in good agreement with those reported in literature.
Introduction There has been much work on fluid-to-particle heat transfer in fluidized beds for a wide range of Reynolds number (0 < Re, < 10,000). Extensive reviews of such data are available in publications by Barker (1965) and Bhattacharyya (1973). In spite of many such studies on fluid-to-particle heat transfer, the results of one range of Reynolds number are often in disagreement with those in the other region. Most of these variations can, however, be attributed to the inaccurate evaluation of the gas and solid temperatures, improper design of experiments, and to the fluidizing characteristics of the bed. In order to evaluate the fluid-to-particle heat transfer coefficient both steady- (Bradshaw and Myers, 1963; Eichhorn and White, 1952; Ferron and Watson, 1962; Frantz, 1961; Juveland, et al., 1964; Kettenring, et al., 1950; Lovell and Karnofsky, 1943; Walton, et al., 1952) and unsteady- (Chang and Wen, 1966; Wamsley and Johanson, 1954; Wen and Chang, 1966) state techniques were used. In the steady-state method, usually the average value of the heat transfer coefficient was calculated assuming that the outlet gas temperature was equal to the bed temperature (Frantz, 1961; Kettenring, et al., 1950). Sometimes the bed temperature was assumed constant, which is different from the inlet and outlet fluid temperatures. In these cases, log (mean temperature difference) was used in the calculation of heat transfer coefficient (Bradshaw and Myers, 1963; Eichhorn and White, 1952; Juveland, et a l , 1964; Lovell and Karnofsky, 1943; Walton, et al., 1952). The solid temperatures were recorded either by inserting bare thermocouples within the bed or by using an optical pyrometer. It is obvious that the temperature showed by a bare thermocouple would neither indicate the solid nor the fluid temperature. Such methods, therelore, involve great error in the solid temperature measurement. In the unsteady-state technique, the solids were tagged by the thermocouples. Hence, the solid and the surrounding gas temperatures were measured simultaneously (Chang and Wen, 1966; Wen and Chang, 1966). Timetemperature profiles were recorded in the cooling cycle by passing cold gas through a previously heated bed of particles. It is obvious that the solid temperature measurement by directly inserting a thermocouple into it is definitely a better approach compared to that by introducing a bare thermocouple directly into the bed. Moreover, the unsteady-state technique provides better estimation of the heat transfer coefficient due to the fact that both the gas and solid temperatures are measured locally. In a fluidized bed, it is established that total heat transfer can be divided into three different modes, namely
fluid-to-particle, particle-to-particle, and bed-to-wall. In the literature, most of the experimental studies were limited to the measurements of the total heat transfer though in some cases it was intended to measure the fluid-to-particle heat transfer only. This was due to the fact that it was very difficult to design the experiments to investigate specifically any selected mode independently. Chang and Wen (1966) and Wen and Chang (1966), for the first time, designed experiments to measure the fluid-to-particle and particle-to-particle heat transfer coefficients, respectively, under transient conditions. However, in all these experiments in heating the solids by hot air and subsequently cooling them by switching to cold air some error is involved due to inertia effects of the flow system. This difficulty in switching from the heating to the cooling cycle can be eliminated by a noncontact heating of the solids via an external source of microwave radiation which only heats the solids but not the fluid. In this way the cooling of the solids can be achieved by simply cutting off the radiation source alone without affecting the fluid flow. In addition, heating by microwave power may offer the following advantages (Bhattacharyya, 1973; Bhattacharyya and Pei, 1973): (i) uniformity in heating of the individual solids; and (ii) constant solid temperature irrespective of the location within the bed. The particle-to-particle heat transfer will, therefore, be eliminated by heating the bed of solids of a uniform size to a constant temperature via microwave power. Then, unsteady-state measurements can be obtained by subsequently cooling them. Due to the uniform size of the particles in the bed even in the cooling cycle all the solids will maintain the same surface temperature and consequently the particle-to-particle heat transfer will be eliminated. Moreover, adiabatic wall conditions will ensure no heat loss through the wall. In the present case, it is therefore intended to measure the exact value of fluid-to-parti.cle heat transfer coefficient following the unsteady-state technique (Chang and Wen 1966; Wen and Chang, 1966) using microwaves as the heating source. Experimental Equipment and Procedure The experimental equipment utilized was a microwave power generator (Eimac, 2.5 kW, 2450 f MHz) connected to a levelled rectangular waveguide network with the test section placed vertically through it (Bhattacharyya, 1973). A schematic diagram of the overall unit is shown in Figure 1. Microwave power was transmitted in a oncethrough process to heat up the bed of solids. Compressed air supplied a t a constant pressure of 45 psig was used as the source of the pneumatic system. Ind. Eng. Chem., Fundam., Vol. 13, No.3, 1974
199
The boundary conditions are
2
U
E
Figure 1. Schematic diagram of the experimental setup.
Through a stabilizing tank the air was then fed into the 2-in. diameter test section through a 15 ft high calming zone of 2-in. i.d. copper tube. To avoid any transmission of vibration the test section was connected to the supply air line with a rubber hose. The test section was made of a vacuum-jacketed Pyrex tubing since Pyrex is transparent to microwave radiation. A special glass to metal connector was designed to connect the metallic pneumatic line with the test section. A sectional diagram of the test section together with the screen and its support is shown in Figure 2. The glass tubing within the waveguide was partially filled with the solids (Fez03 pellets) to form the bed. Some of the solids were then drilled and thermocouples were inserted into the center of the pellets. The hole was later filled up by Fez03 paste. These tagged solids were introduced at various locations inside the bed and were provided with enough thermocouple wire so that under fluidization they could move freely within the bed. The surrounding fluid temperature was measured by inserting a separate covered thermocouple in the approximate vicinity of the particles. These thermocouples were placed inside glass capillaries with the tips just lying within the tube. Thus the thermocouple tips did not touch the solids and measured the gas temperature. In all cases they were made of copper-constantan wires (Thermo-Electric TG30-T) . All the thermocouples were specially designed and connected to filtering circuits to eliminate the ac field pickup due to the presence of microwave radiation (Bhattacharyya, 1973; Bhattacharyya and Pei, 1973). An adiabatic wall condition was ensured by comparing the bed temperature with the inside wall temperature. The inside wall temperature was measured using an infrared ray pyrometer focused on a small platinum coating attached to the inside of the wall. The bed was then fluidized with the introduction of air and the expansion of the bed was measured. After the steady state was attained for the air flow rate as well as the solid and surrounding gas temperatures, the microwave was cut off and the cooling temperature profiles of the tagged solids and the surrounding gas with time were recorded. The above procedure was repeated for different fluid flows for various particle sizes, shapes, densities, and thermal conductivities. The ranges of the investigated parameters are presented in Table I. Analysis The fluid-to-particle heat transfer coefficient, hf,, was calculated from the time-temperature profiles of the solid and the surrounding gas. The elaborate mathematical development is available in the literature (Bhattacharyya, 1973; Chang and Wen, 1966; Wen and Chang, 1966). The dimensionless energy equation of a fluidizing solid is 200
Ind. Eng. Chem., F u n d a m . , Vol. 13, No. 3 , 1974
In the above equation Ts(l, T ) and T ~ ( T are ) the transient temperatures of the solid (at the surface) and fluid, respectively. From eq 1-4, the solid temperature at any time for 7 0 (at the center) is obtained by Laplace transformation (Carlslaw and Jaeger, 1959)
-
rn
TAO.7) = 1
+ 2 N u , C exp(-A,"T) x [An2 + (Nu, - 112] sin A,, [A,' + Nu,(Nu, - l)] n=l
where An are the roots of eq 6. Table I. Range of Parameters of the Present Investigation
Material Diameter, in. Shape
Fe203pellets 0.126, 0.2
Thermal conductivity, Btu/(hr f t O F ) Fluid temperature, Solid temperature, Specific gravity of solids Porosity
OF O F
Velocity, ft/sec Reynolds number, (DsHf/pf)
Modified Reynolds number, [D,Gf/pf (1 - 4 1 Archimedes number,
[D,%P f ( ( f s Ar/ReSm2
Spherical (0.126 in), cylindrical ( 0 . 1 in diameter = height) 0.2830 (spherical 0.126 in. and cylindrical 0.2 in.), 0.3410 (cylindrical 0.2 in. compressed) 80.1-118.4 81.5-163.1 2.0433 (all sizes), 2.1644 (cylindrical 0.2 in. compressed) 0.480 (0.2 in. compressed), 0.600 0.2 in.), 0.723 (0.126 in.) 10.70-16.65 471-1553 1701-3608 1.78 X
lo6 t o 1.6
Pf)/@fZl
0.332-3.54
4
p c
Figure 2. Diagram of test section.
-
i
X lo7
Table 11. Present Data
Spherical 0.126 0.126 0.126 0.126 0.126 0.126
0.723 0.723 0.723 0.723 0,723 0.723
471 491 518 533 578 648
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200
0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0.600
948 1002 1068 1113 1169 1227 1241 1274 1328 1374 1447
0.200 0.200 0.200 0.200 0.200 0,200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0,200
0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480 0.480
1107 1140 1165 1196 1207 1267 1268 1309 1327 1369 1395 1401 1426 1498 1556
16.73 18.43 17.57 16.73 19.32 19.32
0.0387 0.0410 0.0370 0.0342 0.0366 0.0326
1.78 1.80 1.80 1.79 1.82 1.82
0.62 0.57
0.0586 0.0580 0.0549 0.0506 0.0492 0.0478 0.0509 0.0443 0.0425 0.0410 0.0421
13.85 14.08 14.93 14.71 14.91 14.95 14.87 14.80 14.76 14.77 14.90
2.46 2.24 2.09 1.90 1.74 1.59 1.54 1.46 1.34 1.25 1.14
0.0627 0.0632 0.0630 0.0581 0.0554 0.0568 0.0537 0.0501 0.0543 0.0479 0.0470 0.0477 0.0467 0.0438 0.0422
15.97 15.98 15.92 16.08 16.01 15.91 15.92 16.00 15.98 15.98 15.94 15.97 16.00 16.02 15.98
3.52 3.32 3.17 3.04 2.97 2.68 2.68 2.52 2.45 2.30 2.21 2.20 2.13 1.93 1.78
0.51
0.48 0.42 0.33
Cylindrical 49 06 52.90 52.90 50.95 51.91 52.90 57.01 50.95 50.95 50.95 54.92 I
Cylindrical compressed
A, cot A,
=
1 - Nu,
62.55 64.94 66.17 62.55 60.24 64.94 61.39 59.12 64.94 59.12 59.12 60.24 60.24 59.12 59.12
6)
08
E
0480-0723 1 PRESENT DATb 2 BRADSHAW 8 MYERS 119631 3 CHANG 8 WEN (19661 042-0879 4 DE ACETIS a THODOS ii96o1 o 482 5 LEVENSPIEL 8 WALTON 1 1 9 5 4 1 0 4 2 -0862 6 LINDAUER 119671 7 MCCONNACHIE 8 THODOS 119631 0 600 a ROZENTAL 1 1 9 6 2 I 0 576 GUPTA a THODOS ( 1 9 6 3 1 ON, OLSON a LEVENSPIEL 119521
-
Results The comparison of the present data with the literature, in the form of jh us. Re, as shown in Figure 3, indicates that there exist wide variations in values and trends of jh. The results of Bradshaw and Myers (1963), and Chang and Wen (1966) are used for further analysis. Other data available in the literature are not used as this investigation represents only fluid-to-particle heat transfer. The data are first presented in Table I1 and first plotted in the form of (jh)f,/(Ar)0.2 u s . Resm as suggested by Chang and Wen (1966). Their correlation
D., In 0 1 62 06--00323 5
01875-025
0626 0003-0174
- 0019 -0125 _ _ -
-
0625
o oza
COMPRESSED. CYUNDEF
for 900 < Resm < 40,000 and 5.86 x lo6 < Ar < 2.55 X l o 7 includes the data of Bradshaw and Myers (1963). The data show fairly good agreement with their correlation as seen in Figure 4. However, the trend appears to be different. This deviation in trend may not be very prominent (Figure 4) due to the limited range of Resm (1700 < Resrn < 3600). With close inspection it is obvious that the dependence of the j,, factor on the Archimedes number could be higher than that of (Ar)0.2as suggested by Chang and Wen (1966). The present data are not correlated in a similar regression form as shown in eq 7 . Instead, a correlation is attempted in relation to the physical behavior of a fluidized bed. It is realized that a specific ratio of indices in Ar and Re is actually more important than attaching separate indices to them on the basis of a regression analysis. Since the j factor in heat transfer, which is defined as Nu/(RePrf’ 3 ) , includes the viscous force term, the ratio of gravity force to inertia force, defined qualitatively by Ar/Re2 (Boucher and Alves, 1963), will be the important
/ 0 01
‘21Dp=0160in1
1
I02
104
Figure 3. Correlation of literature with experimental data: j , us. Re,.
correlating parameter in such fluidized systems. In addition, the Archimedes number may also be defined as C&e2 so that the ratio Ar/Re2 simply becomes the drag coefficient, CD. This is due to the fact that the gravity force is balanced by the drag force in a fluidizing system. Considering Ar/Resm2, ie., C D ( ~ as the dominant parameter the present data are correlated as
For 0.33 < Ar/Resm2
