= reaction rate, grams/gram catalyst-hour = reaction rate a t stationary point, grams/gram catalysthour = canonical axis, i, i = 1, 2, 3 vector of all of variables, Xi = partial pressure of hydrogen, atm. = partial pressure of n-pentane, atm. = partial pressure of 2-methylbutane, atm. = magnitude of variable x i at stationary point, i = 1, 2, 3 = coded values of variables x i defined in Table I, i = 1, 2, 3 = vector of components, x i a , i = 1, 2, 3 = total pressure, atm. = pressure level of Figure 1, atm.
be required that the attention of the development engineer could be taken from the other equally important engineering matters. However, now "canned" response surface programs exist which can be used with little time investment by the engineer. O n a digital computer, for example, the complete response surface analysis and a partial printing of contours of the response surface (such as that shown in Figure 3) were completed in about 1.5 minutes. Hence, response surface methods would not be a diversion, but rather an important tool to interact with and to guide the chemical engineering judgment in a modeling program within a process development study.
r rs
Nomenclature
literature Cited
B,,
Box, G. E. P., Biometrics 10,No. 1,16 (1954). Box, G. E. P., Hill, l V . J., Technometrim 9 No. 1 , 57 (1967). Box, G. E. P., Youle, P. V., Biometrics 11, No. 3, 287 (1955). Carr, N. L., Ind. Eng. Chem. 52, 391 (1960). Davies, 0. L., "Design and Analysis of Industrial Experiments," 2nd ed., Hafner Publishing Co., New York, 1960. Franklin, N. L., Pinchbeck, P. H., Popper, F., Trans. Inst. Chem. Em. 34. 280 (1956). Franglin, 'N. L.', Pinchbeck, P. H., Popper, F., Trans. Inst. Chem. Eng. 36, 259 (1958). Hunter, l V . G., Mezaki, R., A.I.Ch.E. J . 10,315 (1964). Kittrell, J . R., Hunter, W. G., Watson, C. C., A.I.CI2.E. J . 11, 1051 (1965). Kittrell, J. R., Mezaki, R., lb'atson, C. C., Brit. Chem. Eng. 11, No. 1. 15 11966a). Kittrell,' J. R., Mdzaki, R., \\'atson, C. C., Ind. Eng. Chem. 58, S o . 5, 50 (1966b). Mezaki, R., Kittrell, J. R., Ind. Rng. Chem. 59, No. 5, 63 (1967). Pinchbeck, P. H., Chern. Ene. Sci. 6,105 (1957). T.\'eller, S.,'A.I.Ch.E. J . 2, 53 (1956). Yang, K. H., Hougen, 0. A, Chem. Eng. Progr. 46, No. 3, 146 (1950).
= canonical parameters defining nature of the response surface in transformed coordinates, i = 1, 2, 3
6, = parameter values defining linear effects of the response surface, z = 0, 1, 2, 3 6,, = parameter va1ue.s defining the quadratic and interaction effects of the response surface, i = 1, 2, 3;. j = 1, 2, 3 f = abbreviation of the complete modelf(x; K) in Equation 6 thermodynamic equilibrium constant for reaction of npentane to 2-1nethylbutane, equal to 1.632 equilibrium adsorption constant for hydrogen, atm.-' equilibrium adsorption constant for n-pentane, atm.-' equilibrium adsorption constant for 2-methylbutane, atm.-l vector of all parameters in a model to be estimated from kinetic data forward rate constant, grams/gram catalyst-hour empirical constant in Equations 1 and 2 reaction order with respect to any component A reaction order with respect to any component B partial pressure of any component A partial pressure of any component B components of latent vectors, relating transformed coordinate system and untransformed system
xi x = XI x2 x3 Xia
Ri x, 7 r 7r1
RECEIVED for review June 17, 1966 RESUBMITTED December 28, 1968 ACCEPTED March 16, 1968
GAS-SOLID HEAT TRANSER IN FLUIDIZED BEDS R. S . M A N N A N D L . C. L . F E N G Department of Chemical Engineering, University of Ottawa, Ottawa, Canada
The unsteady-state heat transfer between gas and solid particles was studied in fluidized beds, 2 and 4 inches in diameter. A system of transient heating and cooling of glass beads, silica gel, and alumina between 130" and 288" F. was used. The effect of several variables, particle size (0.004to 0.241l inch), bed settled heights (0.8to 16 inches), particle densities (8.5 to 206 pounds per cu. foot), thermal conductivities [0.013to 1.8 B.t.u./(hr.)(sq. ft.)(' F./ft.)], and air velocities (0.543to 4.347feet per second) on the spaceaveraged heat transfer coefficient, U, was investigated. Using a new approach in interpreting the driving force, a correlation for two ranges of Reynolds numbers, 10 to 60 and 60 to 2200, was developed.
NE
of the inherent properties of fluidized beds is the high
0 rate of heat transfer between the beds and the fluidizing medium. The transfer of heat between particles, fluids, and the surfaces in contact with them is as complex in its facets, and even more so in its mechanisms, as the problems associated with the many phases of fluid flow in such two-phase systems. T h e phenomenon of heat transport in fluidized beds has been the subject of numerous studies (Zenz and Othmer, 1960). Basic equations for fluidization (Leva, 1959), heat transfer from bed to wall and vice versa (Heerden et al., 1951 ; Leva and FVeintraub 1949; Toomey and Johnstone, 1953), and heat transfer in pneumatic fluidized systems (Koble et al., 1951;
Richardson and Ayers, 1959) were comparatively well investigated. The single-par ticle technique (Johnstone et al., 1941), theoretical considerations (Zenz and Othmer, 1960), and the early investigations (Heertjes and McKibbins, 1956; Kettenring et al., 1950; Shakhova and Rychkov, 1957; Walton et al., 1952) did not provide enough information to understand the complex phenomenon of heat transfer in fluid-solid systems. More recent works (Ferron, 1961 ; Frantz, 1961; Fritz, 1956) have brought a new approach to the problem of gas-solid heat transfer in fluidized beds. However, the work of Frantz was limitvd to liquid fluidizing medium and that of Ferron and Fritz to one solid sample only. VOL. 7
NO. 3 J U L Y 1 9 6 8
327
I n the present investigations, heat transfer coefficients for gas-solid systems have been evaluated by introducing a preheated stream of gas into a bed of solids a t an initially low temperature (normally room temperatures) and recording the temperature-time relationships automatically and simultaneously during the heating and cooling of solid particles. With a more precise and accurate measurement, the transient system was used to determine the heat transfer coefficients as a function of superficial gas velocities, bed settled heights, and physical characteristics of solids, such as diameter, density, and thermal conductivity of particles. A correlation between over-all heat transfer coefficient as a function of various characteristic properties of the fluid-solid system circumvents some difficulties, with a view to reconciling past differences and bringing out a better understanding of the heat transfer mechanism and its use as a design parameter.
the flanges of the fluidizing column and the bottom disengaging section. The flanges were grooved at the junction so that the filter plate was fitted into the groove and prevented from moving. The column had a jacket of 6-inch diameter with a jacket space 1 inch in cross section. The jacket air stream was introduced at the bottom of the column through a 1-inch diameter inlet and exhausted at the top. To minimize elutriation, the top of the column was covered with a cap having two layers of screen with a '/r-inch empty space between, filled with glass fiber. Iron-constantan thermocouples, insulated with asbestos glass fiber and silicon, were placed a t several locations in the fluidized bed, in the fluidized column wall, inside the jacket space, and between the inner wall of the insulation and the jacket surface (Figure 2). A Phillips Model PR 3210 A/OO with 12 channels measured the temperatures at different locations. The temperatures were measured 5 minutes from the state of the run, and incorporated into an equation for evaluating the values of heat transfer coefficients. Results
Experimental
A schematic diagram of the apparatus used for heat transfer studies is shown in Figure 1. The apparatus consisted of four main sections: the compressor and the auxiliary metering devices for flow measurements, the filtering, drying, and preheating section, the fluidizing columns, and automatic temperature recorder, coupled with the system of thermocouples. Compressed air at a constant pressure of 120 p.s.i.g. was first passed through oil filters to remove the entrained oil droplets and then through a 4-inch by 3-foot silica gel dryer to remove excess moisture. The pressure-regulated air stream was then split into two streams, one for fluidizing the column and the second to be used as a bath in the jacket. The main stream was metered by one of the two rotameters; the smaller air stream to the jacket was measured by a 1/8-inch inside diameter orifice. Each air stream was introduced into a forced convection type finned tubular heater, which preheated the streams to the required temperatures. Two metal fluidizing columns were used, one 4 inches and the other 2 inches in diameter, installed in such a way that each could be operated independently of the other. The 4inch column was made up of three sections; the bottom part was a conical end, where the air was disengaged from the 3/4inch pipeline into a section of pipe 4 inches in diameter and 12 inches long, to ensure an even distribution of the air stream before entering the fluidized bed. Air then passed through a a/,-inch thick porous stainless steel plate having 20-micron openings of the pores. The porous plate was installed between
filter op
t
Seven samples of the three solids (glass beads, silica gel, and alumina) \+ere used. Characteristic properties of these solids are given in Table I. Different particle diameters a t different inlet temperatures and different bed settled heights were investigated. Particles were screen-analyzed and their diameters and the total area of the particles were evaluated from the equations:
Table 1.
DP? Particle Diameter, Solid Inch Glass beads 0,0039 0.0110 0.0185 Silica gel 0.06569 0.2324
Alumina
m
Glass
Characteristic Properties of Solid Particles
0,0164 0.0411
k8,
Lb./ Cu. Ft. 93.6 93.6 93.6 8.5
p8,
8.5
206 206
wool
Silica
gel
Gkss Wool
Figure 1. 328
l&EC
Schematic diagram of fluidization apparatus
PROCESS DESIGN A N D DEVELOPMENT
1.8 1.8
t II
I
G,>
B.t.u./)Hr.) B.t.u./ (Sq. Ft.) Lbb Mass ( " F./Ft.) F. 0.GO5 0.27 0.605 0.27 0.605 0.27 0.013 0.22 0.013 0.22
Exhaust
b
0.20 0.20
Shape Spheres Spheres Spheres Spherical
Granular Granular
5 minutes and temperature-time curves plotted for each run, which lasted for about 2 hours. Values of dT,/dB in Equation 8 were obtained by measuring the slopes of the curve a t time equal to 60 minutes. T h e value of heat lost by the fluidized bed ( q 1 ) was obtained by the equation :
e
The average temperature of bed and jacket, T j , culated from the empirical equations:
was cal-
>+
+ ATins.) + 0.33 T j ] + 0.382 T8
0.299 L1.66 (Troo,
0.356 [1.66 (Troom
+
+ 0.33 T j ] + 0.241 T ,
A?ins.)
(10)
(11)
and
4 n i c t r l section Figure 2.
represents the heat lost through the insulation. Equation 10 was obtained by combining various resistances and using the literature values of the thermal conductivities of insulation, jacket wall, and reactor wall (Ferron, 1961). T h e heat capacity of the jacket, C5, was determined by a blank run and evaluated by a heat balance equation around
Diagram of fluidized column
6 FVs At = -
and
PSDP
T h e heat transfer coefficient, U , was evaluated by making a time-dependent heat kialance around the fluidized bed as:
W&p,
d TS
= WJp,(Ti
- Tz) - 41 - q 2
(3)
Since no chpmical reaction is taking place, the heat of reaction ue uiscarded: The over-all heat transfer coefficient, U, can be written as: 2.
d T8 UA,AT = FVC -
Glass beads 0.0110"
d0
0
0
where
0.0039"
and
Combining Equations 3, 4, 5 and 6, we obtain
u = -0312 wc At(1
(1
- a)
+,
a)
At(1
+
29
TI - Ts)
and
1
t
I To evaluate the first term on the right-hand side of Equation 8, temperatures of the solid (T,) were measured a t intervals of
2 Vf
Figure 3.
,
l e x p t l . / lvs. a
3 fthec.
I
I
4
5
fluid superficial velocities
VOL. 7
NO. 3
JULY 1968
329
~~~
Table 11.
~
Effect of Mass Velocity on Heat Transfer Coefficient
Lo Inchei 5.2
Dm Inch 0.0039
Sample Glass beads
Silica gel
0.0039
6.9
0.0039
9.9
0.0659
6.8 10
0.2324
11.4 1.7
Alumina
0.0165
3.2 4.59
Glass beads
0.0185
2.25 5
0.011
2.7
x
103,
G,
B.t.u./(Hr.)
Lb./(Hr.) (Sq. Ft.) 452 61 6 813 450 580 580 813 452 500 810 573 761 830 580 568 960 650 960 830 760 830 640 450 1000 1777 2085 996 1500 1777 470 550 47 0 620 388 61 1 670 705 620
(Sq. Ft.)
E.) 6.3
the system without fluidizing the solid. The slope of the tangent to the curve (T,, By, UJ. 0) a t a time interval of 60 minutes and the value ofkins.LinB,= 0.0048
(0
2.37 1.30 3.10 1.97 2.00 1.30 3.13 2.70 1.22 2.30 6.25 2.97 1.20 1.37 1.27 8.43 6.40 7.30 5.80 5.60 8.00 1.50 1.15 5.76 3.48 1.10 1 .oo 8.30 9.10 9.00 8.40 6.80 6.98 6.40 7.80 8.16 11.10
B.t.u. were used. (min.)(sq.ft.) (OF.)
Average values computed from the middle portion of the run (0 = 60 minutes) were used to calculate U. At first it would seem that an average, either arithmetic or integral, of the entire run would be most convenient. However, the variation and thus the error involved in the beginning and end of the run were too great. The value a t 0 = 60 minutes was very close to that of the average of the middle 50% portion of the run. Lexpti values \\ere obtained from transparent columns and plotted against superficial velocities, resulting in different straight-line equations (Figure 3), used in predicting and extrapolating expanded bed heights. I n Figure 4 (obtained from Table 11) is shown the effect of mass velocity on heat transfer coefficient, with the bed settled heights as the parameters. The effect of bed settled heights is shown in Figure 5 , and its effect on the Nusselt group is shown in Figure 6. Figure 7 s h o w the dependency of Nus-
0-'
lo-'
-oGbssbeods=~" L
d2
Alumina m0.0164
am be&=OD185 o Alumina = a0411 A Silica pel =ao6@ 0
k R e = 580
0 G l w s bods = 0.0039" V Silica gel
A' ,,
* 0.06569"
,, = 0.2324"
A Glass beads = 0.01I"
&
A\Re=300-400
Alumina = 0.0164" 0 Glors beads =0.0185"
c
j3
I-
m
3
103
100
500
Figure 4. 330
l&EC
L,,
000
G Ibs/(k.)(sq. ft.) Effect o f mass velocity on heat transfer coefficient PROCESS DESIGN AND DEVELOPMENT
Figure 5. coefficient
feet
Effect of bed-settled height on heat transfer
selt group on the diameter of the particle and the thermal conductivity of the solid. In case of gas-solid fluidized systems, since there are several variables, the experimental data (Table 111) were correlated by use of the functional relationship Nu = f S e , Pr, L,, D,, k , P,)
T h e final correlations were in terms of equations. Reynolds numbers 10 to 100 :
(13)
Using dimensional analysis, Equation 11 was written as Reynolds numbers 60 to 2290 : T h e experimental data (Feng, 1966) covered Reynolds numbers from 10 to 2200. Since the entire range was very wide for accurate correlations, it was broken down into two ranges, 10 to 100 and 60 to 2200. Figure 8 shoivs the relationship between the modified Reynolds number,, fD *
and the PI
Discussion
dimensionless number,
T o derive a satisfactory correlation for heat transfer coefficient in terms of the parameters of the solid-fluid system, three simplifying assumptions were made.
,292
L
A
'O 10
1
-
t
-
I
1
1
l * * l I * '
100
1000
10000
Dv "f ff,
DP
Pi Figure 6. Correlation of (Nusselt number) (Prandtl number) (bed settled heights/diaineter of tube) with Reynolds number
,
teat
Figure 7. Dependency of (Nusselt number) (Prandtl number) on diameter of particles VOL. 7
NO. 3
JULY 1 9 6 8
331
~~
Table 111.
Data for Evaluation of Heat Transfer Coefficient.
ux No.
w,, Lb.
AT, Sq. Ft.
LlDt 4-inch column, glass beads, D, = 0.0039 176.0 166.0 169.3 83.3 3.307 185.5 181.0 148.2 101.0 3.307 194.5 200.0 194.8 123.0 3.307 183.0 174.5 184.0 109.0 3.307 173.0 168.0 169.0 83.6 4.409 188.0 183.0 183.6 100.6 4.409 193.0 193.0 190.2 114.2 4.409 191.0 184.5 185.2 105.7 4.409 178.0 168.0 170.0 90.1 6.614 188.0 186.0 185.0 101.0 6.614 191.0 186.0 190.0 117.5 6.614 179.5 180.0 176.8 98.5 6.614 168.0 160.0 162.5 89.0 8.819 8.819 186.0 186.0 183.8 109
h h h h h h h h h h h h h h
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.646 0.826 1.162 0.879 0.646 0.826 1.162 0.771 0.646 0.826 1.162 0.713 0,646 0.826
1.30 1.30 1.30 1.30 1.725 1.725 1.725 1.725 2.02 2.02 2.02 2.02 3.375 3.375
225.0 249.5 269.0 256 228.0 244.0 262.5 248.5 234.5 251 .O 267.5 244.5 234.5 254.0
176.0 194.5 223.0 206.5 181.0 196.5 210.0 199.5 179.5 200.0 211.5 193.0 168.0 198.0
h h h h h h h h
74 75 76 77 78 79 80 81
1.459 1.858 2.182 1.454 1.858 2.182 2.586 1.454
0.175 0.175 0.175 0.498 0.498 0.498 0.849 0.849
269.0 272.0 266.0 259.0 266.0 269.0 272.5 266.0
251.0 262.5 259.0 259.0 252.5 257.5 259.0 262.5
186.0 193.0 176.0 173.0 180.0 196.0 183.0 190.0
262.5 269.0 275.5 272,5 ADI.
252.5 259.0 266.0 262.5
2-inch column, 184.5 186.0 203.0 206.5 205.0 208.0 200.0 196
h h h h a
1.858 6,480 107 2.161 6.480 108 2.42 6.480 109 2.586 7.980 112 Complete data available from
Alumina, D, = 206.5 190.8 216.0 199.7 180.0 175.3 180.0 176.0 191.0 180.3 203.0 196.3 198.0 187 203.0 193.0
Although the thermal properties of gases, like specific heat, viscosity, and conductivity, increased with increased temperatures, the Prandtl number did not vary significantly when air was used. However, it was introduced to facilitate the use of correlation for different gases. There are neither any data nor any technique available as yet to measure the temperature gradients within single particles of smaller diameters. Since the particle diameters were very small, it has been assumed that the temperature gradients within the particles are negligible compared to the surface film gradients. The correlation has been obtained by dimensional analysis. Although it is not the best method to employ, it is the most appropriate a t the moment for the kind of data involved. The bed temperature is still a matter of controversy. Kettenring et al. (1950) assumed that it was uniform and equal to the equilibrium temperature of the gas leaving the fluidized bed. However, in the present investigations, a screened thermocouple placed a t the exit of the fluidizing column showed considerable variations from temperatures indicated by the thermocouples located within the threshold of expanded bed height. Furthermore, even if the assumption was justified, the exit temperature, Tz, could not be measured accurately. For this reason Tz was not used directly in the present investigations, but was incorporated into the calculations only to obtain the mean temperatures. Walton et al. (1952) assumed the bed temperature to be that I&EC PROCESS DESIGN A N D DEVELOPMENT
0.0411 125.9 132.4 99.4 90.6 96.8 103.4 112.4 108.8
alumina, 184.3 204.2 206.0 197.3
The Prandtl number of the gases did not vary significantly over the temperature range studied. Temperature gradients within the particles were negligible compared with the film gradients. Incomplete but sufficient particle mixing occurred, so that a t any time the temperature of the solid was uniform throughout the bed.
332
103,
B*t.u*/(Hr*) (Sq. Ft.) B.t.u./ Min. ( " P.) qina.,
D, = 26.7 21.1 22.2 23.2
inch 0.551 0.551 0.551 1.653 1.653 1.653 2.755 2.755
inch 0.769 0.700 0.331 0.606 0.756 0.606 0.550 0.770 0.031 0.644 0.594 0.794 0,769 0.900
0.781 0,688 0.419 0.663 0.756 0.675 0.731 0.750 0.856 0.688 0.650 0.613 0,556 0.713
652.3 652.3 652.3 652.3 869.6 869.6 869.6 869.6 1304.5 1304.5 1304.5 1304.5 1739.5 1739.5
0.679 0.463 0.244 0.368 0.375 0.375 0.250 0.275
0.713 0.650 0.563 0.556 0.650 0.593 0.488 0.381
4.7 4.7 4.7 14.1 14.1 14.1 14.1 23.4
0.0596 31.31 0.1112 25.64 0.1125 42.11 0.1385 25.52 0.2108 31.32 .. .~ 0.2785 33.36 0.3754 18.96 0.4313 125.64
0.5375 0.6125 0.5063 0.3500
9.376 9.376 9.376 12.19
0.6713 0.5304 0.5582 0.5833
0.041 1 inch 1.102 0.1125 1.102 0.0812 1.102 0.1429 1.433 0.1375
1.341 1.626 1.978 1.754 1.951 2.643 2.999 2.774 2.374 2.666 3.099 2.599 2.329 2.853
6.30 1.39 ~. 1.36 2.38 3.06 1.97 1.28 2.04 3.13 1.39 1.22 2.76 1.56 2.62
59.16 16.20 ~.~.
41.63 24.42
indicated by a bare thermocouple inserted in the bed. This would give a n intermediate value between that of the solid and fluid. Wamsley and Johanson (1954) obtained the bed temperature indirectly by calculating it from time-temperature history curves of the inlet and outlet gas temperatures. Since the outlet gas temperature was uncertain, the values propagated from the curves were also uncertain. Here, although heat losses were minimized experimentally, no correction terms were involved in the calculations. Heertjes and McKibbins (1956), Rozenthal (1958), and Shakhova and Rychkov (1957) used bare thermocouples, which indicated only intermediate values, to indicate the temperature of the bed. The thermocouples were placed a t different heights above the fluidized threshold, giving a n intermediate value between that of solid and fluid. T o circumvent these difficulties, space-averaged solid temperature, T, was used in the present work. I t was measured by screened thermocouples inserted in the fluidized core, to avoid the possibility of errors due to the collision of cooler particles from the lower regions of the bed with the sensing element. The solids temperature was thus assumed to be in equilibrium with the surrounding fluid. Since there was no possibility of complete mixing, the temperature of the gas throughout the bed could not be the same as that of the outlet gas. Hence, a simple difference of TZand T,(Tz - T,)cannot be used for the driving force. A better way to obtain the value of AT would be from Equation 5. Fritz (1956) indicated that heat transfer data could be better correlated if the driving force AT was defined according to Equation 5. For measuring the gas temperature within the bed, Kettenring used a bare thermocouple, which gave a value somewhere between the temperature of the solid and of the gas. Walton et al. (1952) used a suction thermocouple to measure the temperature of the gas. These measurements could not
Figure 8.
Correlation of heat transfer coefficient against Reynolds number
VOL. 7
NO, 3
JULY 1 9 6 8
333
be considered very reliable, as the flow pattern of the fluid past the particles varied considerably. Wamsley and Johanson (1954) measured the gas temperatures a t the inlet and outlet of the apparatus. Since the outlet temperature was not taken a t the expanded bed height, it could not be considered as truly representing the gas temperature. Rozenthal (1958) and Shakhova (1957) used a bare thermocouple to measure the inlet temperatures of the gas. This temperature was not spaceaveraged and thus could not be considered as truly representing the gas temperature of the fluidized bed. In the present investigations, although the data lvere taken under conditions of considerable turbulence, some incomplete mixing may have occurred. Hence a flow pattern of two phases, dense and dilute, had been assumed, and the temperature driving force, AT, based on this assumption had been calculated from Equation 5. I t is seen from Figure 4 that U was inversely proportional to the mass flow rate for different samples. For the order of magnitude of the mass flow rate used in this investigation, the film theory is applicable. b'hen a particle moves toward the heated area in the bed, the surface of the gas film surrounding the particle becomes thermally in equilibrium uith the temperature of the fluidizing gas. Heat then further penetrates toward the particle by conduction. Since the motion of the particle is very fast, its residence time in the heat transfer region is extremely short; hence, the penetration of heat is very small. When the particles leave the hot zone, the remainder of the heat is transferred back into the gas stream, partly by conduction and partly as a result of attrition of the outer layers of the gas film. The amount of heat transferred to the particles is small and comparatively less than transferred in a steady-state system, because of the low heat capacity of the gas film. I n
such a process, Reynolds number plays a vital role, since it influences the thickness of the gas film, the velocity of the particles, and rate of heat transfer into and out of the gas film. The application of the film theory to explain the phenomenon does not seem to justify the use of a fluidized bed for better heat transfer. LVith the least amount of attrition, a fixed bed would seem better for heat transfer, and the order of magnitude of the mass flow rate has a great deal to do with this anomalous behavior of U . The relationship between over-all heat transfer coefficients and bed settled heights a t various Reynolds numbers is shown in Figure 5. The heat transfer coefficient varies inversely with the bed settled heights; an increase in bed settled height implies an increase in apparent heat transfer area. Thus if the inlet temperature of the gas and the amount of heat transferred are kept constant, U changes inversely with the bed settled height, in conformity with the experimental findings. Figure 7 shows that the heat transfer coefficient, U, expressed in terms of Kusselt number increased with the increased particle diameter. This is consistent uith the findings of M'amsley and Johanson. I t \rould be expected that the larger the particle size,D,, the smaller would be the surface area, A t (Equations 1 and 2). From an empirical point of view, the larger the particle size, the fewer Ivould be the number of particles in the same Lveight of sample (had the sample consisted of small particle sizes) and the smaller the surface area of solids. Since C is inversely proportional to the surface area in heat transfer, it would increase with increased particle sizes. Figure 9 compares the present investigations with others. The present data indicate much lower values of heat transfer coefficients than those of Kettenring et al., Heertjes et al., and Walton et al. This is perhaps due to the difference in the
2
G, LBS./FT.-HR 0
400 I
53
2000 4ooo
800 "
100
~
I
'
~
'
I
"
'
150
200
G,
LBSIFTPHR.
6000 "
'
"
10000 12000 14000 1600C
eo00 "
'
"
~
'
~
'
*
'
~
~
'
~
~
'
250
Figure 9. Over-all heat transfer coefficients vs. mass velocity of fluidizing gas for various investigations 334
I & E C PROCESS D E S I G N A N D DEVELOPMENT
'
-
method of measuring .the temperature, the different mass velocities used, and the different approach. Though higher values of U are indicated from iour data than those obtained by Fritz and Ferron, the trend is more or less similar. The data of Fritz and Ferron were obtained for one fluidizing material (aluminasilica) only and for much loiver mass rate of flow of gas. The present data included a variety and sizes of glass beads. Acknowledgment
The authors are grateful to the National Research Council of Canada and the Ontario Research Foundation for financial assistance. Nomenclature
At
CP, C,,
D, DPj
D, GI kj,
k,
Lelptl. L,
= = = = = = = =
T,, T , T,, Bv.
= = = = = =
TI, Tp U
= =
W,
= = =
41 42
TV,
Xi
e Pa
= =
surface area of entire solid bed, sq. ft. specific heat of fluid, B.t.u.,/(lb. mass)(” F.) specific heat of solid, B.t.u./(lb. mass)(’ F.) particle diameter particle diameter of ith weight fraction of solid tube diameter, ft. fluid mass .velocity, lb./hr. sq. ft. thermal conductivity of fluid and solid, respectively, B.t.u./(hr.)(sq. ft.)(’ F./ft.) expanded bed height, ft. bed settled height, ft. rate of heat loss from fluidized bed, B.t.u./hr. rate of heat loss due to reaction, B.t.u./hr. temperature of fluid and solid, respectively, ” F. mean temperature of room? jacket, reaction hardware, arid insulation, ’ F. inlet and cutlet temperatures, ’ F. space-averaged over-all heat transfer coefficient, B.t.u./(hr.)(sq.ft.) F. fluid flow I ate, Ib./hr. weight of solid fluidized, lb. Tveight fraction of particle passing through ith opening of screen time, minutes particle density, Ib./cu. ft.
literature Cited
Feng, L., Ph.D. thesis, University of Ottawa, 1966. Ferron, J. R., Ph.D. thesis, University of TYisconsin, 1961. Frantz. J. F., Cheni. Eng. Pragr. 57, 35-42 (1961). Fritz, J. C., Ph.D. thesis, University of \Yisconsin, 1956. Heerden, C. Van, Nobel, A. P. P., Krevelen, D. 1%’.Van, Chem. Eng. Sa. 1 (2), 51-66 (1951). Heertjes, P. M., McKibbins, S. TV., Chem. Eng. Sa. 5 , 161 (1956). Johnstone, H. F., Pigford, R. L., Chapin, J. H., Trans. Am. Inst. Chem. Eners. 37. 95 11941). Kettenringr K. N:, Mander’sfield, E. L., Smith, J. M., Chem. Eng. Progr. 46, 139 (1950). Koble, R. A,, Ademino, J. H., Bartkus, E. P., Corrigan, T. E., Ckem. Eng. 58, 174 (1951). Leva. M.. “Fluidization.” McGraw-Hill. New York. 1959. Leva; M.; Tt’eintraub, Grumrner, M., Chem. Eng. Progr. 45, 563-72 (1949). Richardson, 3. F., Ayers, P., Trans. Znst. Chem. Engrs. 37, 314-21 (1959). Rozenthal, E. O., Teplo-i Massoobmen V Protses. Ispareniza, Akad. n h u k S S S R , Energet, Znst. 1958, p. 87. Shakhova, N. A., Rychkov, A. I., Tr. Moskoc. Znst. Khim. Mashinostroeniva 12. 119 119.37). Toome;, R. ’D., Johnstone, H. F., Chem. Eng. Progr. Symp. Ser. 5 , 49, 51-63 (1953). TValton, J. S., Olson, R. S., Levenspiel, O., Znd. Eng. Chem. 44, 1474-80 (1952). TYamsley, \Y. TV., Johanson, L. N., Chem. Eng. Progr. 50, 347 (1954). Zenz, F’, A , , Othmer, D. F., “Fluidization and Fluid Particle System,” pp. 433-7, Reinhold, New York, 1960. RECEIVED for review October 17, 1966 RESUBMITTED hTovember 27, 1967 ACCEPTEDApril 15, 1968 Material supplementary to this article has been deposited as Document No. 9977 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, TVashington, D. C . .4 copy may be secured by citing the document number and by remitting $1.25 for photoprints of $1.25 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress.
PREDICTION OF VAPOR COMPOSITIONS IN VAPOR=LIQUID SYSTEMS I V A R S N E R E T N I E K S
Department of (,’hemica1 Engmeering, Royal Institute of Technology, Stockholm 70, Sweden
A method of calculating vapor-liquid equilibria in multicomponent mixtures is described.
No experimental information of multicomponent mixtures is necessary. The data needed can b e obtained from the binaries. Two models were used to calculate the activity coefficients: the Wilson equation and the Black equation. The Black equation has been extended in order to make it possible to extrapolate to other temperatures. Parameters in the equations are calculated by a nonlinear least-squares fit of observed data. The efficiency of the method when using the Black and Wilson equations in predicting vapor-liquid equilibria has been compared. The efficiency in predicting equilibria at other temperatures, some more than 40” C. from those where datlo was obtained, has also been investigated. The Wilson equation has been more accurate in nearly all cases studied. separation processes, reliable data on equicompositions are needed for good design. For binary systems data are usually available, but at some other temperature. I t is therefore convenient to have a mathematical model \vhich can predict compositions in one phase a t pressures, temperatures, and compositions in the other phase, using data a t other pressures, temperatures, and compositions. N VAPOR-LIQUID
I librium
For multicomponent systems, data are scarce and much experimental work is needed to obtain them. Methods whereby data for a multicomponent mixture can be calculated from binary data only are therefore of great value. As data for different binaries have seldom been determined at the same temperature, the mcthod has to predict the temperature dependence as well. VOL. 7
NO. 3
JULY
1968
335
