Ind. Eng. Chem. Process Des. Dev. 1986, 25, 849-854
shown against the calculated heat loss curves in Figure 8. The temperature curves with heat shield have been calculated based on the assumption that the kiln shell temperatures would remain unchanged. Since the rate of infiltrated air through the annular space varies depending upon the pressure drop and geometrical configuration, the skin temperature of the kiln will change to some extent accordingly. Therefore, the computed values of heat losses from the kiln (400 OF) with heat shield are used for the computation of energy savings instead of the measured values. This will ensure us the conservative side of economic analysis. Combined annual energy savings contributed by the heat shield amount to $34940 ($25 170 from the reduction in convection and radiation losses plus $9770 from the reduced infiltrated air) for a shell temperature of 400 O F in the absence of wind. Contribution of the reduced infiltrated air accounts for 28% of the total energy savings. Rates of energy losses a t various wind velocities are calculated with the wind blowing along the kiln. Fiftypercent of the initial heat loss can be recovered with this simple heat shield. The measured value of energy savings is greater than the calculated value due to the cooling effect of the infiltrated air, and the payoff is generally less than a year. The heat shield also provides a safer working environment around the hot kiln. The shell temperature dropped considerably with the shield, and practically no maintenance work has been required.
Acknowledgment This paper is written in memory of the late Reinert Kvidahl, Resident Manager of FMC Corp., Green River Plant. He was a model chemical engineer who shared the joy of success and responsibility of failure. We thank John W. Coykendall, Robert S. Simokat, James Taylor, and Marc E. Bowman for helpful information.
Nomenclature A = surface area D = kiln diameter d, = air density, lb,/ft3 d, = water density, lb,/ft3 dP = pressure drop due to heat shield, in. of water F = geometric and emissivity factor Gr = Grashof number g = gravitational acceleration, ft/s2 h = heat-transfer coefficient, Btu/(ft2 h OF)
Fluid Dynamics of Gas-Liquid-Solid
849
K = velocity pressure constant (0.93for entry; 0.9 for right angle turn; 0.56 for contraction; and 0.7 for sudden expansion) k = thermal conductivity, Btu/(ft h O F ) L = kiln length under consideration, ft Pr = Prandtl number Q = heat-transfer rate, Btu/h Re = Reynolds number T,= cold absolute temperature, O R Th = hot absolute temperature, O R t, = cold temperature, O F th = hot temperature, O F V = average air velocity, ft/s VP = velocity pressure, ft of flowing fluid Greek Symbols e = emissivity u = Stephan-Boltzmann constant, 0.171 for eq 5 and 10, Btu/(h ft2 OR4) Subscript a = ambient air b = infiltrated bulk air in annular space c = convection e = air entering kiln k = kiln m = mass r = radiation s = shield w = water - = direction of heat flow Literature Cited Blrd, R.; Stewart, W.; Llghtfoot. E. Transport Phenomena; Wiley: New York, 1960; Chapter 6 . Clarke, L.; Davkison, R. Manual for Process Engineering Calculations, 2nd ed.;McGraw-Hill: New York, 1962; p 59. Fan Engineering, 7th ed.; Jorgensen, R., Ed.; Buffalo Forge: Englewood Cliffs, NJ. 1970; p 71. Kim, N. “Heat Shield”; Memo to G. Peverley and R . waggener; FMC Corp.: Green Rlver, WY, Nov 17, 1969. Krelth, F. Prlnclpbs of /feat Transfer; I E P New York, 1973; Chapters 7-9. Myers, A.; &Mer. W. Introductlon to Chemlcal Engineering and Computer CompufaHons; Prentice Hell: Englewood Cllffs, NJ, 1976. Peray, K.; Waddell, J. The Rotary Cement Kiln; Chemical Publishing: New York, 1972; Chapter IV. Chemical Englneers’ Hendbook. 4th ed.; Perry, J., Ed.; McGraw-Hill: New York, 1976; pp 23-66. Plppltt, R. Chem. Eng. Prog. 187& 72(2). 41. Suryanarayana, N. V.; Scofleld, T.; Kleiss, R. E. Trans. ASME 1989, 105. 519-526. Vailant, A. Kiln Operation Optimization and Pollution Abatement; Center for Professional Advancement (CPA): Englewood Cliffs, NJ, 1975. Wendt, J. Applied Combustion Technology; CPA: Englewood Cliffs, NJ, 1978.
Received for review August 27, 1984 Revised manuscript received March 8, 1985 Accepted March 4, 1986
Fluidized Beds
Enrique Costa,’ Antonlo de Lucas, and Pedro Garcia Departamento de Ingenie& Qdmica, Facuhd de Cienclas Qdmicas, UniversMad Complutense, 28040 Madrid, Spain
Correlations for prediction, holdups, porosity, pressure drop, and minimum fluidization velocity in a three-phase fluldization bed with cocurrent flow have been deduced. Two systems were considered for derivation of these correlations: a first system formed by the liquid and the gas (homogeneous flow model) and a second system formed by these two fluids and the solid (drift flux model).
Fluidized beds are widely used as a solid-fluid contact method especially in processes that involve large temperature changes or frequent regeneration of the solid (Van 0196-4305/86/1125-0849$01.50/0
Landeghem, 1980). The expression three-phase fluidization is used to describe fluidization of solid particles by two fluids (IZlstergaard, 1971). Gas and liquid are the 0 1986 American Chemical Society
850
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 4, 1986
-
Figure 1. Diagram of the experimenta !tup: 1,manometric tubes; 2, pressure taps; 3, column; 4, valves; 5; orifice meters; 6, centrifugal pump; 7, reservoir; 8, gas bottle; 9, pressure regulator; 10, sandwich grid.
fluidizing media used in many industrial applications of three-phase fluidization as in catalytic hydrogenation processes, hydrocracking and desulfuration of petroleum fractions, coal hydroliquefaction, Fischer-Tropsch synthesis, fermentations, water treatment, pelletizing operations, etc. (Kim et al., 1975; Begovich, 1978; Soung, 1978; Shah, 1979; Kono, 1980). The principal type of gas-liquid-solid fluidization corresponds to the cocurrent gasliquid flow in which the liquid is the continuous phase and the gas is the disperse phase bubbling in the liquid (Epstein, 1981). An appropriate design of a three-phase fluidized bed requires the estimation of various fluid dynamic parameters as holdups, porosity, pressure drop, and minimum fluidization velocity. 0stergaard (1971), who has been a major contributor to this subject, has presented a review on the fluid mechanics of three-phase fludization, and a great deal of work has been done in the last years in order to obtain correlations for evaluating these parameters. Most of these correlations are very empirical, and there are considerable discrepancies among them (Vail et al., 1970; Dakshinamurty et al., 1971,1972; Akita and Yoshida, 1973; Razumov et al., 1973; Hikita and Kikukawa, 1974; Kim et al., 1972,1975; Bloxom et al., 1975; Khosrowshani et al., 1975; Soung, 1978; Begovich and Watson, 1978a, 1978b). In this paper wider ranges of variables are being considered and experimental data are correlated according to well-established two-phase flow models (Wallis, 1969; and Butterworth and Hewitt, 1977. These correlations could probably be extrepolated better than before to differe_nt systems with similar pattern flow of the two fluids (Garcia, 1984).
Experimental Section A schematic diagram of the experimental set up is shown in Figure 1. Three methyl methacrylate cylindrical columns of different sizes were constructed. Each column is provided with two perforated plates a t the bottom, the space between them being filled with Raschig rings, in order to obtain a uniform distribution of the gas-liquid cocurrent flow. There are 11 pressure taps to determine the axial pressure profile. The gas is fed to the column between the two perforated plates from pressure bottles. The liquid is fed with a centrifugal pump to the bottom of the column under the sandwich grid from a reservoir, and it is then recycled from the top of the column to the reservoir. Gas and liquid flows
were regulated by needle valves and measured by orifice meters. For each experiment, the following variables were selected column diameter, type of solid, static height of the bed, and gas and liquid flow rates. Once a steady state was attained, the temperature of the system was measured and the axial pressure profile was determined from the average values of the 11 manometric readings (Garcia, 1984). Axial pressure profile, together with physical properties of gas, liquid, and solid, and the total mass of solid in the bed allowed the calculations of the fluid dynamic paremeters in the following way. The total pressure drop in the fluidized bed (when the fed liquid is used as manometric liquid) is in eq 1 where
AI’ = p,g(H + Ah) (1) H is the height of the fluidized bed and Ah the difference of the liquid level between a manometric tube connected to the bottom of the bed and a hypothetic manometer that would be placed at the limit of the three-phase fluidized bed, that is, the level where the solids-free two-phase region begins. The values of H and Ah may be obtained as the intersection of two straight lines, one of positive slope representing the pressure drop profile in the constant porosity region of the three-phase fluidized bed and the other negative slope representing the pressure drop profile in the solids-free two-phase region above the bed (Epstein, 1981). The phase F holdup, 9, is defied as the volume fraction of the fluidized bed occupied by this phase and is given for the bed as a whole by volume of the phase F €f = (2) total volume of fluidized bed According to this equation the individual phase holdups are interrelated as €s + €1 €g = 1 (3)
+
From eq 2 and for solid density p s , the solid holdup, ea, will be M B~. /P8 .
=-
(4) AH where A is the cross-sectional area of the column and M , the total mass of solid in the bed. The total axial pressure gradient at any bed level is the bed weight per unit volume at that level €a
(5) and the total pressure drop across the bed of height H , assuming average values for the holdups of the three phases, is then given by
+ ‘lPl + €gPg)gH
=
(6)
When q from eq 3 is substituted into eq 6, the following expression for the gas holdup is obtained
AP
- - %(PB - PI) - PI - gH %Pg - P1
(7)
in which pg can usually be neglected relative to pI. Then, substituting eq 1 into eq 7, we obtain tg
tg
(;
- 1)
-
$
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 4, 1986
The frictional pressure drop in the bed is the total pressure drop corrected for the hydrostatic head of the two-phase fluid APf=AP-pgH (9) where pf is the composite fluid density given by 'gPg + tlPl €1 N - P1 (10) Pf = €
and substitution of eq 1 and 10 into eq 9 gives
APf N
[
(H + Ah) - ?H]gpI t
(11)
According to Euzen et al. (1981), the value of the minimum fluidization liquid velocity, V, was obtained by extrapolation of the curve €-VI to the static bed porosity to, which is considered a good technique given the large range of H, and Vg investigated. The following solids and fluids were used solid particles of glass, aluminum, and benzoic acid covered with a paint film; water and aqueous solutions of (carboxymethy1)cellulose; air, carbon dioxide, helium, and methane. The following variables ranges were considered. Particles: ps = 1200-2700 kg/m3; 4 = 0.87-1; d, = 3-5.9 mm. Liquid: pI = 994-998 kg/m3; pl = 9.7 X 10-4-8.1 X low3kg/(ms); VI = 1.5-9 cm/s. Gas: pg = 0.15-1.6 kg/m3; pg = 10-6-2 X kg/(ms); Vg = 2-16 cm/s. Bed: D = 4.6-15.1 cm; Ho = 11-41 cm.
Results and Discussion Previous Considerations. The gas-liquid-solid fluidized bed was assumed to be constituted by two systems: the first being formed by the liquid and the gas and the second being formed by the mixture of the two fluids and the solid. Taking into account the limiting experimentalconditions ( Vlpl/ Vgpg) = 102-5500 and Vgpg = 0.012-0.18 kg/(m2s), according to Turpin and Huntington's diagram (1967), a bubble flow can be expected in all cases. Therefore, an homogeneous flow model (Butterworth and Hewitt, 1977), was considered to be suitable for the system constituted by the two fluids. For the mixture of the two fluids and the solid, a drift flux model was considered, a model that is widely applied to two-phase fluidization (Wallis, 1969). Two-Fluid System: Homogeneous Flow Model. The type of flow in this model may be regarded as one-dimensional flow in which there is complete interaction between the phases. The analysis consists of using the single-phase equations with suitable mean physical properties (Butterworth and Hewitt, 1977). Therefore, with the liquid as the continuous phase and assuming a constant ratio el/€ for the whole bed VH = VI + v g (12)
(14) where VH is the homogeneous fluid velocity and pH and fiH its density and viscosity, respectively. The total axis pressure gradient for homogeneous flow at any bed level will be FH
= p1
one-dimensional separated flow in which a relative flow of the two phases is considered, in our case, the homogeneous fluid and the solid (Wallis, 1969). According to this model the frictional pressure gradient is given by a Fanning-type equation
and the frictional pressure drop across a bed of height H assuming average value for the porosity, t, is then given by
where fD is the friction factor for a given bed porosity k d A , is the projected surface area of the particles per unit volume in the flow direction. For spherical particles A, is 6 A, = 4dP Accordingly with Wen and Yu (1966) for a single fluidsolid system, the friction factor fD and the friction factor for individual particles fDmare interrelated as (19)
Three-phase Fluidized Bed. Assuming that the buoyed weight of the solid particles is supported by the upward homogeneous fluid drag on these particles, we have
When the liquid velocity increases and the gas velocity remains constant, the ratio e l / € increases and so does the homogeneous fluid density pH, eq 13. Then in contrast to the two-phase fluidized bed, APf/Hdecreases as shown by eq 20. On the other hand, all our experimental data for APf/H agree with those calculated with eq 20, showing the consistency of the data with respect to the proposed homogeneous flow of the gas-liquid system. In the case of three-phase fluidization of nonspherical particles, eq 18 and 19 should be substituted by the following equations: 6 A, = 44d,
Now, from eq 17 and 20 taking into account the expressions of VH, p ~ A,,: and fD (eq 12, 13, 21, and 22, respectively), the following expressions are obtained for the frictional pressure drop and the porosity:
-Uf - - 3(1 - t)fD,(1 H
- a)Pl(Vl + vg)z 44d,tn (1 - 4 A p S - (1 - a)p11 (23)
a being the void volume fraction filled by the gas
an equation that by integration leads to eq 9. Homogeneous Two-Phase Fluid-Solid System: Drift Flux Model. The drift flux model is essentially
851
53
a=-
€
852
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 4, 1986
4t 2t
I
t 2
0
8
6
4
VI
V,
(cm/s)
Figure 2. Exponent n vs. V,. Air-water-glass system: V, = 7.09 cm/s; 6 = 1;d, = 3.04 mm; Ho= 16.07 cm; D = 4.59 cm.
(cmls)
Figure 3. Exponent n vs. Vr Air-water-glass system: V , = 4.62 cm/s; 6 = 1; do = 3.04 CM; Ho = 16.07 cm; D = 4.59 cm.
In the following sections, the parameters fD-9 n, and a (see eq 23-25), the porosity t, and the minimum fluidization velocity Vh, will be considered. Individual Friction Factor, f~,.This coefficient is assumed to be related with the terminal particle Reynolds number Re, by means of equations similar to those proposed for a single fluid-solid system (Rowe, 1961; Schiller and Neumann, 1935) fD. = -(1 24 -t 0.15Re,0.687) for Re, lo3 Re, fD,
= 0.44
for
io3 < Re, < io5
(26)
being the Reynolds number for the homogeneous flow considered: (27) Figure 4. All experimental data of the friction fador f D , vs. terminal particle Reynolds number.
kH
Substitution of eq 13, 14, and 25 into eq 27 gives Vmdp(lRe, =
PI
PI (28)
The terminal velocity of a single particle V, is determined by balancing the gravitational, buoyancy, and drag forces per unit volume of the particles, that is,
and substitution of eq 13, 21, and 25 into eq 29 gives
Exponent II of the Porosity. In order to establish the influence of the gas and liquid velocities on the exponent n of eq 22, a set of experiments was carried out for different constant gas velocities, varying the liquid velocity in each set. When the individual friction factor fD, is calculated by simultaneous solution of eq 26, 28, and 30 for each experiment, eq 23 allowed the calculation of exponent n: r
Plotting n vs. VI for constant V and n vs. V , for constant V, leads to results similar to tiose shown in Figures 2 and 3, respectively. It is observed that n varies linearly, with V, being independent of V,. These results are in accordance with those obtained in the two-phase fluidization case, in which n is independent of the continuous-phase velocity eq 19.
The following equation was obtained with all the experimental data by a nonlinear regression program (Marquardt, 1963) n = 5.7 - 8Vg (32) Vg being expressed in meterslsecond. Figure 4 shows a plot of the individual friction factor fD, vs. Re,. This friction factor was calculated from eq 23 and 32 and the Reynolds number Re, from eq 28 and 30, using the experimental data. As can be seen, data fit quite well in eq 26. Void Volume Fraction Filled by the Gas, a. It has been assumed that the ratio a is related to the velocities of gas and liquid by means of a function similar to that proposed by Achwal and Stepanek (1976) and to the rest of the variables by a potential function. This relation was obtained from the numerous experimental data by a nonlinear regression program (Marquardt, 1963) a = 3.464 X X Vl-0.66(4dp)O.5OP82.30
)
(33)
3.74
1
+ 1.74(VI:vg
H00.43(,,]
-
P,)0.06k~.08~0.23
all the variables being given in SI units. Figure 5 shows the good agreement of the experimental data to this equation, the average deviation being 7.9 % . The higher deviation corresponds to high values of a, for which Vl is near the mininum fluidization velocity. In such conditions the experimental errors were considerably higher. Porosity t. The porosity of the fluidized bed can be predicted with eq 24 and the expressions of fD,, n, and a, eq 26, 32, and 33, respectively.
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 4, 1986
.
2 IO'
IO'
3 IC'
"..
Figure 5. Correlation for the ratio a. Equation 33. All experimental data.
ecalc
853
0.2
0.6
0.4
0 exp
0.8
Figure 7. Calculated vs. experimental porosities. Air--water--glass system: V , = 3-8 cm/s; VI = 1.08-5.24 cm/s; Cp = 1; Ho= 19.4-41 cm; d, = 3.04 cm.
16.1
/'
erp
Figure 6. Calculated vs. experimental porosities. Air-water-glass system: V, = 8-8.51 cm/s; VI= 1.57-8.60 cni/s; Cp = 1; d, = 3.04 mm; D = 4.59 cm.
8 Eerp
Figure 8. Calculated vs. experimental porosities. Gas-water-glass system: V, = 8-8.16 cm/s; VI= 2.39-6.60 cm/s; 4 = 1;H,, = 19.4 cm; d, = 3.04 mm; D = 4.59 cm.
As an example, plots of calculated vs. experimental porosities are represented in Figures 6-8, showing a good approximation of the points to the diagonal. The average deviation of all the data was only 5.2%. The greatest discrepancies, always lower than lo%, correspond to experiments in conditions close to the minimum fluidization velocity or to the terminal velocity. Furthermore, the data corresponding to different static bed heights (Figure 61, various column diameters (Figure 7), and different gases (Figure 8) confirm that these variables do not influence the porosity in the investigated range, since they do not appear in eq 24. Minimum Fluidization Velocity. Equation 24 together with eq 26, 32, and 33, for a value of the porosity equal to that of the static bed to, should predict the minimum fluidization velocity of the liquid for each gas velocity:
all the variables being given in SI units. Figure 9 shows the good accuracy of eq 35, the deviations being always lower than 10%. This equation showed a better prediction with respect to eq 34 in the range of variables investigated.
The values of V , thus calculated were in almost all the cases higher than the experimental ones. This is due to the use of eq 33 to calculate ao,since, as explained before, this equation presented the highest deviation in the region
Conclusions The proposed equations for predicting fluid dynamic parameters of gas-liquid-solid fluidization with cocurrent flow and the disperse gas-phase bubbling in the liquid require a trial-and-error calculation. From the physical gas, liquid, and solid properties, fluid velocities, column diameter, and height and porosity of the
close to the minimum fluidization conditions. However, even though our eq 34 showed a better prediction than the other correlations in the literature (Begovich and Watson, 1978a, 197813; Bloxom et al., 1975), it was considered convenient to deduce a more accurate and totally empirical equation from the great number of experimental data. The following potential correlation was obtained by a nonlinear regression program Vmf,
=
6.969 x
-
1 0 - 4 ~ , - 0 . 3 2 8 ( ~ d , ) 1 . 0 8 6 ( ~ ~ P1)o 8 6 5 0 0 0 4 2I~-0.355
(35)
854
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 4, 1986 z =
distance in upward direction
Greek Symbols a = ratio E ~ / c ,eq 25
for static bed = bed porosity of fluidized bed
a. = a E
to
= bed porosity of static bed
= phase F holdup tg = gas holdup tl = liquid holdup tf
E,
= solid holdup
K~ = gas viscosity, kg/(m s) g H = homogeneous fluid viscosity, kg/(m
0
.
0
30 vg-0.328
I
60
45
( + @ ) l W 6 (p,-p,)0.865
DO042
-0355
PI
Figure 9. Correlation for minimum fluidization liquid velocity. E q u a t i o n 35. A l l experimental data.
static bed, the following parameters are calculated: Vmf, and a with eq 34 or 35 and 33, respectively; f D , and e by solving simultaneously eq 24, 26,28, 30, and 32 with the calculated value of a;with the calculated values of a and e, the holdups of each phase (tg = a€;€1 = (1 - a)€; t, = 1 - t) and the ratio M f / H with eq 23 and 32. These equations, based on the two-phase flow models, can probably be extrapolated more precisely than before to other systems with similar pattern flow of the two fluids, different from those studied here. Nomenclature A = cross-sectional area of the column, m2 A, = projected surface of the particles per unit volume of the particles in the direction of the flow, m-l D = column diameter, m d, = particles diameter, m f D = friction factor f D . = friction factor for individual particle g = acceleration of gravity, m/s2 H = fluidized bed height, m Ho= static bed height, m Ah = difference of the manometric liquid level between two manometric tubes situated at the bottom and H height of the column, m n = porosity exponent, eq 22 M , = total solid mass in bed, kg P = dynamic pressure, N/m2 AP = total pressure drop in the fluidized bed, N/m2 Pf = frictional pressure, N/m2 APf = frictional pressure drop in the fluidized bed, N/m2 Re, = terminal particle Reynolds number V = superficial gas velocity, m/s v”, = superficial homogeneous flow velocity, m/s V , = superficial liquid velocity, m/s Vmf,= minimum fluidization liquid velocity, m/s
s)
= liquid viscosity, kg/(m s) pf = composite fluid viscosity, kg/m3 pg = gas density, kg/m3 pH = homogeneous fluid density, kg/m3 pI = liquid density, kg/m3 ps = solid density, kg/m3
4 = geometric factor of particle
Literature Cited Achwal, S. K.; Stepanek, J. B. Chem. Eng. J. 1878, 72, 69. Akita, K.; Yoshda, F. Ind. Eng. Chem. Process D e s . D e v . 1873, 72, 76. Begovich, J. M. M. E. Thesis, Oak Ridge National Lab, Oak Ridge, TN, 1978: ORNLlMIT 6448. Begovlch, J. M.; Watson, J. S. AIChE J. 1878a. 2 4 , 351. Begovlch, J. M.; Watson, J. S I n Fluidization; Davidson J. L., Keairns D. L., Eds.; Cambridge University Cambridge, England, 1978b; p 190. Bioxom, S. R.; Costa, J. M.; Herranz, J.; McWliliam, G. L.; Roth, S. R.. ORNLlMIT 219, 1975; Oak Ridge National Lab, Oak Ridge, TN. Butterworth, D.; Hewitt, G. F. Two-Phase Flow and Heat Transfer; Oxford University: Oxford, England, 1977. Dakshinamwty, P.; Subrahmanyan, V.; Rao, J. N. Ind. Eng Chem . Process D e s . Dev. 1871, IO, 322. Dakshinamurty, P.; Rao, K. V.; Subbaraju, R. V.; Subrahmanyan, V. Ind. Eng. Chem. Process Des. Dev. 1872, 1 7 , 318. Epstein, N. Can. J. Chem. Eng. 1881, 59, 649. Euzen, J. P.; Laurent, J.; Pentero, A,; Van Landeghem, H. J. Eur. Fluid. 1881, 33. Garcia, P. Tesis Doctoral, Universidad Complutense de Madrid, Madrid, 1984. Hikita, H.; Kikukawa, H. Chem. Eng. J. 1874, 8 , 191. Khosrowshahi, S.; Bioxom, S. R.: Guzman, C.; Shclapfer, R. M. ORNLlMIT 216, 1975; Oak Ridge National Lab, Oak Ridge, TN. Kim, S. D.; Baker, C. G. J.; Bergougnou, M. A. Can. J . Chem. Eng. 1872, 50, 695. Kim, S. D.: Baker, C. G. J.; Bergougnou, M. A. Can. J. Chem. Eng. 1875, 53, 134. Kono, H. Hydrocarbon Process. 1880 (Jan) 123. Marquardt, D. W. J. SOC.Ind. Appl. Math. 1883 (June), 7 1 , 431. Ostergaard, K. I n Fluidization; Davidson J. F., Harrison, D., Eds.; Academic: Cambridge, England, 1971; p 751. Razumov, I. M.; Manshiiln, V. V.; Nemets, L. L. Int. Chem. Eng. 1873, 13, 57. Rowe, P. N. Trans. Inst. Chem. Eng. 1881, 39, 175 Schiller, L.; Neumann, A. V I D Z . 1835, 77, 318. Shah, Y. T. Gas-Liquid-Solid Reactor Design; McGraw-Hill: New York, 1979. Soung, W. Y. Ind. Eng. Chem. Process Des. Dev. 1878, 77, 33. Turpin, J. M.; Huntington, R. L. AIChE J. 1887, 73, 1196. Vail, Y. K.; Manakov, N. K.; Manshilin, V. V. Int. Chem. Eng. 1870, 70,244. Van Landeghem, H. Chem. Eng. Sci. 1880, 35, 1912. Wallis. G. B. One-Dimensional Two-phase Flow; McGraw-Hili: New York, 1969. Wen, C. Y.; Yu, U. H. Chem. Eng. Frog. Symp. Ser. 1866, 62, 100.
.
Received for review M a r c h 21, 1985 Revised manuscript received D e c e m b e r 30, 1985 Accepted M a r c h 3, 1986
