404
Ind. Eng. Chem. Fundam. 1984, 23, 484-489
1.
Y'=3.319 SLOPEc2. 601 R"2=0.9533
found that the equilibrium relationship could be satisfactorily described by a straight line plot of the composition data. The Othmer-Tobias method of describing the conjugate curve was satisfactory for correlation of tie-line data in the systems studied. Determination of equilibrium concentrations by refractive index measurements was satisfactory for the systems studied.
/ 4 ,
/ /
/ /
J.0-
I
Registry No. Furfural, 98-01-1; toluene, 108-88-3; isobutyl acetate,
110-19-0; m e t h y l isobutyl ketone, 108-10-1.
Literature Cited
-
P I
/
i , -i.d
-2.3
o / / /
/
i ,
,
,
,
,
,
I
-1.8
-1.6
iog ( 1
-1.4
-
a)/
-1.2
o
Figure 7. Othmer-Tobias plot of tie-line data for water-furfuralisobutyl acetate a t 30 O C .
more toxic than either of the other two solvents studied (Sax, 1979); this may influence the final choice of solvent for such an extraction process. In ternary liquid systems of low mutual solubility between two pairs of liquids, as in the systems studied, it was
Conway, J. 6.;Philip, J. 6.Ind. Eng. Chem. 1953, 45(5), 1083-5. Croker, J. R. MSc. Thesis, University of New South Wales, Kensington, Australia, 1983. Jaeggie, W. Escber Wyss News 1975, (2).38-51. Khol'kin, Yu. I . Zh. Priki. Khim. 1960, 33(4), 914-9. Knight, 0. S. Trans AIChE 1943, 39, 439-56. Othmer, D. F.; Tobias, P. E. Ind. Eng. Chem. 1942, 34(6), 693-6. Othmer, D. F.; White, R. E.; Trueger, E. Ind. Eng. Chem. 1941, 33(10), 1240-8. Sax, J. I. "Dangerous Propertles of Industrial Materials", 5th ad.; Van Nostrand-Relnhold: New York, 1979. Trimble, F.; Dunlop, A. P. Ind. Eng. Chem., Anal. Ed. 1940, 72(12), 721-2. William, D. L.; Dunlop, A. P. Ind. Eng. Chem. 1948, 40(2), 239-41. Wilson, B. W. "Furfural Production from Bagasse", Proceedings Seminar on Utilisation of Bagasse, Mackay, Australia, Nov 5-6, 1970; pp 48-64. Receiued for review July 19, 1983 Accepted F e b r u a r y 10, 1984
Incipient Fluidization Condition for a Tapered Fluidized Bed Yan-Fu Shl, Y. S. Yu, and L. T. Fan' Department of Chemical Engineering, Kansas State University, Manhattan, Kansas 66506
A model has been proposed for the condition of incipient fluidization in a tapered bed. The model is based on the balance between the forces, includingthe gravitational and fluid f r i i n a l forces, exerted on the fluidized particles and the total effective weight of the particles. Equations have been derived from the model for predicting the critical fluidizing velocity (onset velocity) in terms of the superflclal veloclty at the bottom of the fluidized bed and the maximum pressure drop through the tapered bed. A series of experiments was carried out in two-dimensional, tapered beds with different apex angles using water as the fluidizing medium and silica gel or sand as the fluidized particles. The experimental data indicate that the proposed model is valid and the derived equations are of practical use.
Introduction Straight cylindrical or columnar fluidized beds have been employed extensively in the process industries. Recently, the use of tapered fluidized beds is beginning to receive much attention for biochemical reactions and biological treatment of waste water (see, e.g., Scott and Hancher, 1976; Pitt et al., 1981). Tapered fluidized beds have also been used successfully in chemical reactions, crystallization, and in other areas (see, e.g., Levey et al., 1960; Ishii, 1973; Golubkovich, 1975; Dighe et al., 1981). Features of the tapered fluidized bed, especially its advantages over the columnar fluidized bed, are discussed below. Usually the size distribution of a particle system which can be employed in a columnar fluidized bed need be narrow. If the particle size distribution is too broad, small particles may be entrained and large particles may be defluidized, settling on the distributor. The cross-sectional area of the tapered fluidized bed is enlarged along the bed 0196-4313/84/1023-0484$01.50/0
height from the bottom to the top. Therefore, the velocity of the fluidizing medium is relatively high at the bottom, ensuring fluidization of the large particles, and it is relatively low at the top, preventing entrainment of the small particles. Therefore, we can operate the tapered fluidized bed with particles whose size distribution is wide. This feature is specially important for an operation in which the particle size changes (coal combustion, crystallization, microbial growth, etc.). For an intensely exothermic reaction in the columnar fluidized bed the major fraction of heat is released near the distributor, creating a high-temperature zone and possibly destroying the distributor and sintering the particles. However, in the tapered fluidized bed, the velocity of the fluidizing medium at the bottom of the bed is fairly high. This gives rise to a low particle concentration, thus resulting in a low reaction rate and reduced rate of heat release. Therefore, the generation of a high temperature zone near the distributor can be prevented. 0 1984
American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
C Curve 2
E
U I
Curve 1
l i
c 0
g
n ?I
0
0 0 Vslocily,~
Figure 1. Pressure drop-velocity relations for the cc!umnar and tapered fluidized beds with the same initial packed height.
485
the superficial velocity of the fluidizing medium at the upper surface of the tapered fluidized bed reaches the minimum fluidizing velocity, umf.However, the calculated critical fluidizing velocity in terms of the superficial velocity at the bottom of the bed, u,, and the calculated maximum pressure drop, (-Urn=), based on their model are appreciably greater than the experimental data. Nishi (1979) has proposed that when the pressure drop in the tapered bed is equal to the effective weight of the particles on the unit bottom cross-sectional area, i.e., at point B in Figure 1,the particles would begin to fluidize. However, the observations made in our preliminary experimentshave shown that the particles remain in a static state at point B, and that the critical fluidizing velocity, u,,is higher than the calculated velocity by Nishi's equation, u', at point B. In our preliminaryexperimental study, we have observed that when the fluid velocity increases, the pressure drop through the tapered bed also increases. At a certain velocity, the pressure drop through the bed reaches the maximum and the total particles in the bed are lifted upward by the fluid. At this moment, the particles at the bottom of the bed begin to fluidize; thereafter, the condition of fluidization will extend from the bottom to the top, and the pressure drop will decline fairly sharply, as depicted in Figure 1. This experimental observation motivates us to postulate that fluidization is initiated when the force exerted by the fluidizing medium flowing through the bed is equal to the total effective weight of the material in the bed under the assumption that friction is negligible between the particles and bed walls. For simplicity, we further assume that the lateral velocity of the fluid is sufficiently small to be neglected and that the vertical velocity of the fluid is uniformly distributed on the cross-sectional area. The pressure drop through a packed bed with a differential height of dh is equal to (Ergun, 1952)
The velocity of the fluidizing medium at the bottom of the tapered fluidized bed can be greater than the terminal particle velocity. In this case the tapered fluidized bed can be operated without a distributor (Golubkovich, 1975). For a deep gas-solids columnar fluidized bed, the pressure difference between the bottom and the top is large, and thus the gas tends to expand as it rises through the bed, rendering the upward gas velocity to increase continuously, and reaches the maximum at the top. This will tend to agitate the top of the bed violently; this promotes elutriation of the particles from the bed surface and instability (pressure fluctuations) in the bed. In contrast, the cross-sectional area of the tapered fluidized bed increases upward; this enables it to accommodate the increase in the gas volume, rendering it possible to operate -dP = (Au Bu') dh (1) smoothly with the deep bed (Ridgway, 1965). Because of these advantages, the use of tapered fluidized where beds is rapidly increasing; however, much is unknown concerning characteristics of the tapered fluidized bed. Hydrodynamic features of the tapered fluidized bed are very different from those of the columnar fluidized bed, as can be seen in Figure 1where typical pressure drop1 - '0 Pf velocity relations of the tapered bed (curve 1)and that of B = 1.75-the columnar bed (curve 2) are shown. The superficial eo3 velocity of the fluidizing medium in the tapered bed inThe overall pressure drop across the entire bed height, H, dicated in the figure is based on the bottom cross-sectional is obtained by integrating eq 1, i.e. area. The static bed heights in the two beds are identical. This figure shows that the critical fluidizing velocity in terms of the superficial velocity at the bottom of the bed, (-U)= f H + ( - d P ) = S H + h o ( A + ~Bu') dh (2) pho ho u,, at which the particles on the bottom of the tapered bed begin to fluidize, is greater than the minimum fluidizing velocity, umt. The maximum pressure drop, (-Urn=),For a two-dimensional tapered bed with a thickness of w o and an apex angle of 8, we have (see Figure 2) which is created a t u, in the tapered bed, is greater than the maximum pressure drop, (-APmf), in a columnar bed. 10 ho Therefore, the known hydrodynamic relations for the cou = uo- = uo(3) 1 h lumnar bed cannot be directly applied to the tapered bed. The objectives of this work were to propose a mechaand nistic model for the condition of incipient fluidization in the tapered bed, to derive equations for predicting the (4) critical fluidizing velocity, uc,and the maximum pressure drop, (-AP-), and to verify these equations experimenTherefore (see Appendix A) tally. Theoretical Section Different views exist concerning the incipient fluidization condition of a tapered fluidized bed. Gelperin et al. (1960) have proposed that particles would fluidize when or
+
Ind. Eng. Chem. Fundam., Vol. 23,No. 4, 1984
486
(-AI')= AuO-
0 lo + 2H tan -
10
HlO
+ Buo2
(5b) 0 In 10 Io + 2H tan 2 2 Equations 5a or 5b can be rewritten in dimensionless form, respectively, as 2tan-
The dimensionless form of eq 11 can be expressed as 11
1.75-
2lO2In -
1
10
$,€03
or
112
- l02
Re:
+ 150--
1- €0
210
Re, - Ar = 0 4,2€03 11 + 10
or
1.75--
(1- €0) H '#JSto3
dplo
10
+ 2H tan -2
(6b)
The cross-sectional area of a tapered fluidized bed increases continuouslyfrom the bottom to the top; therefore, the force exerted by the fluidizing fluid on the particles is not directly proportional to the pressure drop. This force in a differential bed height of dh is equal to the product of the pressure drop through it, (-dP), and the cross-sectional area of the bed lw, dF = IwJ-dP) = lwo(Au + Bu') dh (7) Thus, the overall force exerted by the fluidizing fluid on the particles in the entire bed with a height of H is (see Appendix B)
1.75,
1
lo2
In [ ( l o
1 - €0 150-
(-AI',,)
(
B
E In :)u:
+
Re, - Ar = 0 (13b)
= 0 lo + 2H tan -
10
Aut-
0 In 2 tan 2
g -0 - €o)(P, - P f ) W d r ( l l +lo) (10) 2 According to the proposed model, the particles on the bottom begin to fluidize when F = G. Therefore, the critical fluidizing velocity, u,) can be calculated by equating eq 9 and 10. This gives
Re:
Equivalently, replacing uo in eq 5b with u,) we obtain
or
=
0 2
/lo
Replacing uoin eq 5a with u,,the maximum pressure drop through the tapered bed (-AI'-), is calculated as follows
(8)
The overall effective weight of the particles in the bed is (see Appendix C)
10
4ZEo3lo + H tan -
H+ho
Zwo(Au + Bu2) dh
+ 2H tan -2
+ Bu:
10
HlO 0 lo + 2H tan 2 (14b)
The dimensionless forms of these equations are, respectively
and
Eu, = 150--
(1 - eo)2 1 1 $:to3
10
Re d p 2 tan -0 In 2
lo
+ 2H tan -0 2 + 10
+ (AwoldT)u,-
1
-0 - Co)(Ps - P f ) W O H ( h + 1 O k = 0 2 which, in turn, gives -B'+ (B'2 + 4A'C9'J2 u, = 2A ' where
(11)
(12)
Experimental Section A schematic of the experimental facilities is shown in Figure 3. The dimensions of the two-dimensional tapered beds employed are listed in Table I. Notice that the apex angles of the two tapered beds were 20' and 30'. All vessels were made of transparent acrylic resin so that the solids behavior could be observed. The distributor was a stainless steel screen of 48 mesh. Silica gel and sand particles were fluidized. Characteristics of these particles are listed in Table 11. Water at room temperature (approximately 20 "C)was used as the fluidizing medium. The flow rate was measured by
Ind. Eng. Chem. Fundem., Vol. 23, No. 4, 1984
I
487
I
. 1
I
Figure 4. Comparison of the pressure drop through the tapered packed bed computed by eq 5a or 5b with the experimental data.
it Po+ Figure 2. Structure of the tapered fluidized bed.
0.02.i
I 0 : Water rtomlg. tmk @ : Pump @ : Rotameter @ : Tapred fluidized bed Q : Piezometer
Figure 3. Schematic of the experimental setup. Table I. Dimensions of the Two-Dimensional Tapered Beds apex angle dimension 20° 30’ 0.0508 (2”) 10, m 0.0508 (2”) 0.0508 (2”) 0.0508 (2”) wo,m 0.577 (22.7”) 0.378 (14.9”) HT,m 0.254 (10”) 0.254 (10”) L, m Table 11. Properties of Solid Particles 0..
material silica gel sand
d,, mm 1.125 0.711
kg[m3 2590 2590
€mi
0.346 0.386
a rotameter, and the pressure drop was measured by a piezometer. A known weight of particles was poured int~the bed and then the particles were fluidized fully by water at a certain velocity. The loading of particles for each experiment ranged from 1to 3.9 kg. When a stable state of fluidization was established, the values of the velocity and pressure drop were recorded as the velocity was gradually reduced to zero. After the velocity reached zero, the velocity was gradually increased and the values of the velocity and pressure drop were again recorded. From these data the pressure dropvelocity curve was construded. The critical fluidizing velocity, uc,and the maximum pressure drop, (-AP,-),were determined from the curve. The procedure described above gave a reproducible bed porosity even though the packed bed pressure drop is very sensitive to
Figure 5. Comparison between the experimentallymeasured critical fluidizing velocity, (u,),,,, and the calculated fluidizing velocity, (uc)c&d.
the bed porosity, and different degrees of packing of the particles produce different bed porosities. Results and Discussion The experimental results have shown that the pressure dropvelocity diagram of the tapered fluidized bed indeed has the shape of curve 1in Figure 1; in the increasing fluid velocity cycle, the pressure drop reaches a maximum value and then decreases sharply, approaching an essentially constant value. In the decreasing fluid velocity cycle, however, the pressure drop decreases continuously and the peak is not recovered. This is similar to the pressure dropvelocity diagram of the columnar fluidized bed. The difference between the diagram of the tapered bed and that of the columnar bed is that the peak in the former is much sharper than that of the latter. A comparison of eq 5a or 5b with the experimentally measured pressured drop through the tapered packed bed shows that they are in good agreement (Figure 4). Therefore, the assumptions that the lateral velocity of the fluid is negligible and the vertical velocity of the fluid is uniformly distributed seem to be reasonable. Figure 5 compares the critical fluidizing velocity, u,, calculated by eq 12 with the experimental data, and Figure 6 compares the maximum pressure drop, (-Urn=), calculated by eq 14a or 14b with the experimental data. It is evident from these figures that the agreement between the calculated
488
Ind. Eng. Chem. Fundam., Vol. 23,
No. 4, 1984
X
Figure 6. Comparison between the experimentally measured maximum pressure drop, (-APmax)alptland the calculated maximum pressure drop, (-A?'mar)ealcd.
values and experimental data is satisfactory, indicating that the proposed model is valid and the derived equations are of practical use. When the apex angle becomes negligibly small, i.e., 9 0, we have
-
lo2 In[
(lo +
Conclusions 1. The hydrodynamic features of the tapered fluidized bed are very different from those of the columnar fluidized bed; therefore, the known relations for the columnar bed cannot be used in calculating those for the tapered bed. 2. A model has been proposed for the condition of incipient fluidization in the tapered fluidized bed. Equations have been derived from the model for predicting the critical fluidizing velocity, u,, and the maximum pressure drop, (-AI',,). The experimental data obtained in twodimensional, liquid-fluidized tapered beds indicate that the proposed model is valid and the derived equations are of practical use. Acknowledgment The authors wish to thank Dr. C. C. Chang for his assistance. Appendix A. Derivations of Eq 5a and 5b from eq 2 Equation 2 in the text is
(-AF')= J H + h o ( A ~ + Bu2) dh Since 10
1
lim 0
ho h
u = uo- = uo-
we have
2H tan
(A-1)
h0
(-M)=
4
(A-2)
I"'"(+ Au$
Buo2g) h2
dh
ho
H
+ ho +
HhO (H + ho)
= AuohoIn - Buo2
and
h0
10
lim 8--0
lo
8
+ H tan -2
Furthermore, because
=1
Thus, eq 13b reduces to (11 1 . 7 5 7 R e 2+ 1 5 0 '"Re 7 - Ar = 0 48'0
4 8
(16)
we obtain
€0
which is the well-known equation for predicting the minimum fluidizing velocity, ud, of the columnar fluidized bed (Kunii and Levenspiel, 1969). Furthermore, when 8 0, or ll lo, eq 14a or 14b reduces to (-AI',,) = A@ + Bu;H (17)
-
-
and eq 11 reduces to Bu;H + A u - ~(1 - ~ o ) ( p -, p f ) H g = 0 = (1- 'o)(Pa - P =-G wolo = (-Qp,f)
or
(-AP)= Auo-
(18)
or
(-AF',,)
(A-3)
f M
(19) This is also the well-known result that relates the pressure drop a t the minimum fluidizing velocity in the columnar bed, (-APmf), to the effective weight of particles per unit cross-sectional area of the bed. These developments indicate that the proposed model is internally self-consistent. In applying the proposed model to a gas-solids system, a further simplification can be made by letting ( p , - p f ) ps because pa >> pf. Since such a system is usually nonparticulate, some empirical modifications probably need be made on the model equations.
10
1,
+ 2H tan -8
9 In 2 tan 2
+ Buo2
10
1,
Hi0
+ 2H tan -2e
(-4-6) Equation A-5 or A-6 is the same as eq 5a or 5b, respectively, in the text. Appendix B. Derivation of Eq 9 From eq A-2, we have 10
10
1 = -h and u = uoh0 h Substituting these expressions into eq 8 leads to F =
IoH+ho[ 'I
Awouolo+ Bwouo21Jz0; dh
+
= AwouoZoh(~hoBwouo210ho In hlcho
+
H
+ hi
= Aw0uola Bwouo210ho In ___
h0
03-2)
Ind. Eng. Chem. Fundam., Vol. 23, No. 4, 1984
489
F = force exerted by the fluidizing medium on the particles
Since
in the bed, N
G = effective weight of material in the bed, N H = height of the fluidized bed, m
HT = overall height of the tapered-bed section, m
or 03-41 substituting these two equations into eq B-1 yields
This i s eq 9 in the text. Appendix C. Deriviation of Eq 10 From eq A-2 we have 10
1 = -h h0
Substituting this equation into eq 10 gives
ho-=distance between the apex and the bottom of the tapered bed, l 0 / P tan (0/2)1, m L = top length of the tapered bed, m 1 = length of the tapered bed at height h, m lo = length of the bottom of the tapered fluidized bed, m l1 = length of the top of the tapered fluidized bed, m (-AP)= pressure drop, N/m2 (-APd) = pressure drop at the minimum fluidizing velocity in a columnar bed, N/m2 (-Urn,) = maximum pressure drop, N/m2 u = superficial fluid velocity, m/s uo = superficial fluid velocity at the bottom of the bed, m/s u, = critical fluidizing velocity in terms of the superficial velocity at the bottom of the tapered fluidized bed, m/s u d = minimum fluidizing velocity in terms of the superficial velocity at the bottom of the fluidized bed wo = width of the tapered bed, m eo = void fraction of the packed bed I.L = fluid viscosity, Ns/m2 pf = fluid density, kg/m3 ps = solid density, kg/m3 4, = sphericity of the solid particles Ar = Archimedes number, [ ( d , 3 p f g ) / ~ ~ ] [ ( p -, PJ/P~I Eu = Euler number, (-hp)/(pfuo ) Eu, = Euler number at the maximum pressure drop or critical fluidizing velocity, (-APm,)/pfu,2 Re = Reynolds number, dppfuO/p Re, = Reynolds number at the critical fluidizing velocity, dpPfuC/I.L
Literature Cited
From eq B-3 we have HZ0
ho = 11 - 10 Substituting this equation into eq C-2 yields
or
G=
g
-0 - ~o)(P,- P 3 w d ( l i + 10) 2
This is the final expression of eq 10 in the text. Nomenclature d, = particle diameter, m
(C-4)
Dighe, S. V.; Blinn, M. B.; Buggy, J. J.; Krasickl, B. R.; Pierce, B. L. Proceedings of the 16th Intersociety Energy Conversion Engineering Conference, ASME New York, 1981; pp 1059-1067. Ergur), S. Chem. Eng. Rog. 1952, 48, 89. Gelperin, N. I.; Einstein, E. N.; Gelperin, E. N.; Lvova, S. D. Khim. i Tekhn. ropl. I Masel 1960, 5(e), 51. Qoiubkovich, A. V. Khlm. I Nee. Mash. 1975, No. 3. 21. Ishll. T. Chem. Eng. Scl. 1973. 28(3),1121. Kunii, D.; Levenspiel, 0. "FluMlzation Engineering"; Wiiey: New York, 1969. Levey, R. P., Jr.; De La Qarza, A.; Jacobs, S. C.; Heidt, H. M.; Trent, P. E. Chem. Eng. Prog. 1960, 56, 43. Nishi, Y. Kagaku Kogaku Ronbunshu 1979, 5 . 202. Pkt, W. W.; Hancher, C. W.; Patton. B. D. Nucl. Chem. Waste Manage. 1961, 2, 57. RMgway, K. Chem. R o c . Eng. Jurn 1965, 317. Scott, C. D.; Hancher. C. W. Blotechnol. Bloeng. 1976, 18, 1393-1403.
Received for review July 29, 1983 Revised manuscript received April 16, 1984 Accepted April 23, 1984
