JUNJI FURUKAWA and TSUTOMU OHMAE Department of Industrial Chemistry, Kyoto University, Kyoto, Japan
Liquidlike Properties of Fluidized Systems By ingenious analogy with liquid systems, fluidization has been characterized in a way that will prove valuable to an expanding field
MACROSCOPICALLY
fluidized particles . Interparticle Force have free surfaces and fluidities similar In a fluidized system the particles to those of liquids. Microscopical'ly have both random vibrating migration they exhibit vibratory-migration motion and vertical convection. This convecsimilar to that of liquid molecules. tion is a secondary consideration caused These similarities may be used as a basis by gas channeling and does not affect for investigating a fluidized particle degree of fluidization. Moreover, variasystem, and various physical properties tion in column diameter and column of such a system were studied for cominclination are factors influencing this parison with the theory of liquids. type of convection. In liquid fluidizaThe fundamental difference between a tion this secondary convection does not gas and a liquid is the degree of interappear, and only the random vibrating molecular force between molecules. In migration needs to be considered, gases, the molecules have free translation In vibrating migration there are with practically no intermolecular potential fields around the particle. I n forces, but in liquids the molecules are fluidization, frictional forces of fluid on vibrating in a closed potential field the particles work as expansion forces, formed by the surrounding molecules. while gravity forces work as a contraction They migrate in zigzag paths by moving force. These two forces may be coninto gaps between molecules. This sidered to produce attraction and repuldifference in molecular motion accounts sion between particles. These forces may for the differences in thermal expansion, or may not be in equilibrium at a point, viscosity, surface tension, and miscibility thus giving rise to vibration. Then the for gases and liquids. volume vibration of a particle group, a In fluidized particle systems the existmicrofluidized bed consisting of n parence of interparticle forces analogous to ticles, may be considered as the resultant the intermolecular forces of liquids also of n-particle motion. may be characterized by expansion, viscosity, surface tension, and miscibility. . The expansion force on a particle group is the fluid friction force. The In gases and liquids temperature is a pressure drop, AP, is given by Leva's good measure of the extent of thermal equation (72), which is valid for spherical motion. For a fluidized system, a particles corresponding variable must be visualized to characterize the degree of particle AP/L = 20OpU(1 - €)2 motion. Dp2gEs
The, contraction force is the gravitational force on a particle group which is equal to the friction force at equilibrium (77). Then
The potential energy @ of volume vibration (74) caused by the two opposite forces is : % =
l e s v r ( p . - AP/L)dV = orU[AV
B/(V
where
(Y
-
V,)
=
+ C / ( V - VP)'- HI
+
(3)
200 p le/D2g
A = V,V,Z/(V, - V,)3 B = v,z; c = vp3/2; I,
VPZ
(Ve
+
- V,)
= v;/3
VP3
2(Ve - V,)Z
Because V, is the sum of actual volume of n particles, the volume of a microfluidized bed, V, cannot contract below this value. Considering this limit, Equation 3 shows that potential energy has a minimum value zero at equilibrium state V = Ve, and d@/dV = 0, d2@/dV2< 0, and @ = 0. When V > Ve, the increase of the first VOL. 50, NO. 5
MAY 1958
821
t
Microfluidized bed showing nparticle group
4 Fluidization unit was made of glass tube with 6-cm. inner diameter
A.
Microfluidized bed Fluidized bed C. Bed support (bronze screen) D. Distributor (lead shot) E. Paddle F. Load G. Air bubbles H. Nozzle J. Air blowoff K. Withdrawing tubes I. Filter 8.
b
Tubes were attached to fluidization column SO that samples could be withdrawn for miscibility studies
Apparatus for viscometry utilized modified Stormertype steel paddle blades
U
t
v
For surface tension measurements, bubble-producing apparatus had flattened nozzle
'1'
I
term exceeds the decrease of the second and third terms: and ZI increases. For V , < V < V,, the first term decreases as V decreases, but the second and third terms increase greatly, and 9, increases. That is, when the volume of the particle group expands or contracts from the equilibrium volume, the resisting forces cause the volume vibration. If this volume vibration is considered microscopically, interparticle forces occur as interparticle distances r decrease or increase from the equilibrium distance. For a fluidized system, there is no formula expressing voidage, E , as a func-
822
tion of interparticle distance. On the other hand. in a fixed bed, Ergun and Orning (5) considered a cylindrical channel model and proposed the following formula: 0,= [4~/(1- e)l(l/Sv)
[4~/(1 - e)](Up/6) 2 / 3 [ ~ / ( 1-
2
e)]
(0,) (4a)
Then, Equation 3 is expressed as
(4)
The channel diameter. D,, corresponds to the interparticle distance re, and the specific surface, S,, of a spherical particle is expressed by rD,2/ (a/6)(02) = 6/D2 (74). Considering Equation 4 to be applicable to the dense fluidized system, the formula changing voidage to interparticle distance is
INDUSTRIAL A N D ENGINEERING CHEMISTRY
re =
Variable Analogous to Temperature
Temperature is a measure of the extent of molecular motion, and the average
LIOUIDLIKE kinetic energy of molecular vibration is given by kT. In a fluidized system, however, temperature does not increase the particle motion; that is, it affects only particle and fluid densities and fluid viscosity. However, increase of fluid velocity is a direct cause of increased particle motion. Thus the variable analogous to temperature as a measure of extent of particle motion is found from the average kinetic energy of particles. The average kinetic energy (K.E.) of particles may be calculated by using Equation 5. Generally, every type of vibration may be treated as a harmonic oscillation, if the amplitude up is moderate. Potential energy (P.E.) of a harmonic oscillation is given by (9) P.E. = (1/2)cxZ
BEHAVIOR
O F FLUIDIZED
Table 1.
Materials Used
Particle
Experiment Expansion and viscometry
Abs.
density, diam., grams/ Type microns cc. Poly(viny1 acetate) bead 277 1.3 (sphere) 324 Av.
Fluid Density, Type (g./cc.) Air 0.0012
384 588 755
Surface tension
Sand (irregular shape)
324 384 538
2.56
Water
1.0
Miscibility
Sand
160 200 277 324 384 538 640 755
2.56
Water
1.0
Charcoal
755
1.2
(6)
where c = (d2@/dx2),=oand x cos 2rvt
PARTICLES
= up
Then
P.E. = ( 1 / 2 ) c x 2 = (1/2)cu,2 cos2 2rut (7) and kinetic energy of vibration is given by ( 70)
bed. In experimentation, fluid viscosity, p, is not easily changed and pU is changed only by variation of flow rate, U, of fluidizing fluid.
K.E. = ( 1 / 2 ) m v 2 = (1/2)ca,2 sin2 2 r ~ t( 8 )
Experimental
for
studies of expansion, viscosity, surface tension, miscibility, and the effect of p U , the temperaturelike factor. Although the investigation of various fluids and particles is useful and interesting, -it may make the problem more complex, and this investigation was limited in this respect. Kinds of Particles an'd Fluids. v
u =
dx/dt = - 2 r v
Q,
The liquidlike properties of a fluidized system can be characterized by
sin 2 r v t
and m = c ( 2 a v ) z or 2 w = .\/c/m
From Equations 7 and 8 the changes of potential energy and kinetic energy have equal periods and amplitudes except for the n/2 phase difference. Thus the average values over a time interval are equal and can be expressed (70) as
Application of Equation 9 to a fluidized particle system gives the average kinetic energy of particle motion:
KX. = ( 1 / 4 ) (d2@/dx2)aP2 = f I ( r / A ' ) P (0,'/7eZ)[1
+
(Dn/Te)]QpU
(10)
+
In Equation 10, because D,Z/r?[l (D,/re)]a," is almost constant with U (7),
-
K.E. = (const.) MU
(11)
Therefore, p U may be considered to be the variable analogous to temperature as a measure of the exteiit of particle motion. The equation for fluidization which can be derived from Equations 1 and 2 contains this pU: €311
-
= 200pu/g~,2pp
(12)
For example, increase of pLu causes an increase of t and expansion of fluidized
Table 11.
Volume Expansion of Fluidized System
Flow
Flow
Rate of Volume Air (U),Expansion, Cm./Sec. (V/V,/) ( D p = 277 Microns) 5.0 10.0 15.0 20.0 25.0 42.0 71.0 95.0 103.0 105.0 105.5
7.0 1.15 1.30 1.48 1.66 2.25 3.15 4.00 5.00 6.00 6.50
(Dp= 324 Microns) 7.0 9.0 12.5 16.4 21.5 28.0 56.0 90.0 110.0 140.0 142.5
1.00 1.05 1.14 1.23 1.35 1.50 2.16 2.84 3.85 6.55 8.00
Rate of
Volume
Air (v), Expansion, Cm./Seo. (V/Vmj) (D,= 384 Microns)
'
8.6 11.7 17.0 26.0 30.0 65.0 102.0 123.0 153 164 165
1.0 1.06 1.16 1.34 1.41 2.00 2.95 4.00 5.86 8.15 9.0
(D,= 588 Microns) 10.5 18.0 , 29.4 72.5 97.5 145 170 185
1.0 1.113 1.3 2.0 2.35 4.00 6.16 9.5
(D, = 755 Microns) 15.6 25 30 95 133 205 240 254 255
VOL. 50, NO.
1.0 1.15 1.20 1.90 2.37 4.50 7.1 10.5 12.5
s
MAY 1958
823
Table 111. Relationship between Expansion Coefficient and Particle Size
8
Particle Size
Expansion Coefficient
Min. Fluidization Velocity
IS
Microns
Sec./Cm.
(U m l ) Cm./Sec.
R
277 324 384 588
0.0286 0.0220 0.0175 0.0156 0.0105
5.2 7.0 8.8 10.5 15.5
c1
( D P )9
12 Z 6 Z
d
755
8 m 4 8
2
FLOW RATE OF AIR U, CM./SECOND
Figure 1.
Effect of air flow rate on fluidized bed volume
At low flow rates, expansion ratio is proportional to velocity
Using spherical particles of equal size, liquid fluidization-eg., glass powders fluidized by water-provides an idealistic experiment. Spherical glass powder was not used, but it was possible to choose a combination of particles and fluids such that the selection had little effect on analysis of the four variables under consideration. These combinations are shown in Table I. Particles were sized with Tyler sieves at least four times. The over- and undersized particles were not greater than 10%. Consequently in discussing experimental results, variations caused by size distribution are neglected. TVater was selected to provide complete fluidization. When particle density is close to water density, fluidization cannot be attained, and fluidization by air is necessary. When a gas is used, however, a portion of the gas stream does not contribute to fluidization. Therefore, water was used, except for a critical case when air was used. EXPANSION. Although water fluidization has many advantages for expansion studies, it does not provide the necessary regularity of particle shape, and air fluidization of granular poly(vinyl acetate) was used. Granular polymer does not deviate appreciably from a spherical shape and was readily available. The particles were covered with zinc oxide or starch, below 10 microns, to avoid aggregation by electrostatic charge. VISCOSITY.Because shape uniformity is important, granular poly(viny1 acetate) was again selected. SURFACETENSION. Surface tension was studied by bubbling air into a water-sand fluidized system. The number and shape of the rising air bubbles were satisfactorily observed in the transparent zone above the sand.
824
TWO-COMPONENT PARTICLE MISMiscibiiity experiments are conducted in liquid media, because in gas fluidization the convection of particles may be disadvantageous. Apparatus. FLUIDIZING UNIT. A glass tube of 6-cm. inner diameter was used. The particle bed was supported by a 100-mesh bronze screen and a layer of lead shot 5 cm. thick which acted as a gas distributor. EXPANSION TESTS.The side wall of the glass fluidizing tube was marked in millimeters. I n gas fluidization it is CIBILITY.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Table IV. Flow
Rate of Viscosity Air ( V ) ,
Cm./Sec.
h,) I
Poises ( D p = 277 Microns) 4.65 6.2 6.95 9.8 11.6
12 7.4 6.65 5.31 5.01
(Dp= 324 Microns) 5.8 6.92 8.45 11.6 15.3 17.5 35.0 57.0 80.0 100.5 111.5
13.5 10.5 8.7 6.8 5.6 5.3 4.0 2.7 1.58 1.02 0.71
(61)9
7
necessary to keep the length to diameter ratio less than 3 to avoid slugging cffects The volume expansion was read after equilibrium was attained. The expansion ratio is the ratio of fluidized volume to minimum fluidized volume, and the variation of this ratio with flow rat? of air was determined. VISCOSITY.The apparatus shown is a modification of the design used b) Rfatheson (73). The modified Stormertype steel paddle blades (page 822) were 7 X 3.3 X 0.4 cm. By calibrating the viscometer for a certain number of revolutions per minure and several loads. W , with viscosity-known liquids, the viscosity of the fluidized system could be measured. In calibrating formula obtained 17 = 1/50 (TY-90) at 113 r.p.m. SCRFACETEWION.In the bubbleproducing apparatus, the diameter of the nozzle was 2.21 mm. with the edgr accurately flattened out to give a large foaming surface. The bubble volume
Viscosity of Fluidized Bed Flow
Rate of Viscosity -4ir KO, h,), Cm./Sec. Poises ( D p = 384 Microns) 6.89 7.7 10.45 13.03 15.0 17.5 38.0 60.0 72.5 100.0 125.0
22.0 18.5 12.4 9.5 8.5 7.3 4.85 3.4 2.83 1.7 1.4
( D p = 588 Microns) 12.15 15.35 19.8 22.8 25.0
22.5 15.0 10.8 9.4 8.6
( D p = 756 Microns) 17.6 21.3 24.0 27.0 30.1
19.0 14.5 12.0 10.5 9.5
LIQUIDLIKE BEHAVIOR OF FLUIDIZED PARTICLES is proportional to surface tension and nozzle size. Calibration was carried out with a known glycerol solution a t approximately the same viscosity to eliminate most of the nozzle irregularities (75). The bubble rate was about 10 bubbles per minute. MISCIBILITY. Sample-withdrawing tubes were attached to the side wall of the fluidization column. The two solids (sand and carbon or coarse and fine sand) were mixed well and then fluidized in the tube. Equilibrium was attained in a few minutes, and after a certain time a separation could be seen. Convection effects (in gas fluidization) are the most disadvantageous factors in the phase separation. Samples were taken from each layer, and the concentrations were ascertained by drying and sieving.
I
Figure 2. between
[l
Relation and
U'/a
- (Vm,/V)q
/O w
k 5
0.2
Results
I
Expansion. Figure 1 shows the variation of volume expansion against air velocity. Poly(viny1 acetate) particles 0.277, 0.324, 0.384, 0.584, and 0.755 mm. in diameter were fluidized over a range of air flow rates of 5 to 250 cm. per second. At low flow rates the expansion ratio was proportional to velocity but deviated from the linear relationship at higher flow rates (Table 11). The expansion ratio, V/V,f, may be expressed by the empirical Equation 7
v/vm,
=1
+ S,(U - Urn,) +
+ ..
S,'(U - Urn,)'
(13)
Table 111 shows the relationship between coefficient of expansion and particle size D,. The thermal expansion of a liquid may be expressed as (78) V/Vm = 1
+ S(T - T,) + 6'(T
-
T,)'
+ ...
is a straight line for each particle size.
then TI/' = G1[1
The resulting equation is
where n = 3.1 =t0.3. The similarity of Equations 15 and 16 indicates that the principle of corresponding states is valid for fluidized systems. The free volume theory (78) of liquids also may be applied to analyze the expansion of liquids. Here, because 1
av
(=v(n>,=
and the free volume VF = (4/3)n (v1/3 - v r n 1 / 3 )
= (VJVm)
- (V,/Vm
- ( T - Trn)/(To-
Particle Diameter (D,), Microns
(15)
In a fluidized system when the volume expansion approaches a maximum, where d(V/V,,)/dU + a ,it was found (7) that a plot of log (Vmax - V)/(Vmax - Vmj) US. log (Urnax - U)/'(Um*x Urn/)
-
(17)
VI2
G,[I
- (Vrnf/V)1'31
(18)
Equation 18 is similarly not valid for higher velocities. Because Equations 17 and 18 are similar, the expansion of a fluidized system is considered to be caused by the expansion of the free volume between particles. Viscosity. Rheological measurements with the modified Stormer viscometer produced stress-hardening curves typical of those obtained in dispersion rheology investigations. In contrast, the same viscometer gave a linear relationship for glycerol, a Newtonian fluid. This non-Newtonian effect results from com~~~~
Table V. Surface Tension of Fluidized System Flow Rate of VOl. of Density of Water (U), Air Bubble, Fluidized Bed Cm./Sec. Cc. X lo3 (pi),Grams/Cc.
Surface Tension of Bed ( u ) , Dynes/Cm.
324
0.375 0.500 0.625 0.760 0.920 1.02 1.10
33.0 31.1 27.9 23.8 21.5 19.3 16.8
1.75 1.65 1.59 1.52 1.46 1.43 1.38
168 153 130 106 93.5 81.0 68.5
384
0.380 0.520 0.620 0.750 0.890 0.980 1.100
38.5 35.5 33.2 31.0 28.0 26.2 23.0
1.77 1.66 1.635 1.58 1.53 1.47 1.415
200 174 160.5 145 126 115 96
538
0.50 0.60 0.82 0.90 1.15 1.40
44.3 42.5 38.3 35.0 30.2 26.0
1.80 1.70 1.635 1.57 1.47 1.39
235 223 183 162 129 106
- 1) x Tm)]*/1O
- (Vm/V)l/S]
Equation 17 is valid for liquids below the normal boiling point. A corresponding equation from the linear parts of Figure 2 is
~
[l
0.5
(14)
Between the melting point, T,, and the boiling point of a liquid the liquid density is almost linear with temperature, while near the critical temperature, T,, additional terms are needed in Equation 14. A useful approximation (77) for the expansion coefficient, 6, is 1/(3 Tm). In a fluidized system if minimum fluidized flow rate Umf,is considered to correspond to the melting point of a liquid, then the expansion coefficient of fluidized particle 6, = (1/6)(1/Um,). From the principle of corresponding states an empirical equation was found to be (3)
v/vm
0.4
0.3
- (v,//v)"a
VOL. 50, NO. 5
MAY 1958
825
4 = Fe-E,/U
Figure
3. Re-
lation between logarithm
,
of
viscosity and reciprocal rate of flow
so
25 20
/s
/ o s
8
7
6
FLOW RATE U, CM./SECOND
pression of the particles by a rotating shearing stress decreasing interparticle distances (76). This effect cannot be due to turbulence, because the rotation speed is not high enough. The same phenomenon can be observed if sandpaper is placed on a cylindrical rotor. Therefore the shape and size of the stress-producing instrument are not important factors. Measurements were made a t low rotational speeds (below 150 r.p.m.) so that rheological properties were almost linear. Further, rotational speed was kept constant (113 r.p.m.), and the load was adjusted for each run. Table IV shows the effect of particle size and velocity on measured viscosity. These data agree with those of Matheson, Herbst, and Holt (73) and Diekmann (4). By using the Andrade equation for liquids ( 7 ) , which has been interpreted by Eyring ( 8 ) , an equation of viscosity q applicable for a fluidized system may be found,
Table VI. Flow Rate of Air ( U ) ,
Cm./Sec. 1.10 1.44 1.82 2.22 2.84
Compositions of Coexisting Layers
Coarse Particle Concn. Upper layer Lower layer XI, wt. % XZ,wt. yo 538-Micron Sand 277-Micron Sand 7.1 71
+
3.3 2.4 1.97 1.43
538-Micron Sand 0.97 1.27 2.31 3.2
0.70 1.04 1.33 1.52
Composition Ratio, XdX2 0.1 0.0426 0.0295 0.0224 0.0158
+ 324-Micron Sand 57.5 66.5 82.0 87.0 $.
1.46 1.21 0.83 0.66 0.56
640-Micron Sand 1.43 2.04 2.495 3.33
79.5 81.5 88.0 90.5
23.00 15.00 0.74 0.45
640-Micron Sand 1.79 1.94 2.48 2.94 3.25
(20)
The data are represented on a log q vs. 1/u diagram (Figure 3). The correlation is linear over the range of velocities of dense fluidization. For irregular particles, the curves are slightly concave upward. In Equation 20 the value of F ranges from 3 to 3.5 poises; E, corresponds to the viscosity activation energy and is directly proportional to D,Z. The probability of particles overcoming the energy barrier is Particle energy exchange therefore occurs readily as in a group of liquid molecules. For the viscosity of a binary system (coarse and fine particles) log viscosities are additive volumetrically. Kendahl’s relation, Equation 21, is also applicable:
277-Micron Sand 75.5 80.0 85.0 87.0 89.0
+ 324-LMMicronSand 70
9.3 3.6 2.64
1.75 766-hfjcron Charcoal 95.8 91.0 88.9 84.0
0.37 0.226 0.0089 0.00525 0.0195 0.0151 0.0098 0.0076 0.0063
80.1 85.0 88.0
0.132 0.045 0.0309 0.0199
20.2 34.0 40.0 45.0
4.7 2.67 2.21 1.87
+ 200-Micron Sand
765-Micron Charcoal f 160-Micron Sand 0.430 0.467 0.532 0.612 0.680
56.5 52.5 50.0 45.0 42.0
755-Micron Sand 1.77 2.05 2.96
5.1 6.2
12.95 7.5 2.84 1.46 1.19
755-Micron Sand 2.50 3.05 4.26 5.6
39.5 33.5 20.0 12.6
384-Micron Sand 2
-
4 - -6 Vm//V - Vm/
A
Figure 4. Relation between viscosity aniFm,/(V-
Vm,)
826
INDUSTRIAL AND ENGINEERING CHEMISTRY
1.18 1.47 1.83 2.56
21.3 15.0 15.0 10.8
1.8 2.0 2.3 2.7 2.88
31.5 26.2 21.4 16.4 15.0
+ 384-AIicron Sand 65 70 76 82 86
0.199 0.107 0.0372 0.0177 0.0138
68.5 80.0 83.5
0.575 0.446 0.250 0.150
70 74.5 86.1 93.1
0.301 0.200 0.174 0.115
+ 538-Micron Sand 75.0
+ 277-Micron Sand
LIQUIDLIKE B E H A V I O R O F FLUIDIZED PARTICLES (21) 1% T m i r = Y o 1% ?lo f Y b 1% 76 This equation amounts to an averaging of the activation energy for viscous flow. An analogy for liquid molecular migration, as given by Frenkel ( 6 ) ,
s a
(22a)
Figure 5. Effect of water flow rate on coarse particle composition in upper and lower layers
Here increasing flow rate U increases the average migration velocity, W p , rapidly at first and later gradually. The diffusion coefficient, D, of a liquid
XI = weight % coarse particles in upper layer X Z = weight % coarse particles in lower layer
i; = X / t o e - E I R T (22) may be applied to a fluidized system :
ir, = Xf/tfe-*r/u
-
D
T/q
CHARCOAL (755~1- y1ND(.Oqu) SAND(5-38 SAND 32
-
3 @
L LL
0
E /
2 B
s L
may be applied to a fluidized system, and this diffusion coefficient, D f , will be proportional to the reciprocal of the fluid flow rate. The magnitude of this diffusion coefficient is a measure of the extent of mixing in the fluidized bed. It changes more in the low velocity range than in the higher velocity range. In gas fluidization the convection effect also must be considered. Batschinski (2) found that the viscosity of a liquid is inversely proportional to the'total free volume Vp: 7 = K/VF (23) where VF = V - V,,,
3 COMPOSITION OF COARSE PARTICLES, WEIGHT 76
tor for the slightly nonspherical bubble. shape was determined using glycerolwater as the calibration system. The factor is given by u,/b,p, and data are shown in Table V. Increase of fluid rate decreased surface tension, uf,and the data were correlated linearly by Ul/Urnf
I
=
( V m f / v ~ ) ~[1 " -(u(Pm,/P)2'3(Umax
-
urn/)/ ( u r n a x - U m j ) l = - umf)
L / ? / ~ ~ r n ~ L X
(24)
This equation is analogous to the change of liquid surface tension, u ) with temperature (17):
Application of Equation 23 to fluidization gives a satisfactory correlation (Figure 4) when the minimum fluidization volume is taken as V,. Surface Tension. The correction facI
6
r/Vm
- (T -
(V,/V)2'*[1 I
?-,)/(To
-
T m ) l=
I
For a fluidized system a decrease of interparticle forces occurs with an increase of fluid rate U analogous to an increase of temperature, T , for a liquid. Miscibility. Two systems, coarse and fine sand and sand-active carbon, were used for the miscibility study (Table VI). The plot of miscibility us. fluid rate (Figure 5) is similar to a liquid solubility curve. For instance, the coarse and fine sand system is similar to watertrimethylamine; the sand-active carbon system is similar to the water-phenol system. The solubility of a liquid in solution may be expressed (as a zero-order approximation) by (10) log ( X i / X z ) = (Xi- X,)Q / R T (26)
Similarly data for the fluidized systems are represented by (Figures 6 and 7 ) Figure 6. Relation between log X l / X z and reciprocal rate of water flow
log ( X d X Z ) = p
+ qlU
(27)
Solubility characteristics of a liquid system are determined by Q (Equation Figure 7. Relation between log X l / X 2 and reciprocal rate of water flow
Coarse and flne system
Charcoal and sand system I
0.l
0.3
OS 1/ u
0.7
0.8
1.5
2.0
2.5
1/u VOL. 50, NO. 5
MAY 1958
827
2 6 ) , the potential energy to break an A-A or B-B molecular pair to form an A-B molecular pair. When Q is positive, the system is stable in the separated state, but there is a mixing tendency from entropy considerations. The mutual solubility of the liquid system depends therefore on the magnitude of these two tendencies. The energy term decreases with temperature, and the system thus has a higher solubility. This is the behavior of the sand-active carbon fluidized system. When Q is negative, the system is stable in the mixed state, and infinite miscibility should occur. However, when free volume decreases, a separation effect occurs, giving the observed solubility. When molecular motion increases with temperature, the free volume associated with each molecule is greater; this results in lowered mutual solubility for the system. For the fine and coarse sand system, the fine sand decreases the free volume of the coarse sand with increasing fluid velocity, thus causing repulsion forces between particles and limited miscibility, even though there is the tendency toward infinite miscibility from potential energy considerations. If Q is equal to zero, an ideal solution occurs with infinite miscibility. Because limited miscibility does occur in a fluidized system analogous to limited solubility in liquid systems, the existence of interparticle forces caused by potential energy differences between the mixed and unmixed states is indicated. Conclusion
Data on the fluidized systems studied indicate that motion and properties of the particles are caused by interparticle potenrial forces. These interparticle forces may be compared with intermolecular forces in liquid systems. This analogy shows : Particle motion is the result of friction forces of the fluidizing agent and the force of gravity. The product of fluid velocity and viscosity characterizes average particle kinetic energy as temperature expresses the molecular kinetic energy of a liquid. The liquidlike behavior of a fluidized system can be substantiated by expansion, viscosity, surface tension, and miscibility measurements.
D, E E,
F G,
G H K L P
Q R
S,
T T, U V V,
W X a
b c
g k 1,
m n
P q r
t U X
Y
a
= coefficient in Equation 3, VeVpz/ (V, - VPY B = coefficient in Equation 3, VPz C = coefficient in Equation 3, V P 3 / 2 D = self-diffusion coefficient of molecule, sq. cm. per second D, = diameter of channel, cm. D, = self-diffusion coefficient of fluid-
A
828
+
9D p
P
= constant in Equation 5,
8,
expansion coefficient of fluidized particles, seconds per cm. = expansion coefficient of liquid, degree-’ = voidage = viscosity of fluidized system, grams/cm. second = viscosity of liquid, grams/cm. second = mean distance between adjacent molecules or particles, cm. = viscosity of fluidizing fluid, gram/ crn. second = number of vibrations, second-1
61
Nomenclature
ized particle, sq. cm. per second = particle diameter based on screen analysis, cm. = activation energy for viscous flow of liquid, calories = constant in Equation 20; corresponds to activation energy for viscous flow of fluidized particles, cm. per second = coefficient in Equation 20, poises = coefficient in Equation 18, secondl’z = coefficient in equation 17, = coefficient in Eauation 3. V,2 VP2/(v,- V,)$+ Vp3/(ke V,) V,”/2(Ve - V,)Z = coefficient in Equation 23, poises x cc. = bed length, cm. = pressure drop through bed, grams per sq. cm. = potential energy difference between molecular pairs, grams X cm. = gas constant, gram cm. per K. = specific surface of particles, sq. cm. per cc. of particle = temperature, K. = melting point, K. = flow rate of fluidizing fluid, cm. per second = volume occupied by bed, cc. = free volume of liquid molecule, cc. = load of modified Stormer viscometer, grams = concentration of a comaonent (of molecule or of pa’rticle), wt. 7 0 amplitude of volume vibration, cm. volume of bubble, cc. force constant of vibration, (d2@/ dr2)r = re, grams/cm. acceleration of gravity, 980 cm. per sq. second Boltzmann constant length of a microfluidized bed in equilibrium state, cm. reduced mass of oscillator, grams number of particles in a micro fluidized bed constant in Equation 27 constant in Equation 27 distance between adjacent particle, cm. time, seconds velocity of particle displacement, cm. per second displacement of oscillator, cm. volume fraction constant in Equation 3, 200 PI, fDp2g
E
1f 11
x P V
INDUSTRIAL AND ENGINEERING CHEMISTRY
=
= absolute density of fluidized par-
p
pp (r
r
@
(3
ticles, grams per cc. density of solid particle, grams per cc. = surface tension, dynes per cm. = “mean life” of a moleculc or particle in the same equilibrium position, second = potential energy of volume vibration of a microfluidized bed, gram cm. = migration velocity of liquid molecule, cm. per second = migration velocity of particle in fluidized bed, cm. per second =
Subscripts a , b = components in binary system c = critical point of liquid e = equilibrium state of fluidized system f = fluidized bed g = glycerol-water m = melting point of liquid mf = minimum fluidized state = particle = upper layer 2 = lower layer
?
Acknowledgment
The authors express their appreciation to Michio Kurata for his helpful suggestion in preparing this paper. Thanks are also due Masayuki Kawahata for help in translating the manuscript. literature Cited
Andrade, E. A. da C., Phil. M a g . 17,497 (1934). Batschinski, A. J., Z . jhysik Chem. 84, 643 (1913). Bauer, ‘E., Magat, M., Surdin, M., Trans. Faraday Snc. 33, 81 (1937). Diekmann, R., Forsythe, W. L., Jr., IND.ENG.CHEM.45,1174 (1953). Ergun, S., Orning, A. .4.,Ibid., 41, 1179 (1949). Frenkel, J., Trans. Faraday Snc. 33, 58 (193G). Furukawa, J., Ohmae, T., M g y 8 Kagaku Zasshi 55, 3 (1952). Glasstone, S., Laidler, K. J., Eyring, H., “Theory of Rate Processes,” McGraw-Hill, New York, 1941. Guggenheim, E. A., “Mixtures,” p. 36, Oxford Univ. Press, London, 1952. Kotani, M., “Physics of Molecule,” p. 30, Kyoritsu Publishing Co., Tokyo, 1950. Landolt-Bornstein, “PhysikalischChemische Tabellen,” p. 1051, Springer, Berlin, 1923. Leva, M., Grummer, M., Chem. E n g . Progr. 44, 511 (1948). Matheson, G. L., Herbst, W. A., Holt, P. H., IND.ENG.CHEM.41, 1099 (1949). Ohmae, T., Furukawa, J., K6gyB Kagaku Zasshi 57, G (1954). Ibid.,p. 783. Ibid.,p. 788. Ohmori, K., Ono, S., “Chemical Physics,” p. 416, Kyoritsu Publishing Co., Tokyo, 1950. Toda, M., “Structure of Liquids,” pp. 125, 164, 268, Kyoritsu Publishing Co., Tokyo, 1947. RECEIVED for review October 20, 1953 ACCEPTED July 18, 1957
