of plane of constant concentration, cm. 'second = propagation velocity of concentration discontinuity, cm./'second = partial differential operator = voidage, dimensionless = solids density, grams/cc. = propagation velocity
/3
6
b t
p8
literature Cited
(1) Adler, I. L., Happel, J., Cheni. Eng. Progr. Symp. Ser. 58, KO. 35, 98 (1962). (2) Benenati, K. F.: Brosilow, C . B., A.I.Ch.E. J . 8, 359 (1962). (3) DeHaas: K.D.. M S .thesis, Purdue University, 1963. (4) Happel, ,J,: Pfeffer, K.! d.I.Ch.E. J . 6 , 129 (1960). (5) Kynch, G. J., Trans. Faraday SOC.48, 166 (1952). (6) Oliver, D. R., Chem. Eng. Sci. 15, 230 (1961). (7) Richardson. J . F., Zaki, \V.N.,Ibid.. 3, 65 (1954). (8) Shannon, P. 'r.. DeHaas, R. D., Tory, E. M . , Chemical Engineering Symposium. Division of Industrial and Engineering Chemistry, ACS, University of Maryland, College Park, Md., NO\.. 14-15. 1963. (9) Shannon. P. T.. Strouue. E. P.. Torv. E. M.. IND. E s c . CHEM.FUSDAMENT'.UA 2, i o 3 (1963): (10) Stimson, M.. Jeffrey, G. B.,Proc. Roy. Soc. A 111, 110 (1926). (11) Stroupe, E. P., h1.S.. thesis, Purdue University, 1962. (12) Tory, E. M., Ph.D. thesis, Purdue University, 1961. (13) Verhoeven, J., B. Eng. thesis, McMaster University, 1963.
SUBSCRIPTS a = above concentration discontinuity 6 = below concentration discontinuity cr = pertaining to constant rate period i = initial value-i.e., a t time zero t = tangent point on flux curve of chord from initial flux 1 = intersection of flux curve and chord from final flux tangent to curve 2 = a t first inflection point of flux curve 3 = tangent point on flux curve of chord from final flux m = final value-i.e., a t infinite time
\
/
,
I
RECEIVED for review November 9, 1962 h-ovember 5, 1963 RESUBMITTED ACCEPTED March 2, 1964
SUPERSCRIPT * = pertaining to batch settling ( u = 0)
HOMOGENEOUS FLUIDIZATION E
.
R UC KENST E I N
,
Polytechnical Institute, Bucharest, Romania
A physical model is proposed for a homogeneous fluidized bed. An equation of motion is established for one of an ensemble of particles in interaction with a fluid. By means of this equation an expression is established for the variance, ut2, of the void fraction, a quantity connected with the mixing process taking place in the fluidizing agent. An expression for the axial diffusion coefficient is established starting from the model suggested here and using the equation derived for ue. This equation i s in satisfactory agreement with experimental results obtained b y Kramers.
M
a fluidized bed behaves a t low velocity like a liquid, a t somewhat greater velocities like a liquid containing gas bubbles, and finally, a t high velocities, like a boiling liquid. T h e first case corresponds to homogeneous fluidization. I n the last two cases part of the fluid travels through the bed as bubbles (8, 72, 20, 24, 35, 37). If the flow rate of that part of the gas which travels through the bed as bubbles does not increase along the height of the bed (36), the bed behaves like a liquid traversed by gas bubbles; if it increases ( 3 ) ,the bed behaves like a boiling liquid. An attempt has recently been made (28) to explain the appearance of these structures. T h e macroscopic analogy between a homogeneous fluidized bed and a liquid has led Furukawa and Ohmae (70) to propose a "microscopic" theory of the fluidized bed, based on an analogy with the microscopic theories of the liquid state. IVithin this theory the attractive forces bet\veen molecules have as a counterpart the weight of the solid particles, while the repulsive forces correspond to the forces of friction between the fluid and particles. Assuming that the volume of the bed is pulsating. they express the potential of these forces as a function of rhis volume. The transition from global to microscopic is made by equating the hydraulic diameter to the distance bet\\.een rlvo particles. This assumption yields a relation bet\\.een the void fraction and the distance between two parrides. ACROSCOPICALLY,
260
I&EC FUNDAMENTALS
Furukawa and Ohmae assume further that the solid particles are subject to a harmonic oscillatory motion; they then establish an equation for the mean kinetic energy of the particles and consider that this quantity represents in a fluidized bed Xvhat temperature represents for a liquid. .\ similar idea has been suggested by Todes ( 3 4 ) . By using the mean kinetic energy of the solid particles for the temperature in a number of equations valid for the physical constants of a liquid, FurukaLva and Ohmae have obtained equations for certain physical properties of the homogeneous fluidized bed-e.g., viscosity and surface tension. T h e aim of the present paper is to obtain information as to the behavior of a homogeneous fluidized bed without using any analogy, but starting with a certain model and an equation of motion for a particle which is one of an ensemble of particles interacting with a fluid. The model as well as the equation of motion is used to study axial mixing in the fluidizing agent. The Physical Model
The bed becomes fluidized when the pressure drop through the fixed bed multiplied by the cross section of the fluidization column equals the Lveight of the solid particles. For fluid velocities higher than rhe minimum fluidization velocity the bed expands and solid particle motion sets in. The difference in behavior of a fixed and a fluidized bed is due to this motion
of the solid particles. I n the case of a fixed bed, Peclet’s number: defined by the expression
Pe
E
2R
U ~ / ’ EE
(\\here K is the particle radius, u, the velocity in the empty tube, E the mixing coefficient. and e the void fraction), is constant for sufficiently high values of Reynolds number ( 7 , 27, 26). H‘ a fluidized bed \vere only a physical extension of a fixed bed? the mixing rvithin the fluidization agent bvould be caused, just as in the case of a fixed bed, by the motion of the fluid between the fixed solid particles and Peclet’s number should therefore become greater as the void fraction increases (see Equation 2 8 ) . I n fact. Peclet‘s number for radial mixing decreases a t first as e increases, goes through a minimum for E = 0.7, and thcn increases (6. 7 7 ) . This is probably because of the solid particle morion within the bed, which causes such a n increase of the mixing coefficient that Peclet’s number decreases, although an effect similar to that which takes place in a fixed bed tends to increase it. Variations in the coefficient representing axial mixing in a fluid bed cannot be explained in the \yay that has been used for fixed beds. In a gas fluidized bed the solid particle motion takes place throughout the whole bed, and the solid particles mix rapidly. Sutherland (32) has given experimental proof that this rapid mixing is caused by the bubbles. It is therefore reasonable to assume that in the limiting case of a homogeneous fluidized bed the mixing of the solid particles is negligible. [The state of homogeneous fluidization is a limiting one, approximately rcalized only if the liquid flow through the fluidized bed is free from external disturbances ( 7 9 ) .] Therefore in the case of homogeneous fluidization the individual particles d o not range throughout the whole bed, but move short distances around their average positions. \$‘e therefore represent the homogeneous fluidized bed as an ensemble of solid particles moving about certain points considered as the nodes of a n imaginary lattice through \vhose free volume the fluid is flowing. Beriveen a liquid and a fluidized bed there exists a certain parallelism. T h e diffusion coefficient in a liquid and the mixing coefficient of the solid particles in a homogeneous fluidized bed are both very small. T h e mixing coefficient in a liquid traversed by bubbles (29. 37) and that of the solid particles in a nonhomogeneous fluidized bed (32) are, however, large. Let us now characterize this ensemble of particles from the geometrical point of view. A solid particle is associated tt-ith a certain mean volume \L-ithin the fluidized bed, which is given by
“fluid atmosphere” pertaining to a given particle mean an average atmosphere pertaining to a relatively small group of particles; nevertheless the population of this group must be large enough to make it possible to define its void fraction. \.Ve call this quantity the local void fraction. Hence the atmosphere surrounding a particle \vi11 be characterized by the length
(3) where E is the local void fraction 6 varies in time. Its pulsation may be characterized statistically by the variance, u> u,* =
(E
- ern)*
(4)
where E , is the time average of the local void fraction. (In the case of a homogeneous fluidized bed the time average of the local void fraction may be considered equal to its spatial average-i.e., to the over-all void fraction of the fluidized bed.) T h e pulsation of the fluid atmosphere-i.e., of lengths 2-may be characterized statistically by length L.
L is proportional to a mean of the absolute values of the fluctuations of CC. Since the fluctuations of 2 are due to the solid particle motion, it is reasonable to assume that the mean of the absolute values of the solid particles‘ displacements is proportional to L. Equation of Motion of a Solid Particle
According to the model used, the solid particles move about the nodes of the imaginary lattice mentioned above. \Ye now make the simplifying assumption that this motion is unidimensional along a vertical direction. Each solid particle travels a distance L’ about a node of the imaginary lattice: L’ 2 above the node and L’ ‘2 below it. ,4t the points situated a t i L ’ 2 from the node the velocity of the particle is zero. because a t these points the direction of motion is reversed. The period of a cycle (up and dowm motion) is denoted by T. I n establishing the equation for this motion of the particles we neglect interaction by collision between them. \Ye first write the equation of motion of a single particle in a fluid :
m where R is the radius of the particle. If the horizontal and vertical distributions of the particles are equal, the imaginary lattice is a simple cubical one and the volume, V , may be characterized by a single length
4 srn3(1 -
E
r
=
n
~
3
~3
(2)
Brcause of the motion of the solid particles, the ”atmosphere” of each single particle is changing continuously. Unfortunately, it is difficult to characterize the geometry of this atmosphere exactly-. Tvhile it is possible to define the void fraction for a sufficiently large number of particles and therefore an average volume of the atmosphere of fluid. this average void fraction will not apply on a fine scale to the fluid surrounding any single particle. Therefore. in the follo\ring: the Lvords
4 ~
3
nR3p
dw ~
dt
(6)
where v is the velocity of the particle, t the time. g the acceleration of gravity, w the relative velocity of the particle with respect to the fluid: p the fluid density, and m a factor Xvhich, multipled by 4 ‘3 .irRipp.gives the so-called apparent mass. The left side of this equation represents the product of the mass of a particle and its acceleration. The first term on the right side represents the difference between the weight and the buoyant force. The second term represents the resistance of the fluid to the motion of the particle. T h e drag coefficient. C: depends on Reynolds number. in which the velocity is that of the fluid’s relative motion with respect to the particle. Probably C depends on acceleration too. but since this influence is not knoivn. i t \vi11 not be taken into account. The last term on tha right side of the equation represents the reaction of the fluid on the particle. For a perfect fluid m = VOL. 3
NO. 3
AUGUST 1964
261
l t ' 2 (76). For a viscous fluid its value may be still taken as l('2 if w is sufficiently small ( 7 5 ) . In the case of unidimensional motion in the vertical direction, the only one considered for the time being, Equation 6 takes the form :
m
4 3
dut nR3p
~
dt
Therefore
By aid of Equation 11 the void fraction, e,. may be expressed as a function of the velocity of the fluid. u,. Eliminating ( p s - p)g benveen Equations 9 and 10. one obtains:
(7)
\Ye now establish the equation of the motion of a particle belonging to a homogeneous ensemble of particles. First, a n expression has to be found for the resistance of the fluid to the motion of the particle. A remark made by Ranz (23) concerning the pressure drop in a fixed bed makes it possible to overcome this difficulty. Ranz concludes that in the case of a fixed bed the resistance opposed to the motion of the fluid may be calculated by making use of the expression Ivhich holds for a single particle, if in that expression the velocity is replaced by the velocity corresponding to the minimum cross section of the free volume between the particles. The geometrical model used enables us to express the velocity, u1: of the fluid in the minimum cross section as follo\vs :
de
mu,p
-~
tYPs
+ mp) dt ~~
(1 2 )
~~~~
Expanding in series the difference C(u,q - t)' - C(u,q - t)?, retaining only the first-order term, and neglecting 1' as compared to u,p, we obtain
de
mop €'(Ps
+
-
mp)
df
(12')
Equation 12' is now used to obtain a relationship benveen and uc.
i2
Approximate Relationship between v ? and u e Obtained by Use of Equation of Motion
\Ye assume that Equation 7 may be extended to one particle belonging to an ensemble of particles: introducing, however, in the expression of the resistance? the velocity given by Equation 8. Probably this assumption is justified only if throughout their motion the particles remain uniformly distributed. Only under these conditions is it possible to characterize a group consisting of a particle and its neighbors by a single geometrical parameter-the void fraction. Such a situation arises only if u e is sufficiently small. Since, a t least a t present, it seems impossible to establish an equation of motion closer to reality, we continue to use Equation 7 with the above modifications. \Ye furthermore consider that coefficient m does not depend on the void fraction. I n this way Equation 9 is obtained for a particle belonging to a group: (Ps
+
dc mp)
;=
-(Ps
-
p)g
+ 38 c PR -
-
(uoq
-
0)2
Equation 1 2 ' cannot be solved, because \$-edo not know the dependence of the void fraction, e. on time. Some information concerning the relation betiveen 3 and u, ma)- be obtained using a grossly approximate computation method Lvhich consists in estimating each term of Equation 12' in terms of the time T : and the average values: 3 and u c :
3 8(ps
muop
+
e2bs
+
mp)
'de' 'dt,
+
pu,2 mp)R
--
b(Cq2)
(k)e=cm (13') 6,
muop
+ mp)
€m2(~s
(13'') 7
Replacing each term in Equation 12 ' by its approximate value, multiplied by a constant, \ve are led to Equation 14. T h e fluid velocity in the last term on the right side of Equation 9 is expressed by the ratio u , . / E . T h e drag coefficient, C, is a function of Reynolds number formed Lvith the relative velocity, u,q - i. By averaging Equation 9 \vith respect to time, between 0 and T . one obtains 3 P
Seglecting
18
(10)
7 1 ~T
and
262
l&EC FUNDAMENTALS
-kl
(Ps
+
mp)R
f
= em
u< -
\\.here k l and kr are t\vo positive constants of the order of unity. The reason for assigning algebraic signs to each of the three terms of Equation 14 is given in Appendix I. Since
compared to u , c . Equation 10 leads to
Denoting the values of C and p for e = e m . C,: and uo. w e consider in a first approximation that the mean value of the product Cc' is practically equal to Copo2.
-
it follo\\s that
Making use of Expression 15. Equation 14 becomes
Solving this second-degree algebraic equation, \ve obtain
where r
\\.here
Probably (see Appendix 11)
anism of axial mixing \vithin the fluidizing agent i n the case of a homogeneous fluidized bed has remained to a large extent unexplained. 'l'he present qection points out a possible mechanism for the latter process and establishes certain equations for the axial mixing coefficient. \Ve first sho\v that the values found expc.rimentally for coefficient E, (.i.1.9) are. for not too low values of the void fraction. considerably higher than the calculared values. if i t is supposed: by analogy with the fixed bed. that the mixing is a consequence of fluid motion brt\veen the colid pariicles assumed to be fixed. S e x t , taking into account a qua'i-local motion of the qolid particles. \\'e establish an equation for the axial mixing coefficient. This equation agrees satisfactorily Lvith the experimental results of K r a m r r i ( 1 9 ) . ( O n l y Kramers' data are used for comparison. because this is the only investigation in which precautions \vex taken to ensure as homogeneous fluidization as possible.) Seither the fluid's o\cn turbulence (although this may be important for sufficiently high values of the void fraction) nor the 'I'aylor effect ( d . 25. 3.1) (\\-hich becomes incrrasingly important as the velocity distriburion over the cross section drviates from a uniform distribution) is considered. \Ve noiv evaluate the part contributed to the total axial mixing by the motion of the fluid betlveen the solid particles. the latter being assumed fixed. The axial mixing coefficient due to this effect is denoted by El. 'I'his effect is similar to the one occurring in a fixed bed and for this reason the calculation is based on an analogy with the fixed bed. I n the case of a fixed bed and for sufficiently high values of Reyiolds number. the axial mixing coefficient can be computed from Equation 23
(7, 27). E, = Ru
(23)
Equation 23 may be deduced by the aid of Einstein's equation A n Equation for ue
T o obtain separate equations for 7 * and u.: one more relation is required. S o such relation can be derived from our model. 1-herefore. i c e assume that the average kinetic energy density of a particle is proportional to the difference between the kiiietic energy densities of the fluid a t velocities u, and u,.
\\-e have chosen the form of Equation 21 - as a difference of energies. in vie\\ of the fact that a t u,, = u,: zs2 must vanish. Eliminating z 2 betwwn Equations 17 and 21 and using Equation 20 for $, \ve are led to Equation 22 for 6,.
.S series of ecluations expressing certain properties of the fluidized bed may be established b)- use of Equation 22. \.e begin \vith the axial mixing in the fluid acting as fluidizing agent. The expression \vhich \vi11 be established for the axial mixing coeficient conrains the variance of the local void fraction ; Ekjiiation 22 i.; L I to ~compute this quantity.
Axial Mixing Coefficient in a Homogeneous Fluidized Bed
.ilthough iinl~ortaii~progress has been achieved lately concerning tile mrchanism of axial mixing- in a fixed bed [ v e the comprc~hetisive paper by Hofmann ( 1 . 3 ) ] . the mech-
E =
(At)z
2 0
using the square of the particle diameter for the mean value. ~~
( A x ) * , of the square of the displacement and the interval. 2 R/ u . during Lvhicli the fluid floics through a layer of particles for time 8. For a fixed bed Ice have therefore
U
For a first evaluation of the axial mixing coefficient \re use another geometrical model for simplicit)-. different from [hiit used in the first part of this paper. \Ye consider the fluidized bed as being obtained from a fixed bed by separating the horizontal la)-ers of particles. I n other lvords. the distance bet\veen the centers of t\vo successive particles is 2 R in a horizontal direction and 2' i n a vertical direction
\\'e also suppose that thr fluidi7ed bed consist%of a w c c e \ ~ i o t iol' cells of t\\.o kinds: i n k i n g cells having a Iirighc 2 R roti\i\ting of a horizontal layer of particlrs. in \chic11 the mixinq idkeq place as in a fined bed. each follo\\ccl by a particle-free c r l l \vith no mixing. C ' - 2 K high. (If ihe lluitlizcd bed \verr represented b> the geometrical model ii.ed iiiitiall! , \vc cotrld not assume that the mixing in each hOri7olital 1a)t.r of particlr> VOL. 3
NO. 3
AUGUST
1964
263
takes place as it does in a fixed bed and therefore the calculation which follows could not have been carried o u t ) Consequently. by analogy tvith a fixed bed, in the case considered, we have
I
t
t I006
ro4
Einstein's equation leads to
E1
2 -
R2U
a03
2'
1
Eliminating S ' by using Equation 25. we obtain (26):
El
6 = K
Ru(1 - em)
Pel increases with the void fraction, e, and becomes infinitely large for E , = 1. For a fixed bed, Equation 28 leads to the well known expression
=2
(28 I )
Equation 28' is. for R e > 20: in good agreement with the experimental data concerning axial mixing in gases. I n the case of a liquid flowing between the particles of a fixed bed for Reynolds number from 0.01 to 150, Pe is in the range 0.3 to 1
19). For this reason, in the case of liquids the actual value of E , within the stated range of Reynolds numbers is several times higher than the value obtained from Equation 27. In Figure 1 coefficient E1 is plotted against the void fraction, for glass spheres 0 . 5 and 1 m m . in diameter: the fluidizing agent being \later, Comparing the computed values to those obtained experimentally by Kramers (shown in Figure 2 ) . it is found that E l is considerably smaller than the experimental values. Although in a liquid-fluidized bed and in the range of Reynolds numbers \vithin ivhich the experimental determinations (79) have been carried out! the actual values of E , are probably- t\vo to three times larger than those derived from Equation 2'. for not too low values of the void fraction they are considerably smaller than the values obtained by Kramers. T h e mixing in the fluidizing agent is therefore due not only to the floiv of the liquid bet\\.een the solid particles assumed to be fixed. but also to another cause. I n a homogeneous bed the solid particles are not fixed but are subjected to a motion of quasi-local character. Oiling to this motion a mixing process takes place in the fluidizing agent. and by taking this e f k t into account a satisfactory agreement with Kramers' experimental results is obtained. O t h r r causes are also possible. G r o s internal circulation of the fluid and particles is one possible cause. O n the othcr hand. Harrison. Davidson. and Koch (72) estimated the maximum stable size of a ..bubble" and reached the coiic1u;ion that all fluidized beds contain bubbles, but that for liquid-fluidized syatrms they are small (of the same order as thy parriclc diameter). It has also been sho\vn that the 264
I&EC FUNDAMENTALS
1""
(27)
T h e mixing coefficient. E,, first increases with velocity, goes through a maximum, and then decreases and vanishes for Isni = 1. Peclet's number for mixing, formed with coefficient El, is given by
Pe
e
0.4
0.6
&m
I
0.8
Figure 1. Axial mixing coefficient, El (Equation void fraction for glass spheres fluidized with water
27), vs.
homogeneous structure represented by an ensemble of solid particles fixed in the nodes of an imaginary lattice (a model different from the one used in the present paper) is stable for perturbations neither on the scale of one particle (27) nor on a "macroscopic" scale ( 7 4 . If this is the case, any fluidized bed contains bubbles which? obviously: have a n influence on the value of the mixing coefficient. Equation for Axial Mixing Coefficient. $\e' now establish an equation for the axial mixing coefficient. E,, taking into account the quasi-local motion of the solid particles. T h e mixing in the fluidizing agent becomes more intense as the amount of energy. E , dissipated locally per unit time and unit mass is larger. The term "locally" refers to an element of fluid having a width. 1, equal to the scale of turbulence. In this element E is converted into heat by a highly intricate mechanism. For the case of very high Reynolds number this mechanism has been investigated by Kolmogoroff (2. 77, 78> 22). T h e axial mixing coefficient. E,. depends on the amount of energy, E , dissipated per unit time and unit mass of fluid, the length, 1: the fluid density. p , the fluid viscosity, 7 . and the diffusion coefficient, D, of the tracer used in the experimental de terminations. Dimensional considerations lead to
For sufficiently high values of the Reynolds number, effects on a molecular scale have a negligible influence on the mixing coefficient. However, if E, is to be independent of 7 and D, function J must take the form:
Consequently for sufficiently high values of the Reynolds number
E,
(30)
a E1 31'!3
The amount of energy dissipated per unit time within the entire fluidized bed is given by Equation 31.
H = L.A,b
=
u,Shg(p, - p ) ( l
- e,,!)
(31)
If the solid particles are not too far apart. i t may be assumed that the energy dissipated per unit mass of the fluid is dis-
the decrease of n: and above a certain value of e , because of the decrease of E it lowers E,. For this reason the curve of E, CS. E, goes through a maximum. This maximum has not become apparent in Kramers' experiment (79), either because the value of e , for which it appears was higher than the highest value e , = 0.9 reached by the void fraction in those experiments, or because with increasing values of e , the turbulence of the liquid a n d the Taylor efyect became increasingly important. I t is not yet possible to establish an equation for 1 except for a limiting case, in which it is assumed that 1 depends only on the amplitude, L ' . Since a factor reducing I \\-hen e,,, increases is not taken into account: the relationship benveen i and t m thus obtained must lead to a too rapid increase of the mixing coefficient with eN1. If we assume that 1 depends only on L ' , it follo\vs that
i
a
L'
(35)
Consequently
Eliminating
E,
a
u,
between Equations 22 and 30, we obtain
+
( z ~ ~ g R ~ ) ~ (~p~S( -y mp )) *' ~~ [ 1 - ($)2]23X
Using for 63 Equation 3, for 9 Equation 8. and for the drag coefficient, C, Allen's equation
C Figure 2.
Comparison of Equation
__ ....
Equation 3 8 Kramers' d a t a
38 with
=
_
18.5 _
~
Kramers' data
Equation 37 becomes
(79)
tributed quasi-uniformly throughout the bed a n d therefore
[I For not too low valurs of the void fraction the mixing in the fluidizing agent may be due mostly to the motion of the solid particles. If so. i r seems reasonable that, for values of the void fraction not too close i o irs minimum value, length 1 should depend on the amplitude. L ' '2, of the solid particle oscillations, and on the number. n , of oscillators (particles) contained in a unit volume. 1 takes higher values as L' and n increase. T h e number of oscillators per unit volume is given by 11
=
1 - E, 4 - 7R3 3
__
(33)
and L ' by (34) X s the void fraction increases. L' increases but n decreases. For these reasons it is possible that the curve i = / ( e ) goes through a maximum. T h e increase of the void fraction has t\vo opposite consequences: Because of L' it increases the axial mixing coeficietit. and. on the other hand, because of
-
1.21 (1 -
e,)2311.fi
(38)
I n Figure 2 Equation 38 and Kramers' experimental results are compared. ,4fairly satisfactory agreement is obtained if we take k = 3.3. T h e weak maximum shoivn by the experimental curve for t, = 0.67 cannot be explained by Equation 38-i.e.. by considering the mixing due to the motion of rhe solid particles. I t may, hoLvever, be explained in another way. ( I t is possible that the explanation of the weak maxim u m of the experimental curve for 2 R = 0 . 5 m m . and the change in curvature of the other experimental curve should be related to the mixing coefficient E l . ) There is a discrepancy between the curve corresponding to 2 R = 0.5 m m . and rhe curve corresponding to 2 R = 1 mm.. probably due to the fact that Equation 38 holds only for sufficiently high values of Reynolds number. and rhe range of R e p o l d s numbers that appears for particles with a diameter for 2 R = 0.5 m m . corresponds to too low values. T h e positive deviarions found for 2 R = 1 m m and higher values of e,, are due to the fact that the dependence of i on n has not been considered. T h e negative deviations for the lower values of tni are due to the facr that Equation 38 does not take into account the mixing resulting from the flow of the VOL. 3
NO. 3
AUGUST
1964
265
Equation 39 is in satisfactory agreement rvith Kramers’ experimental results referring to axial mixing in a fluidized bed. I t is thercfore sometimes possible to extend the multitubular model to a fluidized bed. This is probably due to the fact that although the solid particles drift apart. the mixing due to the motion of the solid particles compensates for the influence of this drift. Appendix I.
Detailed Derivation of Equation 14
\Ye deduce Equation 14 in another way which! to a certain extent, permits stating more precisely the approximations Lchich have been introduced and clarifying the problem of sign assignment. hlulriply Equation 12’ by o; then
1 d _n%P _ _ ~v - e
p,
+
dt
mp
Integrating this equation bvith respect to time bet\\een 0 arid t (the moment a t which the particle is a t a distance - L’ 2 from the node is taken as initial moment). we obtain
1 Pa
and for the mean value of c2 over period
Figure 3.
mP
c - dt
dt
T
Comparison of Equation 41 with Kramers’ data r
__ .. . .
Equation 41 Kramers’ d a t a
E,
a 11:3Rh4.3
Em O:
1 -
l
T h e two integrals appearing in this equation cannot be evaluated, as we d o not know how the void fraction and or the velocity depend upon time. Consequently these quantities are evaluated by dimensional considerations. being expressed as functions of T and of the mean values, uc, u2, and E , ~ . A qualitative plot of the product, .(e - c n L ) is shown in Figure 4, c. From the curve it may be concluded that
(39)
\There Rh represents the hydraulic radius. Equation 39 has been established by using for the fixed bed the multitubular model (7). I n accordance Lvith this model the fixed bed may be assumed equivalent to a bundle of tubes of radius equal to the hydraulic radius, Ri,:
R,,
.
( I 9)
fluid betlveen the solid particles, which are assumed to be fixed. This effect is not without importance for void fraction values approaching the minimum. It seems: therefore. that the microeddying a t the scale of one particle can explain Kramers’ experimental data. Ruckenstein and Teoreanu (30)have proposed a n expression for the axial mixing coefficient in a fixed bed :
de
- €,)dl
5
0
if the area of each shaded portion of Figure 4. c, is smaller than the area of the preceding unshaded portion. Under these conditions it is obvious that
[l
L’(E
- R
-
e,)
dt
1
dt
0 ; moreover. it seems verv
probable that the shaded a n d unshaded portions do not differ considerably from each other. It \\ill therefore be arsumed that the first integral of Equation I takes a negative valur Dimensional considerations lead to
17
d Figure 4.
b
C
e
Qualitative plot of void fraction, velocity v of particle, v ( e - e,,,), (d/dt)(l/ e ) , and v(d/dt)(l/e)
vs. time
T h e product c(d d t ) ( l E ) appears in the second integral A qualitative plot of the derivatlve (d d t ) ( l E ) is shoun in Figure 4. d. lvhile Figure 4. e. represents the product ~ ( ddt ) ( l 6 ) . From the curve3 it may be concluded that
85 8L 75
Dimensional considerations lead to
7C
65 Finally we obtain Equation 14. T h e quantities k l and A ? . considered as constants in a first approximation. noiv have a clearer meaning:
66
55 5C
\
95
40 T h e approximation u.hich consists in taking k l and k? as constants is similar to that Lvhich appears in the classical theory of turbulence, where the correlation coefficient of the two components of the velocity fluctuations is considered to be constant.
35 30
25
Appendix II.
20
Analysis of an Approximation
T o prove that Inequality 1 9 is satisfied. the quantity
/5
/o is plotted in Figure 5 against the void fraction. e m . C is given by Equation 3. p by Equation 8:and the drag coefficient by Allen's equation
5 0
Figure 5. j group vs. void fraction VOL. 3
NO. 3
AUGUST
1964
267
T h e two curves shoivn in Figure 5 have been drawn for 2 R = 0.5 m m . and 2 R = 1 mm., the fluidizing agent being water. This plot shows t h a t j > 10. From the way in which constants k l , kp, and k~ have been introduced, it may be inferred that they are probably of the same order of magnitude (unity). For this reason it seems probable that Inequality 19 will be satisfied even if the fluid is a liquid. If it is a gas, the ratio pJp takes very high values and Inequality 19 is satisfied. Nomenclature
C D E El
=
drag coefficient
= diffusion coefficient =
mixing coefficient
= axial mixing coefficient due to fluid motion
bettceen solid particles assumed to be fixed g
= acceleration of gravity
h
=
H
= energy dissipated per unit time in bed
k , k‘,
kl,
k l . ks
height of fluidized bed
constants scale of turbulence = lengths defined by Equations 2. 3. and 5 = amplitude of u p and do\+n motion of a particle = distance defined by Equation 25 = coefficient in expression of apparent mass = number of particles per unit volume = pressure drop through bed = radius of solid particles = hydraulic radius = cross section of fluidization column = time = fluid velocity = volumetric flow rate of fluid = G S = value of u, a t incipient fluidization = =
= U”V = velocity of solid particle =
= = = = =
= = =
= = = = = = =
268
l&EC
average temporal value of squared velocity of a particle volume defined by Equation 1 relative velocity between particle and fluid displacement time interval quantity defined by Equation 8 quantity defined by Equation 18 local void fraction mean value of void fraction variance of local void fraction dynamic viscosity of fluid kinematic viscosity of fluid density of fluid density of solid particles dissipated energy per unit time and unit mass of fluid time interval for u p and down motion of a particle
FUNDAMENTALS
literature Cited
(1) Ark: R., Amundson, N. R.. A.I.Ch.E. J . 3, 280 (1957). (2) Batchelor, G. K., Proc. Cambridge Phil. Soc. 43, 533 (1947). (3) Baumgarten, P. K.. Pigford, K. I>., .4.I.Ch.E. J . 6 , 115 (1960). (4) Bischoff, K. B., Levenspiel. O., Chern. En?. Sci.17, 25- (1962). (5) Cairns, E. J . , Prausnitz, J . M , A . 1. Ch. E . J . 6, 400 (1960). (6) Ibid., p. 554. (7) Carman, P. C., “Flow of Gases through Porous Media,” Butterworths, London. 1956. (8) Davidson, J. F., Paul. K. C., Smith. M. J . S.: Duxbury, H . A . , Trans. Inst. Chem. Engrs. (London) 37, 323 (1959). (9) Ebach, E. A , \Vhite, R. K.. A . I. Ch. E. J . 4, 161 (1958). (10) Furukawa, J., Ohmae, T., Ind. Eng. Chem. 50, 821 (1958). (11) Hanratty, T. J., Latinen, G., LVilhelm, K. H . , A . I . Ch. E . J . 2 , 372 (1756). (12) Harrison, D., Davidson, J. F., Koch, J. \V., Trans. Inst. Chem. En,grs. (London) 39, 202 (1961). (13) Hofmann, H.. Chem. EnE. Sci.16. 193 (1961). (14) Jackson. K., Trans. Inst. Chern. E n y s . (London) 41, 13 (1963). (15) Kada, H., Hanratty. T. J., A . I . Ch. E . J . 6, 624 (1960). (16) Kocin, H. E., Kibel. I. N.: Koze. 9.V.. “Teoreticeskaia Ghidrodinamika (Theoretical Hydrodynamics),” Gostehizdat, Moskow, 1755. (17) Kolmogoroff. A. N., Dokl. Akad. .VauX SSSR 31, 538 (1941). (18) Ibzd.. 32, 16 (1741). (17) Kramers, H., Westermann, M. D., Groot, J. H., Dupont, F. A. Third Congress of European Federation of Chemical Engineering, B1, 1762. (20) Leva, M., “Fluidization,” McGraw-Hill, New York, 1959. (21) McHenry, K. LV., Wilhelm. R. H.. A . I. Ch. E . J . 3, 83 (1937). (22) Obuhov, A. M., Iaglom, A. M., Priklad. .Mat. i Mehanica 15, 1 (1951). (23) Ranz, LV. E., Chem. Eng. Progr. 48, 247 (1952). (24) Reboux, F., ”Ph6nom2nes de Fluidisation,” Association Fransaise de Fluidisation. Paris, 1954. (25) Ruckenstein, E., Comfit. Rend. 253, 1166 (1961). (26) Ruckenstein, E., Reo. Chim. (BucharPst) 11, 721 (1960). (27) Ruckenstein, E., Reo. Phys. Akad. R P R 7 , 137 (1962). (28) Ruckenstein, E., Zh. Prikl. Khim. 35, 70 (1962). (29) Ruckenstein, E., Smigelschi, O., Rei’. Chim. (Bucharest) 10, 211 (1959). (30) Ruckenstein, E., Teoreanu, I.. Bul. Inst. Politeh. Bucurrjti 24, No. 1, 101 (1962) (in English). (31) Siemes, LV., Weiss, LV., Chem. Inp. Tech. 29, 727 (1957). (32) Sutherland, K. S., Trans. Inst. Chem. Enprs. (London) 39, 188 (1 961). (33) Taylor, G. I., Proc. Roy. Soc. A219, 186 (1953). (34) Todes. 0. M.. “Metodi i Protesa Khimiceskoi Tekhnology,” p. 65, Izdatelstvo Akademii Nauk SSSK, 1955. (35) Wilhelm, R. H., Kwauk, M., Chem. Eng. Progr. 44, 201 (1948). (36) Yasui, G., Iohansen, L. N., A . I. Ch. E . J . 4, 445 (1958). (37) Zabrodsky, S. S., ”Ghidrodinamika i Teploobmen v Pseudojijenom Sloe (Hydrodynamics and Heat Transfer in a Fluidized Bed) ,” .Gosenergoizdat, Moskow, 1963. RECEIVED for review March 7, 1963 ACCEPTED November 19: 1963
