I
J. M. ANDREWS‘ Humble Oil & Refining
Co., Baytown, Tex.
Kinetic Study o f
Fluidized Solids Entrainment The equations developed here provide a basis for estimating catalyst entrainment in various process designs for both dense phase beds and transfer I ines employment of fluidization principles in chemical engineering is assuming increased importance. Fluidization of solids is being used in many conversion and adsorption processes such as catalytic cracking, hydroforming, fluid coking, ore reduction, and many others. All of these operations employ fluidized dense phase beds, as in the reactors and regenerators of the catalytic cracking process, obtained by flowing vapors, gases, or liquids u p through a bed of solid particles. I n the fluid catalytic cracking process employing 3A catalyst, the vapor velocities are 1 to 4 feet per second. T h e resulting dense bed of fluidized particles obeys all hydrostatic laws and is characterized by a turbulent surface similar to that of a boiling liquid. I n continuous, regenerative processes it is also necessary to circulate catalysts a t high velocities between the reactor and regenerator in transfer lines which may operate at much lower fluid densities. Under these conditions, the fluidized solids no longer resemble a boiling liquid but still have hydrodynamic characteristics similar to those of flowing fluids. I n fluid operations, both‘ the entrainment of solid particles into the escaping fluid over a dense bed and transfer line flow rates are significant factors in the operation of the unit. In the catalytic cracking process, for instance, cyclones and electrical precipitators are necessary to prevent an excessive loss of entrained catalyst from the system, and transfer line performance can limit unit capacities. Therefore, it is desirable to have an understanding of these factors whereby design calculations can be made. Fluidization principles have been comprehensively reviewed (Z), based mainly on the empirical analyses. I t is the purpose of this discussion to set forth T H E
Present address, Computer Systems, Inc.,New York 12, N. Y .
a fundamental approach for solution of both entrainment and transfer line problems.
The present analyses can be employed both as a basis for design of fluid systems and to provide an understanding of the variables involved. Further research is indicated to relate /3, the entrainment rate constant, with physical properties of fluidizing medium and particles. Entrainment Mechanism I n previous entrainment studies, the approach most generally employed has been a consideration of Stokes’ law forces only. However, regardless of whether Stokes’ law predicts a sufficiently high particle velocity, entrainment almost always exists to some extent above a fluidized bed. This contradiction can readily be understood by comparing a simple system containing only a few particles with a complex system containing many particles. Stokes’ law has been confirmed by data obtained from simple systems, and the effective fluidizing forces in such systems are buoyancy and friction whereby the gas or fluid acts on the particle. These forces can be calculated from particle size and density, fluid viscosity and density, and the relative velocities of the gas and particle. I n a simple system, the particle will remain stationary in space if a proper fluid velocity is employed. When the fluid velocity is increased, the particle moves vertically because of the increased fluid friction on the particle. Conversely, decreasing the fluid velocity results in a lowered frictional drag and the particle begins to drop. When more than one particle is present in the system, Stokes’ law forces will prevail, except in instances where collisions occur between two or more particles. An impact from below will impart kinetic energy to a suspended par-
ticle. When this kinetic energy is equal to or greater than the ene;gy required to lift the particle out of the vessel, the particle will be driven from the container. I n this case, entrainment exists despite the fact that Stokes’ law calculations would indicate a stationary state. Furthermore, there is considerable evidence indicating that agglomeration of the catalyst particles occurs in most fluidized systems. One characteristic of most agglomerated systems is a superficial gas velocity higher than the free-fall velocity of the particles; these particles must agglomerate to maintain a dense bed. Once the particles are agglomerated, the agglomerates are subject to the same forces discussed for individual particles above. However, it has been suggested that agglomerates derive their kinetic energy from the individual kinetic energies of the agglomerating particles rather than from agglomerate impacts. Thus, several high velocity individual particles or smaller agglomerates may form a relatively high velocity agglomerate. Either mechanism for the realization of agglomerate escaping energy is acceptable in the ensuing development. The preceding discussion shows that entrainment above a dense bed results from an action sequence involving collision and subsequent replacement of the escaped particles or formation and replacement of high speed agglomerates. The entrainment rate calculation can be simplified by determining which step in the action sequence is rate limiting. T h e velocity of the rate limiting step will then correspond to the entrainment velocity. Once the rate-limiting step and energy associated with this step are determined, the number of particles with the requisite energy can be estimated from statistical analysis. The entrainment rate can be calculated from a knowledge of the velocity for the rate limiting step and the number of particles with this energy. VOL. 52, NO. 1
JANUARY 1960
85
Energy of Entrainment. I n the present approach, it is assumed that the rate of escape from the surface of the dense bed is the limiting step in a steady state system. This assumption is readily justified as the replacement of an escaped surface particle or agglomerate from the dense bed proper requires only a small amount of energy, but escape from the surface to the top of the column requires considerably more kinetic energy. I t is therefore concluded that \\.hereas most of the particles will have sufficient energy to replace an escaped particle in the surface, very few particles will have the necessary energy to escape. T h e above assumption makes the subsequent analysis independent of the conditions prevailing in the dense bed proper. The escape energy for a single particle can be determined from an energy balance as indicated in the following equation relating kinetic and potential energy of the particles ' -mucdu, = mgdx
(1 1
Equation 1 can be integrated to give the following form relating bed outage to particle velocity:
This development ignores the buoyancy of the fluidizing medium, which is a gas in the present application and therefore negligible in comparison to the mass of solid particles. Furthermore, no consideration was given to drag coefficients or friction factors inasmuch as many different equations expressing these factors can be visualized, and there is not sufficient data available at this time to evaluate the various relations. However, the success of the simplified development in correlating the variables may well preclude extensive complication of the present approach. Energy Distribution. Having determined the necessary escaping energy, the fraction of particles containing the requisite energy either as individuals or agglomerates will be estimated statistically. It is assumed that the energy distribution of the particles in the surface of the dense phase follows the MaxwellBoltzmann distribution law: ft
= Pe-&
1 p = -~
E/S
(4)
It may be concluded that 0 is the reciprocal of average particle energy. Not all of the particles with sufficient energy will escape from the surface of the dense phase. A consideration of the geometry of three-dimensional space indicates that there are six vector velocities. These are the usual three dimensions with positive and negative values for each dimension. Only particles with a positive velocity in the vertical dimension will escape from the surface of the dense bed. Because the entire surface can be broken up into an infinite number of cubes, it may be assumed that only one sixth of the particles-in each and therefore all of the cubes comprising the entire surface volume-have the proper energy vector for escape. Rate of Entrainment The fraction of particles with sufficient energy to escape and the necessary velocity have been calculated in the previous development. T h e entrainment rate of the escaping particles is equivalent to their escaping velocity. Therefore, the loss of the zth fraction of particles \rith initial surface kinetic energy equivalent to or greater than the outage x can be calculated from the flow rates, bed densities, and a consideration of energy partition as follo\rs:
but
The total loss from the system can be obtained for all fractions, including those Lvith more than the required energy, by integrating from the minimum escaping energy to infinite energy:
After adding an arbitrary coefficient of conductance: A: and substituting for and p i
The coefficient of conductance will be arbitrarily employed to compensate for factors such as the shape of the vessel outlet, which will affect the percentage of approaching particles that escape. 'I'his equation can be integrated graphically or as a difference between complete and incomplete factorial functions. This equation as developed is theoretically applicable to only one particle size. However. a weighted summation of the individual particle size rates will give the total rate of loss for all particle sizes present in the system, and as shown in the following analysis, Equation 8 is a good approximation for a range of particle sizes. If it is assumed that in a mixture of different catalyst sizes there is no interaction bet\reen different sizes :
(3)
There are several conditions which must prevail before the Maxwell-Boltzmann energy distribution law is applicable. Strady state conditions must be maintained. The total energy of the surface particles must remain constant. T h e particles must be identical and indistinguishable from each other. T h e movement of the particles must be sufficiently unrestricted to follow the laws of probability. The total number of particles in the system must remain unchanged. T h e assumption regarding the equivalence of the individual particles requires that only one particle
86
size be considered at this point. I t is also assumed that the equipartition of energy prevails. The constant appearing in the Maxwell-Boltzmann equation can be evaluated from the material and energy balances ( 7 ) :
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 1.
For constant velocity, entrainment is a function only of
bed outage
FLUIDIZED S O L I D S E N T R A I N M E N T
pld2e(sl-s2)a
+
p8d3e(5r-3z)z
+ ...]
(11)
The exponential terms in the brackets of Equation 11 will approach unity, because the product gmp is small and the differences in this product for two particle sizes will probably be negligible. Therefore, the following approximation can be made:
[Md‘
+
PPd2
+
P3d3
+
,
..
I
(12)
Equation 12 demonstrates that if a weighted mean particle mass is employed this equation is approximately the same as Equation 8, providing 6 is independent of particle mass. The validity of the latter assumption is demonstrated by the data employed in the ensuing evaluation of the entrainment constant. Relationship for Transfer Lines. The prior analysis has been confined to two phase fluidized systems defined as containing both a dense bed phase and a more disperse phase wherein the outage is the distance between the interface of the dense and dispersed phases and the
vessel outlet. I n other sections of equipment, such as in vertically aligned transfer lines and fluid stand-pipes, the fluidized system contains only one phase. I n these systems, there is zero bed outage. Replacing the bed outage of Equation 8 by the energy term equivalent to outage, and then integrating from p1 = 0 to = m (as all particles have sufficient velocity to “escape” a t zero bed outage) yields the following definite integral :
I n Equation 13, I? (1.5) has been replaced by its value of 0.8862, and X has been assigned a value of unity as in a transfer line all particles have escaping Diameter,
Catalyst 3A, narrow cut
3A, wide cut
readily evaluated from the integrated form of the entrainment rate Equation 8. If Equation 8 is differentiated with respect to bed outage and divided by the negative square root of x , the following equations are obtained: 1 dW
-4 2ax = [ X A p P (2g)1/2mg] [e-@%=] 6
Data employed for evaluation of the entrainment rate equation are shown below and in Figure 1.
Microns
Lb. X 108
Vessel Diameter, In.
50 60
0.347
3.8
0.600
1.2
m,
energy and there are no exit problems, Application of Entrainment Equations. Eventually, p may be estimated directly from the physical data of the fluidized solids system. However, a t the present time, it is necessary to determine p from experimental data, and it is not
(14)
P>
Lb./Sq. Ft. 27.0 28.0
Entrainment data are included for a 40- to 60-micron narrow 3A catalyst fraction fluidized with air flowing 1.2 feet per second and for equilibrium, regenerated 3A catalyst which has all sizes from 0 to loo+ microns diameter and is fluidized with air flowing 1.7 feet per second. As discussed later, the particle masses are high because porosity data were not available.
4000 3000 2000
1000 800 600
400
200
100
I
2
3
4
6
810
20
30 40
U f , FT./SEC.
A Figure 2. p and X are determined from the slopes and intercepts of these lines 1. 40- to 60-micron 3A catalyst
0.
4 BED OUTAGE,
FEET
Equil., reg. 3A catalyst
Figure 3. on velocity
For a particular fluid
VOL. 52, NO. 1
p
is dependent only
JANUARY 1960
87
To determine p, smoothed curves were drawn through both sets of data and the slopes of these curves in Figure 1 were obtained a t various values of bed outage. These slopes were then divided by the square root of bed outage and plotted us. bed outage in Figure 2. Equations 12 and 15 show that for either size range catalyst, a plot of
(~.
g)
us. x
should be linear on the semilog coordinates of Figure 2. The conformance of both curves to the anticipated form is excellent. The slope of these lines on semilog paper is equivalent to -6mg. T h e intercept at zero bed outage gives the product of the associated constant XA,, ( 2 g p m p coefficients 12
.
~~
The equation for the 40- to 60-micron catalyst data in Figure 2 was determined to be: in
(- $2)
=
-4.6439 - 0.485~ (16)
In
(-
$z) =
In (0.0096) - 0.485~ (17) From the coefficient of x in Equation 16 or 17 g m p = 0.4850; p = 4.33 X
IO7
From the logarithm term on the right side of Equation 17
(X)(0.0872)(27.0)(8.03)(0.48~0) 6 X = 0.0062
The coefficient of conductance is low indicating high resistance in either the vapor phase or the dense bed surface. The value of h is probably dependent on vessel shape, fluid properties, and outage. However, there are not sufficient data to evaluate these effects or to determine whether the calculated values of X are reasonable for the current systems. The least mean square line for the equilibrium, wide cut regenerated 3A is defined by the following equation : In(-
$g)
=
-7.77 - 0 . 7 1 5 8 ~ (18)
ln (0.000420) - 0.7158~ (19)
6
= 3.70 X 10’;
X
=
0.00203
(20)
Application of Transfer Line Equations. The value for fl can also be determined from transfer line operations, and the following data summarize
88
INDUSTRIAL AND ENGINEERING CHEMISTRY
the transfer line operations included in the present evaluation.
are employed, it will be necessary to obtain experimental data for the evaluation Vessel
Solid 3A (wide cut) Fluid coke H-42, dry mix
51
m,
D, Microns
Sp. Gr.
80 100 80 80
2.41 2.00 1.25 1.75
Several different sizes of vessels were employed in this study. All were vertically arranged transfer lines employing either air or nitrogen as a fluidizing medium. Y o difference could be observed in the two gases for this purpose. All the solids fluidized were regenerated materials. A s indicated above, four different solids were used for this purpose Lvith representative particle diameters of either 80 or 100 microns selected on a weight mean basis. The specific gravities employed in this study were calculated for the solid material rather than for the porous particle as the porosity data were not available. However, as the product of /3m always appears together and as it is believed that all particles had about the same porosity, this effect will cancel out; but when it is possible to do so, /3 should be evaluated \vith the actual particle porosity that was employed. Densities were obtained from static pressure readings with taps located sufficiently far from bends to avoid errors due to acceleration. The reciprocal of the square root of /3 was calculated for each of the operations in the present study from Equation 13. These reciprocals are presented in Figure 3 as a function of gas velocity, which shows that the reciprocal is a unique function of pas velocity for all of the data points employed. The gas velocity used here is the true gas velocity calculated from the specific gravities of the particles and the fluid density to allow for the void volume in the fluid bed. I t is interesting that the values of 6 calculated from the dense bed operations are also included in this correlation, and both dense beds and transfer lines apparently follow the same relationship between p and velocity. As there is no overlap in velocities for the two types of operation, this conclusion is preliminary and subject to confirmation by additional data. If the conclusion proves valid, entrainment data for design purposes can be obtained from transfer line operations in small containers. However, before it can be applied to the larger vessels, it will be necessary to determine the value of the coefficient of conductance for two-phase operations, and this factor may vary widely as a function of vessel geometry. Furthermore, when gases similar to air and nitrogen are employed, the present data and correlations can be employed directly in transfer line and standpipe design calculations. When other gases
Lb. X
Diameter,
lo9
Fluidizing Gas
In. 3.8 3.8
1.421 2.07 0.730
Air, Air
N2 N:
1.0
1.0
1.03
Nz
of /3 and X where needed, although it is possible that laboratory evaluations can be employed directly in the design of larger commercial equipment. Nomenclature A = cross sectional area of vessel, sq. ft, bj
dj
E ft
g
(2g)3’2 p pm for particles with
=
12 diameter equal to j = pm for particles with diameter equal to i = totai energy of fluidized system, ft. l b . = weight fraction of particles with energy € 5 = acceleration of gravity, ft.j(sec.)z
4;
Lj =
e-*xpf
dx for particles
with diameter equal t o j mass of an individual particle, lb. total number of particles in fluidized system p = fluidized density of particles at surface of bed ps = density of individual fluidized particles sj = gpm, for particles \\it11 diameter equal to j t = time, sec. u = velocity of fluidized particles, ft./sec. zei = vessel inventory of catalyst lvith energy e t , lb. Wt = dw,/dt = change in vessel inventory per unit time or entrainment rate under steady state conditions for catalyst with energy e i , lb.:’sec. x = outage, distance from surface of bed to outlet. ft. .v p = entrainment rate constant = E
m
il’
= =
r m
€1
= initial kinetic energy per particle
mui2 -~ , ft. lb. -
X pj
= =
constant coefficient of conductance weight fraction of particles with diameter equal to j
Subscripts catalyst
c
=
i j
= energy level of catalayst =
diameter of particle, ft.
Literature Cited (1) Glasstone, Samuel, “Textbook of Physical Chemistry,” 2nd ed., p. 272, Van Nostrand, New York, 1946. (2) Leva, M., IYeintraub? M., Grummer, M., Pollchik, M., Storch, H. H., U. S. Bur. Mines Bull. 504 (1951). RECEIVED for review M a y 4, 1959 ACCEPTED September 8, 1959
