BOUNDARY LAYER MASS TRANSPORT WITH HETEROGENEOUS CATALYSIS C
.
LEMBIT K USI K
A
N D J 0 H N H A P P E L,
York lTrutersit,, L‘nri*rsr/y Herghts, .Vert York, AV. Y.
A theoretical study was made of gas diffusion rates in particle beds with heterogeneous catalysis. The free surface model (spherical particle surrounded by a spherical envelope) is used with boundary layer theory. In considering the density variation in the boundary layer it is generally necessary to solve two integral equations. By neglecting second-order effects the integral equations are solved to predict dissolution rates in a particle bed for a chemically dilute gas for void volumes of 0.3 to 1.0. These integral equations are also used to analyze the effects of a molar velocity perpendicular to the surface of the particle. The effect of this normal velocity is essentially the same for spherical surfaces as for flat plates and good agreement is obtained wiih simple film theory.
FIXED and fluidized beds have a Lj-ide application in chemical engineering. The design of catalytic reactors, fluid saturators, chromatographic columns, leaching beds, ion exchange columns, adisorbers, etc., involves a knowledge of the interaction between solid and fluid surfaces. HoiL.ever, a good part of the scientific inirestigation i n this field has been of a n empirical nature. T o gain a clearer insight into the subject of mass transfer, a theoretical study \vas undertaken of vapor phase surface-catalyzed reactions in a multiparticle system with forced convection (100 < R e ” t = Re’ < 1000). T h e theoretical determination of the rate of mass transfer is obtained generally b;i solving some simplified form of the momentum and diffusion equations. I n a pioneer treatment of catalytic problrms H ougen and \Vatson used one-dimensional film theory; the relevant equations can be found in several texts (77, 39, d 2 ) . Other authors have elaborated on this by using two- and three-dimensional models with various theories. A tabulation of some of the significant ivork in this field is given in Table 1 , The present work extends the theoretical investigation i n two ways. First, the transport coefficient for small transfer rates is calculated for particles in a bed by using the free surface model. Second, the effect of a high molar velocity perpendicular to the spherical catalytic surface is investigated. T h e results are applicable to arbitrary reactions i n series with diffusion. Procedures
T h e present theoretical analysis of mass transfrr in particle beds is considered by first describing the free surface model which was introduced and successfully used by Happel in slo~v viscous flows to predi’ct pressure drops in a n assemblage of particles (75). I n the :?resent work the model is adapted for use with boundary layer theory. Then boundary layer equations are solved by introducing polynomials to describe the velocity and density distributions (Pohlhausen’s method) (29, 38). I n the genera1 case, one has to solve t\vo coupled total differential equations. For the case of a sphere the boundary layer thickness does not change radically until the separation point is approached ; the surface concentration also changes slowly. By using a constant surface concentration a generalized equation can be obi:ainc:d relating the concentrations to Present address, HQ. U. S. .Army Ordnance, BRL, Aberdeen Proiing Grounds, hld.
Table 1.
Theoretical Mass and Momentum Transport Considered by Various Authors Author Problem or Model Considered Hougen and \Vatson (771 One-dimensional mass transport Navier-Stokes equations with itera.Tensen (201 tive methods Heat-mass transport with potential Ruckenstcin ( 32) flow Heat-mass transport with creeping Friedlander ( 7 I ) motion Xlicklev r / n i . (20 i Flat plate boundary layer equations with blowing or sucking, exact solution Xferk (2-1.25) Flat plate boundary layer equations with blowing or sucking, von Karman integral method Mixon and Carberry (28) Flat plate boundary layer equations neglecting velocity perpendicular to surface hfeksyn (23j. Tomotika Single spheres or cylinders, solution to momentum boundary layer ( 1 7 ) . Schlichting (.?8) equations Garner and Keey (7.21, Boundary layer mass transfer using Akselrud ( 3 ) integral methods Linton and Sutherland Mass transfer using power series (22) Rosner (33-37), Acrivos Boundary layer with chemical reacand Chambre (2, 6,7), tion; integral methods ( 2 1 ) Potter (37) Cohen rt ai. (70) Boundary layer theory assuming concentration depends on only one velocity component Bird rt o / . (-1) One-dimensional and boundary layer equations considering mass velocities perpendicular to surface Recombination of gas on catalytic Chuns and Anderson ( 9 ) platc ;\crivos ( 7 ) Asymptotic boundary layer solutions for bodies of arbitrary shape with suction Carberrv (5) Mass transfer in particle beds using flat plate boundary layer equations Horlscher ( 76) Particle bed mass transfer with a film thickness greater or less than some critical value Griffith 1,7J) Collected formulas for spherical case
the rate of transport. Such a n equation is developed in this paper for a binary system involving arbitrary surface reactions. Model
If the fluid flow near the surface of any particle is unaffected by other particles relatively far away from the surface, any one VOL.
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particle can be isolated with its fluid envelope and considered as a representative unit cell. The idealization of the unit cell to a spherical particle surrounded by a spherical fluid envelope gives the free surface model (75), which is the basis for this study (Figure 1). The size of the fluid envelope is chosen so that the void fraction, e, is given by
The effect of vortexes developed in wake flow (flow after boundary layer separation) is extremely involved. Taneda (40) indicates that spheres in an infinite stream have vortexes which are stably attached up to Re = 300. At higher Reynolds numbers the vortexes tend to shed. I n packed or fluidized beds the information on vortexes is meager. Therefore the effect of vortexes is treated in a necessarily crude manner. If the vortexes are shedding, it is assumed that a time-average vortex size can be used. I t seems reasonable to assume that the effect of vortexes is to cut down on the area available for fluid flow. Therefore the volume of the vortex behind each sphere is subtracted from the total volume of the unit cell in determining the potential velocity outside the boundary layer. A void fraction that considers the vortex volume as “unavailable area for fluid flow” can be defined as:
The vortexes would be expected to shrink in size as the void volume decreases and to vanish altogether a t some low value of the void volume. Most of the packed beds met in practice would have void volumes of 0.3 to 0.4. For the purposes of this paper a bed with a void volume of 0.2 is considered to be very tightly packed and it is assumed that at such a low value of the void volume the vortexes have disappeared. For lack of better information a linear relation is assumed between vortex length and void fraction e for 0.2 6 e 1.0.
, -'
z(lv?C,,),
=-
kr
'Ichmidt number 2
-+
= S.V/ko
hi-
=
velocity potential function (see Equation ?)
=
dimensionless concentration,
CpaeP
modified Reynolds number, R e e radial direction in spherical coordinates surface a-ea per unit volume of catal>st
Ui'll - = ' I 8 a line over a letter indicates a vector = vector dot product = the "del" operator (4)
=
(-j. r
(-)
SI-BSCRIPTS '4 = component . 4of binary system av = average B = component B of binary system = componentj j k = index to represent u,j, d! etc. = values at solid-fluid interface 5 Y = component in x direction = component in y direction I 1 = value at edge of boundary layer = value as 2.V + 0 0 = component j evaluated a t edge of boundary layer 1s = component j evaluated a t solid surface
p
Acknowledgment This research was supported in part by a grant from The Petroleum Research Fund administered bv the .4merican Chemical Society. Grateful acknowledgment is hereby made to the donor of this fund. The aid of the Sational Science Foundation for fello\vship support of C. L. Kusik is greatly appreciated.
,'
1-
= 1-L_ __
Y7
= ratio of convective to total energ\ ( T , - T , ) ZS,C,, __________
1
=
ZC
B,,I as 2.V
=
~
total molar flux in Y and y directions, respectively concentration thickness defined by Equation 14 radius of cross section (Figure 2) perfect gas constant = Pc-lT-' 2a C, Reynolds number = --
*v, -. x.y -
=
a( u / u,)
molar flux vector, moles time-' area-' rnolar flux o f j in x and y directions, respectively fourth-degree polynomial pressure
Y
=
1
transport,
pa
distance measured perpendicular to surface (Fiqurt. 2)
literature Cited (1) Acrivos, A , ?A.I.Ch.E. Journal 6, 412 (1960). (2) Acrivos. A., Chambre, P. L., IND.ENG.CHEM. 49, 6 (1957). (3) Akselrud, G . , Zhur. Fzz. Khim. 27, 1446 (1353). (4) Bird: R. B., Stewart, W. E., Lightfoot, E. N.,"'Transport Phrnomena," Wiley, N e w York, 1960. VOL.
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J. J., A. Z. Ch. E. Journal 6, 460 (1960). P. L., Appl. Sci. Research A6, 96 (1956). P. L., Acrivos, A., J . Appl. Phys. 27, 11 (1956). apman, D. R., Rubesin, M. W., J . Aeron. Sci. 16, 547 (1949). (9) Chung, P. M., Anderson, A. D.: A R S Journal 30, 3 (1960). (10 Cohen, C. B., et al., GMTR 268, Contract AF 18(600)1190, duided Missile Research Division, Ram0 Wooldrige Corp., Los Angela, Calif. (11) Friedlander, S. K., A. I. Ch. E. Journal 3,43 (1957). (12) Garner, F. H., Keey, R. B., Chem. Eng. Sci. 9 , 119 (1958). (13) Gordon, K. F., “ A Cell Model for Mass and Heat Transfer in a Fixed Bed.” Deuartment of Chemical and Metalluraical Engineering, Univer’sity of Michigan, Ann Arbor, MTch., A. I. Ch. E. Washington meeting, 1960. (14) Griffith, R. M., Chem. Eng. Scz. Sci. 12, 198 (1960). (15) Happel, J., A. I. Ch. E. Journal 2, 197 (1958). (15) (16) Hokischer, Zbtd., 4, 300 (1958). Hoelscher, H. E., Zbid., (17) HouFen. 0. A.. Watson. K. M.. “Chemical Process Kinetics.” ’ Part IIf, Chap. 18, Wiley,’New York, 1959. (18) Zbid., Chap. 20. (19) Hsu, H., Bird, R. B., A. Z. Ch. E. Journal 6, 516 (1960). (20) Jensen, V. G., Proc. Roy. Sac. (London) A249, 346 (1959). (21) Lighthill, M. J., Zbid., A202, 359 (1950). (22) Linton, M., Sutherland, K. L., Chem. Eng. Sci. 12, 214 (1960). (23) Meksyn, D., J . Aeron. Space Sci.25, 631-4, 664 (1958). (24) Merk, H. J., Appl. Sci. Research AS, 237 (1959). (25) Zbid., p. 261. ,
I
(26 Mickley, H. S., et al., Natl. Advisory Comm. Aeronaut., k c h . Note 3208 (1954). (27) Milne-Thomson. L. M., “Theoretical Hydrodynamics,” ’ ‘ 4th ed., p. 499, Macmillan, New York, 1960. (28) Mixon, F. O., Carberry, J. J., Chem. Eng. Sci. 13, 30 (1960). (29) Morduchow, M., Natl. Advisory Comm. Aeronaut., Rept. 1245 (1955). (30) Morris, D. N., Smith, J. W., J . Aeron. 9i.20, 805 (1953). (31) Potter, O., Trans. Znst. Chem. Engrs. 36, 415 (1958). (32) Ruckenstein, E., Chem. Eng. Sci. 10, 22 (1959). (33) Rosner, D. E., Aero Chem. Lab., Princeton. N. J., Tech. Pub. 14, (1958). (34) Zbid., 16, (1960). (35) Rosner, D. E., Aero Chem. Lab., Princeton, N. J., TM-12
I
36) Rosner, D. E., A R S Journal 30, 114 (1960). 37) Rosner, D. E., Jet Propulsion 28, 445 (1951). (38) Schlichting, H., “Boundary Layer Theory,” Pergamon Press, New York, 1955. (39) Sherwood, T. K., Pigford, R. L., “Absorption and Extraction,” McGraw-Hill, New York, 1952. (40) Taneda, S., J . Phys. SOC.Japan 2, 10 (1956). (41) Tomotika, S., London, Aero Research Committee Rept. 8r Memo. 1678 (July !‘935). (42) Treybal, R. E., Mass Transfer Operations,” Chap. 2, McGraw Hill, New York, 1955. RECEIVED for review November 17, 1961 ACCEPTEDJune 4, 1962
PROPELLER PUMPING AND SOLIDS FLUIDIZATION IN STIRRED TANKS JOSEPH V. PORCELLI, JR.,’
AND G E O R G E R . M A R R , JR.*
Columbia University, New York, N . Y. Data characterizing propeller pumping, fluid entrainment, and solids fluidization in stirred tanks are presented. It is shown that propeller pumping is given b y the product ND13, independent of Reynolds number and tank geometry in the turbulent range of operation, but that entrainment flow is a function of tank geometry. Particle nuidization is correlated in terms of the settling velocity for a single particle and the total circulation rate of the tank fluid.
N THE CHEMICAL PROCESS INDUSTRIES
the most frequently used
I method for the achievement of solid-liquid contacting is the stirred tank. Despite the widespread use of agitated vessels through the years, little is known concerning the effects of many important variables on the resultant behavior of stirred tank systems. The type of impeller, the geometry of the tank-impeller system, the presence of auxiliary equipment such as baffles or stator rings, and the properties of the liquid and solid phases all influence the behavior of such systems. There are two requirements for satisfactory operation in most solid-liquid contacting systems: adequate suspension of the solid particles and adequate turbulence in the liquid phase. Adequate suspension infers suspension to that degree which ensures uniform process conditions leading to predictable results. Adequate turbulence is that degree of turbulence
Present address, Scientific Design Co., Inc., New York, N. Y.
* Present address, Electronic Associates, Inc., Princeton, N. J. 172
l&EC FUNDAMENTALS
which minimizes mass transfer resistances and maintains the various mass transport processes at acceptable levels. Past research in the field of solids suspension has taken two directions. In many studies, the suspension process was isolated from the mass transfer processes by the utilization of inert (nondissolcing, nonreacting) solids, such as silica sand, in various liquids. Other experimenters have directed their efforts towards the mass transfer aspects, the suspension phenomena being described qualitatively as a secondary study. Neither of the above approaches assumes the barest knowledge of the flow characteristics of stirred tanks. The suspension characteristics of any given stirred tank design are empirically determined, aided at best by the results of dimensional analysis on the system. I t was felt that an approach to the solids suspension problem which includes the understanding of the fluid mechanical characteristics of the stirred tank system and the “laws of settling” could yield simple and general relationships among the physical properties of solid-liquid systems, the impeller-tank geometry, and the solid suspension
