PARTICLE AND FLUID DIFFUSION IN HOMOGENEOUS FLUIDIZATION GERALD
HOUGYTONl
Dejartment of Chemical Engineering, Division of Engineering Research, University of Pittsburgh, Pittsburgh, P a .
A Markoff theory of particle diffusion in homogeneous fluidization is founded on nonlinear Langevin equations and associated quasi-linear and linear stochastic equations, describing the “microscopic” particle-fluid and particle-particle interactions when velocity fluctuations are small and their distribution i s Gaussian, Anisotropy is permitted through directional differences in fluctuation energy and particle-fluid friction. Particle and interstitiul fluid diffusion are found to be symmetric stochastic processes characterized by a single directional diffusivity sensitive to void fraction and particle-fluid properties. Comparisons of theoretical and experimentall diffusivities indicate that considerable anisotropy and inhomogeneity exist during fluidization, attributable to mean velocity distributions and random “macroscopic” disturbances. The stochastic model is then generalized to include fluidized diffusion arising from macroscopic turbulence on the scale of several particle diameters.
term “fluidization” describes a physical situation in a multiparticulate system is supported or transported in a fluid velocity field where, on average, the particlefluid drag forces a t least balance the buoyancy forces. Homogeneous fluidization is believed to occur a t low fluid velocities, particularly when the continuous phase is a liquid. Homogeneity corresponds to the ideal state of uniform dispersion of the particulate phase ; the interstitial fluid turbulence is localized or fine-grained, in that “bubbles” or “slugs” of continuous phase do not form and cause the bed level to rise and fall periodically as in so-called “boiling fluidization” which is observed with gases a t high velocities. I n both homogeneous and boiling fluidization the individual particles have fluctuating positions and velocities causing adjacent particles to squeeze between each (other and migrate throughout the bed. Existing experimental information, summarized by Sutherland (26), indicates that particle mixing increases with fluid velocity and is greatest in the regime of boiling fluidization, where the presence of bubbles presumably causes the gross macroscopic transport of large groups of particles. By proposing a direct analogy between molecular processes in liquids and particle behavior in fluidized beds, Furukawa and Ohmae (7) have suggested that the physical properties of fluidized systems such as bed viscosity, 70, can be estimated by using the mean kinetic energy of the fluidized particles, Xz, in place of absolute temperature in the expression for the corresponding property of molecular liquids. They mention, without elaboration, that the particle diffusivity in homoby simple geneous fluidization should be proportional to A,/?,, analogy with existing ((5,6 ) molecular theories of the liquid state. Ruckenstein (24) has considered the limiting case of extremely low velocities just exceeding the velocity for incipient fluidization, where the particles are presumed not to diffuse but merely to isscillate about fixed equilibrium positions. The particle oscillation, with associated void fraction pulsation, is believed to increase the axial dispersion coefficient of the continuous phase flowing through the interstices. I n the context of molecular systems the fixed lattice model of Ruckenstein is analogous to molecular behavior in solids which exhibit long-range order, with the probability of migration HE
Twhich
Present address, School of Engineering Science, The Florida State University, Tallahassee, Fla.
being small compared with liquids, since the molecules of solids spend most of their time oscillating in a deep potential well a t each lattice point. However, Ruckenstein’s treatment is not strictly theoretical, since it is founded on dimensional and empirical considerations ; its success ultimately depends upon the adjustment of a single empirical constant to fit suitable experimental data for axial fluid dispersion in fluidized systems. A “stochastic-cell” model has currently been formulated (70) which postulates that migration in Newtonian liquids occurs by molecules squeezing between each other in a random continuous manner described by a linear stochastic equation, such that the rate of reorganizatiop in a local molecular cell is small enough for a weak structure to persist over a significant period of time, thereby introducing a n element of local structural organization which has long been known to exist, from x-ray diffraction studies of the condensed state. Because of its flexibility, this approach has recently ( 7 7) been extended to include solids mixing, turbulence, and fluidization by permitting nonlinear dissipation in the stochastic equation. T h e present communication formulates a detailed stochastic model for particle and fluid diffusion during homogeneous fluidization in terms of stochastic equations derived from the nonlinear Langevin equation used previously (72) to describe the behavior of free particles in a n accelerating fluid. Stochastic Model
Because of the somewhat elaborate mathematical formalism required to treat fluidization on a stochastic basis, it is appropriate first to outline the underlying physical concepts. T h e model will be founded on a nonlinear Langevin equation representing a force balance (72) on a single fluidized particle, and accounting directly for both particle-fluid and particle-particle interactions. Even though the particle-fluid interactions may be highly nonlinear, their effects can be linear if the velocity fluctuations are small compared with the mean flow. For small fluctuations the nonlinear Langevin equations for each spatial direction may be reduced to quasilinear and linear stochastic equations describing the random vertical and horizontal velocities, respectively, of single fluidized particles. By treating particle diffusion as a Markoff process and assuming a Gaussian distribution of particle velocities, we then construct a problem in Uhlenbeck-Ornstein VOL. 5
NO. 2 M A Y 1 9 6 6 153
(29) stochastics, the solution of which will yield the particle diffusion equations in velocity and configuration space. The stochastic model will be general enough to permit anisotropic diffusion Mith an unequal partition of energy, phenomena which might be expected in any physical situation where the energy of horizontal particle fluctuation must be acquired indirectly from the energy of the vertical mean flow. While the possibility of directional differences has been recognized ( 75) experimentally, there has apparently been no attempt to incorporate such behavior into a quantitative mathematical treatment of fluidized particle diffusion. The symmetry of the stochastic equations with respect to the interchangeability of particle and fluid velocities will permit us to recast the problem in terms of fluid diffusion, and thereby establish a n hitherto unavailable connection between the particle and interstitial fluid diffusivities. The fluidized diffusion coefficients can then be expressed as a function of void fraction, as determined from measurements (78, 30) of pressure drop and bed density. A comparison of theoretical and experimental (8, 75) diffusivities shows that a “microscopic” model of fluidization, based on single particle behavior as described by a Langevin equation, predicts diffusivities more than an order of magnitude too low, even if the fluctuation energy approaches that of the local mean flow. Since the physical parameters are in a range which renders the microscopic model formally rigorous from the physicomathematical standpoints, we deduce that the unpredictably high diffusion rates arise, not from particle oscillation (24) or particle migration, but from “macroscopic” phenomena such as spatial variations in mean velocity, and random macroscopic turbulence. Then, by taking the wider view (77) that stochastic equations will exist for all multiparticulate systems, regardless of the scale of their internal relative motions, it becomes possible to imbed both microscopic and macroscopic diffusion processes within the same mathematical formalism. Having established conceptual and mathematical connections between homogeneous fluidization and homogeneous turbulence, and armed with a theoretical expression for the turbulent eddy diffusivity, we demonstrate how diffusivities may be obtained from measurements of the decay and intensity of turbulence in fluidized systems. Stochastic Equations for Fluidized Particle$
+
l&EC FUNDAMENTALS
- 61) =
{
fl;
111
-1;
u1
-Ul> 0 - UI < 0
Equation 1 applies only to the vertical velocity component, The corresponding force balances for horizontal components, u2 and u3, are discussed below, since these equations represent special cases of Langevin Equation 1 from which buoyancy forces are absent and there is no mean flow. T o produce acceleration dul/dt in a particle of mass ppVp, it is also necessary to accelerate a n associated mass of surrounding fluid, xpfVp, where x is the dimensionless coefficient of virtual mass. The analogous fluid qcceleration term, (1 4- x)PjVPdul/dt, represents pressure gradients arising from the accelerated fluid displaced by the particle together with the virtual mass of the displaced fluid. The force q1(t), not present in the earlier Langevin equation (72), arises because more than one particle is present in a fluidized bed, and represents the random acceleration force arising from contact collisions with external particles moving in the neighborhood of the one under consideration. The term ql(t) corresponds exactly to the random bombardment force utilized in existing (2, 3, 70-72, 74, 29) stochastic models of molecular and Brownian motion. The frictional resistance of the particle to the external fluid motion is generalized by an nth power drag law such that changes in the direction of the drag force with relative particle-fluid velocity, u1 - U I , can be accounted for by the function signum(ul - ul), the drag coefficient, an, being evaluated a t the conditions of fluidization. In the case of a single particle in an infinite fluid, a, will be a function (76) of particle size and the physical properties of the fluid, whereas in fluidized systems cy, is additionally found (78) to be dependent upon the system void fraction, e. The concept of homogeneous fluidization implies the constancy everywhere of the physical properties of the continuous and discontinuous phases ; the time average properties characteristic of the combined phases, such as void fraction e, must also be independent of position. The gravitational force on a particle, (PI pP) Vpg,will vanish if the particle and fluid have the same density, this buoyancy force being negative if pp > PI and positive if ui, of the particles.
-
The treatment is concerned with a detailed stochastic analysis of a typical batch fluidization system in which gravitation and gross fluid flow are in opposite directions along the same vertical coordinate axis, X I , and the remaining two spatial coordinates, x z and x3, lie in a horizontal plane. Each fluidized particle will have a velocity vector u = ilul i2u2 the components of which (UI,U Z , 243) correspond to the coordinate axes (XI, X P , x 3 ) . Along horizontal coordinates xz and x 3 there will be no components to the time average fluid velocity, while in the vertical direction the mean flow generates enough drag to support the particles against gravity. Thus, even though the time average of each particle velocity component is zero, the velocity field surrounding the particles, and hence the drag characteristics, can vary considerably from the vertical to the horizontal directions, leading to the hitherto unconsidered possibility of anisotropic diffusion of particles in homogeneously fluidized systems. By superposing the vertical forces exerted on a single particle in a fluidized bed, a nonlinear Langevin equation can be generated that is analogous to the force balance formulated earlier (72) to describe the behavior of an isolated particle in an accelerating fluid : 154
where
+
Pp
< P/.
Evidence (72) from considerations of the Brownian motion in suspensoids, as well as the experimental behavior of single accelerated spheres, indicates that the so-called Basset force, Bl(t), will usually be negligible, being introduced to account for transient variations in the velocity distribution around the particle. While the neglect of transient effects is implicit in all Langevin equations hitherto utilized (2, 3, 7 7 , 72, 74, 29) to describe particle-fluid interaction, the absence of the Basset term implies the existence of a quasi-steady-state velocity distribution such that the spatial velocity variations are identical with those for steady state at the same instantaneous relative velocity. The equation proposed by Ruckenstein (24) to describe the oscillatory motion of particles around a fixed lattice position corresponds to the specific situation where exponent n = 2, ql(t) = 0,and Bl(t) = 0 in the Langevin equation formulated earlier (72) by this author. However, Ruckenstein’s force balance additionally omits a term, -pfVpdui/dt, SO that his equation does not yield the solution u1 = U I when pf pp,
q l ( t ) = 0, and B l ( t ) == 0 as expected from physical considerations in the limiting cases where either cy, = 0 or a, > 0 with t + m , T h e origins of the terms in Equation 1, except ql(t), as well as the general properties of linear and nonlinear Langevin equations have been discussed in detail by Houghton
(72). Nonlinear Langeviii Equation 1, representing a superposition of vertical forces, can now be linearized by introducing simplifications that are consistent with a condition of homogeneous fluidization :
I n truncating the binomial expansion leading to Expression 8, all terms of order u12, U I ’ ~ , U I U ~ ’ ,cyn’ul’, ula,’, etc., and higher have been neglected, since the fluctuations are small according to Criteria 2, 4, and 6. If nonlinear Langevin Equation 1 is combined with Relations 2 to 8, the following “quasi-linear” stochastic equation results:
(9) where
Bl(t) = 0
Relations 2 represent the conditions imposed on the vertical fluidization velocity and, as in statistical treatments of turbulent diffusion, the interstitial fluid velocity between the particles, uI, is presumed to be composed of a time average component, al, and a fluctuating component, U I ’ , of much smaller magnitude, such that 8
(7)
81 == (1/2S)J-@ Ul(t)dt
T h e limit of integration, 0, is a time interval that is long compared with the period of the particle oscillations and the relaxation time, 1//31, but short compared with the time for appreciable diffusion in the mean. Conditions 2 also require that the finite mean vertical velocity should not vary with time under conditions of steady homogeneous fluidization. Subsidiary force Balance 3 defines the physical requirement inherent in all batch fluidization phenomena-namely, that the bed of particles is supported against gravitational forces by the drag generated through the existence of a finite average fluid velocity. Since the drag coefficient is a function of void fraction (78), Relation 4 takes into consideration the effect of local void fraction fluctuations, e’, on the drag coefficient, the mean value of which will be E , a t a time average void E’. Restriction 5 implies the fraction of i, where E = a existence of a quasi-stcady-state condition with respect to the instantaneous behavior of each particle, such that drag coefficients obtained (78:i from measurements of steady fluidization velocities can be used to spec’fy coefficient an in Equations 1 and 4. T h a t the fluctuating particle velocity, ul, is of much smaller magnitude than the mean fluidization velocity, as in Inequality 6, is consistent with visual and photographic observations (20, 26) of particles in fluidized beds, while diffusion considerations discussed below lead to the theoretical relax)ppl’2. tionship ( p p x p r ) u i := (1 Because of the limitation lull 1511, and with the aid of Relations 2 and 4, the nonlinear drag term of Langevin Equation l can be approximated by a truncated binomial expansion if it is assumed that the nth power drag law applies to both the mean flow and the fluctuations. An analogous assumption is inherent in existing models of turbulent fluid diffusion, in which it is customary (7, 9 ) to require the Navier-Stokes equations to apply to the eddy fluctuations as well as the mean flow; thus
+
+
anlU1
+
- ullnsgn(ul - crJ
( i , j = 1, 2, 3)
(36)
The use of the mean and variance of Equations 35 and 36, respectively, in Equation 21 then results in the following partial differential equation in velocity space (Fokker-Planck equation) :
>>
That the assumptions embodied in Equations 29 to 34 are justified may now be verified by noting that joint Distribution 29 for P is a solution to Equation 37 as t + m . However, Fokker-Planck Equation 37 may be rendered valid to any desired degree of approximation by taking time averages of Form 7 and ensemble averages of Types 30 and 31 over a finite time interval, 0 = t - to, such that 0 >> l / P k , where PA is the smallest of the directional friction coefficients or 0, > Pt < P j , i, j , k = 1, 2, 3. Thus, for time intervals 0 l / P k , statistical equilibrium can be approximated to any required degree when, according to the ergodic hypothesis ( 4 ) ,the ensemble and temporal averages will be equal so that, for example, = m 2 / 2 , the bar indicating a time average according to Definition 7. Partial differential Equation 37 differs from the Fokker-Planck equation for Brownian and molecular systems (2, 3, 70-72, 74, 29) in that the coefficients are functions of direction in velocity space.
>>
158
I&EC FUNDAMENTALS
. T h e ensemble average ai-' is a positive constant characteristic of the directional intensity of the velocity fluctuations in a particular state of stationary homogeneous 2& = 2&-1 fluidizLtion. Thus: d < u i 2 > / d t and it is not difficult to show that if a n artificial velocity disturbance were introduced, it would decay exponentially into the surrounding fluidized medium until as t + m , d/dt + 0 and ultimately 6 , < u t > . = 6i-i, so that with Relation 60 and recalling that = &-' = constant, we find = 6 i ~ i . The energy decay equation equivalent to stochastic Equation 51 is then:
+
d/dt
+ 2 6i
=
2 B,K~
(i
=
(62)
1, 2, 3)
where K , is the intensity of the velocity fluctuations in a state of stationary turbulence. 13ifferential Equation 62 has the same form as decay equations conventionally adopted ( 7 ) for the interpretation, on a n experimental basis, of the decay of turbulent energy in fluids not containing particles. However, the energy decay equations used in homogeneous turbulence ( 7 ) are empirical in origin, while Equation 62 has been deduced from a stochastic equation, each term of which has physical significance. Thus, by placing a stochastic interpretation on energy decay equations, a connection can be established between homogeneous turbulence and homogeneous fluidization, both conceptually and mathematically, as recently noted by Houghton ( 7 7). For vertically fluidized beds, the total derivative of Equa= d/at 513 tion 62 can be written as d/dt /dxl, i = 1, 2, 3, so that we may determine the decay constant, ai, by measuring the decay of either artificially introduced velocity changes with time at a number of fixed points, or intensity with distance along the vertical axis of fluidization. Perhaps the most convenient method would be that conventionally used for homogeneous fluid turbulence (7), in which a n artificial disturbance modifying the local turbulence intensity would be placed a t a particular cross section (XI = 0) in the fluidized bed and the intensity < v , Z ( x I ) > measured a t various downstream points. Although the intensity varies with distance, it does not vary with time a t each measuring point ( d < v t > / d t = 0), so that decay Equation 59 becomes bldjdx1 2 6, = 2 & K , , i = 1, 2, 3, and the solution is :
+
+
linear decay process in association with a Gaussian velocity distribution. The severe physicomathematical difficulties arising from the possible existence of nonlinear stochastic equations ( I 7) and non-Gaussian velocity distributions will be discussed in future communications. Conclusions
T h e foregoing comprehensive treatment of homogeneous anisotropic fluidization and the detailed comparison of the theory with experiment should stimulate careful and coordinated experimentation on fluidized systems, particularly in developing more direct methods of measuring particle and fluid fluctuation velocities. More attention should be paid to obtaining quantitative estimates of the degree of anisotropy and inhomogeneity in the bed through measurements of the average velocity distribution in the containing vessel, as well as through the development of techniques for detecting the magnitude and decay of large scale disturbances of the scale of several particle diameters. Alternatively, the results might be utilized to design equipment and conditions to approach a state of homogeneous fluidization in which each particle migrates as an independent entity. Nomenclature
constant in general distribution function, dimensions depending upon exponent s random particle acceleration function, cm./sec.2 transient Basset force, g. cm./sec.2 drag coefficient for fluid8gfz(p, ized bed, dimensionless particle diffusivity, sq. cm./sec. 2.7183, base of natural logarithms, dimensionless interstitial fluid diffusivjLy, sq. cm./sec. diffusivity for macroscopic disturbances, sq. cm./ sec. fluid configuration probability distribution function, dimensionless random fluid acceleration function, cm./sec.2 local acceleration due to gravity, cm./sec.* random acceleration function for macroscopic disturbances, cm./sec.2 spatial unit vector, dimensionless turbulent fluid flux, cm./sec. particle diffusion flux, cm./sec. g. sq. cm./ Boltzmann constant, 1.3805 X sec.2 OK. scale of random macroscopic disturbances, cm. (p, x p f ) V p , particle mass plus virtual fluid mass, g. (1 x)pfV,, displaced fluid mass plus virtual mass, g. exponent in particle-fluid drag law, dimensionless 2 Rla,;l / p f , particle Reynolds number, dimensionless particle configuration probability distribution function, dimensionless particle velocity probability distribution function, dimensionless velocity probability distribution function for macroscopic disturbances, dimensionless particle-particle random bombardment function, g. cm./sec.2 ilxi i2xz i3x3, position vector, cm. particle radius, cm. exponent in general distribution function, dimensionless time, sec. absolute temperature, OK. ilul i2uz $3, particle velocity vector, cm./sec. particle velocity component, cm./sec. ilvl i*uZ i 3 u 3 , velocity vector of macroscopic disturbance, cm./sec. velocity component of macroscopic disturbance, cm./sec.
+
+
As in analogous fluid turbulence measurements (7), the average flow, b,, is steady, so tihat the intensity will be temporally stationary a t each cross section and the ensemble averages < u i z ( x l ) > , i = 1, 2, 3, can be replaced by time averages obtained by directionally o:riented probes. T h e decay constant, 6 $ , can be determined by measuring the distance, b,/2ai, at which the excess intensii.y, [ < v , ~ ( x I ) > - K , ] , has decayed to l / e of its initial value, [ - K ~ ] . If the height of the fluidized bed is great enough to approximate x 1 -+ a ,the same intensity measurements would yield the K~ according to Equation 63, so that, with the ai, the experimental information would be complete for the calculation of E i = D, = Et from Expression 61. Such a n experimental approach would provide invaluable information on the scale and directional properties of the random movements in fluidized systems, particularly in conjunction with marked. particle and fluid tracer techniques. The velocity fluctuation measurements should also be used to verify that the stochastic behavior can be approximated by a
+
+
+
+
+
+
VOL. 5
NO. 2
MAY 1966
163
V
= volume, cc.
spatial coordinate, cm. volume fraction of designated particles, dimensionless = volume fraction of designated fluid, dimensionless = =
x
X Y
GREEKSYMBOLS drag coefficient in Langevin Equation 1, dimensions depending on exponent n friction coefficient in particle stochastic Equation 16, set.? friction coefficient in fluid stochastic Equation 45, set.? friction coefficient for macroscopic disturbances of stochastic Equation 59, sec.-l void fraction, dimensionless viscosity, poises fixed time interval in Equation 7 , sec. intensity of macroscopic disturbances, sq. cm./ sec.2 particle energy, g. sq. cm./sec.2 (1 X) (u?(a?). = parameter, dimensionless q / p , kinematic viscosity, sq. cm./sec. variable of integration, sec. 3.1416 density, g./cc. interstitial fluid velocity, cm./sec. transition probability in configuration space, dimensionless coefficient of virtual mass, dimensionless transition probability in velocity space, dimensionless angular frequency of particle-fluid oscillations, rad./sec.
+
= = = = = = =
n
P 0
1 2, 3 m
2
fluidized bed point of incipient fluidization fluid or continuous phase running indices corresponding to coordinates 1, 2, 3 exponent in drag law of Equation 1 particle or discontinuous phase initial condition vertical coordinate horizontal coordinates single particle in a n infinite fluid total for all spatial directions
SUPERSCRIPT = marked particles or fluid
*
MATHEMATICAL SYMBOLS A = finite difference n = product or pi function = !sum = bar or time average defined by Equation 7 I = prime or fluctuating quantity < > = ememble average = derivative, d/dt Literature Cited
(1) Batchelor, G. K., “Homogeneous Turbulence,” Cambridge UniLrersity Press, Cambridge, 1953. (2) Chandrasekhar, S., Reu. .2lod. Phys. 15, 1 (1943). (3) Doob. J. L., Ann. .Math. 43, 351 (1942). (4) Doob, J. L., “Stochastic Processes,” Chapman and Hall, London. 1952. (5) Einstein, A , , Ann. Phvcnk. 1 7 . 549 (190$) (6) Eyring, H., J . Chr‘m. Phys. 4, 283 (1936). (7:I Furukawa. J., Ohmae, T., Ind Eng. Chem. 50, 821 (1958). (8) Hanratty, T. J., Latinen, G., LVilhelm. R. H., A . I . Ch. E . J . 2, 372 11956’). (9) .Hinze, J. 0.: “Turbulence,” McGraw-Hill, New York, 1959. (10) Houghton, G., J . Chem. Phys. 40, 1628 (1964). (11) Ibid., 41, 2208 (1964). (12) Houghton, G., Proc. Roy. Soc. A272, 33 (1963). (13) Houghton, G., unpublished. (14) Kirkwood, J. G., J . Chem. Phys. 14, 180 (1946). (15) Kramers, H., it’estermann, M. D., Groot, J. H., Dupont, F. A. A., Proceedings of Symposium on Interaction between Fluids and Particles (London), June 20-22, 1962, Third Congress of European Federation of Chemical Engineering. (16) Lapple, C. E.: Shepherd, C. B., Ind. Eng. Chem. 32, 605 (1940). (17) Levy, P., “Theorie de l’addition des variables aleatoires,” Gauthier-Villars, Paris, 1937. (18) Lewis, i V . K., Gilliland, E. R., Bauer, \V. C., 2nd. Eng. Chem. 41, 1104 (1949). (19) Longuet-Higgins, H. C., Pople, J. A., J . Chem. Phys. 25, 884 (1956). (20) Massimilla, L., Tt’estwater, J. T.V., A . I . Ch. E. J . 6 , 134 (1960). (21) Ottar, B., Acta Chem. Scand. 9, 344 (1935). (22) Pigford, R. L., Baron, T., IND.EXG. CHEM.FUNDAMENTALS 4, 81 (1965). (23) Poisson, S. D., M i m . Inst. (Paris) 11, 521, 566 (1832). (24) Ruckenstein, E., IND. END. CHEM.FUNDAMENTALS 3, 260 (1964). (25) Stokes, G. G., Proc. Cam6ridge Phil. Soc. 9,8 (1850). (26) Sutherland, K. S., Trans. Brit. Inst. Chem. Engrs. 39, 188 (1961). (27) Taylor, G. I., Proc. London Math. Soc. 20, 196 (1921). (28) Taylor, G. I., Proc. Roy. Sac. A219, 186 (1953). (29) Uhlenbeck, G. E., Ornstein, L. S., Phys. Reu. 36, 823 (1930). (30) Lt’ilhelm, R. H., Kwauk, M., Chem. Eng. Progr. 44,201 (1948). RECEIVED for review May 21, 1965 ACCEPTEDSeptember 20, 1965
INFLUENCE OF MIXING ON ISOTHERMAL REACTOR YIELD AND ADIABATIC REACTOR CONVERSION B E R N A R D G I L L E S P I E I A N D J A M E S J. C A R B E R R Y Department of Chemical Engineering, University of Notre Dame, Notre Dame, Ind.
the fact that backmixing of reacting fluid elements and yield (selectivity) for all but simple isothermal reaction networks, a precise quantitative assessment of finite mixing effects is lacking. Effective comparison of conversion and/or yield is readily established for the plug flow reactor (PFR) and continuously fed stirred tank IVEN
G affects conversion
1
Present address, Socony-Mobil Oil Co., Paulsboro, N. J
164
l&EC FUNDAMENTALS
reactor (CSTR) for given holding time or, as Levenspiel (73) demonstrates, the ratio of volume requirements for these isothermal limiting reactor types can be determined as a function of conversion. Rather complex systems involving nonlinear kinetics may be so analyzed with the understanding that either segregated (macromixed) or nonsegregated (micromixed) flow must be assumed in fashioning the CSTR model (79). Intermediate levels of mixing are more commonly encountered
