260
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
Peclet number; the new packing arrangement acts mainly to improve the liquid distribution. In addition, the new packing arrangement used up approximately 7% less packing than did the conventional packed bed. Acknowledgment One of the authors (S.J.K) wishes to thank the Office of the University Dean at the University of Cincinnati for a Summer Faculty Fellowship provided at the initial stage of this study. Nomenclature C, = concentration of tracer in the dispersion region, kg/m3 d = diameter of packing, m 8 = axial dispersion coefficient, m2/s E = exit age distribution of the column, dimensionless E, = exit age distribution of the dispersed plug-flow region, dimensionless Ed = exit age distribution of dead-water region, ( f c / f d ) X exp(-(T,/fd)%),dimensionless f = fraction of flow through the dispersed plug-flow region only, dimensionless G = gas mass flow rate per unit area of bed, kg/s-m2 Ga = Gallileo number, d;gp2/p2, dimensionless 1 = axial position within the bed, m L = height of packed bed, m M = liquid mass flow rate per unit area of bed, kg/s.m2 P e = Peclet number, u d , / D , dimensionless P e L = axial Peclet number, u L / D , dimensionless ReL = superficial Reynolds number based on mass flow rate, M d /pL, dimensionless Re = geynolds number based on average velocity in the bed, d p ( L / p ) p / p L ,dimensionless
f, = mean residence time for flow of liquid in dispersion region, S
f d = mean residence time for flow of liquid in dead-water region, s u = average velocity of liquid in dispersion region, L/F,, m/s 2 = dimensionless position in the bed, l / L
Greek Letters I.(
= first moment of exit age distribution in the column, s
first moment of exit age distribution in the dispersion region, s pL = viscosity of liquid, kg/m-s p = density of liquid, kg/m3 2 = variance, second moment about mean in the column, s2 0 : = variance, second moment about mean in the dispersion region, s2 % = dimensionless time, t / f c Literature Cited
pc =
Choudhary, M., Szekely. J., Weller, S.W.. AIChE J., 22, 1021, 1028 (1976). Dutkai, E., Ruckenstein, E., Chem. Eng. Sci., 23, 1365 (1968). Khang, S. J., Fitzgerald, T. J., Ind. Eng. Chem. Fundam., 14, 208 (1975). Levenspiel, O., "Chemical Reaction Engineering", Chapter 9, Wiley, New York, N.Y., 1972. Levich, V. G., Markin, V. S.,Chismadzhev, Y. A,, Chem. Eng. Sci., 22, 1357 (1967). Michell, R. W., Furzer, I . A,, Chem. Eng. J., 4, 53 (1972). Pillai, K. K., Chem. Eng. Sci., 32,59 (1977). Ridgeway, K., Tarbuck, K. J., Brit. Chem. Eng., 12, 384 (1967). Roblee, L. H. S.,Baird. R. M., Tierney. J. W., AIChE J., 4, 460 (1958). Stanek. V., Szekely, J., Can. J. Chem. Eng., 50, 9 (1972). Stanek, V., Szekeiy, J., AIChE J., 20, 974 (1974). Szekely, J., Poveromo, J. J., AIChE J., 21, 769 (1975). Wen, C. Y., Fan, L. T., "Model for Flow Systems and Chemical Reactors", Chapter 5, Marcel Dekker, New York, N.Y., 1975.
Received for review July 27, 1978 Accepted March 26, 1979
Magnetic Stabilization of the State of Uniform Fluidization Ronald E. Rosensweig Exxon Research and Engineering Company, Corporate Research Laboratories, Linden, New Jersey 07036
A mathematical stability analysis based on equations of motion augmented for stresses of magnetic polarization predicts that the state of uniform fluidization may be magnetically stabilized against growth of perturbations in voidage. The propagation and decay of voidage fluctuations are related to the physical and magnetic properties of the fluidized system with stabilization controlled by a dimensionless modulus representingthe ratio of magnetic energy to a kinetic energy of the fluidized medium. The stabilized regime exists when the fluid velocity equals or exceeds the velocity of minimum fluidization but is less than the velocity at which transition to bubbling and turbulence occurs. The transition velocity increases with increased magnetizationof the bed particles and may exceed the minimum fluidization velocity by more than a factor of 10.
Introduction Various workers have developed equations of continuity and motion to describe the behavior of a system of particles suspended in a fluid against the force of gravity under the influence of fluid dynamic drag. The equations have a simple formal solution corresponding to the well known state of uniform fluidization. In all previous studies it has been shown by the methods of hydrodynamic stability analysis that the formal solution is unstable to small perturbations of voidage. An early analysis (Rice and Wilhelm, 1958) considered bubble formation in a fluid bed to result from a Rayleigh-Taylor instability a t the bed lower interface but it seems more realistic due to presence of the support grid to consider instability as being excited within the bed 0019-7874/79/1018-0260$01 .OO/O
volume. Initial study predicating volumetric instabilities (Jackson, 1963; Pigford and Baron, 1965) predicted that the ordinary state of uniform fluidization should always be unstable, with instability much more severe, in general, for gas-fluidized beds than for liquid-fluidized beds of comparable particles. More complete equations of motion accounting for resistance to shear have been introduced (Anderson and Jackson, 1968; Murray, 1965) with a result that, although dissipation processes modify the propagation and growth behavior, the state of uniform fluidization nonetheless remains unstable. A review of work in this field is given by Jackson (1970). In the present study it is found that a bed comprised of magnetizable particles subjected to a uniform magnetic field may experience magnetic stresses that are stabilizing C 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 261
quired, these having the form given below. Magnetostatic Field Equations. I t is assumed that anywhere within the flow a local region exists that is large compared to the size of any bed particle but small compared to the distance over which field may vary appreciably. Accordingly, a sensor positioned in the region would detect a mean magnetic field that is well defined, and it is in that sense that local magnetic field is employed here. The same continuum assumption underlies the definitions for velocity fields of the fluid and solids phases. From Maxwell's equations
Magnetizable Particle Bed F i e l d Coil
Current Loop
Bed Support
Support Fluid
Figure 1. Sketch to illustrate the magnetic fluidized bed and nomenc1atu re.
curl H = 0
(5)
div B = 0
(6)
against the growth of voidage. There is technological interest in suppressing bubble formation in fluidized beds since the gas in the bubbles tends to bypass the bed producing lower yields in heterogeneous catalytic reactors and adverse effects on other chemical and physical operations. In addition, with bubble coalescence and growth a bubble dimension may spread across the bed cross section and cause erratic slugging that further lowers the performance potential of the bed and produces entrainment losses of particles (Kunii and Levenspiel, 1969). The magnetic system studied here is sketched in Figure 1, which illustrates nomenclature introduced later. A solenoidal field coil having current loops wound on a bias provides arbitrary orientations of applied fields relative to the flow direction of support fluid. The bed is comprised of magnetizable particles, and one of the objects of this analysis is to determine the influence of the intensity of magnetization on the process. In this analytic theory for magnetic stabilization of fluidized beds, the effort is made to simplify by omission terms that are negligible in magnitude or poorly known and inessential, so they will not obscure the role that magnetism plays in the process. Thus, terms in fluid density including virtual mass terms are dropped in comparison to terms in solids density, and shear resistance within a given phase is neglected while retaining fluidparticle interaction as an essential mechanism of the flow. The basic relationships are given in the following.
B=M+H (7) In (1)to (4) above, the symbols t, u , v represent the local mean values of the voidage, the fluid velocity, and the particle velocity, respectively; ps is the density of solids; g is the acceleration owing to gravity; E and E5 are the local mean values of the stress tensors for the fluid and particle phases; f is drag force per unit volume of bed arising from interaction between the fluid and the particles, H i s the magnetic field strength, B is the magnetic induction, and M is the ferric induction or magnetization. Equation 5 is the expression of Ampere's law when there is no current flow, eq 6 denotes continuity for the field of magnetic induction, and eq 7 is a defining equation indicating that material property M is responsible for the difference between the induction B a n d the field intensity H. Constitutive Relationships Equations 1 through 7 are written in a general form. However, the system of equations cannot be solved as it stands without specification of relationships that express the dependence of f, E, Ea, E", and Mon the flow field, the magnetic field, or other parameters of the system. These are given as the following constitutive relations. Fluid-Particle Interaction Term. I = tP(t, M) ( u - v) + virtual mass term (8)
Basic Relationships The equations proposed by Anderson and Jackson (1967a,b) are adopted as the starting point for this analysis. These consist of two equations of incompressible continuity, one for the fluid and another for the particles together with two vector equations of motion modified by considering the fluid to be a gas, weightless and free of inertia, so that gas density does not appear in the equations. Rationalized CGS units are employed throughout. fluid continuity particle continuity
at
-
at
+ div
t
a ( 1 - t) at
~
u=0
+ div (1 - t ) v = 0
(1)
(2)
fluid equation of motion t div E - f = 0 (3) solids equation of motion av (1- t)p5[ (v. grad)v = (1 - t) div E + f +
+
1
(1 - t)p,g+ div ES + div E" (4)
The solids equation of motion is augmented above with the magnetic stress tensor Emwhich is dependent on the magnetic field, so in addition to the foregoing relationships the equations governing the magnetostatic field are re-
P ( E , M) is a drag coefficient expressing force per unit volume per unit superficial velocity due to relative motion between fluid and particle phases. In the absence of magnetization has been a well studied parameter in fixed bed mechanics. At this point, however, it is not necessary to introduce a specific functional form for p. Later it will be assumed that b(t, M) = 4 m P ( t , 0 )
(9) where 4 mis a constant. More generally there may be a change in the dependence of P on t according to the orientation of M relative to the flow. The virtual mass term while not written explicitly here is proportional to fluid density pf. The term can be omitted consistent with the assumption of a massless gas since the solids densities of interest far exceed those of the gas. There is an added virtue for this procedure of avoiding the problematical choice of a virtual mass coefficient. Fluid Stress Tensor Components. Components of the fluid stress tensor will be taken as f i k = -p& + viscous terms (10) where p is the local mean value of the fluid pressure and &k is the Kronecker delta with values 8ik=0 i # k &=1
i = k
(11)
262
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
The viscous terms will be ignored consistent with the assumption of inviscid flow within a given phase for it is known that instabilities, if they occur, are due to the particle inertial terms. Accordingly, the fluid stress terms in eq 3 and 4 become simply div E = -grad p (12) Solids Stress Tensor Components. tiha
= particle pressure term
+ viscous terms
(13)
Both terms on the right side of (13) will be neglected on the grounds that no sure means are available for their estimation while instability for nonmagnetic systems is predicted in their absence anyway. Hence, this procedure permits determining if magnetization can stabilize the system even under the adverse condition when solids damping is absent. The result of this simplified approach should be definitive, whereas an estimation of the solid stress tensor would be speculative. Hence, in the present treatment, from (13) div E, = 0 (14) and this term drops out of eq 4. Magnetic Stress Tensor. A general magnetic stress tensor applicable to nonlinearly magnetizable media and based on conservation of magnetostatic field energy has been derived by Cowley and Rosensweig (1967) with the identical result obtained independently in another manner by Penfield and Haus (1967). The expression for the tensor components may be stated as
+ magnetostriction term (15) where H denotes the magnitude of H . The magnetostrictive term may be rigorously neglected if the emulsion is magnetized to saturation since then the magnetization per unit mass upon which the term depends is independent of bed voidage variations. For unsaturated beds, a further relationship is required to relate magnetization to voidage and applied field intensity; however, this refinement is neglected herein. It is noted that (15) reduces, as it should, to the wellknown Korteweg-Helmholtz stress tensor for linearly polarizable magnetic media when the permeability p = B / H is constant, and to the classical Maxwell stress tensor for a vacuum when p = 1 (Rosensweig, 1971). The i component of the magnetic force f, per unit volume exerted on a fluid element is given by a t i k m / a x , so from (15) div E" = f, = 1 -[-H grad H + H div B + (Bgrad)] H (16) 47 = -(M1 grad) H 47 where the second equality is obtained using the conditions that div B = 0 of (6), substitution for B from (7), employing the vector identity ( H -grad) H = H grad H - H X (curl H)along with curl H = 0 of (5). Assuming particles of the medium are free to orient magnetically with the direction of the field, then M = ( M / H )H and again employing the vector identity the force can be expressed in terms of magnitudes of the vectors as 1
f, = -M 47r
grad H
Figure 2. Sketch defining magnetic parameters of bed solids.
Magnetic Equation of State. I t has already been assumed that M is collinear with H . Then assuming additionally that the magnitude of the magnetization is proportional to the solids content in a unit volume the equation of state may be written as
H -M- - (1- t) E
M B In (18), M i s magnetization of a unit volume of the bed emulsion while the parameter M , represents magnetization of a unit volume of the magnetic particle material. In (18) H / H is a unit vector oriented in the field direction. It should also be noted that Ma depends on magnetic field intensity H. For simplicity, any dependence of magnetization on particle shape is neglected.
Ma = M,(W
(19)
Rigorously, the field intensity H in (19) should be taken as the value within the magnetic particle. In general, this value differs from the continuum value associated with that region of the bed. Again, when the particles are magnetically saturated there is no error. Figure 2 represents a magnetization curve for magnetically soft material, i.e., material having negligible hysteresis, remanence, or coercivity. The material magnetizes nonlinearly and saturates at a value denoted M,, as field intensity is increased to a high value. Only the first quadrant properties are important for this analysis. Since perturbations in uniformity of the bed may alter the local value of field intensity and direction, it will be necessary later to develop an expression for the attendant perturbation of the magnetization vector. Additional magnetic parameters, the tangent slope, and the chord slope will be required for this purpose. As Figure 2 illustrates, the definitions of these parameters are chord susceptibility =
M,,O HO
xo = -
(20)
(a2),
tangent susceptibility = 2 = -
(21)
For the unperturbed bed, the quantity Mo representing magnetization per unit volume of the bed is thus given as Mo = (1 - to)M,,o = (1 - ~ x & o . Solution for Steady State with Uniform Magnetic Field A simple solution of these equations is one representing a steady state of uniform fluidization and uniform magnetization in which the local mean particle velocity is
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
everywhere zero, the fluid velocity is constant in both space and time and directed vertically upward, and the voidage is uniform in space and independent of time. These conditions may be stated as u=uo=iuo
a €1
at
+ u o i grad + t1
_ -a E 1 at
+ (1 -
to)
to
div u1 = 0
263
(1')
div vl = 0
v = v,=o t
H=
=
to
E & = go
M=M,,=idM,
B=B,=&Bo
(22)
where i is the unit vector in the upward vertical direction, iois a unit vector in the direction of the applied field, and uo, to, Ho, and Mo are constants. These satisfy the continuity and magnetostatic field equations identically, while the fluid equation of motion reduces to grad p o + P(to)uoi= 0
(23)
When eq 23 is combined with the solids equation of motion to eliminate p o the result is, recognizing that g = -gi and grad Ho= 0 - A 1 - to)Ps = 0
PO(E0)UO
(24)
When P is known as a function of e, eq 24 determines the uniform voidage corresponding to any given fluid velocity uo and eq 23 subsequently permits the uniform fluid pressure gradient to be calculated. I t is seen from the above that the presence of the uniform applied magnetic field contributes no explicit terms to the steady solution regardless of the field direction or magnitude.
Stability of the Steady-State Solution The stability of the steady-state solution against small perturbations can now be examined by hydrodynamic stability technique. Voidage perturbations, if they grow, may be taken as possible precursors of bubbles in fluidized beds. The presence of voidage perturbations will perturb the uniform applied magnetic field so it may be anticipated that magnetic body forces will be generated and play a role in determining stability. Derivation of the Linearized Equations. The uniform fluidization solution is augmented by a small perturbation, writing €
=
€0
+ €1
+ v1 = v1
M = & + Ml = &Mo+Ml
+ B1= & B o + B1
+ 2HoH1.&)1/2
+ HI-& M = Mo + Mi.& = Ho
(26)
written correct to first order. Form of the Linearized Equations for Plane Waves. A prerequisite for solving the simultaneous set of eq 1' through 4' is relating the magnetic term of eq 4' to independent parameters of the system. Equations 5,6, and 7 give the basic relationships governing the magnetostatic field. The final three equations of (24) introduced definitions of the small perturbations of magnetic field and now it is desired to obtain field equations for these perturbation quantities. This may be done by combining these relationships to eliminate the steady components of field, producing the following set of simultaneous equations curl Hl = 0 div Bl= 0
B1= Ml + Hl Eliminating B1, this set of equations reduces to the following pair curl Hl = 0 (27)
Next, the equation of state must be expressed in terms of the perturbation quantities. From the definition of 2 given as (21), it follows that a small variation 6H in magnetic field intensity produces a change 6M,,, in magnetization of the bed solids given as 6Ms,o= ij6H. The variation 6H from (26) is given as 6H = H - Ho = Hl.io. Thus, 6MS,,= i j H1.ioand so the magnitude Ms,o
H=I-I,+Hl=&Ho+Hi B = B,
= (Ho2
div Hl = -div Ml
P = Po + P1 u = uo+ u1= iu,+ u1
v = vo
1 4ir where eq 4' results from combination with eq 3' to eliminate the pressure gradient. The suffix zero is used to indicate that a quantity is evaluated a t conditions corresponding to the formal uniform fluidization solution whose perturbation is studied. Po = @(to) and the notation Po' is used to denote the derivative Po' = (d@/dt),. In formulating the magnetic final term of (49, use was made of the Pythagorean relationships H = I& + HlI
-Mo grad ( H1.$ (4')
(25)
and the perturbations are assumed both small and smoothly varying in space and time, so that any derivatives are also small. i denotes the unit vector in the direction of flow and iodenotes the unit vector in the direction of the applied field. The expressions of (25) are substituted into the equations of continuity and motion with all terms of order greater than one in small quantities neglected. In this manner, a set of linear partial differential equations in the perturbations is generated. The linear equations obtained from eq 1to 4 in this way, upon combination with eq 12, 14, 17 and the steady solution given by eq 24, are
Ms,o(H)
= Ms,o(Ho) + 6M,,o
= XOHO + 2H1.4 Using this relationship to substitute for Ms,oin (18), then subtracting the unperturbed form of (18) from the general form of (18), recognizing that io= Ho/Hoand retaining first-order terms gives the following relationship for the vector perturbation-of-state equation.
264
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
It is seen from (28) that even though Mremains parallel to H , the vectors M1and H1are not parallel. Taking the divergence of (28) and combining with (27) now gives an equation that determines the field in the presence of the voidage perturbations
from (1’) and (2’1, eliminating Po using the steady solution (241, using (36) and dividing by p s , there is obtained 1-2to-€o(l-to)-
Ma,02 = 0 (37) 4*Pa where z i o = €u0 denotes the fluidizing velocity on a “superficial” basis, the direction of i has been denoted the x direction, and the definition of a is a(1 -
grad(i,,.HJ + 6. grad el (29) Obtaining a general solution of (29) valid for any distribution of voidage will be circumvented since the general procedure in studying the motions of the fluid and the solid will be to assume that initially a train of waves of solids concentration of infinitesimally small amplitude is suddenly produced in the fluid mixture. A spectrum of such small disturbances will always be present as “noise” resulting from statistical fluctuations, flow irregularities, and other sources. The development of a single Fourier component or wave of the noise may be considered in isolation owing to the linearity of the equations. Then the motion of a wave through the mixture and its change in amplitude with respect to time may be computed analytically. If the amplitude increases exponentially, the uniform density distribution is said to be unstable to that disturbance mode. Thus, consider plane waves of the form €1 = t1E
Hl = HIE
a =
€O)--V2€1
cos2 0 1 + (1 - to)(i - COS^
o)xo +
- Eo) cos2 e
(38)
Equation 37 duplicates Anderson and Jackson’s eq 19 (1968) with vanishing fluid density and substitution of the magnetic term for the coefficient of V2c1. Equation 37 is applicable to general plane wave voidage perturbations in three spatial dimensons in which mean flow is directed along the x axis and applied field is oriented at angle 8 with the direction of the train of plane waves in voidage. General Solutions for the Voidage Perturbations. It is noted, due to the form of the plane wave expression eq 32, that wavelength is given by X = 2a/k
(39)
where
k = I4
where
E = exp(st) exp(ik.x)
(32) with tl the amplitude of perturbation in voidage, A, a constant vector to be established by the analysis, and k the wave number of the plane wave disturbance. The Ampere’s law condition of (27) has not yet been imposed on the solution. From (31) and (32) curl Hl = E curl Bl - h1X grad E = -iEHl X k
so with (27) it follows that HIXk=O (33) Hence, it has been proven that Bl is a vector parallel to the wave number vector k. It may now be seen that the term iqHl occurring in (29) may be replaced by EH1 cos 0 with Hl the magnitude of the vector H , and 8 defined as 0 = cos-1 (i,,.&)
1
Po’ a€, -Po ax
(34)
where ko is the unit vector in the direction of k. With substitutions of el andAH1from (30) and (31) into (29) a solution is found for H1as cos 0 Ms,O€l 1 + (1 - € o ) ( l - cos2 8)xo + k(1 - E O ) cos2 8 (35) With (35), the magnetic term of (4’) becomes 1 -Mo grad (Hl.i,,)= 47r (1- €o)(COS2 8)Ms,02/47r grad t l (36) 1 + (1 - €0)(1- cos2 0)xo + k(1 - €0) cos2 8
-8-1 -
It is now possible to obtain a determinate partial differential equation in the voidage perturbation el. Taking the divergence of (49,substituting for div u 1 and div v 1
In general, s is a complex number which may be written s=[-iq (41) where i = (-l)1’2. The imaginary part of s determines V,, the propagation velocity of the waves, according to = ,/I4 (42)
v,
while the real part determines the rate of growth or decay of the waves with time. If 5 is positive, the disturbance grows and the state of uniform fluidization is unstable, while if [ is negative, the disturbance decays and the uniform state is stable. A growth distance can be defined also as Ax = o/tl4
(43)
which gives the distance of travel in which the amplitude of the wave grows or decays by a factor e, the growth or decay being distinguished by the sign of 5. Substitution of the plane wave expression (eq 30) for el into eq 37 then yields an algebraic equation c1[As2 + Ps + (f + ibPC)] = 0 (44) where A=-
€0
(45)
1 - €0
p = - - €0 g 1 - €0 Lio
Po’ c = 1 - 2€0- €o(l - €0) PO
UO
b = -k
(47)
COS Y
€0
(49)
Ind. Eng. Chem. Fundam., Val. 18, No. 3, 1979 265
A , P, and E depend on properties of the unperturbed system while b and f depend on the wave vector k whose magnitude was denoted k and y is the angle between the direction of flow and k. Equation 44 is satisfied if either =0
(50)
or
As2 + Ps + (f + ibPC) = 0 (51) If eq 51 is satisfied, there exists a nontrivial solution (el # 0) of eq 1’to 4’. A nontrivial solution of eq 1’to 4’ may also exist when eq 51 is not satisfied, provided that c1 = 0 corresponding to a partricle distribution that remains uniform and unperturbed. From the continuity eq 1’and 2’ it then follows that div u1 = div vl = 0 (52) or representing u 1 and v1 by u1 = el exp(st) exp(ik.x) (53) vl = q1 exp(st) exp(ik.x) (54) it follows that (55) k.21 = k.fi = 0 showing that the velocity perturbations are transverse to the wave vector as was pointed out for the nonmagnetic flow by Anderson and Jackson (1967a). These transverse waves generate no disturbances in the voidage and hence are not of immediate interest. Accordingly, attention will be limited to solutions corresponding to the roots of eq 51. As eq 51 is quadratic in s, the real and imaginary parts of its two roots can be readily obtained and are given by
p [ -1 *
= 2A
1 d -
(56)
and
where 2 = 4 -A f P2 AbC q = 4 p
Po
2 too -
€0)
so that
c = 3 - 2to
y = cos-1 (Ek)
tl
Po’ -_ _
(59)
In eq 56 and 57 the choice of negative sign before the radical corresponds a t once to waves that decay and hence need not be further considered here. Before proceeding further, the quantity C of eq 47 will be made definite by assuming a model for the particleto-fluid drag coefficient &. For this purpose it will be illustrative to adopt the Carman-Kozeny relationship representing an extrapolation from laminar flow through a bed of packed particles, for which
(62) Magnetic stabilization is achieved if E < 0 or neutral stability is obtained when 5 = 0. Accordingly, from eq 56 taking the positive choice of sign neutral stability requires 1 z > -42 (63) 4 Substituting for z and q in eq 63 and then for A , C, b, and f from (45), (47), (48), and (49) gives the following criterion of stability as a central result of this theory N,N, > 1 (unstable) = 1 (neutrally stable)
(64)
< 1 (stable) where (65)
N , is a dimensionless group representing a ratio of kinetic energy to magnetic energy, and
where y as defined previously is the angle between the direction of flow and the direction of the wave train so that cos y = i-ko. N a t u r e of t h e Predicted Behavior Examination of (66) shows that, when all other parameters in it are specified, the maximum value of N , occurs when the angle y is zero; that is, a disturbance wave oriented along the flow direction poses the greatest threat to upsetting the bed’s stability. This threat is best met when 0 is zero, that is when the orientation of the applied magnetic field is collinear with the direction of flow for then cos 0 is unity and N , has its least value. Hence, the analysis has determined the optimum orientation of the applied magnetic field to be along the axis of flow. It is noted that the state of uniform fluidization according to the analysis cannot be stabilized by a transversely applied magnetic field. This may be seen from (66) for when 0 = 7 1 2 , the value of N, becomes infinite and no finite value of N, or speed may be found that can satisfy the criterion (64). In fact, a lesser amount of stabilization in transversely applied magnetic field has been found in subsequent testing, and the topic deserves further study. With field oriented collinear with the flow direction, Le., in the preferred direction, the expression for N , given by (66) simplifies to the following in which xo is absent.
The criterion given by (64) and the specialized form N , of (67) when combined may be plotted to display transition values of N, as shown in Figure 3. Here K is a constant factor depending on properties of the fluid and the particles. Then from eq 60 ignoring any influence of magnetic field gradients on K
N, =
-
€0)
4a(3 - 2tOY [l + (1-421
(68)
266
Ind. Eng. Chern. Fundarn., Vol. 18, No. 3, 1979 10-2
-
/
1 5 ' L
lo-&
'
.1
1
'
.k
'
.I '
I -.e-
I
NmNv = 0
1!0
Figure 3. Dependence of the magnetic modulus on voidage and susceptibility at neutral stability for magnetically stabilized beds of infinite extent, y = 8 = 0.
The plot of this relationship illustrates the dependence of the magnetic modulus N , on voidage and susceptibility. Any one curve of constant susceptibility divides the operating area into two territories. A bed operating at any point with coordinates below the curve fluidizes stably without bubble formation; beds operating above the curve are unstable towards bubbling. It is seen that low susceptibility is desired to achieve more readily stabilized beds. A saturated bed medium has zero susceptibility and requires the lowest level of magnetization at a fixed speed or sustains the highest speed a t a given level of magnetization while remaining stable. With 2 = 0, (68) simplifies to N , = c2(1- t o ) / 4 d 3 - 2 ~ and ~ the ) function ~ exhibits a calculus maximum at to = 0.8139. From (64) and (65) the maximum point corresponds to N , = 5.21 X When a Richardson-Zaki drag law is used in place of the Carman-Kozeny expression, N , increases with to but displays no calculus maximum. The maximum is probably an artifact, perhaps arising from use of the CarmanKozeny expression a t conditions beyond its realm of applicability. Next it is desired to examine the prediction of theory regarding growth and decay rates of disturbances in unstable and stable regions, respectively. To accomplish this task in the most general way, eq 56 will be put into the normalized form. The result is the following expression involving but three dimensionless groups, namely, f,, k,, and the group N,N,. 1 f , = -1 -x
+
,hi
(69) New definitions introduced into (69) are (reduced growth factor)
(70)
(reduced wave number)
(71)
k k, = g / €,no2 t,
2 = -(3
-
2co) cos y
(72)
€0
Here f,, the normalized growth factor, according to (69) depends only on reduced magnetization and reduced wave number. The relationship is plotted in Figure 4. The curve for N,Nv equal infinity demonstrates the universal instability of nonmagnetized fluidized solids, a result that is independent of the scale of the disturbance. The other
Figure 4. Dependence of normalized growth factor on scale of disturbance and presence of magnetization for uniformly magnetized fluidized beds of infinite extent.
curves illustrate the gradual stabilizing influence attendant to increasing the value of the reduced magnetization (smaller value of NJV.,). When the parameter diminishes to N,Nv = 1, the normalized growth factor assumes negative values for wave mode and the system exhibits stability. As shown by the behavior of f , a t large wave number, a small scale disturbance exhibits the fastest time rate of change in amplitude. As the magnetic parameter is increased without bound, the decay rate approaches a limiting value of f , = -1. This corresponds to a decay time constant of 2 Uo/g, the flight duration of a particle projected upwards in the gravity field a t initial speed Uw Phase velocity may be found from the combination of (42) and (57). Expressed in normalized form this becomes
(73) The definition of V , appearing in (73) is
V, =
VP 7 UO
(reduced phase velocity)
(74)
In order to further reduce the number of variables, attention is focused on the speed of axially directed waves y = 0 with the unperturbed voidage specified constant at eo = 0.8139. With these conditions 6, = 3.372 and V , becomes the following function of k, and N,Nv. 1.192 =k.
v,
(7 = 0,
€0
= 0.8139) (75)
Equation 75 is plotted as Figure 5 wherein it is seen that for all stabilized beds the wave propagation speed is much faster than the fluidizing speed a,. Also, increase of magnetization is seen to increase the phase velocity, indicating that the faster decaying modes travel more rapidly through the medium. Up to this point, the predictions of the model have been independent of particle size although it has been indicated that the voidage realized in any particular flow will depend upon the particle size and other parameters. Now an
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 100
Nm Nu = 1/2
Nm N,r = 1
-
I
t/
Uo -
’
2.0 ~
I
1 1
I
0
3.0
I
i i
STAELY FLUIDIZE0
O
I 400
I I
c
UNFLUlDlZEU
I 200
I
4
/ lot / L
1.0
I
LNSTAELY FLUlDl7EU
t
I 4,
4
1
267
= 0.35
x= 1 1 600
iI
800
k
(g,L,021
Figure 5. Dependence of phase velocity on magnetization and wave number for magnetized fluidized beds of infinite extent with Q = 0.8139 and y = 0.
additional assumption will be introduced in order to obtain a quantitative relationship between transition speed and particle size. The new expression that is needed is the dependence for K that appears in (60). Following the form of the Carman-Kozeny relationship, K may be represented as (76) where p is the fluid viscosity and Dp the particle size. The factor 4 ,,,in accord with eq 9 is introduced to account for particle alignment effects or other influence exerted by the applied magnetic field that would affect the particle-tofluid drag. Introducing (76) into (60) and eliminating Po with the steady-state force balance given by (24) results in the expression
Figure 6. Phenomenological behavior of magnetically stabilized, fluidized bed predicted by the theoretical treatment. Magnetic stabilization produces a wide range in which the bed medium is quiescently levitated and free of bubbles, fluctuations, or turbulence. This stabilized range exists between minimum fluidization velocity Li, and transition velocity ut of the neutral stability curve. Susceptibility 2 is taken constant at 1 with f d = 0.35.
Carman-Kozeny relationship is subtly inappropriate when the bed is magnetized, and contributes importantly to the difficulty. The essential difficulty persists when using a Richardson-Zaki drag correlation. Additional study is required to clarify this point. Figure 6 gives the prediction of theory for the dependence of transition velocity on magnetization of the bed particles. Gas superficial velocity u0 is normalized to minimum fluidization velocity urn,and magnetization is represented nondimensionally also. The plot is developed by assigning a value t d to the dumped bed or unfluidized bed voidage. From (24) and (60) with eo = Ed when uo =
(77) With y = 0 and 0 = 0, the criterion NmNV= 1 of (64) may be expressed as
This expresses normalized speed, the group on the left side, as a function of void function to and susceptibility 2. Eliminating u0 between (77) and (78) then gives an expression for reduced particle size in the form
-
( 1 5 0 ~ r n ) ’ ~2 (t0)3/4 1 DP ( 4 ~ ) l / ~ t-~2(~3~ ) ’ / ~ [ (1 1 +- t0)2]1/4 (p2~s,02/g2ps3)1/4 (79) Equations 78 and 79 taken together determine the reduced speed as a function of the reduced particle size. Constant values of 2 and r$rn may be assigned and values of to inserted into (78) and (79) to separately compute corresponding values of normalized speed and particle size. With constant susceptibility and for a given solid fluidized with a given fluid there is found a characteristic particle size yielding a maximum of throughput speed. Consequently, transition velocity decreases with increase of particle size at particle sizes greater than the characteristic size (but see below). Also, it is found that increase of susceptibility reduces the speed at which transition occurs, a trend previously deduced from Figure 3. Experimental tests with C 0.8 (Rosensweig, 1979) yield decreasing transition velocity with increasing particle size for constant susceptibility, a trend which is a t variance with values computed in the manner described above. It is suspected that the voidage dependence given by the
so that values of cio/z.irn may be computed for given values of 6.In Figure 6, €d is taken constant at 0.35. For the same values of to, values of N , are computed from (67) upon assigning a value of susceptibility 2. 2 is taken constant at 1 in Figure 6. From the criterion of (64) (NrnN,= 1) and the definition (65) the normalized magnetization of Figure 6 is given by (81) The values of normalized velocity vs. normalized magnetization are plotted as the upper curve in Figure 6 and represent the neutral stability curve separating the region of unstable fluidization from the region of stably fluidized flow. The horizontal line for uO/&, = 1represents the theoretical equilibrium solution for onset of fluidization. From (23) and (24) this curve is uninfluenced by magnetization of the bed solids. Thus the theory predicts the existence of a wide area located between the curve of incipient fluidization or levitation and the curve where transition to the bubbling, fluctuating state of flow occurs. In this area the bed is in a state of quiescence. It is noted that the theory predicts stabilization may be achieved a t velocities exceeding the minimum fluidization speed by large factors.
Conclusion According to the theory, magnetization is able to prevent instabilities in the state of uniform fluidization. The applied magnetic field is most advantageously oriented in the direction of the bed axis, i.e., along the direction of flow. Other features predicted by the theory describe the
268
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
growth or decay rate of voidage waves and the speed of wave propagation. A particularly interesting aspect of the behavior is the three-phase diagram illustrated by Figure 6. In analogy to a physical chemical phase diagram for a pure component, the unfluidized bed represents a solid phase, the stabilized region is the liquid phase, and the bubbling bed represents the vapor. Gas velocity plays the role of temperature as an agitating influence and applied magnetic field corresponds to pressure. The minimum fluidization line, analogous to melting temperature, is little affected by field (pressure). The stability theory is seen to predict the liquid-vapor equilibrium line; this produces an intersection with the melting line having the topology of a triple point. Experimental bubble point curves obtained by increasing gas flow rate at constant field would correspond to “solid-vapor transition” (sublimation) to the left of the triple point, and theoretically exhibit a break a t the triple point. A trend toward zero slope for M approaching zero is noticeable in experimental data (see Example 12 and Figure 6 of U S . Patent No. 4 115927 of Sept 26, 1978). The uniform applied magnetic field exerts no net force on the bed as a whole. However, the field acting on the uniform bed does establish a state of uniform stress. Magnetics is used here not as a muscle but as a controlling influence which is brought into play when a nonuniformity introduced in the bed emulsion in turn disturbs the uniformity of the state of stress; the result is the appearance of local forces tending to restore the bed to its initial state of uniformity. Since magnetized particles may attract each other, it is not obvious at the outset that a stabilized bed will resemble a fluid, particularly in the absence of bubble agitation. However, experimental study confirms the existence of a magnetically stabilized region having fluid behavior (Rosensweig, 1979). With moderate magnetization, particulates of the bed are quiescently levitated, the bed expands homogeneously with an increase in gas speed, light objects float, heavy objects sink, and the bed medium is flowable. When the bed particles are highly magnetized, the medium develops an appreciable yield stress due to micro-scale attraction between particles, flowability is impaired, and so a modified description of bed stability and behavior may be necessary. Thus, it could be argued with some logic that the assumption of a gradient field force density employed in this work has meaning only if it derives from a scale that is large compared to the distance between particles. In comparison, it is interesting to note the paper of Mutsers and Rietema (1977) wherein assumption of cohesive force between particles leads to a stabilizing term that algebraically is analogous to the last term of eq 4 given by eq 36. From this point of view the present study gives a definite relationship for the elasticity coefficient E in their work. Finally, recent experimental measurements indicate that yield stress approaches zero in these beds as fluid speed increases toward the transition value. This fact lends support to stability theory in which short range forces between particles are neglected.
Acknowledgment Appreciation is due to M. Berger and R. L. Espino for their support throughout this work and to J. J. Schlaer for assistance with the experimental details. Nomenclature B = magnitude of the magnetic induction, G Bo = magnitude of the magnetic induction in the uniform medium, G
B = vector magnetic induction, G Bo= vector magnetic induction in the uniform bed medium, G
B1= perturbation of vector magnetic induction, G (Bl= B - Bo) Dp = particle diameter, cm E = harmonic wave function defined by eq 32 E = fluid stress tensor, dyn/cm2 E” = magnetic stress tensor, dyn/cm2 EE = solids stress tensor, dyn/cm2 f = particle-to-fluid drag force per unit volume, dyn/cm3 f, = magnetic body force, dyn/cm3 g = acceleration due to gravity, 980 cm/s2 g = vector acceleration of gravity, cm/sz H = magnitude of magnetic field, Oe Ho= magnitude of magnetic field in the uniform bed medium, Oe H = vector magnetic field, Oe H o = vector magnetic field in the uniform bed medium, Oe H1= perturbation of vector magnetic field, H1= H - Ho, Oe H1= vector amplitude of perturbed magnetic field distribution function, Oe i = the imaginary number, (-1)1/2 i = unit vector in the direction of flow io = unit vector in the direction of magnetic field Ho k = wave number magnitude, cm-’ k = vector wave number, cm-’ ko = unit vector in direction of k k , = dimensionless wave number defined by eq 72 K = constant appearing in drag expression of eq 60,dyn s/cm4 M = magnitude of magnetization for the bed medium, G Mo = magnitude of magnetization for the uniform bed medium, G M , = magnitude of magnetization for the bed solids, G = magnitude of magnetization for the bed solids in the uniform bed, G M8,g= saturation magnetization of the bed solids, G M = vector magnetization of the bed medium, G M o = vector magnetization of the uniform bed medium, G (Mo = Bo - H,) M1= perturbation of vector magnetization, G (Ml = M - Mo) N , = dimensionless group representing a ratio of kinetic energy to magnetic field energy, N m = P , Z ~ O ~ / M ~ . O ~ N , = dimensionless group defined by eq 66 p = fluid pressure, dyn/cm2 p o = fluid pressure in uniform bed, dyn/cm2 p1 = perturbation of fluid pressure, dyn/cm2 (pl= p - pol s = complex frequency defined by eq 32 and 41, s-l t = time, s uo = fluid speed in the bed medium, cm/s a, = value of a. at transition, cm/s a, = empty column fluid speed, cm/s Zim = value of a, at incipient fluidization, cm/s u = fluid velocity in the bed medium, cm/s uo = fluid velocity in the uniform bed, cm/s u 1 = perturbation of fluid velocity, cm/s (u1 = u - uo) fi = vector amplitude of perturbed-fluid velocity-distribution function, cm/s u = speed of solids phase, cm/s v = velocity of solids phase, cm/s vo = velocity of solids phase in the uniform bed, cm/s v1 = perturbation of solids’ velocity, cm/s ( V I = v - vo) Vl = vector amplitude of perturbed-solidsvelocity-distribution function, cm/s V,, = phase velocity, i.e., speed of travelling wave, cm/s
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979
VI = dimensionless phase velocity, VI = Vp/ao
x = position vector, cm
normal oriented in the i direction, dyn/cm2 = element of magnetic stress tensor, dyn/cm2 tiks= element of solids’ stress tensor, dyn/cm2 ps = mass density of the solids, g/cm3 $ m = magnetic anisotropy factor (no units); see eq 76 xo = chord susceptibility based on properties of the magnetic solids, G/Oe 2 = differential susceptibility based on properties of the magnetic solids, G/Oe tikm
Ax = growth distance defined by eq 43
Greek Letters a = term defined by eq 38 (no units) @ = particle-to-fluid drag coefficient, dyn s/cm4
= value of /3 in the uniform bed, dyn s/cm4 tc? = rate of change of @ with respect to voidage e, evaluated for e = eo, @( = (d@/dt),,dyn s/cm4 y = angle between direction of unperturbed flow u oand the wave vector k,rad e = volume fraction of voids, i.e., the voidage td = volume fraction of voids in the dumped bed eo = volume fraction of voids in the uniform bed el = perturbation of the voidage, tl = t - to iil = amplitude of the voidage distribution function eI = term defined by eq 73 (no units) 11 = magnitude of the imaginary part of the complex frequency s, s-1
6 = angle between the applied field vector H o and the wave
number vector k h = wavelength of the disturbance pattern, cm p
= fluid viscosity, g/cm-s, P
t = growth factor representing real part of the complex frequency s, s-l
Literature Cited Anderson, T. B.; Jackson, R. Ind. Eng. Chem. Fundam. 1967a, 6, 478. Anderson, T. B.; Jackson, R. Ind. Eng. Chem. Fundam. 1967b, 6, 527. Anderson, T. B.; Jackson, R. Ind. Eng. Chem. Fundam. 1968, 7, 12. Cowley, M. D.; Rosenswelg, R. E. J . FluMMech. 1987, 30, 671. Jackson, R. Trans. Inst. Chem. Eng. 1963, 41, 13. Jackson, R. Chem. Eng. Rog. Symp. Ser. No. 1051970, 66, 3. Kunii, D.; Levenspiel, 0. “Fluldiratlon Engineering”, Wlley: New York, 1969; PP 3,9. Murray, J. D. J . F/uM Mech. 1965, 27, 465. Mutsers, S. M. P.; Rietema, K. Powder Techno/. 1977, 78, 239. Penfield, P., Jr.; Haus, H. A. ”Electrodynamics of Moving Medla”, Research Monograph No. 40; The M.I.T.Press: Cambridge, Mass., 1967; Table C.2, p 255. Pigford, R. L.; Baron, T. Ind. Eng. Chem. Fundam. 1965, 4 , 81. Rice, W. J.; Wllhelm, R. H. AIChEJ. 1956, 4 , 423. Rosenswelg, R. E. “Fwohy&&ymmics”, in “Encycbpaedc Dictbnery of physics," Suppl. 4, Pergammon: 1971; pp 111-117. Rosenswelg, R. E. Science 1979, 24, 57.
tr = dimensionless growth factor defined by eq 70 Eik
260
Received f o r review September 13, 1978 Accepted March 7, 1979
= element of fluid stress tensor representing component of stress in k direction experienced by a surface having its
Steady-State Decoupling of Distillation Columns T. J. McAvoy Department of Chemlcal Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
Steady-state sensitivity of one-way decoupling to errors in decoupler gain is defined using Bristol’s relative gain array, and applications to distillation are given. Columns controlled via manipulation of reflux, and boilup, conventional control, are prone to sensitivity problems whereas material balance controlled columns are not. In some columns sensitivities can be so high that one-way decoupling appears to be impossible to achieve. Past studies on distillation decoupling are examined in the l i t of the sensitivity results presented. One-way decoupling of conventional h m u r i t y columns is shown to be approximately equivalent to material balance control.
Introduction A number of authors have discussed the subject of decoupling distillation composition control loops (Luyben, 1970; Luyben and Vinante, 1972; Toijala and Fagervik, 1972; Wood and Berry, 1973; Schwanke et al., 1977). In all of these studies complete, two-way decoupling was attempted. In a recent article (Jafarey et al., 1978) it is shown that steady-state two-way decoupling probably cannot be achieved in distillation systems and that only one-way decoupling seems feasible. In this paper the subject of steady-state decoupler sensitivity is discussed. In any real world decoupling application, errors in decoupler gains are inevitable. Decoupler sensitivity is concerned with the practical question of how much interaction resulb from errors in decoupler gains. It is shown that even one-way steady-state decoupling may not be achievable in certain distillation columns. Initial, incisive results on decoupler sensitivity were published by Shinskey (1977a). His approach was based on the relative gain array, RGA (Bristol, 1966), and made use of a model published by Toijala and Fagervik (1972). Shinskey showed that columns controlled via manipulation 0019-7874/79/1018-0269$01 .OO/O
of reflux and boilup, conventional control, are more likely to have sensitivity problems than material balance controlled columns. After outlining the results and limitations of Shinskey’s sensitivity approach, an alternate approach is discussed. This alternate approach also shows that conventionally controlled columns will be the most sensitive to decoupler errors. By using new analytical results for the RGA (Jafarey et al., 1979), it is shown that the most sensitive conventionally controlled columns are those with high reflux ratios and/or high product purities. The new approach to decoupler sensitivity also indicates, in agreement with Shinskey (1977a), that material balance controlled columns may not have sensitivity problems. It is shown that past studies on decoupling conventionally controlled columns, both experimental and simulational, have dealt primarily with low sensitivity columns. Lastly, it is shown that one-way decoupling of high-purity, conventionally controlled columns is approximately equivalent to material balance control. Decoupler Sensitivity To discuss the concept of decoupler sensitivity the system shown in Figure 1 will be used. In the analysis 0 1979 American Chemical Society
