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BOUNDARY VALUE PROBLEMS IN ELECTROFLUIDIZED AND
MAGNETICALLY STABILIZED BEDS
A.A. Moissisa; Markus Zahna
a
High Voltage Research Laboratory Laboratory for Electromagnetic and Electronic Systems
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,
Cambridge, Massachusetts
To cite this Article Moissis, A.A. and Zahn, Markus(1988) 'BOUNDARY VALUE PROBLEMS IN ELECTROFLUIDIZED
AND MAGNETICALLY STABILIZED BEDS', Chemical Engineering Communications, 67: 1, 181 — 204
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BOUNDARY VALUE PROBLEMS IN
ELECTROFLUIDIZED AND MAGNETICALLY
STABILIZED BEDS
A.A. MOISSIS and MARKUS ZAHN
High Voltage Research Laboratory
Laboratory for Electromagnetic and Electronic Systems
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
(Received March 14, 1987; in final form August 11, 1987)
The stress balance boundary condition at the free surface of a one dimensional electrofluidized or
magnetically stabilized bed is reviewed. Conditions for zero interfacial force balance at the top of the
bed are investigated. For the case of a uniformly fluidized bed, it is shown that a nonzero particle
collision pressure is necessary for a force free bed surface. For the case where the collision pressure is
neglected, it is shown that the electrofluidized or magnetically stabilized bed must be nonuniform at
equilibrium. Stability criteria for a semi-infinite bed show that a critical field strength is necessary for
stabilization. Below this critical condition the bed is convectively unstable. The temporal and spatial
responses of a stabilized bed to small signal variations in superficial gas velocity, electrode position,
and applied potential are calculated. Specific attention is directed to sinusoidal time variations in
superficial gas velocity, where at low frequencies a standing wave character results for the fluidized
bed variables.
KEYWORDS
Electrofluidization
Magnetically stabilized beds
Fluidization
Stabilization
Boundary value problems.
INTRODUCTION
Hydrodynamic analysis has demonstrated that the state of uniform fluidization is
unstable to small signal perturbations [1]. For a gas-solid fluidized bed these
instabilities result in the formation of gas bubbles. Recent work has shown by
analysis and experiment that these bubbles can be eliminated if a suitably strong
magnetic or electric field collinear with the direction of the gas flow is applied to a
bed of highly magnetizable or polarizable particles [2-10]. Unlike magnetic
systems which only have magnetization forces, electric field systems can also have
free charge forces described by Coulomb's law. Such space charge effects cause
particle chaining resulting in loss of fluidity. To eliminate space charge effects in
the electric case, alternating high voltages must be applied at a frequency much
greater than the largest reciprocal dielectric relaxation time of the system. Since
the polarization forces on dielectric particles will act in exactly the same way as
magnetization forces on magnetizable particles, the analysis is presented here for
beds of polarizable particles but may also be applied directly to beds of
magnetizable particles if we replace the electric field E by the magnetic field H
and the bed permittivity e by the bed magnetic permeability u. This paper
181
A.A. MOISSIS AND M. ZAHN
182
presents the governing equations, general equilibrium and perturbation solutions,
and stability conditions together with a formulation of appropriate boundary
conditions for the various parameters used in a two phase model of a gas fluidized
particle bed stabilized against bubbling by an electric field.
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GOVERNING EQUATIONS
Hydrodynamics
By introducing the voidage cp(r, r), a 'smoothed' variable defined as the volume
fraction of fluid in a volume element of the fluidized bed, we write conservation
of mass for the fluid and particle phases
ata (CPPt) + V· (CPPt u ) = 0
a
at [(1 -
.
(fluid phase)
cP )Ps] + V . [(1- cP )Psv] = 0
(solid phase)
(1)
(2)
where Pt and Ps are the fluid and particle phase mass densities, while u and v are
the interstitial fluid and particle phase velocities respectively. For incompressible
phases, where Pt and Ps are constants, (1) and (2) may be added giving
v . [cpu + (1 - cP )v] = 0
(3)
Since we are concerned with the fluidization of a solid by a gas with negligible
mass density and viscosity when compared to the solid particles, we write the
conservation of momentum for gas and particles in the approximate form
-v,(cpJ}i)-p(cp)[u-v]=O (fluid phase)
Ps(1- CP) [
av
]
at + (v- V)v
_
(4)
_
= V· [(1- CP)Ts]- V· [(1- CP)P,.I]
+ (1- CP)Psg- J}VCP + CPP(CP)[u-v] + Fe (solid phase) (5)
where Pt is the fluid phase pressure, P, is the solid phase pressure, P(CP) is the
Darcy drag coefficient, which for laminar flow is assumed to depend on voidage
cP, Ts is the particle viscous stress tensor, and Fe is the body force density acting
on the particle phase due to the electric field. If we assume that each phase
behaves like a Newtonian fluid we may describe the particle viscous stress tensor
as
(6)
where lis is the effective particle shear viscosity and k, is the effective particle bulk
viscosity.
The difference between the solid phase and fluid phase pressure is termed the
collision pressure in the fluidization literature [11J,
PcCCP, E)
= P, - J}
(7)
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ELECTROFLUIDIZED BEDS
183
The collision pressure accounts for the stress transmitted to the particle phase due
to the interaction between individual particles. In a fluidized bed particles are not
in permanent contact with their neighbors, however a particle may still interact
with its neighbors by colliding with them; this interaction can contribute to the
force on particles in a non-uniform bed in which a differential volume of particles
experiences more collisions on one surface than on its opposite surface; hence the
collision pressure will depend on the voidage cjJ [11). For the case of an
electrofluidized bed the polarized particles may interact due to forces of electrical
origin without necessarily colliding; hence we will assume that the collision
pressure is not only a function of the local voidage, but also of the local electric
fleld E.
The effective particle shear viscosity /is' used in (6) will also depend on the
applied electric field. Measurements conducted by Rhee [8) using a Couette
viscometer with an applied AC radial electric field showed that for Rochelle salt
particles the effective shear viscosity /is in SI units (Kg/m . s) may be expressed by
u, = 441.4 exp[ -2. 24UsolUm! ]exp[1.22Ed
(8)
where Uso is the equilibrium superficial gas velocity, Um! is the minimum
fluidization velocity and E 1 is the normalized electric field
E, = (Ep E 2/ Ps dpg)112
(9)
while dp and Ep are the particle diameter and the permittivity respectively. The
fluid phase and solid phase momentum balance equations may be combined:
psCl- cjJ)
[aatv+ (v - V)v]= -V· [(1- cjJ)Pc]I-+ l3(cjJ)[u - v]
+ V . [(1 -
cjJ )T,)- (1 - cjJ )p,gix
+ Fe (combined) (10)
The fluid and solid phase pressures are thus eliminated in favor of the collision
pressure P:
Electric Fields and Forces
Maxwell's equations in the electroquasistatic limit with no volume charge are
v X E =O~E= -V<I>
(11)
V·D=O
(12)
D
= E(cjJ)E
(13)
where E is the curl-free electric field which can thus be represented as the
negative gradient of a scalar potential <1>, D is the displacement field, and E(cjJ) is
the effective medium macroscopic bed permittivity which is some voidage
weighted average of particle and fluid permittivities. For simplicity, we assume
that the particles are electrically linear so that the effective permittivity f( cjJ) does
not depend on field strength.
A.A. MOISSIS AND M. ZAHN
184
The principle of virtual work gives the force density and stress tensor as [6-10],
[12]
(14)
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(15)
THE STRESS BOUNDARY CONDITION FOR A ONE DIMENSIONAL
ELECTROFLUIDIZED BED
In order to simplify our problem we assume a one dimensional configuration,
illustrated in Figure 1. In this one dimensional problem all variables are assumed
to be x dependent only, with all vectors (E, v, u, g) x directed. If we make the
assumption that the macroscopic electrofluidized bed parameters are finite
everywhere, the fluid phase momentum balance Eq. (4) gives us a condition for
the fluid pressure continuity at the time varying top surface of the bed (x = L(t»,
Pr(L(t), t) = 0
(16)
where we assume that ambient pressure is zero. In a uniform bed the equilibrium
fluid pressure is found to vary linearly with position through the bed. The
combined momentum balance Eq. (10) gives us a condition of force equilibrium
at the top surface of the bed
(1 - t/> )Pc 11._ - Ps1/(l - t/» :: 11._ + Tx..,(x
= L+) - TxAx = L) = 0
(17)
where L_ is right below the top surface of the bed, while L+ is right above. The
viscous coefficient' 1/ is defined as
1/=
k s + (4/3)Jls
Ps
~
-
______
f
.....
(18)
high
voltage
Vo coswt
_ _ _ /electrode
TO)
1_/ f
electrically
grounded
distributor
f"
super ficiol gas
velocity
Us
FIGURE lOne dimensional e1ectrotluidized bed geometry.
ELECTROFLUIDIZED BEDS
185
EQUILIBRIUM FORCE BALANCE
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Nonzero Particle Collision Pressure (Uniform Fluidization)
Most previous fluidized bed analysts have assumed the particle and fluid phase
pressures to be equal, so that the collision pressure is zero. In static equilibrium
(v = 0) (17) then requires the jump in the Maxwell stress tensor across the top
surface of the electrofluidized bed to be zero.
If we use the expression for the Maxwell stress tensor given by (15) with unity
voidage outside the bed, we find that, with the assumption of a uniform steady
state voidage, the effective permittivity of the fluidized bed which gives no jump
in the Maxwell stress tensor across the interface is
EpEO
E(cfJ) = Eo(1 _ cfJ) + EpcfJ
(series)
(19)
This effective permittivity corresponds to a series model for the electrofluidized
bed [6] where the particles of permittivity Ep are totally packed without voids in
layers perpendicular to the applied electric field. Capacitance measurements have
suggested [6] that this model does not describe well the effective permittivity of
the bed. In fact these measurements seem to agree with a Lorentz Sphere Unit
Cell model for the effective permittivity,
E( cfJ) =
Eo[3E +2cfJ(Eo-E»)
p
p
3Eo + cfJ(Ep - Eo)
(Lorentz sphere)
(20)
It appears therefore that condition (17) with zero collision pressure results in an
inconsistency in the equilibrium stress balance at the top surface of the
one-dimensional bed if the series permittivity model of (19) is not appropriate. A
nonzero collision pressure appears to resolve this inconsistency by balancing the
jump in Maxwell stress tensor in (17) at the top of the bed for any other
permittivity model.
Zero Particle Collision Pressure (Nonuniform Equilibrium Fluidization)
With the particle collision pressure P; assumed negligible,
Pc=O
(21)
the condition of force equilibrium at the top surface of the bed (17) can still be
satisfied if we require the jump in Maxwell stress to be zero. This then requires
the voidage to be unity at the top of the bed,
cfJ(L(t), t)
=1
(22)
and we allow the bed to be nonuniform. For steady state conditions a/at = 0 and
v = 0, so the governing equations (1), (4), (5) for one dimensional flow with
PI = 0 reduce to
d(cfJu)/dx = 0 or u = UsN
(23)
dl}/dx
= -f3(cfJ)u
f3(cfJ)u - (1- cfJ)Psg +F., =0
(24)
(25)
A.A. MOlSSIS AND M. ZAHN
186
where Us is the superficial gas velocity. Equations (24) and (25), combined with
the expression for the force density of electric origin, leads to a differential
equation for cf> [6].
f3( cf> >Us
Psg - cf>(I- cf»
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dcf>
dx
1
2eo£o dcf>2
(26)
2[1]
2
2d
e( cf»
As a model for the interfacial drag we use the Carman-Kozeny drag law,
f3( cf> )
=
1501lt (I - cf> )2
(cf>sdp)2
cf>2
(27)
where cf>s denotes the particle sphericity and Ilt the effective fluid viscosity.
Boundary condition (22) combined with the differential Eq. (26) allows numerical
integration backwards from x = L, where cf>(x = L) = 1. For the modest electric
fields used in our experiments (y=eo£UpsgL«I) Zahn and Rhee predicted
that the voidage is essentially uniform over most of the bed volume, and then
rises to unity over a thin transition region near the top (Figure 2). As the electric
field strength increases, the width of this transition region was calculated to
increase. The values of y = 1l0M~f p.gl: corresponding to the magnetically
stabilized analysis [5] were found to be significantly higher (y =0.82) than the
values calculated for our electrofluidized bed experiment (y = 0.05) because the
magnetic force for easily achievable laboratory magnetic fields (B 1000 Gauss)
greatly exceeds the electric force for easily achievable laboratory electric fields
(breakdown limited in air to values less than 30 kV fcm). Consequently, the
thickness of the transition layer at the top of the magnetic bed is significantly
=
1.0,---------------,
y
0.8 -
=
.0
E~o
~ =0.0!5
1!50'1U'
{o.r e-
o =t¢. dp)2 p. Q = 0.;3 - - -
Xo.
6 _
-e-
Independent
I
~~.::~l~
o4 _
Lorenlz.b
'Oh'" mod., / '
.
0.2-
I
I
I
I
O:---::-'::------:"-,-------,:!.:----,L-----'
o
0.2
04
0.6
0.6
1.0
xl l
FIGURE 2 Nonuniform voidage stratification necessary to satisfy zero surface force boundary
condition of unity voidage at the top of the bed with collinear electric field and flow. Parameter Q' is a
measure of the drag force while y is a measure of the relative electric force, both compared to particle
weight. In electrofluidized bed experiments, Q' typically varied from 0.12 at minimum fluidization 10
0.3; y varied from zero to 0.05 16].
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ELECfROFLUIDIZED BEDS
187
larger than in the electric case This bed nonuniformity explains anomalous yield
stress measurements in a magnetizable fluidized bed which approaches zero at the
top bed surface, as the bed approaches unity voidage [5].
Due to this nonuniformity in voidage, the equilibrium fluid pressure distribution will no longer be linear. The pressure will vary linearly with position in the
bulk region where the voidage is essentially uniform, however in the transition
region near the top of the bed the pressure change with position will be more
abrupt. Consequently, the pressure drop across the bed will be less than the
weight per unit area of the bed because the particles at the top of the bed are
partially supported by the electric (or magnetic) force. Experimental pressure
profiles of magnetically stabilized beds did exhibit this theoretical trend [5].
GENERAL SOLUTIONS FOR PERTURBATIONS
For the electrofluidized bed, because the parameter y is so small, the bed is
essentially uniform over its entire length, except for the thin transition region at
the top. Alternatively, we may allow the collision pressure to account for stress
equilibrium at the top so that the equilibrium voidage distribution is strictly
uniform everywhere. For either reason our continuing analysis assumes an
equilibrium state of uniform fluidization where voidage cj>o is constant.
One Dimensional Problem
For the governing equations presented above, we thus assume the simplest
equilibrium solution of uniform fluidization with constant interstitial gas velocity
uoix, void age cj>u. with stationary particles (v = 0). With voidage cj> constant, c( cj»
and E are constant so that the electric force density Fe in (14) is zero. Then (24)
and (25) reduce to
dP.
(28)
-(3(cj»Usfcj> = -(1- cj>o)Psg
c:=
which tells us that the static fluid pressure in the electrofluidized bed will support
the weight of the particles. For the one-dimensional problem of Figure 1, the
electrically grounded distributor base corresponds to x = 0, L(t) is the bed height,
while the high voltage electrode at sinusoidal rms voltage Vo is placed above the
bed at x = T(t), where T> L. For such a one dimensional geometry we find that
at equilibrium the fluid pressure and electrostatic potential distributions are given
by
p. (x) =
cj>o)Psg(Lo-x) (O<x < L o)
fO
0
(L o< x < To)
(29)
[(1-
Co
VoX
cuLo + c( cj>o)[ To - L o]
<l>o(x) =
[
(O<x<L o)
coLoVo + c( cj> )[x - LoJVo (
)
Lo<x<To
coLo + c( cj>o)[To - L u]
(30)
188
A.A. MOISSIS AND M. ZAHN
In order to determine the response of the one-dimensional electrofluidized bed
system to small signal excitations we introduce perturbed (primed) variables
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I/>(x, t) = 1/>0 + I/> '(x, t)
vex, t) = v'(x, t)ix
E(x, t) = Eoix + e'(x, t)ix
<I>(x, t) = <l>o(x) + <I>'(x, t)
u(x, t) = Uoix + u'(x, t)ix
If(X, t) = PjO(x) + P;'(x, t)
D(t) = Doix + d'(t)ix
L(t) = L o + L'(t)
(31)
The perturbation in the electric displacement d' will not be a function of x due to
the fact that the electrofluidized bed is assumed to be charge free by (12). The
perturbed collision pressure of (7) may thus be expanded to linear terms about its
equilibrium value
(32)
Furthermore, from the constitutive law (13) we may relate the perturbations e'
and d'
d'(t) = e(l/>o)e'(x, t) + Eo: ; 10 I/> '(x, t)
(33)
If we then differentiate (10) with respect to x and we use (1), (2), (6) and
(11)-(15) we obtain a single linear constant coefficient differential equation in
perturbation voidage 1/>'
821/>' 181/>' Vw 81/>' V~ 821/>'
a31/>'
8t2 +~at+~ 8x -N 2 ax 2 -7Jax2at=O
(34)
where
(35)
(36)
(37)
with
2
A(1/>0) =
W-
2(
d [ e(1/>0)
1 ]
1/>0)e 1/>0) dl/>2
(38)
U,« is the superficial gas velocity at equilibrium. From (34)-(38) we see that N 2 is
a measure of the ratio of drag to electrical forces. It should be pointed out that
assuming zero collision pressure in this analysis will not alter the form of the
differential equation describing the perturbations in voidage; only N will change.
General Solution for Perturbation Voidage
The governing Eq. (34) may be written in nondimensional form
a2;P a;p a;p 1 a2;P _ a3 ;P
a? + at + ai - N 2 ai 2 - 7J ai 2 at = 0
(39)
ELECTROFLUIDIZED BEDS
189
where we define
ij = 7j/V~t",
ro = g/(c/Jouo),
L = (ro/Vw)L',
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4>=c/J'
c/Jp ,
t = rot,
i = (ro/Vw)x
t; = (ro/Vw)L o
u=~u'
ii=(I-c/Jo) v '
c/JpVw
c/Jp Vw '
- = c/Jo(1 - c/Jo)ro,
n. (3 V Z p,
'f'p 0
w
d=
P
[E
0
dE
do
I
'f' 0
n.
'Yp
(40)
]-1 d'
ci> = E(c/JO) [ Eo:; 10 c/Jp(Vw/ro)f <1>'
where c/Jp is an arbitrary normalizing constant for the voidage. Equation (39) has
a general solution
4J(i, t) = [4>1 sin kzi + cPz cos kzi]ekIXeS; + c + BeN'x
(41)
_
NZ
k - .,..-:---.:..."..,;;--::
(42)
where
1 -
and
_
kz =
and 8,
2[1 + NZijs]
I - N4
V4(1 + NZijs)Z
Nt(st + s)
1 + NZijs
(43)
t are constants independent of time and space.
Stability of Semi-infinite Bed
Previous work solved for the stability of an electrofluidized bed with and without
particle viscosity by solving for the natural frequency s. For finite length beds it
was necessary to specify boundary conditions on the voidage at the top and
bottom of the bed. Without an applied electric field, the natural frequency s was
found to have a positive real part so that any disturbance grew exponentially with
time and the system was said to be unstable. It is thought that such instability
results in bubbles. As the electric field was increased, a critical value was reached
where the real part of s was zero; for larger electric field values the real part of s
became negative, so that any disturbance decayed with time. The bed was then
said to be stabilized. The analysis which will follow in the next sections
generalizes earlier work by treating the case of a bounded bed with boundary
conditions 011 the other fluidized bed variables u, v, Pf and <1>.
Previous stability analysis on a semi-infinitely long bed has not determined
whether an unstable electrofluidized bed is convectively or absolutely unstable. In
an unstable medium a disturbance (say a pulse) may evolve in time and space in
two different ways: The disturbance may grow and propagate away from its
source, so that at a fixed point in space the disturbance will decay with time; the
instability is then a convective one. Alternatively, the disturbance may grow
encompassing more and more space, so that at a fixed point in space the
disturbance grows with time: the instability is then absolute. Work has shown that
190
A.A. MOISSIS AND M. ZAHN
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an unbounded (viscous or inviscid) bed is unstable [2,3,6,8], a bounded inviscid
bed is stable, while a bounded viscous bed is unstable [5,7]. These are general
characteristics of convective instability stabilized by boundaries or destabilized by
losses. In what follows we will show that the instability of an unbounded bed is
convective using the stability approach developed by A. Bers which is generally
applied to plasmas [13].
We assume that the voidage iP has a space and time dependence of the general
form (which is consistent with the form used by Bers in [13])
(44)
where k and ware both complex. Using this assumption in (39) we obtain the
dispersion relation for the system:
D(k, w) = w2+ j(1 + ijP)w - Uk + k 2/N2) = 0
(45)
In order for the system to be stable, all complex frequencies wei(,) corresponding
to real wavenumbers k = kr must lie in the bottom half complex w plane (Wi < 0)
so that any disturbance decays with time. Solving (45) for wei(,) we obtain
solutions of the form
w(k ) = -}(1 + ijk;)
2
r
2)]
[u 'J11 _ 4Uk(1 ++ijk;)2
k:/N
r
We define
Po=
(46)
1 + ijk;
(47)
2
so that w(kr) may be expressed as
wei(,) = -jpo ± Y-P5 + k;/N2 + jk r
(48)
We may obtain the imaginary part of W, Wi, using
1
Wi =2][51 - w·]
(49)
where w· denotes the complex conjugate of the complex frequency
using (48) in (49) we obtain
Wi = -Po
±~ [ -
[-P5 + k;/N 2] + Y[ -P5 + k;/N 2]2 +
k;T
w.
Hence,
12
(50)
For wei(,) in the bottom complex w plane we require Wi(i(,) <0, or equivalently,
Po> ± ~ [ - [-P5 + k;/N2] + Y[ -P5 + k;/N2]2 + k;
r
(51)
Now Po > 0 and the term in the square root is also real and positive. If we square
both sides of the inequality and reduce we obtain the simple condition,
N 2< (1 + ijk;)2
(52)
This is the condition necessary for the stability of a semi-infinite bed. For the
inviscid bed (ij = 0) this condition becomes:
N 2 < 1 (inviscid bed)
(53)
ELECfROFLUIDIZED BEDS
191
which agrees with earlier inviscid analysis [8]. For the case of a viscous bed
(nonzero ij) the right hand side of the inequality (52) reaches its minimum for
kr = 0; thus the condition for stability of the viscous bed for all wavenumbers will
also be
N2 < 1
(viscous bed)
(54)
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2
Therefore for N > 1 there is a range of real wavenumbers k, for which the
corresponding wi(kr ) > 0 so that any disturbance grows with time and the system
is unstable. We now investigate whether the electrofluidized bed instability is
convective or absolute.
We begin once again with the dispersion relation (45). If we now solve the
dispersion relation for k( w) we find:
]2
_ _
j
~[
j
4w( w + j)
2k(w) = jijw -1/N 2±
jfJw -1/N 2 -jijw -1/N2
The above expression gives us two separate curves for
Wi~ +00 we obtain
_
~-.
2k(Wi~ +00) = ±2j --:!-
(55)
k (w). In the limit where
(56)
t)
indicating that the two curves for k(w) lie on opposite sides of the real k axis. As
the imaginary part of w is reduced these two curves eventually approach each
other and touch at a point referred to as a pinch point [13]. The instabilities will
be absolute jf we obtain pinch points in the upper half complex w plane. In order
to obtain the pinch points of the dispersion relation D(k, w) we set
-
Dtk, w) = 0
aD\
and - ak
= 0
(57)
wconst
Conditions (57) lead to a fourth order polynomial equation for W,
4fJ2w4 + 4ij[fJj - 2/N2]W 3 + (4/N 2)[I/N2 - 2fJilw2
+ [4j/N 4-jfJ +2fJ]w -1/N 2=0 (58)
For the inviscid case (fJ = 0) this simplifies to
w 2 + j w - N 2/4=0
(59)
which has solutions for W,
_
j
2
1,~
w = - - ± - v lV - - 1
2
(60)
For an unstable inviscid system (N 2 ) 1) the instabilities are thus convective, since
no pinch points exist in the upper half w plane.
We have thus determined that a semi-infinite electrofluidized bed will be stable
2
for N < 1. The instabilities of an inviscid fluidized bed were found to be
convective. To determine whether the instabilities of a viscous bed are convective
or absolute, one would have to solve for the four roots (pinch points) of w in
2
(58). If (for N > 1) all pinch points are found in the lower half w plane (Wi < 0),
the instability is convective. Any pinch points in the upper half w plane (Wi> 0)
would represent an absolute instability.
192
A.A. MOISSIS AND M. ZAHN
General Solutions for Other Electrofluidized Bed Parameters
Given the solution for the perturbation in voidage ;p of (41) we may then express
all other variables, namely the interstitial velocities, pressure and electrostatic
potential, in terms of ;Po Thus, from the mass conservation equation for the fluid
phase (1), we may express the interstitial gas velocity ii in terms of the voidage ;po
ail
ax
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-=
alP (1'_.
alP
---=-at
(61)
ax'
Furthermore, from the mass conservation equation for the solid phase (2), we
may also express the interstitial particle velocity ii in terms of the voidage ;p:
aii alP
ax = ai
(62)
In order to find an expression for the fluid phase pressure distribution in terms of
the voidage, we look at the normalized momentum balance equation for the fluid
phase, which is derived from (4),
~
ax + (1 - I/>o)il - I/>oii + [(1'(1 - 21/>0) - 1]1/>- = 0
If we differentiate w.r.t.
of
;p we obtain
x and we use (61) and (62) to eliminate il and
;j2p
alP
alP
~ = ai - [(1'1/>0 + 1] ax
(63)
ii in favor
(64)
Finally, in order to express the electric potential in terms of the voidage, we use
(33) expressed in the form
aet>'
-
ax
=
Eo: ; 10 I/>'(x, t) - d'(t)
-e'(x, t)
e( 1/>0)
(33)
In normalized terms this takes the form,
ali>
-
ax = I/> -
-
d
(65)
The non-dimensional perturbation displacement field d, defined in (40), depends
on voidage as well as, on perturbations in applied voltage and in electrode
position.
Using the general voidage solution of (41) in (61), (62), (64), and (65) we
obtain general solutions for the electrofluidized bed parameters. By integrating
(61) we obtain an expression for the perturbation in the interstitial fluid velocity
ELECTROFLUIDIZED BEDS
193
where a(i) is a time-dependent constant in space which comes about from the
integration in i. In a similar manner, from (62) we find an expression for the
perturbation in the mean particle velocity
_
_
se ii
_...
_
1. 2 [(k,4>, + k 24>2)sin k2i + (k,4>z - kz4>,)cos k2i]e k ,X (67)
A
v(i, t) = vet) +P
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1+
_
_...
_...
_
_
2
Here again v(i) is a constant in space that results from the integration in i.
From (64) we find that the perturbation in fluid pressure takes the form,
__ [[s[cPI(kf-kD+2k,k2cP2]
Pt(x, t)=
kf+q
-.
_.]
+ (1 + «4>o)(kl4>1 + k 24>2)
-
sin k 2x +
- (1 + «4>0)(k 2cPI - I., cP2) ]cos k 2i]
[S[cPzCkT- kD-2k,k 2cP d
k- 2 -2
1 + kz
~k:e;~
+ xp,(i) + P2(i) + (1 + «4>0)(B/N 2)e
N 2X
(68)
where PI(i), P2(i) are constants of the integrations in space. The constants of
integration u(i), v(i), p,(i) and P2(i) are in fact related. If we use (41) and
(66)-(68) in the fluid phase momentum balance equation (4) and the combined
momentum balance equation (10) we obtain compatibility equations
p,(i) + (1 - 4>o)a(i) - 4>ov(i) + [«(1- 24>0) -1]C = 0
dv(t)
rz::
(1 - 4>o)u(t) - 4>ov(t) + [«(1 - 4>0) -
-
I]C
(69)
(70)
Finally from (65) we find that the perturbations in electrostatic potential will take
the form,
«(>(x, i) = ei'[[(cP,k, + cP zk2)sin k 2x
.. _
.. _
_
ek 1X
+ (4)2 k, - 4>,k 2)cos kzx] -2 -2
k l +kz
2)e N 2X
- d(i)i + (jj / N
+ Cx
(71)
Boundary/Jump Conditions
In order to solve for the perturbations in voidage, fluid and solid phase velocities
as well as fluid pressure and electrostatic potential we also need to introduce
boundary/jump conditions. Using (3) we may relate the particle and fluid
velocities to the superficial gas velocity
4>u + (1- 4»v = Us
(72)
in terms of (normalized) perturbations this gives
as(t) =
«C + u(i) + v(i)
(73)
194
A.A. MOISSIS AND M. ZAHN
We assume that the particles at the bottom of the bed are constrained by the
ground electrode to remain stationary
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v(x = 0, t) = v'(x = 0, t) =
°
(74)
Continuity of the electric displacement vector D and of the electric potential 4>
across the time varying top surface of the bed (by (11)-(12» allows us to relate
the perturbation in the electric displacement to the perturbation in bed height L'
as well as to the perturbations in the amplitude of the applied voltage V' and the
position of the top electrode T' [6]
d'(t)
= EoR(cf>o)L'(t) - GMo)V'(t) - E OG2(cf>0)T'(t)
(75)
where for the specific geometry of Figure 1 we obtain
1 - cf>o
] [E(r/lO)[E(cf>O) - Eo] + Eo dE I ]
EoL o + E(cf>o)[7o - L o]
1 - cf>o
dcf> 0
EoE(cf>o)
E(cf»
G1(cf>0) = EuL u + e(cf>u)[T _ L ;
G2(cf» = ~ G1(cf»
o
o]
R(cf>o) = [
(76)
Finally the stress balance requirements at the top surface of the bed were given
by (16) and (17).
RESPONSES TO SMALL SIGNAL EXCITATIONS
Nonzero Collision Pressure
We now focus on the problem of a generalized small signal excitation in the
superficial gas velocity, or in the applied voltage, or in the position of the top
electrode (with particular emphasis on perturbations in superficial gas velocity).
The excitations in superficial gas velocity, voltage and top electrode position are
given in the general form:
usCi) = UsDC + Us exp(si)
v(i) = vDC + if exp(si)
(77)
t(i) = t DC + t exp(si)
where s is the driving frequency (s = jw for sinusoidal excitation) while UsDC'
VDC , t DC correspond to the DC components of the excitations. From the analysis
presented thus far the following equations may be derived:
The boundary condition of zero mean particle velocity at the bottom of the bed
(74) applied to (67) gives:
v(i)
=
(78)
The boundary conditions for the electromagnetic fields give us the condition
(75) which takes the nondimensional form
(79)
ELECfROFLUIDIZED BEDS
195
where
(80)
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(81)
From the boundary condition of zero fluid pressure at the top of the bed (16)
we obtain using (68),
cA
3
pl + C.Ap2
+ Lopli) + P2(i) + Coll = CsL(i)
(82)
where
(83)
(84)
(85)
- - - JAJ(L
- - , t)
- - ii -afbl
J2d(t)
o
at
with
=
i.
(86)
0
s, = _1_[(1 _ cjJo) aPe I + aTx x I _aTu I ]
e( cjJo)
de
J2 = Eo dcjJ
aE
0
I0 (Jd p, V 2w);
aE Lo+
aE
(87)
Lo-
(88)
Bed particle mass conservation implies that the integral of (1 - cjJ) over the entire
(time varying) length of the bed is a constant. This leads to the condition [6],
L(i) = J.e'ifb! + Jse,ifb2 + J6 (; + 8 1 B
(89)
(90)
(91)
A.A. MOISSIS AND M. ZAHN
196
Finally, the combined mass conservation equation of (3) gave us (73), while
were expressed by
compatibility relations between f>tCi), u(i), v(i), and
(69)-(70). We may express the steady state values of these variables in response
to the excitations of (77) in terms of a DC and a time varying term at the
frequency of excitation s
t
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Ii (i) = Uoc + Ct exp(Si);
L(i)
= L oc + L exp(si);
PI(i)
= PIDC + PI exp(si);
+ fl exp(si)
d(i) = doc + exp(si)
pz(i) =P2DC + pz exp(si)
v(i) = voc
a
(92)
In particular we will focus on the case of an excitation in the superficial gas
velocity with zero change in the applied voltage (V' = 0) or the top electrode
position (T' = 0). Using (92) in (69)-(70), (73), (78), (82), (86), and (89) we
obtain a system of eight equations for the eight DC terms, voc, UOC' PIDC, PZOC'
t, E, doc, and L oc and a system of eight equations for the eight coefficients of
the time varying steady state terms, Ct, fl, PI' Pz, cP" cPz,
and L. In order to
solve these systems we need to know the dependence of the collision pressure P;
on the electric field E and on the voidage </>. Using (75) we conclude that at
equilibrium
a,
P (A.
c 'f'O>
E) = _ E~ [e( </>0)[e(</>0) - eo]
0
2eo
1 - </>0
+
de I ]
eo d</> 0
(93)
Given the partial derivatives of the collision pressure Pc with respect to the
void age and the electric field, which enter in (32) and hence in (37), (17), (88)
and also (86), we would thus be able to solve for the first order perturbations in
Pc and therefore to determine the response of the system. Since we do not know
the exact dependence of the collision pressure Pc ( </>, E) to the voidage and the
electric field, we cannot proceed until the dependence of Pc on </> and E is
established. It should be pointed out that in order to find Pc (</>, E) in general it is
incorrect to simply replace </>0 by </> and Eo by E in (93). Just because functions
are equal in equilibrium, does not imply that they will be equal for perturbations
about this equilibrium. The solutions to the system of eight equations for the DC
terms suggest an additional constraint on the collision pressure. Solving the
system of equations consisting of the DC parts of (69)-(70), (73), (78)-(79), (82),
(86), (89) we find
c = (1 -
</>o)usoc
R 1lz16 - 13
_
B = 1 exp[NZLo]- R l lzBI (1 - </>o)usoc
UDC = [1 - a(1 - </>O)]UsDC
L oc = 16 (1 - </>o)usoc + BI B
dDC = R,1 6 (1 ~ </>o)usoc + R,B,E
VDC
=0
(94)
3
P IDC
= a</>o(1 -
</>o)usoc
P2DC = [GsBI - G6 ]E
(95)
We therefore find that the DC components of the responses to an excitation in
superficial gas velocity are proportional to the excitation UsDC. As should be
expected, the mean particle velocity of the response is found to have a DC
ELECfROFLUIDIZED BEDS
197
component of zero. These solutions would correspond to the steady state part of
the response of a stable electrofluidized bed system to a step in superficial gas
velocity of amplitude UsDC' For such a step excitation in superficial gas velocity
we would find from (41) in the limit t~oo
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cP(i, i~ (0) = iJe N2x+ C
(96)
which implies that the new steady state void age distribution could be nonuniform.
Experimental observations, as well as previous fluidized bed analysis suggest that
the steady state voidage inside an electrofluidized bed is uniform. This implies
that
iJ = 0
(97)
For the case of a conventional fluidized bed where the electric field is zero, the
need for making any such assumption did not arise. For zero electric field, the
coupling constant N 2 goes to infinity and consequently the steady state solution of
the differential equation for the perturbed voidage in (41) gives us uniform
voidage without any assumptions necessary. Setting B equal to zero is also
consistent with our original assumption that at equilibrium the voidage is
uniform. From equation (94) we see that in order for iJ to be zero,
R )J2J6 - J3 = 0
(98)
which is a constraining relation for the derivatives of the collision pressure, since
these enter in the expressions for J2 and J3 • If we now use the expressions for J2
and J3 given by (88) with N 2 given by (36)-(38) we obtain
(
QI <Po, Ell) 8~1
8E 0 + (1- <Po) 8~1
8<p 0 = Qi<Po, Eo)
(99)
where QI(<PO, Eo) and Q2(t/JO, Eo) are generally nonzero and depend on the
equilibrium voidage and electric field
(100)
(101)
We therefore find that the assumption of a uniform steady state electrofluidized
bed void age imposes constraints on the dependence of the collision pressure on
the void age and the electric fleld.
Zero Collision Pressure
Even though we cannot proceed in the nonzero collision pressure analysis any
further at this time, it is still interesting to examine the consequenses of a zero
collision pressure analysis that neglects the inconsistencies in equilibrium stress
balance. Several analysts of fluidized and electrofluidized beds have in fact
198
A.A. MOISSIS AND M. ZAHN
neglected this stress balance inconsistency in their analysis. A zero collision
pressure analysis, although incomplete, does help us to understand the frequency
dependence of the spatial distributions of the various parameters used in a
macroscopic model of an electrofluidized bed. It also gives us some insight to the
effects of varying the applied voltage, effective viscosity, as well as electric field.
We thus assume in this section that
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(102)
with N given by (37). It should be pointed out that if we were to use the series
model (19) for the effective permittivity of the electrofluidized bed, we would find
no electromechanical coupling, since in that case A (<I» in (38) is zero and
therefore N is infinite, which is equivalent to setting the applied electric field to
zero. Such a bed would not be stabilized against bubbling.
With Pc = 0 (and with a model other than the series model used for the
permittivity), instead of (17) we use an arbitrary boundary condition that sets the
perturbation in voidage to be zero at the top of the bed,
<I>'(L, t)
=0
(103)
We may thus solve the system (69), (70), (73), (78), (79), (82), (89), (103) for the
coefficients of the time varying terms 11, fI, PI, Pz,
L, <PI and <Pz. We may then
use these in (41), (49), (51), (52), and (55) in order to obtain expressions for the
time and space dependencies of <P(i, t), u(i, t), ii(i, t), P(i, t), and <i>(i, t).
These distributions are plotted for various frequencies in Figures 3-7 for a
applied voltage Vo = 5 kV, equilibrium bed height L o = 2 em, with top electrode
position To = 2.4 cm and with the ratio R = Uso/ Urn! = 1.2. The equilibrium
interstitial gas velocity is Uo = 0.45 m/s while the equilibrium pressure at the
a,
f 00.0 I Hz
f 00.003 Hz
~
~
"6-
z-:
"6-
0'::-
o
-"
x/Lo
<p~a.
- - = 0.23,
u~/Uso
La
0
2cm
FIGURE 3 Spatial distribution of the perturbation in voidage in response to a 2% sinusoidal
excitation in superficial gas velocity about its equilibrium value.
ELECTROFLUIDIZED BEDS
199
f'O.OIHz
f ' 0.03Hz
~
"
_E
::l
.'
::l
OLf,0.03Hz
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-'
o
<flo
f ' 0 I Hz
IV-o L-
---'
OL-
-'
o
o
u :-nax
u~ / Usa
0.45mfs, lo'2cm
FIGURE 4 Spatial distribution of the perturbation in interstitial gas velocity in response to a 2%
sinusoidal excitation in superficial gas velocity about its equilibrium value.
bottom of the bed is 196 Pa. The effective bed viscosity used was given by the
empirical formula (8) with Ep = 100, Ps = 1.79 X 103 Kg/rn", dp = 550 Jim for
Rochelle salt. The Lorentz sphere model was used to describe the effective bed
permittivity. The frequency response of the amplitude of the expansion in bed
length was found to have a low pass characteristic whereby the bed height change
becomes very small above a cutoff frequency which depends on applied electric
field and effective viscosity [9].
f '0.003 Hz
f '0.01 Hz
~
"
• E
,>
>
O~~===J
o
x/La
o ~===:::::=:J
o
v'
~ '0.00185, lo ' 2cm
us/ Usa
FIGURE 5 Spatial distribution of the perturbation in particle velocity in response to a 2% sinusoidal
excitation in superficial gas velocity about its equilibrium value.
200
A.A. MOISSIS AND M. ZAHN
f· 0.03Hz
f • 0.01 Hz
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f • 0.03Hz
0':-
-"
o
p'
f ma x
Us I u~o
195 Pnscols , La ' 2cm
FIGURE 6 Spatial distribution of the perturbation in fluid pressure in response to a 2% sinusoidal
excitation in superficial gas velocity about its equilibrium value.
The results presented above were obtained with the assumption that the
electrofluidized bed is stable so that a sinusoidal steady state can be achieved.
This requires that Eo exceed the critical value for stabilization. Determination of
this critical value was presented earlier for a bed of semi-infinite extent and
remains for near future work for a finite length bed.
6
"".01 Hz
K
C
- E
~
-0&
FIGURE 7 Spatial distribution of the perturbation in electric potential in response to a 2%
sinusoidal excitation in superficial gas velocity about its equilibrium value.
ELECfROFLUIDIZED BEDS
201
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CONCLUSIONS
The introduction of a collision pressure term in the governing equations for the
stabilized electrofluidized bed appears to resolve past inconsistencies in the force
balance at the top surface of a uniform fluidized bed. Given the dependence of
the collision pressure on the voidage and on the electric field, we would be able to
determine the response of a one dimensional fluidized bed stabilized against
bubbling by an electric field to a sinusoidal small signal excitation in the
superficial gas velocity. This dependence of P; on the voidage and electric field
has not been established at this point.
A simplified zero collision pressure analysis that neglects the inconsistency in
the stress balance suggested that the response of the bed voidage to a sinusoidal
small signal excitation in superficial gas velocity has a standing wave character,
with the apparent wavelength increasing with the frequency of excitation. For
frequencies above a certain cutoff, diffusive effects appeared to dominate the
response; consequently, for high frequencies the voidage perturbation decreased
with frequency and the standing wave pattern was lost. The frequency response of
the amplitude of the expansion in bed length was found to have a low pass
characteristic.
NOMENCLATURE
Variables subscripted with zero (e.g. <Po, Eo, uo, ... as in Eq. (31» indicate
equilibrium variables.
Variables superscripted with a prime (e.g. <p', v', e', ... as in Eq. (31» indicate
perturbation variables.
Variables superscripted with a tilde (e.g, 4>, of, t, p, ... as in Eqs. (39)-(40»
indicate nondimensional variables.
Variables subscripted with DC (e.g. usoc, Vo c ,
as in Eq. (77» indicate
DC components of the perturbation variables.
Variables with superscript carets (e.g. Us> V, t, ... as in Eq. (77» indicate
complex amplitudes of perturbations.
Variables with subscript max (e.g. <P;"ax> u;".., v;".x> p/ m .,,, <1>;" ax in Figs. (3)-(7»
indicate the maximum value of the perturbation.
t-: ...
A( <Po)
B)
jj,
C
d'
dp
D(k, w)
D
= e(<p)E
voidage, permittivity parameter defined in Eq. (38)
parameter defined in Eq. (91)
integration constants for perturbation voidage solutions introduced in Eq. (41)
perturbation displacement field defined in Eq. (31)
particle diameter
dispersion relation defined in Eq. (45)
displacement field defined in Eq. (13)
202
e'
E
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Eo
E, = [Ep E 2/ (Ps dpg)]'12
g
G t ( CPo), G2( CPo)
GJC CPo). G4 ( CPo),
Gs( CPo), G6 ( CPo)
j
A.A. MOISSIS AND M. ZAHN
perturbation electric field defined in Eq. (31)
electric field defined in Eq. (11)
rms electric field due to applied voltage
normalized electric field that affects particle viscosity
defined in Eq. (9)
electric force density on particles defined in Eq. (14)
acceleration of gravity
parameters defined in Eq. (76)
parameters defined in Eqs. (83)-(85)
identity tensor, unit dyadic
j
yC1
i.. J2 • J3
parameters defined in Eqs. (86-88)
parameters defined in Eqs. (90-91)
nondimensional wavenumber parameters defined in
Eqs. (42)-(43)
real part of the non-dimensional wavenumber
effective particle bulk viscosity introduced in Eq. (6)
bed length
magnetization of magnetically saturated particle
non-dimensional electric field and collision pressure
parameter defined in Eq, (37)
collision pressure defined in Eq. (7)
perturbation fluid pressure
parameter defined in Eq. (47)
fluid pressure
solid pressure
parameters defined in Eqs. (99)-(101)
J4 , Js • J6
kt , k2
Pc=P.-Pt
PI
Po
PI
r,
Qt(CPo, Eo).
Q2(CPO, Eo)
ro = g I( CPouo)
R( CPo)
R I( CPo), R 2( CPo),
R3 ( CPo)
s
normalizing time parameter defined in Eq. (40)
parameter defined in Eq. (76)
parameters defined in Eqs. (80)-(81)
nondimensional complex frequency for perturbation
variables introduced in Eq. (41)
time
Maxwell stress tensor components defined in Eq. (15)
solid's viscous stress tensor defined in Eq. (6)
ELECTROFLUIDIZED BEDS
height of electrode
distributor
T(I)
above
203
electrically
grounded
Maxwell stress tensor component of force perpendicular to top surface of bed used in Eq. (17)
interstitial fluid velocity
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minimum fluidization velocity
superficial gas velocity
particle velocity
applied voltage amplitude
velocity parameter defined in Eq. (36)
spatial coordinate
x
non-dimensional flow parameter defined in Fig. (2)
defined in Eq. (61)
Darcy drag coefficient used in Eq. (4) and given for
Carmau-Kozcny law in Eq. (27)
y
{
e oE5!(psgL )
lloM~/(Psg L)
= nondimensional electric or magnetic force
parameter used in Fig. (2)
o i. »i. = Kronecker delta
{1 1=
J
permittivity of free space
particle permittivity
effective medium permittivity given for series model in
Eq. (19) and Lorentz sphere model in Eq. (20)
e( cf»
IJ = k,
+ (4/3)lls
viscous coefficient defined in Eq. (18)
V
del vector operator
Ilo
III
magnetic permeability of free space
effective fluid viscosity in Carman-Kozeny drag law of
Eq. (27)
Ils
effective particle shear viscosity in Eq. (6)
.. = cf>ouo/g
cf>
= Uso/ g
time constant defined in Eq. (35)
voidage
tP,. tPz
nondimensional perturbation voidage amplitudes defined in Eq. (41)
arbitrary normalizing constant for void age defined in
Eq. (40)
particle sphericity used in Carrnan-Kozeny law of Eq.
(27)
PI
fluid mass density
204
p,
<I>
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w
A.A. MOISSIS AND M. ZAHN
solid mass density
scalar electric potential defined in Eq. (11)
angular frequency of applied voltage in Fig. (1)
non-dimensional complex frequency for perturbations
with real part W, and imaginary part Wi
complex conjugate of W
REFERENCES
I. Anderson, T.B., and Jackson, R., "A Fluid Mechanical Description of Fluidized Beds-Stability
of the State of Uniform Fluidization," Ind. Eng. Chern. Fundamentals, 6,527 (1967).
2. Rosensweig, R.E., "Fluidization: Hydrodynamic Stabilization with a Magnetic Field," Science,
204,57-60, Apr. 6 (1979).
3. Rosensweig, R.E., "Magnetic Stabilization of the State of Uniform Fluidization," Ind. Eng.
Chern. Fundamentals, 18,260-269, Aug. (1980).
4. Rosensweig, R.E., Ferrohydrodynamics, Chap. 9, Cambridge Univ. Press, Cambridge (1985).
5. Rosensweig, R.E., Zahn, M., Lee, W.K., and Hagan, P.S., "Theory and Experiments in the
Mechanics of Magnetically Stabilized Fluidized Solids," Theory of Dispersed Multiphase Flow, R.
Meyer Ed., Academic Press (1983).
6. Zahn, M., and Rhee, S.W., "Electric Field Effects on the Equilibrium and Small Signal
Stabilization of Electrotluidized Beds," IEEE Trans. Ind. Appl., IA·ZO, 137-147, Jan./Feb.
(1984).
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