7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Study on the Critical Stable Region of Magnetically Fluidized Beds
Based on Voids Fluctuation Analysis
Keting Gui and Guihuan Yao
School of Energy and Environment, Southeast University
Nanjing, 210096, China, ktgui~seu. edu. cn
Keywords: Magnetically fluidized beds, Critical stable region, Voids fluctuation
Abstract
With the aid of the voids fluctuation analysis, a criterion of critical stable region of MFBs is derived in this paper. Our
criterion has two differences compared with that of Rosensweig. First, there is a region of critical stability instead of a
sharp boundary located between stable and unstable fluidization, which accords with the experimental results well.
Second, the criterion includes two additional parameters of fluidization, i.e., the particle terminal velocity u, and the
equivalent density p *, which are not considered in the criterion of Rosensweig and included in the analysis of Geldart
about fluidization.
In order to substantiate the critical stable region derived from the voids fluctuation analysis, an experiment on the
stability of MFBs is reported. It is shown that the critical stable region derived based on the voids fluctuation analysis,
corresponds well with the experimental results. The correspondence between theory and experiments indicates that the
voids fluctuation analysis is a useful approach to analyze the gassolid flows in MFBs.
between the stable fluidization and the unstable one.
However, because of the random property of gassolid
flows in MFBs, the experimental results reported by
some other authors show that it is not a sharp boundary
but a critical stable region that is located between the
stable and unstable fluidization (Cohen et al., 1991). In
this paper, therefore, we analyze voids fluctuation of
MFBs by a wavetheory approach, and derive a criterion
of the critical stable region based on the assumption of
the stable propagation of voids fluctuation in the bed.
Finally we also show that such a theoretical criterion
captures the main features of the critical stable region
that arises during our experiments.
Nomenclature
C the relative velocity of kinetic wave
Dm dimensionless group represents the ratio of
the kinetic energy and the magnetic energy
D/gs) dimensionless groups related to the X,
DA)3 dimensionless groups related to the u,
Introduction
The magnetically fluidized beds (MFBs) constitute a new
technology in the application of fluidization. As the
external magnetic fields suppress gas bubbles and
improve the contact between gas and solids in the beds,
MFBs have found many potential applications in industry,
such as in filtration (Wang et al., 2008: Meng et al.,
2003), separation (Fan et al., 2002; Hristor et al., 2007)
and synthesis reactions (Atwater et al., 2003; Graham et
al., 2006; Hao et al., 2008: Liu et al., 2009) most of
which requires that MFBs achieve stable fluidization
with the aid of magnetic fields. This type of MFBs has
been called magnetically stabilized fluidized beds (MSBs)
by some authors (Liu et al., 1991: Ganzha & Saxena,
2000). Research work on MSBs has been reported in the
literature (Rosensweig 1979; Lin & Leu, 2001; Zeng et
al., 2008), among which the most distinguished one is
that by Rosensweig. His criterion of the magnetically
stable fluidization sets up a sharp theoretical boundary
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The Void Fluctuation in MFBs
A TwoPhase M~odel ofMlFBs
In order to study the void fluctuation in MFBs, we need
to model MFBs with a voids fluctuation equation. Such
an equation can be derived from the following twophase
model of MFBs, which was developed by Gui et
al.(1997)
+(u1e) = 0
a 4.
a~l> (1
ati
particles,~ nf,, is the magneti force reule fro the
ex tera magneti fiemldandE g ise thle iacceleration owin
togra veity. Susctepritsicj)1,,3 veoiyd enote the threes
components in the direction X,EZ. The parameters of Ei,y
nf and nf,,, are calculated as follows, which are also
developed by Gui et al. (1997)
E,=+ p/Y (2)
nJ; = (1 e) (u, vI) (1 e)p,g, (3)
2Po,(1e)M? Se
n f,,, = (4)
S3+(32e)Xs 4
In equations (2)(4), p, is the gas pressure, p is the
viscosity and do is the particle diameter. b4; denotes the
Kmnocker symbol and S(() is the correcting coefficient of
drag force due to voids. Po represents the magnetic
COnductivity while gXmodels the magnetic susceptibility
of solid and M, models the solid magnetization.
Substituting (2)(4) into equation (1), we obtain
do the particle diameter
E the stress tensor of fluid
EfX(t)] average value of voids fluctuant signal X(t)
Fd dominant frequency of the autocorrelation
function of the voids fluctuant signal X(t)
f external forces acted on gas and solids phases
G,, dimensionless number related to the magnetic
energy and voids
G, dimensionless number related to the terminal
velocity of particles
g the acceleration owing to gravity
H the magnetic field intensity
M, the solid magnetization
N the number of Xi samples
1%, dimensionless group representing the ratio of
the kinetic energy and the magnetic energy.
N, dimensionless group but related to voids
nf the drag force between gas and particles
n/m the magnetic force resulted from the external
magnetic field
R,(t) the autocorrelation function ofX(t)
Uw, the continue wave velocity of voids
U, the kinetic wave velocity of voids
u,, the terminal velocity of particles
aw the relative velocity of continue wave
Wt the propagation velocity of voids
w the relative velocity of voids propagation
Xj the voids fluctuant signal
Greek letters
6;, the Kmnocker symbol
avoids
viscosityy of fluid
au the magnetic conductivity
$ the equivalent density
Sthe correcting coefficient of drag force due to
voids
asquareerror of the voids fluctuant signal X(t)
xs the magnetic susceptibility
to the frequency of voids wave
Subscripts
i, j the three components in the direction x,EZ
+u +E =0
8t r, i
Be ae 84
v,+ (1 E) = 0
8t Br Br
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Se' de' du
+U, e  0
dt r, dr
8 V,+(1so)=0
dt r, dr
Su ul B v'
p +U p, '+V 
St d r St d r
'+u =
t aU
at ar,
(u, v,) (ps pg)gl pl e s ()
do 3 +(3 1 .` By
In equations (7)(8), the terms on the right hand sides
represent the external forces acted on gas and solids in
MFBs. If we let ; and fs; denote these two external
forces in two equations respectively and subtract (8)
from (7), we obtain
(13)
(14)
sf, (15)
By taking the partial differentiation of equation (15) with
respect to r; and the partial differentiation of equations
(13) and (14) with respect to t and rj respectively, and
rearranging the terms, we obtain the following equation.
dt2 o  o r + Bt so s
8rr2E' pU,UIU p,VV _= d(sf)
+ (16)
Similarly, taking the partial differentiation of equation
(12) with respect to r;, we have
~(+ +~U E V (17)
According to the results in wave theory, the force JA
resulted from a small perturbation can be considered as
the function of the voidage e, the gas velocity u; and the
total flux j; of gas and solid phases (Wallis,1969).
Mathematically,
p, +u 
Clearly, the external force /~ is a function of ui, vi, a and
&/&; Mathematically, we have
f, = J;(u,,vr,,sS (10)
Void Fluctuant Equations in MIFBs
Consider the situation in which a small perturbation
takes place in gassolid system. The variables in equation
(5), (6) and (9) can be expressed as the sum of an
average value and a small turbulent value.
Mathematically, we write
J; = f ( j,,, U, ur)
(18)
UI + u,
e +e
o
FI + Sfi
The relationship between the total flux ji and the voidage
a can be expressed as
u, =
S=
S=
f, +
j, = eu, + (1 e)v,
(19)
Hence, following from equations (18) and (19), we have
= (U, VI) +
= e +(20)
af, af,
= (1 so )
av aj2
Substituting equation (20) into equation (17), we obtain
In the equation (11), SA represents the effect of a small
perturbation on gassolid system, which can be
calculated as
6f e+ + vJ + (12)
dE du y d(de'/Br ) dr
Substituting equation (11) into equations of (5), (6) and
(9), and ignoring all the small quantities of an order
greater than one, we obtain
p, +
St Sr
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
8(8f ) 1 af; Se de' Sf a
+(UI so( d6,( )
dr, s du, dt dr, e S,
+7Ea, )7;a: (21)
Note that in equation (21), the coefficient of the gradient
&' / & coincides with the formula to calculate the
velocity of continue wave, denoted with Uw,, by Wallis
1969, which is written as
U,, = U, e~ = U +E el (22)
Substituting equation (21) into equation (16), we obtain
the voids fluctuation equation of the MFBs as follows.
P2 e s 2e' p UpV
8~st2 o+P o)2 + o s o
Br,8rrd so s
dso BPu, t B, 8888)Br8?
In the above equation, let
P, P
P* = "+ (24)
U + ,I (25)
A = +(2)
P So u 7
where Uo defined by equation (25) can be regarded as
the average velocity of kinetic wave (wallis,1969), which
satisfies the following equation.
CICJ = Uo2Uo, A2J =
8 2a e 2a e 2
+ 2U + A
dt 2 oI r~d U x
+ t +u "~dd' Sr,
(29)
We term Equation (29) as the voids fluctuant equation
derived from the twophase model of MFBs. This partial
differential equation is not straightforward to be solved
by the analytic method. Hence we introduce and discuss
a simplification of equation (29) in the next section.
The Critical Stable Region of MFBs
The Condition of Stabilized Fluidization
In order to further discuss the stability of MFBs with the
VOids fluctuant equation, we first need to simplify
equation (29) as an analytical solution to it is hard to
obtain. Note that because the direction of main flows in
the vertically fluidized bed is perpendicular to horizontal
level, we can assume that the variables r;=rj=z, and thus
the suffix i(j) can be omitted in equation(29). With these
simplifications, equation (29) becomes
82s d29 d29 R e ds),,,
+ 2U + A + + U =(0
St2 o dzt d2 wt"dJv\v
For this simplified version of the voids fluctuant
equation, the general form of its solution is
oSat+,co(t 31
Equation (31) shows that e' is the voids fluctuation
resulted from the voids perturbation which propagates
along the direction Z. In equation (31), the symbols w, W
and a represent the frequency, the propagation velocity
and the amplifying factor of amplitude of the voids
fluctuation, respectively. The value a should be negative
to guarantee that the amplitude of voids fluctuation is not
unlimited with respect to time. The relative velocity of
voids fluctuation, w, and that of continue wave, aw, can
be defined by subtracting the average velocity of kinetic
wave Uo from the propagation velocity W and the
continue wave velocity Uw, respectively. Mathematically
(UI VI)(UJ V )
With equations (24)(27), equation (23) can then be
written as
(32)
(33)
uw = U, Uc
Substituting equation (31) into (30) and separating the
real part from the imaginary part, we obtain the
expressions of a and W2
af2
( +
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
a= 1) (34)
B2 W2 u w
rn = (3 5)
4 w2 w2 C
The square of the frequency, i.e., or, in equation (35)
should be positive. Hence, the square of the relative
velocity of voids fluctuation, tr, must be located
between the velocity of kinetic wave C" and that of
continue wave us, If uwZ > Cz, then s, > w > C. Hence,
uzAv in equation (34) is always larger than 1 and thus a
is positive. This means that the perturbation of the voids
fluctuation grows without any limitation and the state of
fluidization is unstable. If us, < CZ, then C > w> us..
Hence, uzAv in equation (34) is smaller than 1 and thus a
is negative. This implies that the disturbance of voids
decays in the propagation and the fluidized beds will
remain in a stable state.
In summary, the velocities of the kinetic wave and the
continue wave can be considered as the criterion to
determine the stability of MFBs. Specifically, if CZ > us, ,
MFBs is in stable fluidization. Otherwise, it may be in an
unstable state.
Determination of the Mlarginally Stable Zone
According to the analysis above, the condition of
stabilized fluidization can be expressed as
u = u,s;(1 o)
where u, is the terminal velocity of a particle. u, can be
calculated by the Stokes resistance equation because the
size of particles in MFBs is small. n in equation(39) is
the RichardsonZaki constant, which is in the range of
25 (Foscolo et al., 1984). Substituting equation (39) into
equation (22), the velocity of continue wave in MFBs
can be calculated as
U = UuE(n[ni so(nl+ 1)]
(40)
Another important parameter in (38) is &/fk/A The
force related to &/i2 in MFBs is the magnetic force
expressed in equation (4). So 4 / (&A) can be
calculated as
af, 2 uo (1 so)M5
(41)
d(ds/dz) 31 1+ 2(1 so )
Substituting equation (40) and (41) into (38), and
dividing U" at both sides of (38), we obtain a stability
criterion in dimensionless form
U~2 ~/$~ pU+ pV
1 2uo (1 so)M? g ,
U~p 3 +(3 2E )X soU 1pV s
C u
(42)
Substituting equation (28) and (33) into equation (36)
fields
2U Uo A U > 0
(37)
3 + (3 2Eo)Z,
D, (X ) =
2(1 so)
D, (ti) ti q _~ [So (n + 1)]
D = P "
(43)
(44)
(45)
By further substituting the formulas for U,,, Uo and A
into equation (37), we obtain an expression of the stable
condition
2 U s u )( ,U lp,V j
1 if PgU2 psV2
p' l(eh so
UI + so 6 > 0 (38)
In the fluidized beds, the following relationship exists
between the gas velocity u and the voids e (Foscolo et
al., 1984),
where Dd(g) and D,(u are dimensionless groups related
to the susceptibility parameter X,, and the terminal
velocity u, respectively. D,, is also a dimensionless group
but represents the ratio between the kinetic energy and
the magnetic energy. Then the equation (42) can be
simplified into
S+ E,E:I [I R(n + 1)j
U2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
2 P Pp
E 2p so, 1so
E 2p so 1. so
D,(,)2 um > 0
The symbol V in equation (46) represents the average
velocity of particles. Since the characteristics of particleS
or gas in different fluidized beds are not the same, the
average velocity V of particles may be anywhere
between 0 and a maximum value Vmax in an arb~itrary
fluidized bed. According to the results from the
numerical simulation of the twophase flows in MFBs,
the particle velocity Vis observed to be less than the half
of gas velocity U (Gui et al., 1997). Hence we make the
assumption that
0 20 40 610 80 100
o,
Figure 1: The
fluidized bed
marginally stable zone of magnetically
To simplify the expression of equations (48) and (49), let
Dm
G,,,=
[De(Zs)1
Gu, = De(u,) + 1
(50)
(51)
Vmx=
(47)
To simplify the stability criterion (46), let us consider
two extreme conditions of V> and V4U/2.
When V40, in the situations where p* >> p the
stability criterion becomes
D,, > [D, (Zs)] [De (u,)+ 1] (48)
When V4U/2, the stability criterion becomes
D,,, [Dcls)] De~u,] +Deu,)+ (49)
Let D,, be the abscissa and U/U,,, be the ordinate. The
critical curve C of stable fluidization when V> is
drawn in Fig.1 based on equation (48). The critical stable
curve C when V4U/2 is also drawn in Fig.1 according
to equation (49). The range on the left hand side of curve
C, represents the unstable fluidized range, and that on the
right hand side of C is the stable one. The area between
the two curves can be regarded as the critical stable
region of the MFBs. The region of critical stability
reported based on the experiments by some authors is
also shown as the shaded area in Fig. 1 (Cohen et al.,
1991). It is easy to see that the theoretical stable region
between the curves C, and C coincides with the
experimental one.
G,, in Equation (50) can be treated as the dimensionless
number related to the magnetic energy and voids.
Similarly, we regard the termGu in equation (51) as the
dimensionless number related to the terminal velocity of
particles. With these two dimensionless numbers, the
criterion of critical stable region of MFBs can be
expressed as
stable
 critical stabl>(2
unstable
Gm, > Gu,,
U f
G > G, > (G~ G,, +
2 3 U
Gm, <(G~,, G,+ ) 
4 U,,,
4U
We compare the stability criterion of MFBs derived by
Rosensweig from the basic equations of MFBs, written
as (Rosensweig, 1979),
N,,N, <1
N,,N = 1
NN,> 1
stable
critical stable
unstable
(53)
where
[1+ D,(t)]2 +
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
the magnetic field is along the axis of fluidized bed and
the stable fluidization can be obtained easily
(Rosensweig, 1979).
The voids fluctuant signals X(t) in magnetically fluidized
bed are measured using a minicapacitance probe with
the influence of the magnetic field eliminated (Gui et
al.,1994). The squareerror e' and the autocorrelation
function Rxx'(t) of the voids fluctuant signal X(t), are
calculated by following equations respectively.
SpsU
A,,, (54)
PoUf;
47r(3 2so)
Ny = [1+1+(1so)Z (55)
a, (1 so)
In equation (54), N,, is a dimensionless group
representing the ratio between the kinetic energy and the
magnetic energy. N, in equation (55) is also a
dimensionless group but related to voids.
Let us compare the criterion (53) with the one (52).
There are two differences between the two criteria. First,
criterion (53) does not lead to a region of critical stability.
Instead it only produces a sharp boundary between the
stable and unstable fluidization, which is located in the
region of critical stability shown in Fig.1 by a dotted line.
Second, two additional parameters, i.e., the particle
terminal velocity u, and the equivalent density p are
taken into account in the criterion (52). These two
parameters are important when considering the analysis
of Geldart about fluidization in general, there are two
important factors influencing the stability of fluidized
beds. One is the difference of densities between the two
phases and the other is the size of particles (Geldart,
1973). In criterion (52), the difference of the densities p,
and p, are reflected by the equivalent density p The
effects of the particle size are also captured by the
particle terminal velocity u, in criterion (52). The
dimensionless groups N,, and N, in Rosensweig's
criterion, however, only involve the density p, of the
solid particles, but fail to recognize the influence of the
gas density p, and that of the particle sizes.
Experiments
Experimental Facilities and Data Processing
In order to substantiate the critical stable region derived
from the voids fluctuation analysis, an experiment on the
stability of MFBs is reported in this paper. A schematic
diagram of the experimental facilities is shown in Fig. 2.
The bed column fabricated from Plexiglass is 100mm in
diameter and the height of the fixed bed is 115mm. The
ferromagnetic particles are made from cast iron and the
average size of the particles is 0.56 mm. To generate the
external magnetic field, a coil with 400mm in length is
used and the bed is located in the middle of the coil in
which the magnetic flux is homogenous. The direction of
S1"
(56)
1
KX'(lt) XX,_,
~N j+1
(j = 0,1,..........20) (57)
1Digital collection system; 2Capacitance meter:
3Capacitance probe: 4Fluidized beds: 5Ampere
meter; 6Direct supply: 7Coil: 8Valve: 9Orifice
flowmeter: 10Blower
Figure 2: Schematic diagram of the experimental
facilities
In equation (56) and (57), X;'s are the measured voids
fluctuant signal indexed by the subscript i~j) in the
increasing order of their values, N is the number of Xi
samples.. EfX(t)]is the average value of Xi.
Based on the stochastic analysis to the voids fluctuant
signal in MFBs, Gui et.al propose the experimental
criterion of stable fluidization in MFBs (Gui et al., 1994).
This criterion involves two characteristic parameters.
One is the dominant frequency Fd Of the autocorrelation
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
slowly with the increase in H at first, and then falls
rapidly after reaching a maximum point and finally tends
to zero. In contrast, Fig. 3 shows that a decreases
monotonically at a steady rate as H increases. This
indicates that when H increases at the early stage, the
volume of bubbles become smaller and the quantity of
bubbles increases. It implies that the magnetic field can
cause bubbles splitting. Moreover, when the magnetic
intensity H attains a certain value, for instance, to H'
considering the curve 1 (U/U,,,72.05) as shown in Fig. 4,
Fd reaches the maximum value, but a continues to fall to
a relatively small value. In other words, if we increase
the magnetic intensity further, Fd decreases steeply while
a keeps on decreasing. This indicates that the bubbles
almost disappear and the bed turns to the particulate
fluidization at the moment when the voids fluctuation
becomes very small in both amplitude and frequency.
Consequently, the extreme point in which Fd reaches the
maximum value shown in Fig. 4, corresponds to the
critical transition point from the aggregate fluidization to
the particulate fluidization. This transition point can be
used to distinguish the stable and unstable fluidization in
our experiment.
We draw the 5 transition points of Fig. 4 in Fig. 1 and
obtain 5 points located in the critical stable region
determined by our theoretical analysis based on the voids
fluctuation analysis. Such a coincidence between the
theoretical region of critical stability of MFBs and the
experimental results substantiates the validity of our
stability criterion derived in this paper.
Conclusion
In this paper, we analyze the voids fluctuation of MFBs
with the wave theory and derived the criterion of critical
stability of the bed. The criterion to determine the critical
stable region captures the effects of the parameters p, and
pg as well as that of the particle size, which are shown to
be important to the stability of gassolid fluidization
according to Geldert's analysis. We further show that the
theoretical critical stable region derived from the voids
fluctuation analysis corresponds well with our
experimental results. Such a correspondence between
theory and experiments proves that the voids fluctuation
analysis is a useful approach to analyze the gassolid
flows in MFBs.
function of the voids fluctuant signal X(t), which can be
obtained from the reciprocal of delay time where the first
peck of R,'(t) beyond null appears. The other parameter
is the squareerror CF of X(t).
Experimental Criterion of Stable Fluidization in
MIFBs
X~ I
15 4 6
H /kAXm'
U/U,,,: 12.05:21.88:31 .78 41.62:51.44
Figure 3: Relationship between $ and H
2.5 
2.0 \ 4
t 5
1.5
1.0 
0.5
0 2 4 6 8
H/kAXm'
U/U,,,: 12.05:21.88:31 .78:41.62:51.44
Figure 4: Relationship between Fd and H
Fig. 3 and Fig. 4 illustrate the mean square value $ and
the dominant frequency Fd againSt the magnetic intensity
H respectively. The five curves in each of the figures
correspond to five different gas superficial velocities.
The variation pattern of Fd aS magnetic intensity H
increases is illustrated in Fig.4. Specifically, Fd TisCS
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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Acknowledgements
The authors would like to thank the National Natural
Science Foundation of China for financial support of this
work. (50576013)
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