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Levitation of discrete particles in oscillating liquids
Chemical Engineering Science, Vol. 45, No. 2, pp. 483 4W, Printedin Great Britain.
LEVITATION
1990.
ooo!-2x9/90 s3.00 +o.oo 0 1990 Pergamoo Press plc
OF DISCRETE PARTICLES LIQUIDS
YONG DENG Institute of Chemical Metallurgy, Chinese (Received 25 July
and MOOSON Academy
KWAUK’
of Sciences,
1988; acceptedfir
IN OSCILLATING
People’s Republic of China
Beijing,
publication
19 May
1989)
Abstract-Wet ion-exchange resins (pp= 1.17 g/cm3, d=0.0435, 0.0401, 0.0334 or 0.0290 cm) were suspended, while vibrating about a fixed position, by either symmetrically or asymmetrically oscillating water. By taking into account convection acceleration, a proposed mathematical model [eq. (15)], was found to predict adequately the levitation frequency. It showed also that the effect of pressure gradient was more significant than drag force at causing a particle to levitate. Glass beads were also levitated in a 55% aqueous
glycerol.
INTRODUCTION
With the development of fluidization classical Ruidized beds are assuming
technology, nongreater import-
ance. In a jig-fluidized bed, Guo et al. (1984) deoxidized iron ore directly with soIid coal, thus making gas-mixing, gas-circulating and gas-heating unnecessary, while product (iron metal) separation can be realized continuously along with deoxidization. Theoretical and experimental investigations have shown that vertical oscillation of liquid might retard the motion of particles or bubbles or even make them move in a direction opposed to gravity. Many researchers, such as Al-taweel and Carley (1971a, b), Feinman (1966), Herringe and Flint (1974), Houghton (1963), Jameson and Davidson (1966), Schijneborn (1975) and Tunstall and Houghton (1968) have long been interested in this phenomenon, since it would have a profound effect on the hold-up and heat or mass transfer characteristics of such a two-phase system. Levitation was defined by Houghton (1966) as a stable condition in which a particle responds to the oscillating fluid in such a way that the influence of finite buoyancy forces is completely neutralized so that the particle oscillates about a fixed position. Feinman (1966) conducted an experimental study on levitation under sinusoidal oscillations and found that there existed two types of levitation, viz. nonrising levitation and rising levitation, as predicted theoretically earlier by Houghton (1963). Feinman’s experimental data were later correlated by Krantz et al. (1973) to yield the following semi-empirical equa-
Liang and Kwauk (198 1) found that particles denser than the surrounding fluid would move upwards when there existed an air plug between the water column and the piston generated sinusoidal oscillation. They believed that compression of this plug tended to make the water oscillation asymmetric. SchGneborn (1975) studied the effect of vortex shedding on the motion of a particle in an oscillating liquid. He believed that levitation would be impossible in the case of sinusoidal oscillations. Bailey (1974) analysed the motion of a particle in a rapidly oscillating flow and obtained a similar conclusion. The object of the present work was to identify the conditions under which levitation could occur and to propose a mathematical model capable of describing this phenomenon. It is hoped this would not only form a basis for the intelligent design of jig-fluidized bed such as used by Guo, but also may lead to the development of new uses of this type of fluidized reactor. THEORETICAL
tion:
(a + C,)‘a’d GIQ-
X
= o.370
‘Author
period was longer would levitate.
to whom correspondence
than
J3
dr’ + F,.
(2)
The term on the left-hand
side of the equation is the inertial force. The first term on the right is the drag force according to Stokes’ law. The second term is due
Van Oeveren and Houghton (1971) studied the case of asymmetric oscillations and found that when the time for the downstroke for the upstroke particles
s
’ d/+/dt’ - du,/dt’
0
(1)
llg
ANALYSIS
In 1947, Tchen proposed the following equation to describe a particle diffusing in unsteady turbulence:
that
to the pressure gradient in the fluid surrounding the particle caused by fluid acceleration. The third term is, the force accelerating the apparent mass of the particle. The fourth term is the Basset term, which takes
should be addressed. 483
YONG DENG and MOOSON KWAUK
484
account of the effect of the deviation of the flow pattern from steady state. The last term is the external force. With their experimental results, At-taweel and Carley (1971 b) reached the following form of Basset term in a sinusoidally oscillating flow field:
where CD follows eq. (4), and the coefficients C, and C, were given by Odar (1966): c,
= 1.05 -
C, = 2.88 +
0.066
(7)
AC2 + 0.12 3.12 (AC + 1)3
in which AC is an instantaneous acceleration -KK,Jnw/2(sinot-coswt) - K, Jnw/Z
(sin cot + cos cot)].
Substitution into eq. (2) yields, however, a linear equation, indicating that eq. (2) is incapable of accounting for the retarding effect of oscillation on the free fall velocity of particles. Thus the term of the drag force- needs to be modified. If the frequency of oscillation is small, the flow field may be considered as an otherwise stationary one in which a particle falls at the speed (uI - up). Therefore, the drag force can be expressed with Newton’s equation with drag efficient CD possessing the following values: z(l
+0.15Re0.687)
(C,>O.44)
(4)
(the rest)
0.44
AC=
(3)
and the Basset term can be neglected. Equation (2) thus becomes the so-called quasi-steady state model used by many authors, such as Boyadzhiev (1973):
(UP - Ur)* dldu,/dt - du,/dtl-
-fWP,-
Pf)S
in which the drag coefficient Cd was assumed to be a constant, 0.3. Tentative calculations with the above three models, eqs (5), (6) and (lo), showed that particles would not be able to levitate if the oscillation frequency was less than 50 Hz under the present experimental conditions. Corrsin and Lumley (1956) pointed out that if the fluid velocity is dependent on both time and space, the effect of convection acceleration should be included, and the pressure gradient in turbulent flow becomes:
DU
au+(u)&“V%
-PPr
at
[
(5)
According to Schiineborn (1975), this quasisteady state model can predict the average fall velocity of particles only for small values of frequency relative to the natural vortex shedding frequency of the particle. By substituting the unsteady drag given by Odar (1966) in eq. (2), certain authors, such as Schiineborn (1975) and Herringe (1976), derived another model to describe the retarding effect of the oscillating liquid on the particle:
;d’pp%= -CD;d2pflup-U&J,-uf)
+~d’p$+C*;d’pf($f$J
s 0
JZ~
dt’ (6)
(11)
(1) the fluid flow is approximately steady state; (2) the effect of the apparent mass of the particle may be neglected; (3) only the effect of one dimensional convection acceleration is taken into account, and the pressure gradient becomes:
ax
’ du,jdt’-du,/dt’
1
J
Thus the term of pressure gradient, the second on the right of eq. (2), should be modified, especially in the case for which the density of the fluid is comparable to that of the particle, and the value A/d is large. To simplify the problem in order to formulate a practical model to predict the levitation frequency of the particle, we assume that:
dP _=_
X
(9)
Schiineborn (1975) pointed out, however, that the validity of eq. (6) for predicting the average fall velocity is also limited. Another model was proposed later by Liang and Kwauk (t981):
-Pfz=
-~dvPP-_pfh?.
number:
du
"
(
au
-+%KVax2 at
azu
>.
(12)
In the case of small frequency and large A/d value, the first assumption is reasonable. Thus we can neglect the Basset term and calculate the drag force with Newton’s equation. Hjelmfelt and Mockros (1966) pointed out that, if the oscillation frequency was not sufficiently small, the
Levitation
of discrete
particles in oscillating
error caused by neglecting the effect of the apparent mass would be smaller than that caused by neglecting the Basset term. Since, in addition, the coefficient of the apparent mass cannot be accurately determined, we chose to neglect that effect. Further, in a steady state flow field, if the particle is immobile, the fluid velocity may be expressed in a variable-separated form, which, at y = 0 and z= 0, can be written as: u(t, x) = u,(t).
u,(x).
And if the particle is free to move, velocity may be expressed as: u(t, x) = Car(r) Substitution
(13) then the fluid
%(f)l u,(x) + up(t).
485
liquids
Crank and rocker mechanism
Water tank 4 70 +
Plastic tube
I.D. 19
-= 330
U - shaped Plexiglas tube \
(14)
1
into eq. (12) yields:
Levitation d/section Copper wire net
_ __
Water int ak& Fig. 1. A diagram
- y(uz - UP)
(PP - Pr)9(13
Because u is the fluid velocity near the pa .rticle, eq. (15) becomes an ordinary differential equation in terms of up and t so long as we know the values of u,, du,/dx and d2u,/dx2 at x=0. Therefore, eq. (15) can be accepted as the final model involving three parameters, u,, du,/dx and d2u,/dx2.
Fig. 2. Principle
device.
adjustment of both the amplitude A and the asymmetry factor k, (called by Van Overen and Houghton “duration factor”) of oscillation, as described below. In Fig. 2, the length of crank a, the length of rocker b and the eccentric distance e, are suitably designed to be easily adjustable. When the motor rotates clockwise uniformly, the time for the hinge E travelling from E 2 (corresponding to the lowest position of piston) to E 1 (corresponding to the highest position of piston) is longer than that back from E 1 to E 2 because of the eccentricity e. And if e = 0, the oscillation will be symmetric. The following can be easily derived: X+4
EXPERIMENTAL
The present experiments were carried out in a Ushaped Plexiglas tube with fluid oscillation generated by a mechanical device, as shown in Fig. 1. In this device, a crank and rocker mechanism converts the rotation of a motor to vertical oscillations of the piston which drives the liquid in the U-shaped tube. The rotational speed of the motor is controlled by a ZTD-5A silicon controlled governer, so as to adjust the oscillation frequency. The crank and rocker mechanism permits easy
of the experimental
k, = --_
- arccose
#J = 2uccos&
b-u
(16) (17)
ur = uwsin cot du, dt=
of the oscillation
-
ufd
aw cos ot(u
[b2-(usinwt-e2)2]‘/2
cos cot +
u2 b2a2
uw2 sin wt (a sin cot -e) [b2-(usinwt-e)2]“2 cos2 ut
-[b2-(asinwt-e)2]3’2’
generator.
sin Wt - e)
(18)
486
YONG
DENG
and
Experiments were carried out at room temperature according to the following procedure. First, put about 10 particles on the copper wire net, start the motor and gradually increase its speed. When the frequency is large enough to raise the particles to the middle of the tube, carefully adjust the speed of the motor to let the particle vibrate about a fixed position. When the net displacement of this particle becomes less than 0.1 cm in half a minute, determine the rotational speed of the motor with an SZG-20 digital tachometer to obtain the levitation frequency. Every effort was made to avoid horizontal vibration of the U-shaped tube in order to minimize exper-
MOOSON
KWAUK
imental error, which is generally small because of the rather big amplitude of vertical oscillation. RESULTS
AND
DISCUSSION
Two groups of experiments were performed, using wet ion-exchange resin particles in water, and using glass beads in 55% aqueous glycerol. Before each experiment, the resins were soaked in water for at least 24 hours. Their measured density was 1.17 g/cm3, and four particle sizes averaging 0.0435, 0.0401, 0.0334 and 0.0290 cm were used. Figure 3 shows the behavior of particles in an oscillating flow field. When the frequency is low, the
260
270
280
280
303
(a) f = 3.08
Hz.
jb)f
=4.82
Hz.
90
Fig. 3.
Resin particles in oscillating
water flow, A = 2 cm, k, = 1.2, d upper tank).
=0.0435 cm (scale
shows the distance to
Levitation
of discrete
particles
particles merely jump up and then back down to the copper wire net, accompanying the oscillating liquid, Fig. 3(a). As the frequency increases, the particles can gradually remain in the liquid without falling down to the net for certain lengths of time, as shown in Fig. 3(b). When the frequency attains certain value, the particles will be in the state of levitation, as shown in Fig. 3(c). If the frequency increases further, the particles will travel towards the upper water tank. The levitation frequency was measured under different oscillating conditions (A = 1.0, 1.5,2.0 and 2.5 cm; k, = 1.0, 1.2, 1.4 and I .6) for the four sizes of resin particles. With these experimental data, the three parameters, u,, du,/dx and d’u,/dx’, in the mathematical model, eq. (15), were determined accordingly and correlated to particle size by the least square
15
d = 0.0435
in oscillating
liquids
487
method to give the following u, = 1 l.ld
empirical equations:
+ 0.536
dux - - 39.5d + 2.20 dx
d2u, = dx2
(19)
- 1.15 x 106d + 7.25 x 104-
We can see from Fig. 4 that the model can adequately predict the levitation frequency with a maximum error of 18%. And if the oscillation amplitude A or asymmetry factor k, becomes larger, or if the particle size becomes smaller, the levitation frequency is reduced correspondingly. With this model, the trajectories of a resin particle
15
cm
d = 0.0435 cm
\i \
A= 1.0
cm
10
f
5
0
J
2
1
1
1.2
A (cm)
1.4
1.6
ko
ia
f
i.2! 5
0.30
0.35
0.40
0.45
Go
d (mm) Fig. 4. Comparison
of the model and experimental
data of the resin-water
system.
1.8
YONG
488
Fig. 5. Trajectories
DENG
and
MOOSON
KWAUK
of a resin particle in oscillating water flow, calculated with eq. (15), A = 2 cm, k, = 1.2, d = 0.0435 cm.
under oscillation of different frequencies were calculated. The results are shown in Fig. 5. The experiments also verified that particles may be levitated by symmetric oscillations. However, Boyadzhiev’s (1973) analysis with the quasisteady state model, eq. (5), shows that a particle may be caused to levitate only in the extreme case whenf-r co. This contradiction of theory with experiments explains that eq. (5) alone does not suffice to describe the levitation phenomenon. Based on the analysis of SchGneborn (1975) on particle movement in a sinusoidally oscillating flow field, if the frequency is high, free vortex shedding of the particle would be suppressed so that the fluid drag is reduced. And in this case, eq. (6) predicts particle movement more accurately than eq. (5). However, if oscillating amplitude is large, the effect of convective acceleration may also need to be included in describing particle motion. To compare the effect of the drag force with that of the pressure gradient inducing levitation, the ratio of the average drag force to the external (gravitational) force was calculated while a particle was under levitation. Then the ratio of the average pressure gradient effect to the external force can be easily obtained from the relation: F,+F,=IF,l.
0
1
d = 0.0435
cm
k, = 1.6
FP
FD lFEl
’
O
IFEI
ta I
I
-1
1
2 A
3
(cm)
k,, = 1.6
(20)
The results, as shown in Fig. 6, indicate that all values of FD/ IFE 1 are less than 0.5, thus making all values of Fp/IFEI greater than 0.5, implying that the effect of pressure gradient is more significant than that of drag force at causing a particle to levitate under the present experimental conditions. Furthermore, as the value of FD / 1FE 1becomes negative in the case of symmetric oscillation (k 0 = 1.O), the drag force is actually driving the particle downwards, leaving the pressure gradient the only operative force to induce levitation. In Fig. 6, an increase in the asymmetry factor k, results in an increase in the ratio FD/I FEI, indicating
FD __
IFE
I
0 /FE
-1
2
1 A
Fig. 6. Comparison
3
(cm)
of the effects inducing levitation.
I
Levitation of discrete particles in oscillating liquids
489
15
15
A = 1.32 cm
10
f 5
(b)
(al
I 1
0
2
a
3
A (cm)
1
1.2
I
I
1.4
1.0
1.8
ko
Fig. 7. Experimental results of glass beads in 55% aqueous glycerol.
that its effect to reduce levitation frequency is accomplished mainly through making the drag force, rather than the pressure gradient, larger. And an increase in the oscillation amplitude A does not always result in an increase in the ratio FP/j FE 1, meaning that it raises both the drag force and the pressure gradient effect, thus reducing levitation frequency. Figure 7 shows the experimental results of glass beads in 55% aqueous glycerol. The density of the glass beads was 2.80 g/cm 3 and the average diameter, 0.0464 cm. The viscosity of the solution was calculated to be 0.0988 P. This figure proves that more viscous fluid can make denser particles levitate more easily. The experimental data of the two systems are correlated with the following semi-empirical equation: 0.11
x
(;>“”
(A$) -“.34(!z)““.
(5) The present theoretical analysis and experimental results serve to pave the way for studying particle levitation in gas-solid systems, which are more complex because of the compresibility of gases.
NOTATION a A AC
b CA CD CH d e I
(21)
CONCLUSIONS (1) Experiments verified that levitation of particles may occur in both symmetrically and asymetrically oscillating liquids. {2) The mathematical model, eq. (15), which was obtained from the particle diffusion equation in unsteady turbulence by considering the effect of onedimensional convection acceleration, could adequately predict particle levitation frequency. (3) Under the present experimental conditions of the resin-water system, pressure gradient effect offers greater contributions than drag force to particle levitation. And in the case of symmetric oscillations, drag force causes a particle to move downwards. (4) Liquids with high viscosities make it easier to levitate particles with larger densities.
FD FE FP 9 ko P Re r T u Uf UP ux X c1 Y
length of the crank, cm amplitude of fluid oscillation, cm acceleration number length of the rocker, cm coefficient of the apparent mass, eq. (7) drag coefficient, eq. (4) coefficient of the Basset term, eq (8) particle diameter, cm eccentricity of the crank and rocker mechanism, cm levitation frequency or freoscillation quency, s - l time-average of the drag term in eq. (15), dyn external force, dyn time-average of the pressure gradient term in eq. (15), dyn acceleration due to gravity, cm/s* asymmetry factor of fluid oscillation pressure, dyn/cm’ Reynolds number time, s period of the oscillation, s velocity of the fluid, cm/s velocity of the piston, cm/s velocity of the particle, cm/s function of x in the variable-separated forms, eqs (13) and (14), of the fluid velocity coordinate, cm absolute viscosity of the fluid, dyn s/cm* kinematic viscosity of the fluid, cm*/s
YONG DENG and MOOSON KWALJK
490
4 cp
density of the fluid, g/cm3 density of the particIe, g/cm3 angle D,OD, in Fig. 2 relative density ratio (p,/pt)
w
w=ZrLf,
Pf PP
s-
* REFERENCES
Al-taweel, A. M.andCarley, J. F., 1971a, A.1.Ch.E. Symp. Ser. 67, 114-123. Al-taweel, A. M.andCarley, J. F., 1971b, A.1.Ch.E. Symp. Ser. 67, 124-131. Bailey, J. E., 1974, Chem. Engng Sci. 29, 767-773. Boyadzhiev, L., 1973, J. Fluid Mech. 57, 545-548. Corrsin, S. and Lumley, J., 1956, Appl. Sci. Kes. 6A, 114-116. Feinman, J., 1966, Dissertation Abstracts 26, 3806. Guo, Q., Xia, L., Song, B. and Liu, S., 1984, Proc. 3rd Nat. Fluidization Co@ (in Chinese), pp. 353-365. Herringe, R. A., 1976, Chem. Engng J. 11,89-99.
Herringe, R. A. and Flint, L. R., 1974, Proc. 5th Australasian Conf Hydraulics and Fluid Mech. Vol. 2, pp. 103-110. Hinze, J. O., 1959, Turbulence. McGraw-Hill, New York. Hjelmfelt, A. T. and Mockros, L. F., 1966, Appl. Sci. Res. 16, 149-161. Houghton, G., 1963, Proc. R. Sot. A272, 3343. Houghton, G.. 1966, Can. J. Chem. Ennncr - I 44, 9C-95. Jam&on, G. J. and Davidson, J. F., 1966, Chem. Engng Sci. 21, 29-34. Krantz, W. B., Carley, J. F. and Al-taweel, A. M., 1973, Ind. Engng Chem. Fundam. 12, 391-396. Liang, B. and Kwauk, M., 1981, M.S. Thesis (in Chinese), Institute of Chem. Metall., Chinese Academy of Sciences. Odar, F., 1966, .I. Fluid Mech. 25, 591-592. Schiineborn, P. R., 1975, Inter. J. Multiphase Flow 2, 307-3 17. Tunstall, E. B. and Houghton, G., 1968, Chem. Engng Sci. 23, 1067-1081. Van Oeveren, R. M. and Houghton, G., 1971, Chem. Enyng Sci. 26, 1958-1961.
LEVITATION
1990.
ooo!-2x9/90 s3.00 +o.oo 0 1990 Pergamoo Press plc
OF DISCRETE PARTICLES LIQUIDS
YONG DENG Institute of Chemical Metallurgy, Chinese (Received 25 July
and MOOSON Academy
KWAUK’
of Sciences,
1988; acceptedfir
IN OSCILLATING
People’s Republic of China
Beijing,
publication
19 May
1989)
Abstract-Wet ion-exchange resins (pp= 1.17 g/cm3, d=0.0435, 0.0401, 0.0334 or 0.0290 cm) were suspended, while vibrating about a fixed position, by either symmetrically or asymmetrically oscillating water. By taking into account convection acceleration, a proposed mathematical model [eq. (15)], was found to predict adequately the levitation frequency. It showed also that the effect of pressure gradient was more significant than drag force at causing a particle to levitate. Glass beads were also levitated in a 55% aqueous
glycerol.
INTRODUCTION
With the development of fluidization classical Ruidized beds are assuming
technology, nongreater import-
ance. In a jig-fluidized bed, Guo et al. (1984) deoxidized iron ore directly with soIid coal, thus making gas-mixing, gas-circulating and gas-heating unnecessary, while product (iron metal) separation can be realized continuously along with deoxidization. Theoretical and experimental investigations have shown that vertical oscillation of liquid might retard the motion of particles or bubbles or even make them move in a direction opposed to gravity. Many researchers, such as Al-taweel and Carley (1971a, b), Feinman (1966), Herringe and Flint (1974), Houghton (1963), Jameson and Davidson (1966), Schijneborn (1975) and Tunstall and Houghton (1968) have long been interested in this phenomenon, since it would have a profound effect on the hold-up and heat or mass transfer characteristics of such a two-phase system. Levitation was defined by Houghton (1966) as a stable condition in which a particle responds to the oscillating fluid in such a way that the influence of finite buoyancy forces is completely neutralized so that the particle oscillates about a fixed position. Feinman (1966) conducted an experimental study on levitation under sinusoidal oscillations and found that there existed two types of levitation, viz. nonrising levitation and rising levitation, as predicted theoretically earlier by Houghton (1963). Feinman’s experimental data were later correlated by Krantz et al. (1973) to yield the following semi-empirical equa-
Liang and Kwauk (198 1) found that particles denser than the surrounding fluid would move upwards when there existed an air plug between the water column and the piston generated sinusoidal oscillation. They believed that compression of this plug tended to make the water oscillation asymmetric. SchGneborn (1975) studied the effect of vortex shedding on the motion of a particle in an oscillating liquid. He believed that levitation would be impossible in the case of sinusoidal oscillations. Bailey (1974) analysed the motion of a particle in a rapidly oscillating flow and obtained a similar conclusion. The object of the present work was to identify the conditions under which levitation could occur and to propose a mathematical model capable of describing this phenomenon. It is hoped this would not only form a basis for the intelligent design of jig-fluidized bed such as used by Guo, but also may lead to the development of new uses of this type of fluidized reactor. THEORETICAL
tion:
(a + C,)‘a’d GIQ-
X
= o.370
‘Author
period was longer would levitate.
to whom correspondence
than
J3
dr’ + F,.
(2)
The term on the left-hand
side of the equation is the inertial force. The first term on the right is the drag force according to Stokes’ law. The second term is due
Van Oeveren and Houghton (1971) studied the case of asymmetric oscillations and found that when the time for the downstroke for the upstroke particles
s
’ d/+/dt’ - du,/dt’
0
(1)
llg
ANALYSIS
In 1947, Tchen proposed the following equation to describe a particle diffusing in unsteady turbulence:
that
to the pressure gradient in the fluid surrounding the particle caused by fluid acceleration. The third term is, the force accelerating the apparent mass of the particle. The fourth term is the Basset term, which takes
should be addressed. 483
YONG DENG and MOOSON KWAUK
484
account of the effect of the deviation of the flow pattern from steady state. The last term is the external force. With their experimental results, At-taweel and Carley (1971 b) reached the following form of Basset term in a sinusoidally oscillating flow field:
where CD follows eq. (4), and the coefficients C, and C, were given by Odar (1966): c,
= 1.05 -
C, = 2.88 +
0.066
(7)
AC2 + 0.12 3.12 (AC + 1)3
in which AC is an instantaneous acceleration -KK,Jnw/2(sinot-coswt) - K, Jnw/Z
(sin cot + cos cot)].
Substitution into eq. (2) yields, however, a linear equation, indicating that eq. (2) is incapable of accounting for the retarding effect of oscillation on the free fall velocity of particles. Thus the term of the drag force- needs to be modified. If the frequency of oscillation is small, the flow field may be considered as an otherwise stationary one in which a particle falls at the speed (uI - up). Therefore, the drag force can be expressed with Newton’s equation with drag efficient CD possessing the following values: z(l
+0.15Re0.687)
(C,>O.44)
(4)
(the rest)
0.44
AC=
(3)
and the Basset term can be neglected. Equation (2) thus becomes the so-called quasi-steady state model used by many authors, such as Boyadzhiev (1973):
(UP - Ur)* dldu,/dt - du,/dtl-
-fWP,-
Pf)S
in which the drag coefficient Cd was assumed to be a constant, 0.3. Tentative calculations with the above three models, eqs (5), (6) and (lo), showed that particles would not be able to levitate if the oscillation frequency was less than 50 Hz under the present experimental conditions. Corrsin and Lumley (1956) pointed out that if the fluid velocity is dependent on both time and space, the effect of convection acceleration should be included, and the pressure gradient in turbulent flow becomes:
DU
au+(u)&“V%
-PPr
at
[
(5)
According to Schiineborn (1975), this quasisteady state model can predict the average fall velocity of particles only for small values of frequency relative to the natural vortex shedding frequency of the particle. By substituting the unsteady drag given by Odar (1966) in eq. (2), certain authors, such as Schiineborn (1975) and Herringe (1976), derived another model to describe the retarding effect of the oscillating liquid on the particle:
;d’pp%= -CD;d2pflup-U&J,-uf)
+~d’p$+C*;d’pf($f$J
s 0
JZ~
dt’ (6)
(11)
(1) the fluid flow is approximately steady state; (2) the effect of the apparent mass of the particle may be neglected; (3) only the effect of one dimensional convection acceleration is taken into account, and the pressure gradient becomes:
ax
’ du,jdt’-du,/dt’
1
J
Thus the term of pressure gradient, the second on the right of eq. (2), should be modified, especially in the case for which the density of the fluid is comparable to that of the particle, and the value A/d is large. To simplify the problem in order to formulate a practical model to predict the levitation frequency of the particle, we assume that:
dP _=_
X
(9)
Schiineborn (1975) pointed out, however, that the validity of eq. (6) for predicting the average fall velocity is also limited. Another model was proposed later by Liang and Kwauk (t981):
-Pfz=
-~dvPP-_pfh?.
number:
du
"
(
au
-+%KVax2 at
azu
>.
(12)
In the case of small frequency and large A/d value, the first assumption is reasonable. Thus we can neglect the Basset term and calculate the drag force with Newton’s equation. Hjelmfelt and Mockros (1966) pointed out that, if the oscillation frequency was not sufficiently small, the
Levitation
of discrete
particles in oscillating
error caused by neglecting the effect of the apparent mass would be smaller than that caused by neglecting the Basset term. Since, in addition, the coefficient of the apparent mass cannot be accurately determined, we chose to neglect that effect. Further, in a steady state flow field, if the particle is immobile, the fluid velocity may be expressed in a variable-separated form, which, at y = 0 and z= 0, can be written as: u(t, x) = u,(t).
u,(x).
And if the particle is free to move, velocity may be expressed as: u(t, x) = Car(r) Substitution
(13) then the fluid
%(f)l u,(x) + up(t).
485
liquids
Crank and rocker mechanism
Water tank 4 70 +
Plastic tube
I.D. 19
-= 330
U - shaped Plexiglas tube \
(14)
1
into eq. (12) yields:
Levitation d/section Copper wire net
_ __
Water int ak& Fig. 1. A diagram
- y(uz - UP)
(PP - Pr)9(13
Because u is the fluid velocity near the pa .rticle, eq. (15) becomes an ordinary differential equation in terms of up and t so long as we know the values of u,, du,/dx and d2u,/dx2 at x=0. Therefore, eq. (15) can be accepted as the final model involving three parameters, u,, du,/dx and d2u,/dx2.
Fig. 2. Principle
device.
adjustment of both the amplitude A and the asymmetry factor k, (called by Van Overen and Houghton “duration factor”) of oscillation, as described below. In Fig. 2, the length of crank a, the length of rocker b and the eccentric distance e, are suitably designed to be easily adjustable. When the motor rotates clockwise uniformly, the time for the hinge E travelling from E 2 (corresponding to the lowest position of piston) to E 1 (corresponding to the highest position of piston) is longer than that back from E 1 to E 2 because of the eccentricity e. And if e = 0, the oscillation will be symmetric. The following can be easily derived: X+4
EXPERIMENTAL
The present experiments were carried out in a Ushaped Plexiglas tube with fluid oscillation generated by a mechanical device, as shown in Fig. 1. In this device, a crank and rocker mechanism converts the rotation of a motor to vertical oscillations of the piston which drives the liquid in the U-shaped tube. The rotational speed of the motor is controlled by a ZTD-5A silicon controlled governer, so as to adjust the oscillation frequency. The crank and rocker mechanism permits easy
of the experimental
k, = --_
- arccose
#J = 2uccos&
b-u
(16) (17)
ur = uwsin cot du, dt=
of the oscillation
-
ufd
aw cos ot(u
[b2-(usinwt-e2)2]‘/2
cos cot +
u2 b2a2
uw2 sin wt (a sin cot -e) [b2-(usinwt-e)2]“2 cos2 ut
-[b2-(asinwt-e)2]3’2’
generator.
sin Wt - e)
(18)
486
YONG
DENG
and
Experiments were carried out at room temperature according to the following procedure. First, put about 10 particles on the copper wire net, start the motor and gradually increase its speed. When the frequency is large enough to raise the particles to the middle of the tube, carefully adjust the speed of the motor to let the particle vibrate about a fixed position. When the net displacement of this particle becomes less than 0.1 cm in half a minute, determine the rotational speed of the motor with an SZG-20 digital tachometer to obtain the levitation frequency. Every effort was made to avoid horizontal vibration of the U-shaped tube in order to minimize exper-
MOOSON
KWAUK
imental error, which is generally small because of the rather big amplitude of vertical oscillation. RESULTS
AND
DISCUSSION
Two groups of experiments were performed, using wet ion-exchange resin particles in water, and using glass beads in 55% aqueous glycerol. Before each experiment, the resins were soaked in water for at least 24 hours. Their measured density was 1.17 g/cm3, and four particle sizes averaging 0.0435, 0.0401, 0.0334 and 0.0290 cm were used. Figure 3 shows the behavior of particles in an oscillating flow field. When the frequency is low, the
260
270
280
280
303
(a) f = 3.08
Hz.
jb)f
=4.82
Hz.
90
Fig. 3.
Resin particles in oscillating
water flow, A = 2 cm, k, = 1.2, d upper tank).
=0.0435 cm (scale
shows the distance to
Levitation
of discrete
particles
particles merely jump up and then back down to the copper wire net, accompanying the oscillating liquid, Fig. 3(a). As the frequency increases, the particles can gradually remain in the liquid without falling down to the net for certain lengths of time, as shown in Fig. 3(b). When the frequency attains certain value, the particles will be in the state of levitation, as shown in Fig. 3(c). If the frequency increases further, the particles will travel towards the upper water tank. The levitation frequency was measured under different oscillating conditions (A = 1.0, 1.5,2.0 and 2.5 cm; k, = 1.0, 1.2, 1.4 and I .6) for the four sizes of resin particles. With these experimental data, the three parameters, u,, du,/dx and d’u,/dx’, in the mathematical model, eq. (15), were determined accordingly and correlated to particle size by the least square
15
d = 0.0435
in oscillating
liquids
487
method to give the following u, = 1 l.ld
empirical equations:
+ 0.536
dux - - 39.5d + 2.20 dx
d2u, = dx2
(19)
- 1.15 x 106d + 7.25 x 104-
We can see from Fig. 4 that the model can adequately predict the levitation frequency with a maximum error of 18%. And if the oscillation amplitude A or asymmetry factor k, becomes larger, or if the particle size becomes smaller, the levitation frequency is reduced correspondingly. With this model, the trajectories of a resin particle
15
cm
d = 0.0435 cm
\i \
A= 1.0
cm
10
f
5
0
J
2
1
1
1.2
A (cm)
1.4
1.6
ko
ia
f
i.2! 5
0.30
0.35
0.40
0.45
Go
d (mm) Fig. 4. Comparison
of the model and experimental
data of the resin-water
system.
1.8
YONG
488
Fig. 5. Trajectories
DENG
and
MOOSON
KWAUK
of a resin particle in oscillating water flow, calculated with eq. (15), A = 2 cm, k, = 1.2, d = 0.0435 cm.
under oscillation of different frequencies were calculated. The results are shown in Fig. 5. The experiments also verified that particles may be levitated by symmetric oscillations. However, Boyadzhiev’s (1973) analysis with the quasisteady state model, eq. (5), shows that a particle may be caused to levitate only in the extreme case whenf-r co. This contradiction of theory with experiments explains that eq. (5) alone does not suffice to describe the levitation phenomenon. Based on the analysis of SchGneborn (1975) on particle movement in a sinusoidally oscillating flow field, if the frequency is high, free vortex shedding of the particle would be suppressed so that the fluid drag is reduced. And in this case, eq. (6) predicts particle movement more accurately than eq. (5). However, if oscillating amplitude is large, the effect of convective acceleration may also need to be included in describing particle motion. To compare the effect of the drag force with that of the pressure gradient inducing levitation, the ratio of the average drag force to the external (gravitational) force was calculated while a particle was under levitation. Then the ratio of the average pressure gradient effect to the external force can be easily obtained from the relation: F,+F,=IF,l.
0
1
d = 0.0435
cm
k, = 1.6
FP
FD lFEl
’
O
IFEI
ta I
I
-1
1
2 A
3
(cm)
k,, = 1.6
(20)
The results, as shown in Fig. 6, indicate that all values of FD/ IFE 1 are less than 0.5, thus making all values of Fp/IFEI greater than 0.5, implying that the effect of pressure gradient is more significant than that of drag force at causing a particle to levitate under the present experimental conditions. Furthermore, as the value of FD / 1FE 1becomes negative in the case of symmetric oscillation (k 0 = 1.O), the drag force is actually driving the particle downwards, leaving the pressure gradient the only operative force to induce levitation. In Fig. 6, an increase in the asymmetry factor k, results in an increase in the ratio FD/I FEI, indicating
FD __
IFE
I
0 /FE
-1
2
1 A
Fig. 6. Comparison
3
(cm)
of the effects inducing levitation.
I
Levitation of discrete particles in oscillating liquids
489
15
15
A = 1.32 cm
10
f 5
(b)
(al
I 1
0
2
a
3
A (cm)
1
1.2
I
I
1.4
1.0
1.8
ko
Fig. 7. Experimental results of glass beads in 55% aqueous glycerol.
that its effect to reduce levitation frequency is accomplished mainly through making the drag force, rather than the pressure gradient, larger. And an increase in the oscillation amplitude A does not always result in an increase in the ratio FP/j FE 1, meaning that it raises both the drag force and the pressure gradient effect, thus reducing levitation frequency. Figure 7 shows the experimental results of glass beads in 55% aqueous glycerol. The density of the glass beads was 2.80 g/cm 3 and the average diameter, 0.0464 cm. The viscosity of the solution was calculated to be 0.0988 P. This figure proves that more viscous fluid can make denser particles levitate more easily. The experimental data of the two systems are correlated with the following semi-empirical equation: 0.11
x
(;>“”
(A$) -“.34(!z)““.
(5) The present theoretical analysis and experimental results serve to pave the way for studying particle levitation in gas-solid systems, which are more complex because of the compresibility of gases.
NOTATION a A AC
b CA CD CH d e I
(21)
CONCLUSIONS (1) Experiments verified that levitation of particles may occur in both symmetrically and asymetrically oscillating liquids. {2) The mathematical model, eq. (15), which was obtained from the particle diffusion equation in unsteady turbulence by considering the effect of onedimensional convection acceleration, could adequately predict particle levitation frequency. (3) Under the present experimental conditions of the resin-water system, pressure gradient effect offers greater contributions than drag force to particle levitation. And in the case of symmetric oscillations, drag force causes a particle to move downwards. (4) Liquids with high viscosities make it easier to levitate particles with larger densities.
FD FE FP 9 ko P Re r T u Uf UP ux X c1 Y
length of the crank, cm amplitude of fluid oscillation, cm acceleration number length of the rocker, cm coefficient of the apparent mass, eq. (7) drag coefficient, eq. (4) coefficient of the Basset term, eq (8) particle diameter, cm eccentricity of the crank and rocker mechanism, cm levitation frequency or freoscillation quency, s - l time-average of the drag term in eq. (15), dyn external force, dyn time-average of the pressure gradient term in eq. (15), dyn acceleration due to gravity, cm/s* asymmetry factor of fluid oscillation pressure, dyn/cm’ Reynolds number time, s period of the oscillation, s velocity of the fluid, cm/s velocity of the piston, cm/s velocity of the particle, cm/s function of x in the variable-separated forms, eqs (13) and (14), of the fluid velocity coordinate, cm absolute viscosity of the fluid, dyn s/cm* kinematic viscosity of the fluid, cm*/s
YONG DENG and MOOSON KWALJK
490
4 cp
density of the fluid, g/cm3 density of the particIe, g/cm3 angle D,OD, in Fig. 2 relative density ratio (p,/pt)
w
w=ZrLf,
Pf PP
s-
* REFERENCES
Al-taweel, A. M.andCarley, J. F., 1971a, A.1.Ch.E. Symp. Ser. 67, 114-123. Al-taweel, A. M.andCarley, J. F., 1971b, A.1.Ch.E. Symp. Ser. 67, 124-131. Bailey, J. E., 1974, Chem. Engng Sci. 29, 767-773. Boyadzhiev, L., 1973, J. Fluid Mech. 57, 545-548. Corrsin, S. and Lumley, J., 1956, Appl. Sci. Kes. 6A, 114-116. Feinman, J., 1966, Dissertation Abstracts 26, 3806. Guo, Q., Xia, L., Song, B. and Liu, S., 1984, Proc. 3rd Nat. Fluidization Co@ (in Chinese), pp. 353-365. Herringe, R. A., 1976, Chem. Engng J. 11,89-99.
Herringe, R. A. and Flint, L. R., 1974, Proc. 5th Australasian Conf Hydraulics and Fluid Mech. Vol. 2, pp. 103-110. Hinze, J. O., 1959, Turbulence. McGraw-Hill, New York. Hjelmfelt, A. T. and Mockros, L. F., 1966, Appl. Sci. Res. 16, 149-161. Houghton, G., 1963, Proc. R. Sot. A272, 3343. Houghton, G.. 1966, Can. J. Chem. Ennncr - I 44, 9C-95. Jam&on, G. J. and Davidson, J. F., 1966, Chem. Engng Sci. 21, 29-34. Krantz, W. B., Carley, J. F. and Al-taweel, A. M., 1973, Ind. Engng Chem. Fundam. 12, 391-396. Liang, B. and Kwauk, M., 1981, M.S. Thesis (in Chinese), Institute of Chem. Metall., Chinese Academy of Sciences. Odar, F., 1966, .I. Fluid Mech. 25, 591-592. Schiineborn, P. R., 1975, Inter. J. Multiphase Flow 2, 307-3 17. Tunstall, E. B. and Houghton, G., 1968, Chem. Engng Sci. 23, 1067-1081. Van Oeveren, R. M. and Houghton, G., 1971, Chem. Enyng Sci. 26, 1958-1961.